accelerating pde-constrained optimization...

IntroductionOptimization via Adaptive Model Reduction

Large-Scale Constrained OptimizationConclusion

Accelerating PDE-Constrained Optimization Problemsusing Adaptive Reduced-Order Models

Matthew J Zahr

Advisor Charbel FarhatComputational and Mathematical Engineering

Stanford University

Lawrence Livermore National Laboratory Livermore CADecember 7 2015

Zahr PDE-Constrained Optimization with Adaptive ROMs



Multiphysics Optimization Key Player in Next-Gen Problems

Current interest in computational physics reaches far beyond analysis of a singleconfiguration of a physical system into design (shape and topology1) controland uncertainty quantification

14

Better simulation is critical to engine design and calibration optimization

bull Three levels of simulation with different purposes

‒ Predictive combustion combustion optimization and methods development

‒ Full engine simulation engine system optimization and model-based controls

‒ Full vehicle simulation technology interactions component optimization and supervisory controls

bull Each scale requires different level of fidelity

‒ High fidelity combustion on vehicle scales computationally impossible (at the moment)

bull Exponential increase in parameter space translates to exponential increase in simulation space and computational requirements

TAKEAWAY Need for faster simulation faster optimization methods and reduced models for on-board controls

Engine System

EM Launcher Micro-Aerial Vehicle

1Emergence of additive manufacturing technologies has made topology optimizationincreasingly relevant particularly in DOE




Topology Optimization and Additive Manufacturing2

Emergence of AM has made TO anincreasingly relevant topic

AM+TO lead to highly efficient designsthat could not be realized previously

Challenges smooth topologies requirevery fine meshes and modeling ofcomplex manufacturing process

2MIT Technology Review Top 10 Technological Breakthrough 2013




Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design

PDE-Constrained Optimization I

Goal Rapidly solve PDE-constrained optimization problem of the form

minimizeuisinRnu microisinRnmicro

J (u micro)

subject to r(u micro) = 0

where

r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation

J Rnu times Rnmicro rarr R is the objective function

u isin Rnu is the PDE state vector

micro isin Rnmicro is the vector of parameters

red indicates a large-scale quantity O(mesh)





Nested Approach to PDE-Constrained Optimization

Virtually all expense emanates from primaldual PDE solvers

Primal PDE Dual PDE

Optimizer







Primal PDE Dual PDE

Optimizer

micro







Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)





Projection-Based Model Reduction to Reduce PDE Size

Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace

u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where

Φu =[φ1u middot middot middot φku

u

]isin Rnutimesku is the reduced basis

ur isin Rku are the reduced coordinates of unu ku

Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku

ΨuTr(Φuur micro) = 0





Connection to Finite Element Method Hierarchical Subspaces

S

S - infinite-dimensional trial space

Sh - (large) finite-dimensional trial space

Skh - (small) finite-dimensional trial space

Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S





Few Global Data-Driven Basis Functions v Many Local Ones

Instead of using traditional localshape functions (eg FEM) useglobal shape functions

Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes


var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false




Definition of Φu Data-Driven Reduction

State-Sensitivity Proper Orthogonal Decomposition (POD)

Collect state and sensitivity snapshots by sampling HDM

X =[u(micro1) u(micro2) middot middot middot u(micron)

]Y =

[partupartmicro (micro1) partu

partmicro (micro2) middot middot middot partupartmicro (micron)

]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually

ΦX = POD(X)

ΦY = POD(Y )

Concatenate to get reduced-order basis

Φu =[ΦX ΦY

]





Definition of Ψu Minimum-Residual ROM

Least-Squares Petrov-Galerkin (LSPG)3 projection

Ψu =partr

partuΦu

Minimum-Residual Property

A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)

minimizeurisinRku

||r(Φuur micro)||Θ

Implications

Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4

LSPG possesses minimum-residual property

3[Bui-Thanh et al 2008]4[Fahl 2001]





Definition of partur

partmicro Minimum-Residual Reduced Sensitivities

Traditional sensitivity analysis

partur

partmicro=minus

Nsumj=1

rjΦuT partrjpartupartu

Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1

rjΦuT part2rjpartupartmicro

+

(partr

partuΦu

)Tpartr

partmicro

+ Guaranteed to give rise to exact derivatives of ROM quantities of interest

- Requires 2nd derivatives of r

- Φupartur

partmicro not guaranteed to be good approximate to full sensitivity partupartmicro







Minimum-residual sensitivity analysis

partur

partmicro= arg min

a||Φuaminus

partu

partmicro||Θ = minus

[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro

+ Minimum-residual property - Φupartur

partmicrois Θ-optimal solution to

partu

partmicroin Φu

+ Does not require 2nd derivatives of r

-partur

partmicro6= partur

partmicro ie it is not the true ROM sensitivity





Offline-Online Approach to Optimization

Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space

HDM HDM middot middot middot HDM HDM ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

Breakdown of Computational Effort






Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space

HDM HDM middot middot middot HDM HDM

ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M






Numerical Demonstration Offline-Online Breakdown

Parameter reduction (Φmicro)

apriori spatial clusteringkmicro = 200

Greedy Training

5000 candidate points (LHS)50 snapshotsError indicator ||r(Φuur Φmicromicror||)

State reduction (Φu)

PODku = 25Polynomialization acceleration

Material Basis






Optimal Solution (ROM) Optimal Solution (HDM)

HDM Solution ROB Construction Greedy Algorithm ROM Optimization

284times 103 s 548times 104 s 167times 105 s 30 s

126 2436 7437 001

HDM Optimization 197times 104 s





ROM-Based Trust-Region Framework for Optimization

Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO

M middot middot middot

RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M

middot middot middot







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Nonlinear Trust-Region Framework with Adaptive Model Reduction

Collect snapshots from HDM at sparse sampling of the parameter space

Build ROB Φu from sparse training

Solve optimization problem

minimizeurisinRku microisinRnmicro

J (Φuur micro)

subject to ΦTur(Φuur micro) = 0

||r(Φuur micro)|| le ∆

Use solution of above problem to enrich training adapt ∆ using standardtrust-region methods and repeat until convergence






Ingredients of Proposed Approach [Zahr and Farhat 2014]

Minimum-residual ROM (LSPG) and minimum-error sensitivities

J (u micro) = J (Φuur micro) anddJdmicro

(u micro) =dJdmicro

(Φuur micro) for training

parameters micro

Reduced optimization (sub)problem

minimizeurisinRnu microisinRnmicro

J (Φuur micro)

subject to ΨuTr(Φuur micro) = 0

||r(Φuur micro)||22 le ε

Efficiently update ROB with additional snapshots or new translation vector

Without re-computing SVD of entire snapshot matrix

Adaptive selection of εrarr trust-region approach





Compressible Inviscid Airfoil Inverse Design

Pressure discrepancy minimization (Euler equations)

NACA0012 Initial RAE2822 Target

Pressure field for airfoil configurations at Minfin = 05 α = 00





ROM-Constrained Optimization Solver Recovers Target

0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05

Distance along airfoil

-Cp

InitialTarget

HDM-based optimizationROM-based optimization

0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine





ROM Solver Requires 4times Fewer HDM Queries

0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101

Number of HDM queries

Ob

ject

ive

Fu

nct

ion






At the Cost of ROM Queries

0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2

Reduced optimization iterations

Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





Next Shape Optimization of Full Aircraft (CRM)

ROMs are fast accurate and require limited resources

HDM solution (Drag = 142336kN) ROM solution (Drag = 142304kN)

HDM 70times 106 DOF 2hr on 1024 Intel Xeon E5-2698 v3 cores (23GHz)

ROM 170s on 2 Intel i7 cores (18GHz)

Relative error in drag 0022

CPU-time speedup greater than 215times 104

Wall-time speedup greater than 42

Washabaugh Zahr Farhat (AIAA 2016)




Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever

PDE-Constrained Optimization II



J (u micro)


c(u micro) ge 0

where



c Rnu times Rnmicro rarr Rnc are the side constraints







Problem Setup

25

40

16000 8-node brick elements 77760 dofs

Total Lagrangian form finite strain StVK 5

St Venant-Kirchhoff material

Sparse Cholesky linear solver (CHOLMOD6)

Newton-Raphson nonlinear solver

Minimum compliance optimization problem


fextTu

subject to V (micro) le 1

2V0

r(u micro) = 0

Gradient computations Adjoint method

Optimizer SNOPT [Gill et al 2002]

5[Bonet and Wood 1997 Belytschko et al 2000]6[Chen et al 2008]





Restrict Parameter Space to Low-Dimensional Subspace

Restrict parameter to a low-dimensional subspace

micro asymp Φmicromicror

Φmicro =[φ1micro middot middot middot φ

kmicromicro

]isin Rnmicrotimeskmicro is the reduced basis

micror isin Rkmicro are the reduced coordinates of micronmicro kmicro

Substitute restriction into reduced-order model to obtain

ΦuTr(Φuur Φmicromicror) = 0

Related work[Maute and Ramm 1995 Lieberman et al 2010 Constantine et al 2014]






micro-space Background mesh






micro-space Macroelements





Optimality Conditions to Adapt Reduced-Order Basis Φmicro

Selection of Φmicro amounts to arestriction of the parameter space

Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)

Adaptation based on first-orderoptimality conditions of HDMoptimization problem














Lagrangian

L(micro λ) = J (u(micro) micro)minus λT c(u(micro) micro)

Karush-Kuhn Tucker (KKT) Conditions7

nablamicroL(micro λ) = 0

λ ge 0

λici(u(micro) micro) = 0

c(u(micro)micro) ge 0

7[Nocedal and Wright 2006]





Lagrangian Gradient Refinement Indicator

From Lagrange multiplier estimates only KKT condition not satisfiedautomatically


Use |nablamicroL(micro λ)| as indicator for refinement of discretization of micro-space

micro |nablamicroL(micro λ)|





Constraints may lead to infeasible sub-problems

Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]

minimizeurisinRku microrisinRkmicro

J (Φuur Φmicromicror)

subject to c(Φuur Φmicromicror) ge 0

r(Φuur Φmicromicror) = 0

||r(Φuur Φmicromicror)|| le ∆



























Elastic constraints to circumvent infeasible subproblems

Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)

minimizeurisinRku microrisinRkmicro tisinRnc

J (Φuur Φmicromicror)minus γtT1

subject to c(Φuur Φmicromicror) ge t



t le 0












t le 0












t le 0












t le 0












t le 0





Compliance Minimization 2D Cantilever

25

40


Total Lagrangian form finite strain StVK8






fextTu


2V0

r(u micro) = 0



Maximum ROM size ku le 5






Order of Magnitude Speedup to Suboptimal Solution

HDM CNQTR-MOR + Φmicro adaptivity

HDM Solution HDM Gradient HDM Optimization

7458s (450) 4018s (411) 8284s

HDMElapsed time = 19761s

HDM Solution HDM Gradient ROB Construction ROM Optimization

1049s (64) 88s (9) 727s (56) 39s (3676)

CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x





Better Solution after 64 HDM Evaluations


CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)

Reasonable option to warm-start HDM topology optimization





Macro-element Evolution

Iteration 0 (1000)






Iteration 1 (977)





CNQTR-MOR + Φmicro adaptivity





Ongoing Research ProjectsFuture Research

High-Order Methods for Optimization of Conservation Laws

Derived implemented fully discrete adjoint method for globally high-orderdiscretization of conservation laws on deforming domains

Incorporation of time-periodicity constraints

minimizeU micro

W(U micro)

subject to Jx(U micro) = 25

U(x 0) = U(x)

partU

partt+nabla middot F (U nablaU) = 0

minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU







Energetically Optimal Flapping under Thrust Constraint

Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00

Initial Optimal ControlOptimal

ShapeControl

Collaborators Per-Olof Persson (UCB LBNL) Jon Wilkening (UCB LBNL)






Faster Computational Physics Adaptive Data-Driven Discretization

proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b

(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure

Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l

ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns

At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured

Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation

3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g

8

Methods to transform features in global basis functions - minimize relianceon local shape functions

Linear algebra for sparse operators with a few dense rows and columns

Integration mesh to mitigate ldquovariational crimesrdquo





Faster Solvers Adaptive Reduction of High-Dimensional Optimization

minimizemicro

f(micro)

subject to c(micro) = 0

minimizey

f(Φmicromicror)

subject to c(Φmicromicror) = 0

(a) Sub-optimal solrsquon (b) |nablamicroL(Φmicromicrorλ)| (c) Optimal solution

Prove global convergence and develop into general constrained optimizer

Further develop into topology optimization solver - overcome checkerboarding





Fewer Queries Second-Order Methods for Accelerated Convergence

Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves

SensitivityAdjoint Method for Computing Hessian

d2Jdmicrojdmicrok

=part2J

partmicrojpartmicrok+

part2Jpartmicrojpartu

partu

partmicrok+

partu

partmicroj

T part2Jpartupartmicrok

+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu

partmicrojotimes partu

partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj

Fast multiple right-hand side linear solver by building data-driven subspace

for image ofpartr

partu

minus1

partr

partu

minusT

Similar to Krylov methods that use a-priori analytical subspace





Approaching Many-Query Extreme-Scale Computational Physics

Framework introduced for accelerating PDE-constrainedoptimization problem with side constraints andlarge-dimensional parameter space

Speedup attained via adaptive reduction of state spaceand parameter space

Concepts borrowed from constrained optimization theory

Applied to aerodynamic design and topology optimization

Order of magnitude speedup speedup observedCompetitive warm-start method





Acknowledgement





References I

Barbic J and James D (2007)

Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models

In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association

Barrault M Maday Y Nguyen N C and Patera A T (2004)

An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations

Comptes Rendus Mathematique 339(9)667ndash672

Belytschko T Liu W Moran B et al (2000)

Nonlinear finite elements for continua and structures volume 26

Wiley New York

Bonet J and Wood R (1997)

Nonlinear continuum mechanics for finite element analysis

Cambridge university press

Bui-Thanh T Willcox K and Ghattas O (2008)

Model reduction for large-scale systems with high-dimensional parametric input space

SIAM Journal on Scientific Computing 30(6)3270ndash3288





References II

Carlberg K Bou-Mosleh C and Farhat C (2011)

Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations

International Journal for Numerical Methods in Engineering 86(2)155ndash181

Chapman T Collins P Avery P and Farhat C (2015)

Accelerated mesh sampling for model hyper reduction

International Journal for Numerical Methods in Engineering

Chaturantabut S and Sorensen D C (2010)

Nonlinear model reduction via discrete empirical interpolation


Chen Y Davis T A Hager W W and Rajamanickam S (2008)

Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate

ACM Transactions on Mathematical Software (TOMS) 35(3)22

Constantine P G Dow E and Wang Q (2014)

Active subspace methods in theory and practice Applications to kriging surfaces

SIAM Journal on Scientific Computing 36(4)A1500ndashA1524





References III

Fahl M (2001)

Trust-region methods for flow control based on reduced order modelling

PhD thesis Universitatsbibliothek

Gill P E Murray W and Saunders M A (2002)

Snopt An sqp algorithm for large-scale constrained optimization

SIAM journal on optimization 12(4)979ndash1006

Lawson C L and Hanson R J (1974)

Solving least squares problems volume 161

SIAM

Lieberman C Willcox K and Ghattas O (2010)

Parameter and state model reduction for large-scale statistical inverse problems


Maute K and Ramm E (1995)

Adaptive topology optimization

Structural optimization 10(2)100ndash112





References IV

Nguyen N and Peraire J (2008)

An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations

International journal for numerical methods in engineering 76(1)27ndash55

Nocedal J and Wright S (2006)

Numerical optimization series in operations research and financial engineering

Springer

Rewienski M J (2003)

A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems

PhD thesis Citeseer

Zahr M J and Farhat C (2014)

Progressive construction of a parametric reduced-order model for pde-constrainedoptimization






Standard Difficulty Binary Solutions

Relaxed Penalized Problem Setup


fextTu


2V0

r(u microp) = 0

micro isin [0 1]kmicro

Effect of Penalization

Ke larr (microe)pKe

Ke eth element stiffness matrix

(a) Without penalization

(b) With penalization








fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM

From parameter restriction microp = (Φmicromicror)p

Precomputation relies on separability of Φmicro and micror

Separability maintained if (Φmicromicror)p = Φmicromicropr

Sufficient condition columns of Φmicro have non-overlapping non-zeros





Efficient Evaluation of Nonlinear Terms

Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available

rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0

To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion

[rr(ur micror)]i = D0im(micror)m + D1

ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm

+ D3ijklm(ur times ur times ur times micror)jklm = 0

where

D3ijklm =

part3rtpartuppartuqpartus

(u φmmicro )(φi

u times φju times φk

u times φlu)tpqs

Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]





Lagrange Multiplier Estimate

Lagrange Multiplier Constraint Pairs

λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br

Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute

nablamicroL(Φmicromicror λ τ ) =partJpartmicro

(Φuur Φmicromicror)minus partc

partmicro(Φuur Φmicromicror)T λminusAT τ

Lagrange Multiplier Estimates

λ = λr

τ = arg minτge0

∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro


partmicro(Φuur Φmicromicror)T λ

)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]





Standard Difficulty Checkerboarding

Gradient Filtering Nodal Projection

Minimum length scale rmin

Gradient Filtering 10

partJpartmicrok

=

sumjisinSk

HkjmicroipartJpartmicroi

microk

sumjisinSk

Hkj

Nodal Projection

microk =

sumjisinSk τ jHjksumjisinSk Hjk

(a) Without projectionfiltering

(b) With projection

10Hki = rmin minus dist(k i)









partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM

Nonlocality introduced through projectionfiltering

microe influences volume fraction of all elements within rmin of elementnode e

Clashes with requirement on Φmicro of columns with non-overlapping non-zeros

Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin

Next Helmholtz filtering

Gradient of Lagrangian Updated Macroelements


Introduction

Optimization via Adaptive Model Reduction

Model Order Reduction

Non-Quadratic Trust-Region Solver

Shape Optimization Airfoil Design

Large-Scale Constrained Optimization

Reduction of High-Dimensional Parameter Space

Elastic Nonlinear Constraints

Topology Optimization 2D Cantilever

Conclusion

Ongoing Research Projects

Future Research

fdrm0

fdrm1

fdrm2

fdrm3



Multiphysics Optimization Key Player in Next-Gen Problems

Current interest in computational physics reaches far beyond analysis of a singleconfiguration of a physical system into design (shape and topology1) controland uncertainty quantification

14

Better simulation is critical to engine design and calibration optimization

bull Three levels of simulation with different purposes

‒ Predictive combustion combustion optimization and methods development

‒ Full engine simulation engine system optimization and model-based controls

‒ Full vehicle simulation technology interactions component optimization and supervisory controls

bull Each scale requires different level of fidelity

‒ High fidelity combustion on vehicle scales computationally impossible (at the moment)

bull Exponential increase in parameter space translates to exponential increase in simulation space and computational requirements

TAKEAWAY Need for faster simulation faster optimization methods and reduced models for on-board controls

Engine System

EM Launcher Micro-Aerial Vehicle

1Emergence of additive manufacturing technologies has made topology optimizationincreasingly relevant particularly in DOE
















J (u micro)


where












Primal PDE Dual PDE

Optimizer







Primal PDE Dual PDE

Optimizer

micro







Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3















J (u micro)


where












Primal PDE Dual PDE

Optimizer







Primal PDE Dual PDE

Optimizer

micro







Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Primal PDE Dual PDE

Optimizer







Primal PDE Dual PDE

Optimizer

micro







Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Primal PDE Dual PDE

Optimizer

micro







Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Primal PDE Dual PDE

Optimizer

J (u micro)







Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Primal PDE Dual PDE

Optimizer

J (u micro) micro

u







Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Primal PDE Dual PDE

Optimizer

J (u micro)

dJdmicro (u micro)







u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






u asymp Φuurpartu

partmicroasymp Φu

partur

partmicro

where


u










S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





S




Skh sub Sh sub S






S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





S

Sh




Skh sub Sh sub S






S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





S

Sh

Skh




Skh sub Sh sub S

















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3
















]Y =




ΦX = POD(X)

ΦY = POD(Y )


Φu =[ΦX ΦY

]







Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3






Ψu =partr

partuΦu



minimizeurisinRku


Implications











partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







partur

partmicro=minus

Nsumj=1


Φu +

(partr

partuΦu

)Tpartr

partuΦu

minus1

Nsumj=1


+

(partr

partuΦu

)Tpartr

partmicro



- Φupartur









partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







partur

partmicro= arg min

a||Φuaminus

partu


[(partr

partuΦu

)Tpartr

partuΦu

]minus1(partr

partuΦu

)Tpartr

partmicro



partu

partmicroin Φu


-partur

partmicro6= partur







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


ROB

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M







Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Offline

HDM

HDM

HDM

HDM

CompressROB Φ

ROM

Optimizer

Schematic

micro-space


RO

M

RO

M

RO

M

RO

M

RO

M

RO

M

RO

M









Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







Greedy Training




Material Basis









126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3








126 2436 7437 001







Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROB

RO

M

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO

M








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO








Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Compress

HDM

HDM

HDM

ROB Φ

ROM

Optimizer

Schematic

micro-space

HDM ROB

RO

M

RO

M

RO


RO

M

RO

M

HDM ROBR

OM

RO

M

RO


RO

M

RO













J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3










J (Φuur micro)












(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3








(u micro) =dJdmicro


parameters micro



J (Φuur micro)



















0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3













0 01 02 03 04 05 06 07 08 09 1

minus1

minus05

0

05


-Cp

InitialTarget


0

02

04

06

Dis

tan

ceT

ran

sver

seto

Cen

terl

ine






0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





0 5 10 15 20 25 3010minus15

10minus11

10minus7

10minus3

101


Ob

ject

ive

Fu

nct

ion







0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





0 20 40 60 80 100 120 140 16010minus18

10minus14

10minus10

10minus6

10minus2


Ob

ject

ive

Fu

nct

ion

HDM sample0

20

40

60

RO

Msi

ze





















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




















J (u micro)


c(u micro) ge 0

where










Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




Problem Setup

25

40








fextTu


2V0

r(u micro) = 0












kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3








kmicromicro







































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3

































Lagrangian




λ ge 0

























































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





















































t le 0












t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3











t le 0












t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3











t le 0












t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3











t le 0












t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3











t le 0






25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





25

40








fextTu


2V0

r(u micro) = 0












7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







7458s (450) 4018s (411) 8284s



1049s (64) 88s (9) 727s (56) 39s (3676)















Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3













Iteration 0 (1000)






Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Iteration 1 (977)














minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3













minimizeU micro

W(U micro)


U(x 0) = U(x)

partU


minimizeU micro

W(U micro)


U(x 0) = U(x T )

partU








Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Energy =94095839014e+00

Thrust =176604514000e-01

Energy =49475637668e+00

Thrust =250000000000e+00

Energy =46110004198e+00

Thrust =250000000000e+00


ShapeControl















8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3












8









minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





minimizemicro

f(micro)


minimizey

f(Φmicromicror)












d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







d2Jdmicrojdmicrok

=part2J



partu

partmicrok+

partu

partmicroj


+partu

partmicroj

T part2Jpartupartu

partu

partmicrok

minus partJpartu

partr

partu

minus1 [ part2r


part2r

partmicrojpartu

partu

partmicrok+

part2r

partmicrokpartu

partu

partmicroj+

part2r

partupartupartu


partmicrok

]where

partu

partmicroj=

partr

partu

minus1 partr

partmicroj


for image ofpartr

partu

minus1

partr

partu

minusT
















Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3














Acknowledgement





References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




References I









Wiley New York











References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




References II




















References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




References III

Fahl M (2001)








SIAM











References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3




References IV






Springer



PhD thesis Citeseer











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe











fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3







fextTu


2V0

r(u microp) = 0



Ke larr (microe)pKe









Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3





Implication for ROM
















where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3











where

D3ijklm =


(u φmmicro )(φi


u times φlu)tpqs














λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3












λ = λr

τ = arg minτge0













partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3








partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection










partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3








partJpartmicrok

=

sumjisinSk


microk

sumjisinSk

Hkj

Nodal Projection

microk =



(b) With projection

















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3















Implication for ROM








Introduction









Conclusion


Future Research

fdrm0

fdrm1

fdrm2

fdrm3

accelerating pde-constrained optimization...

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