partitioned modeling of coupled problems and … · gert rebel (goodyear tire), yasu miyazaki...

Partitioned Modeling of Coupled Problems and Applications: New Interpretations ofLagrange Multipliers

Carlos A. Felippa and K. C. ParkCenter for Aerospace StructuresUniversity of Colorado at BoulderBoulder, CO 80309

WCCM-VII, Los Angeles, 17-21 July 2006.

Partitioned modeling of coupled systems is to to transform a tightly coupled model

into a loosely coupled system (pleasing to your eyes?):

Classification of Coupling in Multiphysics

Tight Coupling:Through differing physical phenomena

occupying the same space;

Via hierarchical and/or multi-level variables;

Loose Coupling:Through constitutive relations(?);

Along spatial interfaces - Today’s topic

Coupled physics Roger Ohayon (CNAM, Paris), Denis Caillerie (Grenoble), Michael Ross(CU)

Contact-friction problems Gert Rebel (Goodyear Tire), Yasu Miyazaki (Nihon University, Japan) Jose Gonzales (Sevilla, Spain), Luis Solano (Sevilla)Reduced-order modeling and FEM-related work Damijan Markovic(ENS Cachan), Yonghwa Park (Samsung), Kendall Pierson (Sandia National Lab)Membranous space structures Hiraku Sakamoto(CU and MIT), Yasu Miyazaki (Nihon University) Sebastian Kreissl (Tech. Univ. Minich)Health monitoring of structural systems Ken Alvin (Sandia National Lab), Haru Namba (Shimizu Corp, Japan) Greg Reich (Wright Aeronautical Lab), Xue Yue (Mathlab)MEMS Yonghwa Park (Samsung), Gyeongho Kim (KAIST), Timothy Straube(NASA/Johnson)Structural optimization E.I. Jung (KAIST), Youn-sik Park (KAIST)

Outline of Presentation:

A précis of the method of classical Lagrangemultipliers(the CLM method)

Can we localize Lagrange’s multipliers?

Properties of the method of localized Lagrange multipliers(LLM)

Interface treatment via the LLM frames

Treatment of interface heterogeneities

Applications:Inverse problems for damage identificationContact-impact problemsLocalized vibration control strategyReduced-order model synthesisOptimization of vibration problemsMulti-physics simulationBEM-FEM coupling

La méthode de quantités indéterminées andLagrange’s philosophy on mechanics:

“On ne trouvera point de Figures dans cet Ouvrage. Les methodsque j’y expose ne demandent ni constructions, ni raisonnemensgéomtriques ou méchaniques, mis seulement des opérationsalgébriques, assujetties à une marche réguliere et uniforme. . .”(One will find no pictures in this volume. . .)- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

An old Asian saying:

“A picture is worth a thousand words (百聞不如一見)”may offer additional insight into the method of Lagrangemultipliers. . .

Suppose that you are asked to model a Navajodancing with the following night chant:

“The Earth touches me And I touch the Earth That is how Man can walk.”

Formulation according to Lagrange’s world ofmechanics: “Les methods que j’y expose. . ., mis seulementdes opérations algébriques . . .”

Done and what else is there to be discussed?

The method of classical Lagrange multipliers (CLM) yields a unique constraint condition when involving only two interfaces:

(7)

Example: the use of all three constraints leads to one redundant constraints, causing singularity.

However, the CLM method yields either non-unique or redundant constraint conditions when involving more than two interfaces:

(8)

Example: the use of 2 of 3 constraints leads to non-singular constraint equations, but they are not unique; in fact, there are 3 possible non-singular constraint pairs from which one must choose one particular pairs.

Classical Lagrange multipliers - cont’d

(9)

Classical λ-Method (the CLM method) connects directly from one substructural interface node to that of another interface node:

Note that one has to choose 3 rank-sufficient conditions fromamong 6 possible conditions.

Modeling of Multiply Connected Truss Elements forInverse Problems

Observe that node 27 consists of 8 elements. Model update for node 27 must consider all of them.

When model update or damage is detected in node 27,it could be all or some or just one of the eight elements.

If conventional partitioning that would befit the CLM method were employed,

What interface procedure does the finite element method utilize when assembling elements and/or partitioning the assembled structure ?

Answer: The FEM assembly and partitioning do not utilizethe method of classical Lagrange multipliers!

What does the FEM assembly and partitioning utilize then?

Answer: It utilizes the method of localized Lagrange multipliers.

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A précis of the method of classical Lagrange multipliers(CLM)






The Earthtouches me, And I touch

the Earth

That’s how man can walk.

Suppose the earth is humanized as Mother Earth, and express the level of touches figuratively by the size of a Greek letter:

Now you are asked to model a Navajo dancing with thefollowing night chant:

“The Earth touches me And I touch the Earth That is how Man can walk.”

Which of the following two cartoons would lead to a good model?

Global interface

Localized interface

Classical Notion of Lagrange Multipliers

Partitioning node, f, is lost!

f

New Interpretation of Lagrange Multipliers

Partitioning node, f, is preserved!

f

Assembled

Classical Partitioning

Localized Partitioning

Localization of Classical Lagrange Multipliers

Split!

Localization is achieved by introducing a frame node, f

Split!

Loc

aliz

atio

n Pr

oces

s

Loca

lizat

ion

Fram

e Fe

atur

es

Why increase the number of Lagrange multipliers?

Excerpts from what William Hamilton read at RoyalSociety of London in 1834:

``While science is advancing in one direction by theimprovement of physical laws, it may advance inanother direction also by the invention ofmathematical methods.’’ …

``This difficulty is therefore at least transferred fromthe integration of many equations of one class to theintegration of two of another.

f

Why increase the number of Lagrange multipliers? -- cont’d

``Even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from ...’’

Here, Hamilton refers to the canonical transformation of N equations of motion to 2N equations:

The localized modeling, as in Hamilton’s canonical equations, also increases the interface unknowns from N to 2N multipliers for two interfaces. Now the question is:

Is the localized modeling just an intellectual pleasure(exercise) or does it offer also a practical utility?

Localized Lagrange Multipliers

A visualization of FEM assembly and connectivityrelation in every FEM code:

Partitioning the assembled 4 rod-element system involvesfour constraint conditions as shown below:

Observe all four partitioned nodes refer to the one global node u5.These four constraint equations are unique and rank-sufficient.

For two interfaces, the number of multipliers with the localizedλ-method is twice that of the classical λ-method. Does that meanthe computational cost of the localized λ-method would be doublethat of the classical λ-method ?

Answer: Essentially, the computational cost of the two methodsare equivalent. Here’s why:

Outline of Presentation:A precis of the method of classical Lagrange multipliers(CLM)


Properties of the method of localized Lagrangemultipliers(LLM)




We will focus on two issues:

(1) Non-Matching Interface Grids, and (2) Heterogeneous Interfaces.

Advantages of localized Lagrange multipliers? - continued

The continuum expressions of the two constraint functional indicate that all the interface variables have to be discretized, including the Lagrange multipliers, with the exception of matching node interfaces.

Let us now examine the case of non-matching interfaces.

(A mixed interpolation!)

Mixed formulations can lead to mixed results! Or live with the LBB hoops . . .

Localized λ-method, if co-located pairs of (u(j), λ(j)) are used, transforms the mixed method into a displacement-like method, viz., into that of the interpolation of the frame displacement, uf.

Modeling of Multiply Connected Truss Elements

Observe that node 27 consists of 8 elements. Model update for node 27 must consider all of them.

When model update or damage is detected in node 27,it could be all or some or just one of the eight elements.

If conventional partitioning were employed,

If localized partitioning were used,




Interface treatment via the LLM framesTreatment of interface heterogeneities


Loca

lizat

ion

Fram

e Fe

atur

es

Essential Features of Present Interface Algorithm

Determine the frame nodes via a localized discretization procedure, whereas in classical procedure one must interpolate the interface forces globally.

Localized Global (classical)

Algorithm for Determination of Frame Nodes

Step 1: Compute the interface loads that correspond to a constant stress stateof the subdomains.

Interface loads (not scaled) that correspond to a constant stress state

Algorithm for Determination of Frame Nodes -- cont'd

Step 2: Map the interface forces (Lagrange multipliers) onto the frame.

Mapping Interface loads onto the frame.

Algorithm for Determination of Frame Nodes -- cont'd

Step 3: Compute the forces and moments along the frame at arbitrary frame locations.

Computing Forces and Moments at a point on the frame.

Nonmatching-Node Interface with Linear and QuadraticDiscretizations





Treatment of interface heterogeneitiesApplications:

Inverse problems for damage identificationContact-impact problemsLocalized vibration control strategyReduced-order model synthesisOptimization of vibration problemsMulti-physics simulationBEM-FEM coupling

Localization Example for Engine Support Structure Problem

Damage indication based on global flexibility changes

Damage indication based localized flexibility changes

Scale Model of Nuclear Containment Vessel

Simplified Model of the Test Article

Damage Indication based on global nodal stiffness changes

Damage Indication based on localised flexibility changes

Three Dimensional Contact Patch Test Problem


A precis of the method of classical Lagrange multipliers(CLM)






Hybrid Localized/Global Active Vibration Control

Hybrid Localized/Global Active Vibration Control- cont’d

Design Optimization via Structural Modification

Acoustic Fluid-Structure Interactions (PhD Thesis by Michael Ross)Pr

oble

mM

odel

Gravity Dam FRFOutput DOF Output DOF

Input DOFInput DOF

Nor

mal

ized

Vib

ratio

n M

agni

tude

Alone

Assembled

Nor

mal

ized

Vib

ratio

n M

agni

tude

Nor

mal

ized

Vib

ratio

n M

agni

tude

Nor

mal

ized

Vib

ratio

n M

agni

tude

Compared with

FEM-Boundary Integral Coupling via Non-matching Interfaces

Basic TheoryFelippa, C. A. and Park, K. C., “Synthesis Tools for Structural Dynamics and Partitioned Analysis of CoupledSystems,” Multi-physics and Multi-scale Computer Models in Non-linear Analysis and Optimal Design of EngineeringStructures Under Extreme Conditions (NATO ARW PST.ARW980268), ed. A. Ibrahimbegovic and B.Brank, University of Ljubliana, 2004, 50-110.

K. C. Park, C. A. Felippa and G. Rebel, (2002), ”A Simple Algorithm for Localized Construction of NonmatchingStructural Interfaces,” International Journal of Numerical Methods in Engineering, 2002; 53:2117-2142.

Felippa, C. A. and Park, K. C., “The construction of free-free flexibility matrices for multilevel structural analysis,”Computer Methods in Applied Mechanics and Engineering, 191(19-20) (2002) 2111-2140.

K. C. Park, C. A. Felippa and G. Rebel, (2001), ”Interfacing Nonmatching FEM Meshes: The Zero Moment Rule,”in: Trends in Computational Structural Mechanics, ed. by W. A. Wahl, K.-U. Bletzinger and K. Schweizerhof,CIMNE, Barcelona, Spain, 2001, p.355-367.

Park, K. C. and Felippa, C. A., “A Variatioanl Principle for the Formulation of Partitioned Structural Systems,”International Journal of Numerical Methods in Engineering, vol. 47, 2000, 395-418.

Felippa, C. A., Park, K. C. and Justino, M.R., “The Construction of Free-Free Flexibility Matrices as GeneralizedStiffness Inverses,” Computers & Structures, vol.68 (1998), 411-418.

Park, K. C. and Felippa, C. A., “A Variational Framework for Solution Method Developments in StructuralMechanics,”Journal of Applied Mechanics, March 1998, Vol. 65/1, 242-249.

C. A. Felippa and K. C. Park, “A direct flexibility method,” Computer Methods in Applied Mechanics andEngineering, 149 (1997) 319-337.

Parallel ComputingGumaste, Udayan and Park, K. C. (2000), “Interfacing an explicit nonlinear finite element code with an implicitparallel solution algorithm,” to be presented at the International Congress on Computational Engineering Sciences,August 5-8, 2000, Los Angeles, CA.

Gumaste, Udayan, Park, K. C. and Alvin, K. F. , “A Family of Implicit Partitioned Time Integration Algorithmsfor Parallel Analysis of Heterogeneous Structural Systems,” Computational Mechanics: an International Journal,24 (2000) 6, 463-475.

Park, K. C., Gumaste, Udayan, and Felippa, C. A., “A Localized Version of the Method of Lagrange Multipliersand its Applications,” Computational Mechanics: an International Journal, 24 (2000) 6, 476-490.

Park, K. C., Justino, M. R, Jr. and Felippa, C. A., “An Algebraically Partitioned FETI Method for ParallelStructural Analysis: Algorithm Description,” International Journal of Numerical Methods in Engineering, 40,2717-2737 (1997).

Justino, M. R, Jr., Park, K. C. and Felippa, C. A., “An Algebraically Partitioned FETI Method for Parallel StructuralAnalysis:Implementation and Numerical Performance Evaluation,” International Journal of Numerical Methodsin Engineering, 40, 2739-2758 (1997).

System Identification and Inverse ProblemsXue Yue and K. C. Park (2002), ”Modeling of Joints and Interfaces,” in : Modeling and Simulation-Based LifeCycle Engineering, K. Chong, S. Saigal, S. Thynell and H. Morgan (des.), Spon Press, London, pp.60-75.

Reich, G.W., Park, K. C. and Namba, H. (2001), ”Health Monitoring of a Reinforced Concrete ContainmentVesselby Localized Methods,” Proc. of the Third International Workshop on Structural Health Monitoring, TechnomicPublishing Company, Inc., 2001

Reich, G.W. and Park, K. C. (2001), “A Theory for Strain-Based Structural System Identification,” in: Journal ofApplied Mechanics, 68(4), 521-527.

Reich, G.W. and Park, K. C., “On the Use of Substructural Transmission Zeros for Structural Health Monitoring,”AIAA Journal, Vol. 38, No. 6, 2000, 1040-1046.

Alvin, K. F. and Park, K. C., “Extraction of Substructural Flexibilities from Global Frequencies and Mode Shapes,”AIAA Journal, vol. 37, no.11, 1999, p. 1444-1451.

Park, K. C., Reich, G. W. and Alvin, K. F. “Structural Damage Detection Using Localized Flexibilities,” Journalof Intelligent Material Systems and Structures, Vol. 9, No. 11, 1998, pp. 911-919.

Park, K. C. and Felippa, C. A., “A Flexibility-Based Inverse Algorithm for Identification of Structural JointProperties,” to appear in ASME Symposium on Computational Methods on Inverse Problems, 15-20 November1998, Anaheim, CA.

Reich, G. W. and Park, K. C., “Structural Health Monitoring via Structural Localization,” Proc. 1998 AIAASDM Conference, Paper No. AIAA-98-1892, April 20-24 1998, Long Beach, CA.

Park, K. C., Reich, G. W. and K. F. Alvin, “Damage Detection Using Localized Flexibilities,” in : StructuralHealth Monitoring, Current Status and Perspectives, ed. F-K Chang, Technomic Pub., 1997, 125-139.

Coupled ProblemsPark, K. C., Felippa, C. A. and Ohayon, R., “Reduced-Order Partitioned Modeling of Coupled Systems: Formulationand Computational Algorithms,” Multi-physics and Multi-scale Computer Models in Non-linear Analysisand Optimal Design of Engineering Structures Under Extreme Conditions (NATO ARW PST.ARW980268), ed. A.Ibrahimbegovic and B. Brank, University of Ljubliana, 2004, 267-289.

Park, K. C., Felippa, C. A. and Ohayon, R., “Partitioned Formulation of Internal Fluid-Structure Interaction Problemsvia Localized Lagrange Multipliers,” Computer Methods in Applied Mechanics and Engineering, 190(24-25),2001, 2989-3007.

Park, K.C., Felippa, C. A. and Ohayon, R. (2001), “Localized Formulation of Multibody Systems,” in: ComputationalAspects of Nonlinear Systems with Large Rigid Body Motion (ed. J. Ambrosio and M. Kleiber), NATOScience Series, IOS Press, p.253-274.

Contact-Impact ProblemsY. Miyazaki and K. C. Park, ”A formulation of conserving impact system based on localized Lagrange multipliers,”to appear in International Journal of Numerical Methods in Engineering, 2006.

G. Rebel, K. C. Park and C. A. Felippa (2002), ”A Contact Formulation Based on Localised Lagrange Multipliers:Formulation and Application to Two-dimensional Problems,” International Journal of Numerical methods inEngineering, 2002; 54:263-297.

G. Rebel and K. C. Park, Application of the Localised Lagrange Multiplier Method to a 3D Contact Patch TestProc. 2002 AIAA SDM Conference, Paper No. AIAA-2002-1577, 22-26 April 2002, Denver, CO.

Vibration ControlH. Sakamoto, K. C. Park, and Y. Miyazaki, “Distributed and localized active vibration isolation in membranestructures, submitted to Journal of Spacecraft and Rockets, 2005.

Hiraku Sakamoto, K.C. Park and Yasuyuki Miyazaki, ”Distributed Localized Vibration Control of MembraneStructuresUsingPiezoelectricActuators,”PaperNo. AIAA-2005-2114, Proc. the 46th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference (SDM), 18-21 April 2005, Austin, TX.

Park, K. C., Kim, N. I., and Reich, G.W., ”A Theory of LocalizedVibration Control via PartitionedLQRSynthesis,”Paper No. 3984-63, Proc. 2000 Smart Structures and Materials Conference: Mathematics and Control in SmartStructures, Newport Beach, CA, March 6-9, 2000.

Reduced-Order ModelingD. Markovic and K. C. Park, Reduction of substructural interface degrees of freedom in flexibility-based componentmode synthesis, submitted to International Journal of Numerical Methods in Engineering, 2005.

K. C. Park, ”Partitioned formulation with localized Lagrange multipliers and its applications,” in: StructuralDynamics (Eurodyn 2005), Millpress, Roterdam, 2005, pp. 67-76.

D. Markovic and K. C. Park, ”Reduction of Interface Degrees of Freedom in Flexibility-Based Component ModeSynthesis,” Proc. 5th EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7-12,2005, pp. 900-907.

Park K. C. and Park, Yong Hwa, ”Partitioned Component Mode Synthesis via A Flexibility Approach,” AIAAJournal, 2004, vol.42, no.6, 1236-1245.

BEM-BEM and BEM-FEM ModelingJ. A. Gonz´alez, K. C. Park and C. A. Felippa, FEM and BEM coupling in elastostatics using localized Lagrangemultipliers, submitted to International Journal of Numerical Methods in Engineering, 2005.

J. A. Gonz´alez, K. C. Park and C. A. Felippa, ”Partitioned formulation of frictional contact problems,” to appearin Comm. Num. Meth. Engr., 2005.

MISC TopicsEui-Il Jung, Youn-Sik Park and K. C. Park, ”Structural Dynamics Modification via Reorientation of ModificationElements, Finite Element Analysis and Design, 42(1),2005, 50-70.

Park, Y.H and Park, K. C., “Anchor Loss Evaluation of MEMS Resonators - I: Energy Loss Mechanism throughSubstrate Wave Propagation,” Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, 238-247.

Park, Y.H and Park, K. C., “Anchor Loss Evaluation of MEMS Resonators - II: Coupled Substrate-REsonatorSimulation and Validation,” Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, 248-257.

Park, K. C., “Partitioned Solution of Reduced Integrated Finite Element Equations,” Computers & Structures, 74(2000) 281-292.

Conclusions:

The underlying concept of the method of localized Lagrangemultipliers (the LLM method) and several applications arepresented.

Its construction is unique for multiply constrained cases;

If the interface nodal variables and localized multipliers are colocated, the interface functional is transformed into a displacement functional, hence bypassing the mixed formulation challenge.

It offers a natural framework for heterogeneity regularization.

Conclusions - continued:

It should noted that the coupling via the LLM method has beenapplied so far to spatial interfaces, or loosely coupled systems.

In order to apply the LLM methods to treat strongly coupled (e.g., field coupling, constitutive, multi-level, etc.)problems, the coupling phenomena have to be transformed into a loosely coupled problems. We are presently engaged to develop transformation procedures that can recast strong couplings into loose couplings. (Your participation is welcome!)

partitioned modeling of coupled problems and … · gert rebel (goodyear tire), yasu miyazaki...

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