formulation of a discontinuous galerkin method for unstructured … · 2017. 3. 3. · unstructured...

Formulation of a discontinuous Galerkin method for

unstructured causal grids in spacetime and linear dispersive

electromagnetic mediaReza Abedi

Mechanical, Aerospace & Biomedical Engineering

University of Tennessee Space Institute (UTSI) / Knoxville (UTK)

Saba Mudaliar

Air Force Research Laboratory

Comparison of DG and CFEM methods

Consider FEs for a scalar field and polynomial order p = 2

p dofDG/CFEM

1 4

2 2.25

3 1.78

4 1.56

5 1.44

High order polynomial DG competitive

Disjoint basis

1. Balance laws at the element level

2. More flexible h-, hp-adaptivity

3. Less communication between

elements

Better parallel performance

CFEM: transition element DG

Comparison of DG and CFEM methods:

Dynamic problems

Discontinuities are preserved or generated from smooth initial conditions!

t = 0,

smooth

solutio

nt > 0,

shock has

formed

Burger’s equation (nonlinear)

Global numerical oscillations COMSOL

2. Hyperbolic problems: resolving shocks / discontinuities

1. Parabolic & Hyperbolic problems: O(N) solution complexity

CFEM DG

O(N) O(N1.5) d = 2

O(N2) d = 3

Explicit solver

Example:

10x finer mesh (1000x elements in 3D)

Cost:

DG: 103x

CFEM: 106-107x

Spacetime Discontinuous Galerkin Finite Element method:

1. Discontinuous Galerkin Method

2. Direct discretization of spacetime

3. Solution of hyperbolic PDEs

4. Use of patch-wise causal meshes

A local, O(N), asynchronous solution scheme

Discontinuous Galerkin (DG) Finite

Element Methods

Weakly enforce conservation jump conditions

(e.g., Rankine–Hugoniot)

Can recover balance properties at the element level (vs global domain)

Support for nonconforming meshes and

Arbitrary changes in element polynomial orderno transition elements

needed

Arbitrary

change in size

and polynomial

order

Superior performance for resolving

discontinuities (discrete solution space better

resembles the continuum solution space)

Sample DG solutions

with no evident

numerical

artifacts numerical artifacts generally

spoil continuous FE solutions

in the presence of shocks

In contrast to conventional finite element methods, DG

methods use discontinuous basis functions and

Direct discretization of spacetime

Replaces a separate time integration;

no global time step constraint

Unstructured meshes in spacetime

No tangling in moving boundaries

Arbitrarily high and local order of

accuracy in time

Unambiguous numerical framework

for boundary conditions

shock capturing more

expensive, less accurate

Shock tracking in spacetime:

more accurate and efficient

Results by Scott Miller

http://www.personal.psu.edu/stm134/

Spacetime Discontinuous Galerkin (SDG)

Finite Element Method

Local solution property

O(N) complexity (solution cost scales linearly vs. number of elements N)

Asynchronous patch-by-patch solver

DG + spacetime meshing + causal meshes for hyperbolic problems:

Elements labeled 1 can be solved in

parallel from initial conditions; elements

2 can be solved from their inflow

element 1 solutions and so forth.

SDG

Time marching

Time marching or the use of extruded

meshes imposes a global coupling that

is not intrinsic to a hyperbolic problem

- incoming characteristics on red boundaries

- outgoing characteristics on green boundaries

- The element can be solved as soon as inflow

data on red boundary is obtained

- partial ordering & local solution property

- elements of the same level can be solved

in parallel

Tent Pitcher:

Causal spacetime meshing

causality constraint

tent–pitching sequence

Given a space mesh, Tent Pitcher

constructs a spacetime mesh such

that the slope of every facet on a

sequence of advancing fronts is

bounded by a causality constraint

Similar to CFL condition, except

entirely local and not related to

stability (required for scalability)

time

Tent Pitcher:

Patch–by–patch meshing

meshing and solution are interleaved

patches (‘tents’) of tetrahedra are solve immediately O(N) property

rich parallel structure: patches can be created and solved in parallel

tent–pitching sequence

Advantages of Spacetime discontinuous Galerkin

(SDG) Finite element method

1. Arbitrarily high temporal order of accuracy

• Achieving high temporal orders in semi-discrete methods (CFEMs and

DGs) is very challenging as the solution is only given at discrete times.

• Perhaps the most successful method for achieving high order of accuracy

in semi-discrete methods is the Taylor series of solution in time and

subsequent use of Cauchy-Kovalewski or Lax-Wendroff procedure (FEM

space derivatives time derivatives). However, this method becomes

increasingly challenging particularly for nonlinear problems.

• High temporal order adversely affect stable time step size for explicit DG

methods (e.g. or worse for RKDG and ADER-DG methods).

• Spacetime (CFEM and DG) methods, on the other hand can achieve

arbitrarily high temporal order of accuracy as the solution in time is

directly discretized by FEM.

2. Asynchronous / no global time step

• Geometry-induced stiffness results from simulating domains with

drastically varying geometric features. Causes are:

• Multiscale geometric features

• Transition and boundary layers

• Poor element quality (e.g. slivers)

• Adaptive meshes driven by FEM discretization errors.

Time step is limited by smallest elements for explicit methods

Explicit: Efficient / stability concerns

Implicit: Unconditionally stable

Time-marching methods

Improvements:

Implicit-Explicit (IMEX) methods increase the time step by geometry

splitting (implicit method for small elements) or operator splitting.

Local time-stepping (LTS): subcycling for smaller elements enables

using larger global time steps

SDGFEM Small elements locally have smaller progress in

time (no global time step constrains)

None of the complicated “improvements” of time

marching methods needed

SDGFEM graciously and

efficiently handles highly

multiscale domains

A. Taube, M. Dumbser, C.D. Munz

and R. Schneider, A high-order

discontinuous Galerkin method with

time accurate

local time stepping for the Maxwell

equations, Int. J. Numer. Model.

2009; 22:77–103

2. Asynchronous / no global time step

tim

e

3. Spacetime grids and Moving interfaces

• Problems with moving interfaces:

* Solid-fluid interaction * Non-linear free surface water waves

* helicopter rotors /forward fight * flaps and slats on wings and piston engines

• Derivation of a conservative scheme is very challenging:

• Even Arbitrary Lagrangian Eulerian (ALE) methods do not automatically

satisfy certain geometric conservation laws.

• Spacetime mesh adaptive operations

Enable mesh smoothing and adaptive

operations Without projection errors of

semi-discrete methods.

4. Adaptive mesh operations

• Local-effect adaptivity: no need for reanalysis of the entire domain

• Arbitrary order and size in time:

• Adaptive operations in spacetime:

Example:

LTS by

Dumbser,

Munz, Toro,

Lorcher, et. al.

SDGADER-DG with LTS

- Front-tracking better than shock capturing

- hp-adaptivity better than h-adaptivity

Sod’s shock tube problem

Results by Scott Miller

Shock capturing: 473K elements Shock tracking: 446 elements

http://www.personal.psu.edu/stm134/

4. Adaptive mesh operationshighly multiscale grids in spacetime

These meshes for a crack-tip wave scattering problem are generated by

adaptive operations. Refinement ratio smaller than 10-4.

click to play movie click to play movie

Color: log(strain energy); Height: velocity Time in up direction

Shock visualization link

cracktip-soln-top.mp4movies/cracktip-spacetime-side.mp4https://www.youtube.com/watch?v=hqvGWd0S_rwhttps://www.youtube.com/watch?v=6kh4fp5fJt0shockVisualization_crack1e10.mov

5. Riemann-solution free scheme

• Reimann solutions are often complicated, expensive, and even difficult to

derive particularly for nonlinear and anisotropic materials.

Example: Simple linear elastodynamic problem

( regions III and IV)

• Riemann solutions required for inter-element noncausal

boundaries

• If interior facets are eliminated we obtain a Riemann-

solver free method

• Riemann-solution free scheme can also be more efficient

active elements

predecessor

elements

active element =

list of int. cells

predecessor

elements

integration

cells

Single-element

patch

Outstanding base properties of serial mode

O(N) Complexity

Favors highest polynomial order

Favors multi-field over single-field FEMs

Asynchronous

Nested hierarchical structure for HPC:

1.patches, 2.elements/cells, 3.quadrature points

4,5.rows & columns of matrices

Domain decomposition at patch level:

Near perfect scaling for non-adaptive case

95% scaling for strong adaptive refinement

Diffusion-like asynchronous load balancing

Multi-threading & Vectorization UIUC

Multi-threading

LU solve

Number of OpenMP ThreadsW

all

Tim

e (

s)

assembly

physics

more efficient

CG: BatheSDG

6. parallel computing (asynchronous structure)

http://mechanical.illinois.edu/directory/faculty/r-haber

Other Applications

Multiscale & Probabilistic Fracture

Probabilistic crack nucleation.

Exact tracking of crack interfaces.

Structural Health Monitoring

Multiscale and noise free

solution of scattering enables

detection of defects at

unprecedented resolutions.

Dynamic Contact/Fracture

SDGFEM eliminates common

artifacts at contact transitions

High resolution slip-stick-separation waves

Click here to play movie




movie

YouTube link

YouTube link

YouTube link

YouTube linkYouTube link

movie

Fluid mechanics: Euler’s equation Hyperbolic thermal model Solid mechanics

movies/Multiscale_ForwardAnalysis.mp4movies/brake2e3_tf_420_720_HS500_cm3e6_DS_400.mov.mp4movies/CircularCrack_Path_Damage.mp4movies/SolutionDependentCrackPath.mp4movies/shockVisualization_crack1e10.mp4https://www.youtube.com/watch?v=EX2Vsw50ygMhttps://www.youtube.com/watch?v=IxoS7_bBpHchttps://www.youtube.com/watch?v=sTYTqe0trLUhttps://www.youtube.com/watch?v=NxGapy31rOQhttps://www.youtube.com/watch?v=X5_fRZlBhSwMachReflection.movmovies/MachReflection.mp4

Time Domain Electromagnetics:

Discontinuous Galerkin Methods

• Time Domain (TD) vs. Frequency Domain (FD) solvers:

• Transient problems TD

• Steady state and frequency response:

• Small problem size FD

• Large problem size TD (TDDG ≈ O(N)FD = O (Na), a ≥ 1.5

• Material nonlinearities are better modeled in TD

• Entire spectrum obtained by one TD simulation of

broadband signalBusch:2009

Electromagnetics formulation in spacetime:

Fields, fluxes and sources

• Electromagnetic fields

• Electromagnetic (total) flux densities

• Constitutive equation

,

Inductive Conductive Dispersive

Spacetime electromagnetic flux

Spacetime electromagnetic flux density

Acting on time-like (“vertical”) boundaries

Spacetime flux

Acting on space-like (“horizontal”) boundaries

Comparison with d-form fluxes in spacetime

(d – 1)-form fluxes in spacetime for EM problem

Comparison with other differential form

formulation of EM problem

Source: Peter Russer, Exterior Differential Forms in Teaching Electromagnetics, 2004.

Our electromagnetic fields have an extra dt

Balance laws in spacetime

Source terms: current and charge densities

Integral form of Maxwell equations

Spacetime electromagnetic flux density

(f is a 1-form)

Stokes’ theorem

Strong form of Maxwell equations

(Provides PDEs)

(Provides Boundary conditions and

interface jump conditions)

Diffuse part

Jump part

Strong Form to weak form and FEM

formulation

a. Maxwell’s balance laws (diffuse part of the strong form)

b. Jump equations (BCs, interfaces, etc.)

- Weighted residual statement (WRS)

- Weak statement (WKS)

Stokes’ theorem

Auxiliary Differential Equations:

Elimination of convolutions in TD

• Electromagnetic equations in FD

• Frequency domain constitutive relation for an isotropic media:

• Pull-back in time domain involves a convolution:

• Solution: Elimination of convolution integral by introducing additional fields.


• Assume electrical permittivity uses a Debye dispersion model,

• By the introduction of Auxiliary Field P we get,

• That is convolution term is eliminated by the addition of the field P(k).


Examples

Time Domain Electromagnetics SDG:

Formulation of PML for dispersive media

• Perfectly matched layer for bi-anisotropic dispersive mediaPML stretching dispersive relations

2 Levels of ADEs needed in TD

ConductiveInductive DispersiveLevel

Base

L1

L2

P

M

L

Constitutive model

The recursive formulation of ADEs enables automatic formation of PML

equations for dispersive media in TD

Teixeira, Chew, General closed-form PML constitutive

tensors to match arbitrary bianisotropic and dispersive

linear media, IEEE Microw. Guid. Wave Lett., 1998


Balance of energy / proof of numerical stability

• Energy stability proof (dissipative method)

• Sketch of the proof:

• Bilinear form from the weighted residual statement:

• By showing

and manipulation of bilinear form we can show:

(interchanging * and ^ on

fields and flux densities)


Balance of energy / proof of numerical stability

That is

But,

= 0 outflow

> 0 inflow

> 0 non-causal

(using Riemann

values, etc.)

Numerical results

Convergence studies:

Smooth solution (Elastodynamics)

• Smooth analytical solution

(harmonic function)

• 1D problem

• D: Elastodynamic numerical energy dissipation.

High convergence rates are achieved due to high spatial and temporal orders of

elements.

Convergence plot

2D scattering problem / Nonadaptive meshing

𝐻𝑧 = cos(𝜋

2

𝑥

𝑑𝑥)cos(

𝜋

2

𝑦

𝑑𝑦)

Electric permittivity:

𝜀𝑖 = 1𝜀𝑜 = 10

Magnetic permeability:

𝜇 = 1

Initial condition:

Transverse electric formulation

movies/hybrid_2016_10_10_NA3Sq001_p3.movmovies/hybrid_2016_10_10_NA3Sq001_p3_x264.mp4

2D scattering problem adaptive meshing

t = 0.18 t = 0.50t = 0.70

t = 0.95 t = 1.30 t = 2.35

movies/sq3_tol1e10.mp4movies/sq3_tol1e10.mov

Comparison of Adaptive / nonadaptive schemes

Nonadaptive Adaptive

Initial spatial

Mesh

25K elements 46 elements

movies/EM_CompAdaptNonAdapt.png

Comparison of Adaptive / nonadaptive schemes

Nonadaptive Adaptive

93 Hours

Numerical dissipation 8x10-472 Hours

Numerical dissipation 1x10-4

Initial Mesh:

83K elements

Even finer nonadaptive

simulation

> 500 Hours

Numerical dissipation 2x10-4

(still larger than adaptive one)

Characterization of

Dispersive media

S-parameters for a slab

Material Properties

Transmission / reflection coefficients

Unit cell to dispersive response

(S-parameters) retrieval method

1. (Computationally) solve for t, r:

2. Inverse solution for Z, c

(non-uniqueness)s

Computation of S-parameters

Frequency Domain (FD)

• For each frequency w one

FD simulation is done

• For high frequencies very

fine meshes are required.

• Solution is globally coupled

(Elliptic PDE)

Large problem size FD = O (Na), a ≥ 1.5

Computation of S-parameters

Time Domain (TD)

Gaussian pulse

Properties:

• Ultrashort duration pulses

Broadband frequency content

• (Almost) zero solutions for

initial and final times

facilitates Fourier analysis

Time Domain vs. Frequency Domain

• Time Domain (TD) vs. Frequency Domain (FD) solvers:

• Transient problems TD

• Steady state and frequency response:

• Small problem size FD

• Large problem size TD (TDDG ≈ O(N) FD = O (Na), a ≥ 1.5

• Material nonlinearities are better modeled in TD

• Entire spectrum obtained by one TD simulation of

broadband signalBusch:2009

highly multiscale grids in spacetime

These meshes for a crack-tip wave scattering problem are generated by

adaptive operations. Refinement ratio smaller than 10-4.

click to play movie click to play movie

Color: log(strain energy); Height: velocity Time in up direction

Shock visualization link

Motivation:

Adaptive SDG method to model dispersive mediatim

e

movies/cracktip-soln-top.mp4movies/cracktip-spacetime-side.mp4https://www.youtube.com/watch?v=hqvGWd0S_rwhttps://www.youtube.com/watch?v=6kh4fp5fJt0shockVisualization_crack1e10.mov

Test problem:

Retrieval method for solid slab

𝐸 = 1, = 1

Initial space mesh

𝐸 = 1, = 1𝐸 = 10, = 1

Movie: Solution

movies/inc_Test_x264.mp4movies/inc_Test.movmovies/inc_TestSln.mov

Test problem:

Retrieval method for electromagnetics problem

Transmission and reflection coefficients

Conductive mid-layer

Test problem:


Conductive mid-layer

Test problem:


3-layer set-up

Retrieval method for 2D unit cells

Ying Wu, Yun Lai, and Zhao-Qing Zhang. Effective

medium theory for elastic metamaterials in two

dimensions. Physical Review B, 2007.

lead

rubber

epoxy

Spatial mesh for the SDG method

Retrieval method for 2D unit cells

Movie: Solution, Mesh
movies/inc_LRE_mesh_n2.mp4movies/inc_LRE_mesh_n2.movmovies/inc_LRE_sln_n2.mp4movies/inc_LRE_sln_n2.movmovies/inc_LRE_sln_n2.movmovies/inc_LRE_mesh_n2.mov

formulation of a discontinuous galerkin method for unstructured … · 2017. 3. 3. · unstructured...

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