Download pdf - Higher-Order Discrete Calculus Methods · Higher-Order Discrete Calculus Methods. Higher-Order Discrete Calculus 22nd Dundee B. Perot 06/28/2007 Realistic Parallel, Moving Mesh, Complex

Higher-Order Discrete Calculus

22nd Dundee

B. Perot 06/28/2007

J. Blair Perot

V. Subramanian

Higher-Order Discrete Calculus Methods


22nd Dundee

B. Perot 06/28/2007

Realistic

Parallel, Moving Mesh, Complex Geometry, …

Slide 1Slide 1

Practical,Practical, CostCost--effective, Physically Accurateeffective, Physically Accurate

��


22nd Dundee

B. Perot 06/28/2007

Context

DC Methods are a subset of many other classical approaches

Slide 2Slide 2

Finite Difference Finite Element

Finite Volume Meshless

Mimetic SOMMimetic SOM

StaggeredStaggered

Edge/FaceEdge/Face

Natural Natural NeighborsNeighbors

Discrete Calculus Methods


22nd Dundee

B. Perot 06/28/2007

Exact Discretization

Solution:Solution:Requires Approximations/ErrorRequires Approximations/Error

Discretization Approximation

Slide 3Slide 3

•• But all approximation errors occur in the constitutive But all approximation errors occur in the constitutive equations (in material properties).equations (in material properties).

•• All numerical errors appear with the modeling errors.All numerical errors appear with the modeling errors.

Discretization:Discretization:Continuous PDE => Finite Dimensional Matrix ProblemContinuous PDE => Finite Dimensional Matrix Problem

This Can Be Done ExactlyThis Can Be Done Exactly

≠


22nd Dundee

B. Perot 06/28/2007

Example

The Physical (Continuous) System

Physical Equation (Heat Equation)Physical Equation (Heat Equation)

Components of the Physical EquationComponents of the Physical Equation

( )cTk T

t

ρ∂= ∇ ⋅ ∇

∂

0it

∂ + ∇ ⋅ =∂

q Conservation of EnergyConservation of Energy

T= ∇g Definition of GradientDefinition of Gradient

PhysicsPhysics

MathMath

k= −q g Fourier’s LawFourier’s Law

i cTρ= Perfectly Caloric MaterialPerfectly Caloric MaterialMaterial Material ApproximationApproximation

Slide 4Slide 4


22nd Dundee

B. Perot 06/28/2007

The Numerical (Discrete) System

Example

Exact Discretization of Physics and CalculusExact Discretization of Physics and Calculus1| | 0n n

fc c ff

idV idV dt dA+ − + ⋅ =�� q n ��

2 1n ned T T⋅ = −� g l

1 0n nc c f

I I Q+ − + =D ��

� e ng T= G

Numerically ExactNumerically Exact

Numerical ApproximationsNumerical Approximations

1 efQ M g= −�

2c nI M T=�

�

�

f

e

A

eLfQ k g= − �

�

c c nI cV Tρ=� �

Numerically ApproximateNumerically Approximate

Slide 5Slide 5


22nd Dundee

B. Perot 06/28/2007

Same Mesh – More Unknowns

Higher Order Discrete Calculus

Higher Order Exact GradientHigher Order Exact Gradient

2 1n nedged T T⋅ = −� g l

�

Applicable to Any Applicable to Any PolyhedronPolyhedron

Slide 6Slide 6

2

2 2 1 1 1( ) ( )

n

e e n n n nedge nd T T Tdl⋅ ⋅ = ⋅ − −� �t x g l t x x

efaceedges

dA Tdl× = �� n g t

[2]( )

edge

n

eedgee

face

dT

dTdl

dA

� �⋅� �

� �� ⋅ ⋅ =� ��

� �×� ��

�

��

�

g l

t x g l G

n g

nT

Td� �


22nd Dundee

B. Perot 06/28/2007

Higher Order Discrete Calculus

Higher Order Exact DivergenceHigher Order Exact Divergence

••Need one equation for each edge.Need one equation for each edge.

••Integrate over the very thin CVs surrounding the dual faces.Integrate over the very thin CVs surrounding the dual faces.

••Take the ‘thin’ limit carefully Take the ‘thin’ limit carefully –– so thin elements align.so thin elements align.

Slide 7Slide 7

( )0f

ffcnf e

edgefaces

dA dl∂ ⋅

∂ + ⋅ =�� q n

n q�

��

c

dx

f

[ ] 0f ff ledge edgecells faces

dA dl⊥ ⊥

⊥ ⊥∇ ⋅ + =� �� q q n�

�

oror


22nd Dundee

B. Perot 06/28/2007

Basis Functions not Required

Reconstruction

HaveHave

Assume qAssume q==--kkg is linear on g is linear on primary mesh cells.primary mesh cells.

Slide 8Slide 8

e

e

f

d

d

dA

� ��

⋅� �� ×�

�

�

�

g l

xg l

n g

�

( )f

f

ff

nf

e

dA

dA

d

∂ ⋅∂

� �⋅� ��

×� ��

�

�

�

q n

q n

q l

�

�

��

�

�

NeedNeed


22nd Dundee

B. Perot 06/28/2007

Results

Good Cost/Accuracy for ‘Smooth’ Solutions

Slide 9Slide 9


22nd Dundee

B. Perot 06/28/2007

Conclusions

•• Exact discretizationExact discretization of of PDEsPDEs is possible and is possible and stronglystrongly encouraged.encouraged.

• Excellent numerical/mathematical properties are NOT restricted to FE methods.NOT restricted to FE methods.

• Applicable to ANY polygon mesh (including ANY polygon mesh (including meshlessmeshless), no explicit basis functions, no need to ), no explicit basis functions, no need to build matrices.build matrices.

Slide 10Slide 10


22nd Dundee

B. Perot 06/28/2007

JCP Publications

• Subramanian, V., and Perot, J. B. Subramanian, V., and Perot, J. B. Higher Order Mimetic Higher Order Mimetic Methods for Unstructured MeshesMethods for Unstructured Meshes, , J. J. ComputComput. Phys.. Phys., , 219219, 68, 68--85, 200685, 2006

•• Perot, J. B., and Subramanian, V. Perot, J. B., and Subramanian, V. Discrete Calculus Discrete Calculus Methods for DiffusionMethods for Diffusion, , J. J. ComputComput. Phys.. Phys., , 224224 (1), 59(1), 59--81, 2007.81, 2007.

• Subramanian, V., and Perot, J. B. Subramanian, V., and Perot, J. B. A Discrete Calculus A Discrete Calculus Analysis of the KellerAnalysis of the Keller--Box Scheme and a Box Scheme and a Generalization of the Method to Arbitrary MeshesGeneralization of the Method to Arbitrary Meshes, , Accepted to Accepted to J. J. ComputComput. Phys.. Phys., 2007., 2007.

Slide 11Slide 11


22nd Dundee

B. Perot 06/28/2007


22nd Dundee

B. Perot 06/28/2007

Discrete Calculus Operators

Gradient Operator Gradient Operator GG ( )2 1n n nT T T− �G� � �

Divergence Operator Divergence Operator DD f ff

Q Q �

�� D

Slide 6Slide 6

Curl Operator Curl Operator CC e ee

s s �

�� C

Rotation Operator Rotation Operator RR f ff

U U �

�� R

T= −G D

T=C R

0 constantφ φ∇ = � =

{ } { }0n nT T c= � =G� �

( ) 0∇ ∇× =�

0=DC

( ) 0∇×∇ =

0=CG

Discrete Calculus Operators mimic Continuous Operators


22nd Dundee

B. Perot 06/28/2007

The Discrete Calculus Approach

Continuous vs. Discrete System

PhysicsPhysics NumericsNumerics

0it

∂ + ∇ =∂

q�

T= ∇g

PhysicallyPhysically

ExactExact

k= −q g

i cTρ=PhysicallyPhysically

ApproximateApproximate

NumericallyNumerically

ApproximateApproximate

ef f

e

gQ k A

L= − �

�

c c nI cV Tρ=�

( )cTk T

t

ρ∂= ∇ ∇

∂�

( )I fc nn

AcV Tk T

t L

ρ∂ �= −� ∂ � �

D G�

�

e ng GT=� �

�

NumericallyNumerically

ExactExact

1 0+ − + =Dn nc c fI I Q

Slide 7Slide 7


22nd Dundee

B. Perot 06/28/2007

Discrete Calculus Methods

DC Approach is a methodology – not a particular method!

Slide 9Slide 9

FaceFace--Based MethodBased Method,f fT Q(KB)(KB)

Higher Order MethodHigher Order Method,n eT T

CellCell--Based MethodBased Method,n fT Q�(Mixed)(Mixed)

NodeNode--Based MethodBased MethodnT


22nd Dundee

B. Perot 06/28/2007

Physical Accuracy

Linearly Complete as well as Physically Realistic

Discontinuous Diffusion: Heat Flow at an AngleDiscontinuous Diffusion: Heat Flow at an Angle4 0 0.51 0.5 1

xk

x

< <�= � < <�

1 0 0.54 0.5 0.5 1exact

x y xT

x y x

+ + ≤ ≤�= � + − ≤ ≤�

Slide 10Slide 10


22nd Dundee

B. Perot 06/28/2007

Cost of DC Methods

More Cost-Effective than Traditional Methods

Slide 12Slide 12

For any accuracy For any accuracy level, DC is more level, DC is more costcost--effective than effective than Finite VolumeFinite Volume


22nd Dundee

B. Perot 06/28/2007

Implications

Methods that capture physics well

Slide 3Slide 3

Exact Discretization:Exact Discretization:•• Discrete Operators (Div, Grad, Curl) Discrete Operators (Div, Grad, Curl)

behave just like the continuous operators.behave just like the continuous operators.•• Mimetic.Mimetic.•• Discrete de Discrete de RhamRham Complex (algebraic topology).Complex (algebraic topology).•• , etc, etc•• Physics is always captured Physics is always captured

(conservation, wave propagation, max principal, …).(conservation, wave propagation, max principal, …).•• No spurious modes.No spurious modes.•• No surprises. No surprises.

( ) 0∇ ×∇ =


22nd Dundee

B. Perot 06/28/2007

Incompressible Navier-Stokes Tests


22nd Dundee

B. Perot 06/28/2007

Total cost for a desired accuracy level

High Matrix Condition Numbers Adversely Impacts the Cost

Slide 8 / 16Slide 8 / 16

FaceFace--based DC based DC converges more converges more slowly than Finite slowly than Finite VolumeVolume


22nd Dundee

B. Perot 06/28/2007

Discrete Calculus Operators

Discrete Gradient OperatorDiscrete Gradient Operator ( )n e→� �

1 2 1e n ng T T= −� � �

2 2 2bc

e n fg T T= − +� �

3 2 3bc

e n fg T T= − +� �

4 1 4bc

e n fg T T= − +� �

5 1 5bc

e n fg T T= − +� �

�

1 10 10 11 01 0

G

−� �� −� �� = −� �−� �� −�

�

Discrete Divergence OperatorDiscrete Divergence Operator ( )f c→

1 4 51f f f fcDQ Q Q Q= + +

1 2 32f f f fcDQ Q Q Q= − + +

�1 0 0 1 11 1 1 0 0

D� �

= � �−�

TG D= −�


22nd Dundee

B. Perot 06/28/2007

More Discrete Calculus Operators

1 1 0 01 0 0 10 1 0 11 0 1 0

0 1 1 0

C

−� �� −� �� = −� �−� �� −�

( )e f→ ��Discrete Rotation OperatorDiscrete Rotation Operator

Discrete Curl OperatorDiscrete Curl Operator ( )e f→

1 1 0 1 01 0 1 0 1

0 0 0 1 10 1 1 0 0

C

−� �� − −� �=� �−� �−�

�

TC C= �