Compatible Spatial Discretizations for Partial

Differential Equations

May 14, 2004

Compatible Reconstruction of Vectors

Blair Perot

Dept. of Mechanical & Industrial Engineering

Compatible Discretizations Vector Components are Primary

dAv n

Normal components (Face Elements)

Tangential components (Edge Elements)

Heat FluxMagnetic FluxVelocity Flux

Temperature GradientElectric FieldVorticity

ld

Why Vector Components

Physics Mathematics Numerics

MeasurementsContinuity RequirementsBoundary Conditions

Unknowns should contain Geometry/Orientation InformationUnknowns should contain Geometry/Orientation Information

Differential FormsGauss/Stokes Theorems

Absence of Spurious ModesMimetic Properties

So Why Vector Reconstruction ?

Convection

Adaptation

Formulation of Local Conservation Laws (momentum, kinetic energy, vorticity/circulation)

Construction of Hodge star operators

Nonlinear constitutive relations

( ) vv

Convection/Adaptation

Video Clip

Video Clip

M-adaptationM-adaptation

Conservation Wish to have discrete analogs of vector laws.

Conservation of Linear Momentum Conservation of Kinetic Energy

1 1 ( )fUT T Tf ctR V R R V N U p

D a G

Component Equations

1 1 ( )fUT T Tf ctNR V R NR V N U N p

D a G

Linear Momentum ˆ

( )

( )c

b btcell s boundary

V Np u a

1ˆ c fV RUu 1ˆ c fV RUu 3-Form ?3-Form ?

dAU f nu

Hodge Star Operators

0i

cell Afaces

dA q n

0k T

Have (tangential)

Need (normal)

Have (tangential)

Need (normal)

Discrete Hodge Star Interpolate / Integrate Least Squares

T1

T2

k T q

*dA H T d q n l

T d l

dAq n

Compatible/Mimetic Discretization

T d l

TCE v

TCS v k T q

nT2n eG 2e fC

02f cD

cellP

2e nD 2f eCSdV

2c fGnodeDedgeD

de Rham-like complexde Rham-like complex

T1

T2

dAq n

NotationNotation

ExactConnectivity MatricesTransposes

ExactConnectivity MatricesTransposes

TtT 2

TktCT

Dual Meshes

Circumcenter (Voronio)Circumcenter (Voronio)

T1

T2

T1

T2

T1

T2

T1

T2

Centroid (center of gravity)Centroid (center of gravity)

FE (smeared)FE (smeared)MedianMedian

0k T wdV 0dAdx q n

FE Reconstruction 2D: Raviart-Thomas 3D: Nedelec

( ) ( ) ( )n nb v x a x x x

Compute Coefficients in the Interpolation Compute Integrals

No dual – because FE is an average over all duals. Quadrature rule is a way of weighting the duals.

(which is how you can get other methods)

No dual – because FE is an average over all duals. Quadrature rule is a way of weighting the duals.

(which is how you can get other methods)

Face

Edge( ) ( ) ( )n n v x c x d x x

SOM Reconstruction

T d l

Shashkov, et al. Reconstruct local node values then interpolate

Arbitrary Polygons When Incompressible and Simplex = FE interpolation

nT2n eG 2e fC

02f cD

cellP

2e nD 2f eCSdV

2c fGnodeDedgeDdAq n

N L S RDiscrete Hodge StarsDiscrete Hodge Stars

Voronio Reconstruction

CoVolume method (when simplices). Used in ‘meshless’ methods (material science) Locally Conservative (N.S. momemtum and KE).

ffAf LdA k T d q n l

Discrete Maximum Principal in 3Dfor Delaunay mesh (not true for FE).Discrete Maximum Principal in 3Dfor Delaunay mesh (not true for FE).

Diagonal Hodge star operator

(due to local orthogonality)

T

Staggered Mesh Reconstruction

Conserves Momentum and Kinetic Energy. Arbitrary mesh connectivity. No locally orthogonality between mesh and dual. Hodge is now sparse sym pos def matrix. dA M T d q n l

Dilitation = constant

Face normal velocity is constant

facescell

fCGc

CGfVc U}{1 xxv

Vector Reconstruction

T d lnT

2n eG 2e fC0

2f cDcellPconst 0

2e nD 2f eCSdV

2c fGnodeD const0 edgeD

Expand the Hodge star operationExpand the Hodge star operation

T1

T2

dAq n

nT

nqcq

cT

Nonlinear Constitutive Relations are no problemNonlinear Constitutive Relations are no problem

Other MethodsT1

T2

Methods Differ in:Interpolation AssumptionsIntegration Assumptions

Methods Differ in:Interpolation AssumptionsIntegration Assumptions

CVFEM Linear in elements (sharp dual) Local conservation

Classic FEM Linear in element (spread dual)

Discontinuous Galerkin / Finite Volume Reconstruct in the Voronio Cell

Staggered Mesh Reconstruction

,( ) ( )i j jcellfaces

dV x v dV dA v x v xv n

Symmetric Pos. Def. sparse discrete Hodge star operator

11 ( )CG CG CGc f c f fV

cellfaces

U V U v x x X

( )CG CG CG T CGf c c c c c

facecells

d v l x x v X v

1( )Tcd V dA v l X X v n

X has same sparsity pattern as D

CGcv

Uf

CGcv

InterpolateInterpolate

IntegrateIntegrate

Staggered Mesh Conservation

)( 1ft

T UV XX

,CG

i f fjcell cellfaces faces

V x dV dA A xn x n

Exact Geometric IdentitiesExact Geometric Identities

,0 1 j fcell cellfaces faces

dV dA A n n

Time Derivative in N.S.Time Derivative in N.S.

cellctcellftftT

ff

facescell

VUUVA )()()( 1 vXXXn

Conservation Properties Voronoi Method

Conserves KE Rotational Form -- Conserves Vorticity Divergence Form – Conserves Momentum Cartesian Mesh – Conserves Both

Staggered Mesh Method Conserves KE Divergence Form – Conserves Momentum

facescell

fCGc

CGfc UV }{ xxv

ff

facescell

fcc ULV nv

Define a discrete vector potential So always.

CT of the momentum equation eliminates pressure (except on the boundaries where it is an explicit BC)

Resulting system is: Symmetric pos def (rather than indefinite) Exactly incompressible Fewer unknowns

Incompressible Flow

0fUDef sU C

Conclusions Physical PDE systems can be discretized (made

finite) exactly. Only constitutive equations require numerical (and physical) approximation.

Vector reconstruction is useful for:convection, adaptation, conservation, Hodge star construction, nonlinear material properties.

Hodge star operators have internal structure that is useful and related to interpolation/integration.

www.ecs.umass.edu/mie/faculty/perot