compatible spatial discretizations for partial differential equations may 14, 2004 compatible...
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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