The Relationship between Discrete Calculus Methods and other Mimetic Approaches

Transcription

1 The Relationship between Discrete Calculus Methods and other Mimetic Approaches Blair Perot Theoretical and Computational Fluid Dynamics Laboratory Woudschoten Conerence Oct October 7, 2011

2 Background Hardware: GPUs, FPGAs, HPC, Algorithms Numerical Methods: Unstructured Staggered mesh methods, Fractional step methods, Discrete Calculus Methods. Turbulence Modeling: Turbulent Potentials, Eddy Collision Model Applications: Wind Turbines, DNS, Super-hydrophobic suraces, droplets. 2

3 Mimetic Methods FE: Raviart-Thomas/Nedelec/Whitney Algebraic Topology Electromagnetics FV: Staggered Mesh Methods Many local conservation properties Fluid Dynamics FD: Keller Box Multi-symplectic Wave Eqns FD: SOM Box Robust Heat Eqn NN: Non-Sibsonian Meshless methods Time-dependent domains Solid mechanics 3

4 Numerical Methods Mimetic Methods Finite Dierence SOM Edge/Face Finite Element Finite Volume Staggered Natural Neighbors Meshless All Consistent Numerical Methods 4

5 Question Is there any relationships between the various mimetic methods? (1) Yes many (all?) can be derived as discrete calculus methods. (2) Yes they tend to use the same basis unctions. 5

6 Incompressible Fluid Dynamics n1 t n 1 n ( F) l G e e n t c DU n 1 0 u u dt d p U Need to relate these two u e U un da ue u dl

7 FE Basis Functions Heat Flux Magnetic Flux Velocity Flux Face Elements Nedelec/RT/Whitney un da Constant normal velocity on each ace h u ( x) u x 0 D n Interpolant with continuity o the normal lux h 0 D n L u n u n Constant divergence u h D 7

8 FE Hodge 2-orms to 1 -orms h u ( x) u x Find the constants given the data (4x4) u 1 Evaluate the integral The basis unction determines the relationship between the two velocities 0 D n 0 1 L n D 1 n A ue u h dl 1 U 1 8

9 StagMesh Interpolation ( u x u ) dv ( u x ) dv ( x ) u n da i, i j j i j, i j i i U un da Gauss Theorem (Again) Assume constant lux on ace Assume constant divergence SM = implicit basis unctions u V U ( x cg x cg ) c c aces u c 1 RU V c 9

10 StagMesh Hodge 2-orms to 1 -orms Average Cell velocity u c 1 RU V c u e T Ru c or incompressible u e T 1 V ( R R) c U Explicit Formula or the same matrix relationship (Hodge*) as FE Symmetric Generalizable to polyhedra The intermediate is a (cell average) velocity vector. (Momentum, KE) 10

11 FD Interpolation SOM unda Use 3 ace values at each vertex Average vectors to center Works on (almost) any polygon Assumes constant on ace Assumes constant divergence CoVolume Use Least Squares Nu c 1 A U Also same cell velocity 11

12 FE Basis Functions 1-orms u Temperature Gradient Electric Field Velocity dl Edge Elements Nedelec/RT/Witney h 0 1 n1 u ( x) u wx Interpolant with continuity o the tangential components Constant tangential velocity on each edge u h t e u 0 1 ( w L e ) t e n1 Constant vorticity u h w 12

13 StagMesh Interpolation ( ax) u ( un xw n) da xu dl ue udl SM = Rampant use o Stokes Theorem Stokes Theorem Assume constant along edge Assume constant vorticity u cg n A u ( x cg x cg ) e e edges uxdv unda 13

14 Summary: Basis Functions FE uses Explicit Basis Functions SM uses Stokes Theorem This approach can be applied to arbitrary polygons. SOM uses Discrete Inner Products highly discontinuous/anisotropic materials and arbitrary polygons Many approaches to achieving the same underlying interpolation (Hodge *). 14

15 Test Functions Median Dual Voronoi Dual Dual mesh is not unique. FV = top hat FE = tent unctions (unique or Galerkin) 15

16 FE / FV relationship FV = one CV FE = weighted average o CV Mattiussi C, 1997, An analysis o inite volume, inite element, and inite dierence methods using some concepts rom algebraic topology, J. Computational Physics, 133:

17 FE Discrete Calculus Weighted Exact Discretization w( u) dv wu nda ( w) udv u( w) dv Smeared Flux U D w U aces w Compare D udv unda U aces aces 17

18 FE Exact Discretization udv unda aces ( v) nda vdl T dl T T 2 1 edges w udv u( w) dv cells ( v) ( w) dv v( nw) da aces T ( nw) da T( w) dl edges 18

19 Higher-Order Exact Gradient edge edge T d T T l n2 n1 n2 n2 n1 n1 e n2 xt dl ( x T x T ) t Tdl n1 T n Td n2 n1 Tdl n2 x Td l 1 n ntda ntda t ace e Tdl edges 20 outputs / tet 10 inputs / tet Tdl edge T G 0 n Tn [2] x T dl G Gxn tei edge Tdl Tdl edge 0 T edge ntda ace 19

20 Higher-Order Exact Curl ( v) nda vdl edges Note: Mimetic x( v) nda xv dl n vda edges vdv n vda aces [2] [2] C G 0 15 outputs / tet a v vdv 20 inputs / tet ( v) nda v dl ace edge [2] x( v) nda C xv dl ace edge ( v) dv n vda cell ace x( v) nda ( v) nda n2 v d l 1 n2 xv d l n1 n vda n 20

21 Review FE: Explicit basis unctions Fixed geometry precise proos Implicit dual mesh Semi-implicit Hodge* StagMesh: Implicit basis unctions Arbitrary polygons. Explicit dual mesh Explicit Factored Hodge* 21

22 Summary Underlying assumptions about the solution (basis unctions) are oten the same. The Test Functions are dierent. Test unctions aect the metric (geom). Higher Order is achieved by noticing that you can exactly discretize dierent ways and with dierent moments Anyone can do it. 22