An Approach to Higher- Order Mimetic Finite Volume Schemes

Transcription

1 An Approach to Higher- Order Mimetic Finite Volume Schemes Blair Perot

2 Context Mimetic Methods Finite Dierence Mimetic SOM Finite Element Edge/Face Staggered Finite Volume Meshless Natural Neighbors DC Methods are a subset o many other classical approaches

3 Discrete Calculus Exact Discretization: Continuous PDE Finite Dimensional Matrix Problem This Can (and should?) Be Done Exactly Approximate Solution: Exact System has More Unknowns than Equations. Reducing the unknowns requires approximation (error). All numerical errors appear with the physical modeling errors. In the constitutive equations (or in the material properties). Discretization does not require Approximation Slide 1

4 Mimetic Example Slide 2

5 Example (Continuous) System Physical Equation (Heat Equation) ct kt t Components o the Physical Equation i q 0 t Conservation o Energy g T Deinition o Gradient Physics Math Material Approximation qkg Fourier s Law i ct Perectly Caloric Material Slide 3

6 The Numerical (Discrete) System Exact Discretization o Physics and Calculus. n 1 n idv idv dt da 0 c c qn g d l T T e n2 n1 I I DQ n 1 n c c ge GTn Numerical Approximation o Constitutive Eqns. Q M g 1 e A Q k L g e e Numerically Exact 0 I c M T 2 n I cv T c c n Numerically Approximate Slide 4

7 The Staggered/Dual Mesh Viewpoint I I DQ n 1 n c c 0 g e GT n I idv n1 n1 c c I cv T c c n T n T n T n Q dt da qn Dual Mesh A Q k L g e e g d l e g e Primary Mesh T n Slide 5

8 Method Choices Choice o the Dual Mesh. Median Dual Voronoi Dual Choice o which mesh to use. Node centered temperature. Cell centered temperature. Choice o interpolation. piecewise polynomial reconstruction in cells. piecewise reconstruction in dual cells weighted interpolations (FE). Slide 6

9 Higher Order Mimetic FV Option 1: Use a wider stencil in the interpolation step. Boundaries: One-sided interpolation (stability). Parallel implementation: Diicult to partition cleanly. Inelegant on Unstructured Mesh: Resort to least squares. Option 2: Use more unknowns per cell. This is the FE approach to higher order. Related to Raviart-Thomas/Nedelec ace/edge elements. Slide 7

10 Higher Order Gradient 3 Exact Expressions or any gradient, edge edge ace T dl T T n2 n1 n2 n2 n1 n1 n2 xt dl ( x T x T ) Tdl ntda edges Tdl n1 g T gdl G 0 edge Tn Tn [2] xg dl Gxn I G edge Tdl Tdl edge edge 0 C ngda ace Slide 8 T n Td n2 gdl n1 n2 xg n1 n g da dl Enough or Quadratic Approx [2] G c 0

11 Higher Order Curl 3 Exact Expressions or any curl, ( v) nda v dl edges x( v) nda xv dl n vda edges vdv n vda aces a v adv xa n da an da n2 vdl n1 n2 xv dl n1 n v da Note: Mimetic C G [2] [2] 0 anda v dl ace edge [2] xa nda C xv dl ace edge dv da a cell n v ace Slide 9

12 Higher Order Divergence 2 Exact Expressions or any divergence, ( w) dv w nda aces x( w) dv x( w n) da wdv aces Enough or Quadratic Approx Note: Mimetic [2] [2] D C 0 s w adv an da ace sdv [2] D xa nda ace xsdv adv cell xa n da an da xwdv wdv Slide 10

13 de Rham Complex c H H(curl) H(div) L2 T n g e c [2] G [2] C [2] D [2] [2] [2] D C G c D I c Q Need to Look at the Energy Eqn. Slide 11

14 Higher Order Energy Equation Need Same Number o Eqns as Unknowns (NN, NE) CV approach T n Td Node Eqn: Same as beore. CV about the node Edge Eqn: Use CV about the edge Will use this observation or the edge CV qdv qdv [ q n da] edge cells ec jump edge aces Slide 12

15 Reconstruction Have e g dl e xg dl gnda Need c qn da qdv qn da Middle unknowns are redundant Have NE + 2NF : Need NE + 2NF Slide 13

16 Approximation 10 unknowns on each tetrahedron Or 6 unknowns on each triangle assume T quadratic polynomial then g, q is linear polynomial q is constant in cells T n Td Td T n Assume g varies linearly along each edge to get end values. 3 gradient components in each corner deine g in each corner. Average to aces to get normal heat lux. (exact i linear on aces). Sum over aces to get the divergence (exact i q is linear in cell). Slide 14

17 Comparison Option 1: Basis Functions (FE-like) Assume a polynomial (basis unctions) Solve or polynomial coeicients (using Td, T n.) 10x10 matrix inversion in 3D (in each cell). Integrate (evaluate polynomial at Gauss Points). Option 2: Direct (CV-like) No mapping No basis unctions No matrix inversion No Gauss integration points Slide 15

18 System Overview Divergence is not the -Transpose o the Gradient. (but close: one o-diagonal block is dierent) System is still Positive Deinite. Exact or piecewise-quadratic solutions. No problems with discontinuous material properties. No problems with high aspect ratio tetrahedra. Slide 16

19 Heat Equation Fixed temperature at the ends, zero lux elsewhere. Slide 18

20 Results HO LO Mimetic Regular CV Good Cost/Accuracy or Smooth Solutions Slide 19

21 Conclusions Exact Discretization / Approximate Material HO Mimetic CV can use more unknowns per cell. HO Mimetic CV uses moments or unknowns. Extra equations are due to extra CV regions. Slide 20