Discrete Calculus Methods and their Application to Fluid Dynamics Blair Perot Theoretical and Computational Fluid Dynamics Laboratory October 6, 2011 Woudschoten Conference Oct. 2011 Context Mathematical Tools Order of Accuracy Stability Consistency Physical Requirements Conservation (mass, momentum, total energy) Secondary Conservation (kinetic energy, vorticity, entropy) Unphysical Modes (pressure) Wave propagation (direction) Eigenmodes/Resonance 2 Many Mimetic Methods HO Staggered Staggered Mesh Nedelec FE Support Operators Methods Keller Box Schemes Methods that capture physics well. 3 Why ? Why do some methods capture physics well? They use exact discretization What do they have in common? All are Discrete Calculus Methods Can you make a numerical method so that is mimetic from its design? Yes (I think) 4 Discretization Take a continuous problem to a finite dimensional one. 5 Discrete Calculus: Part 1 Exact Discretization a a t b 0 A B r b Infinite Dimensional Finite Dimensional Partial Differential Eqn. Matrix Problem Basic unknowns are integral quantities. Collect infinite data into finite groups. 6 Discrete Calculus: Part 2 Solution requires Approximation A B aa r A B r CD bb 0 Underdetermined Unique Square Relate discrete unknowns to each other. This relation is a material law. Also related to an dual mesh interpolation. Also related to inner products 7 Implications Exact Discretization means that: • Calculus is exact. • Physics is exact. • Method is mimetic. Discrete Solution Requires: • Approximation of material laws. • Interpolation between meshes. • Assumptions about the solution. 8 Example: Heat Equation T 2 t T ()CT Conservation of Energy t q q kT Fourier’s Heat Flux Law Figure out what should NOT be approximated and what is already an approximation (Tonti). 9 Step 1 Heat Eqn: Fully Separated Heat Equation i Conservation of Energy (Physics) t q iT C Perfectly Caloric Material (Mat.) qgk Fourier’s Heat Flux Law (Mat.) g T Def. of Gradient (Math) Discretize Physics and Math - exactly. Approximate Material laws - using interpolation. 10 Exact Discretization IIQn1 n out Gauss’ Theorem (with time) cc f faces Like FV: Not very original. tn1 In1 idV Q out dtqn out dA c f nn1 c t f vv gn122g d l|| n 1 T d l n 1 T n 1 T n 1 evv v21 v 11 Math: still exact 11 Step 2 Exact Discrete System I n1 c i Q n t q ID00f Ic 00IG g n1 0 g T e n1 Tv • Exact • Over-determined • Uncoupled • Unknowns on different meshes 12 Step 3 Vertex Centered Mimetic g (C1) Vertex Centered (C2) Median dual mesh Q n1 i CT IMCTc c c v qgk kt n11 n n n Qf 2 ()BB gee g Approximation Part 13 Solvable Vertex System n1 n ID 00Ic Iv IM00 Q 0 C f 00 IG g n1 0 e 00IBkt n1 n1 kt Bnng 2 Tv 2 e n1 ntk n 1 n 1 tk n n MDBGDBGCTITT v c ()()22 v v • Single unknown (T at vertices). • Looks like no dual mesh was used. • B is Hodge * operator. 14 Other Possibilities Mesh: Use polygons as the primary mesh. (quad/hex meshes, particle methods, SOM) Approx: Use other basis functions for interpolation. (Rational polynomials, Natural Neighbors, Fourier) Exact: Use different exact discretizations (FE, Keller/Priessman Box Schemes) 15 Cell Centered Mimetic Q Cell Centered Median dual mesh g Raviart-Thomas for Q n1 i CT IMCTc c c c qgk t ()gn11 g n A n Q 2 e e1/ kc f 16 Cell System ID 00n1 n Ic Ic IM00 Q 0 C f 00 IG g n1 0 e 00AIn1 t T n1 t g n 1/kc 2 c 2 e MD n1 n C Tc Ic n1 GA2 Q g n t 1/kc f e • Symmetric system of unknowns • Not reducible 17 Comparison Mimetic 10x more accurate or 1/10th cost. 18 Fluid Example - Droplets Moving Mesh Unstructured Surface Tension 19 Example – 3D Droplets Moving Mesh Large distortions Complex BCs 20 DNS 21 Summary Exact discretization (or 2 step construction) leads to mimetic methods. Place numerical approximations with the physical approximation. Discrete Calculus analysis is accessible to all computational scientist. (Gauss’ Theorem). Discrete Calculus Methods are not a type of numerical method. The approach can produce some FV, FE, FD, and other methods. www.ecs.umass.edu/mie/tcfd/Publications.html 22 23 .