Conforming Vector Interpolation Functions for Polyhedral Meshes

Conforming Vector Interpolation Functions for Polyhedral Meshes Andrew Gillette joint work with Chandrajit Bajaj and Alexander Rand Department of Mathematics Institute of Computational Engineering and Sciences University of Texas at Austin, USA http://www.math.utexas.edu/users/agillette Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University / 29

Interpolation in Graphics vs. Simulation Interpolation of vector fields required for geometric design. No natural constraints on interpolation properties. Some exploration of scalar interpolation over polyhedra. Coupled vector fields related by integral and differential equations. Physical nature of problem offers natural discretizations of variables and boundary conditions. Discrete Exterior Calculus suggests a need for vector interpolation over polyhedra. Goal: Develop a theory of vector interpolation over polyhedra conforming to physical requirements with provable error estimates. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 2 / 29

Motivation Many authors (Bossavit, Hiptmair, Shashkov,...) have recognized the natural interplay between primal and dual domain meshes for discretization of physical equations. Potential benefits of a theory based on interpolation over dual meshes: Accuracy vs. speed tradeoffs available between primal and dual methods. 2 Error estimates for dual interpolation methods analogous to standard estimates. 3 Validation of primal-based results with dual-based discretization methods. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 3 / 29

Outline Background on Vector Interpolation 2 Novel Discretizations Using Polyhedral Vector Interpolation 3 Error Estimates for Polyhedral Vector Interpolation Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 4 / 29

H(Curl) versus H(Div) Throughout, we will consider a model problem from magnetostatics: Domain: Contractible 3-manifold Ω R 3 with boundary Γ Variables: b h Input: Equations: j (magnetic field / magnetic induction) (magnetizing field / auxiliary magnetic field) (current density field) div b = 0, b = h, curl h = j Boundary Conditions: Γ written as a disjoint union Γ e Γ h such that ˆn b = 0 on Γ e, ˆn h = 0 on Γ h. While b and h are both discretized as vector fields, they lie in different function spaces: h H(curl ) := { ( 3 ( ) } 3 v L (Ω)) 2 s.t. v L 2 (Ω) b H(div ) := { ( } 3 v L (Ω)) 2 s.t. v L 2 (Ω) omputational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 6 / 29

Local Conformity Constraints Functional continuity can be enforced on a mesh T by imposing certain constraints at each face F = T T 2, involving the normals to the mesh elements T, T 2 : H(curl ) := { ( 3 ( ) } 3 v L (Ω)) 2 s.t. v L 2 (Ω) h H(curl ) h T ˆn + h T2 ˆn 2 = 0, F T H(div ) := { ( } 3 v L (Ω)) 2 s.t. v L 2 (Ω) b H(div ) b T ˆn + b T2 ˆn 2 = 0, F T These constraints hold for primal meshes (T i =tetrahedra) and dual meshes (T i =polyhedra). Goal: Solve for h and b as functions defined piecewise over T, guaranteed to satisfy the applicable conformity constraints. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 7 / 29

Whitney Elements for Primal Meshes The Whitney elements provide a simple and canonical way to construct piecewise functions over a primal mesh T in H(curl ) or H(div ): y 0 Start with linear barycentric coordinates: λ i (v j ) = δ ij x y λ i d.o.f. per vertex x Define for each edge v i v j :: η ij := λ i λ j λ j λ i <y, x> < y, x> 2 Define for each face v i v j v k : η ijk := λ i λ j λ k + λ j λ k λ i +λ k λ i λ j < y, x> η ij d.o.f. per edge < x, y, z >, < x, y, z > < x, y, z >, < x, y, z > η ijk d.o.f. per face omputational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 8 / 29

Whitney Elements for Primal Meshes In 3D, it can be shown that the η ij satisfy the H(curl ) constraints and the η ijk satisfy the H(div ) constraints. See, e.g. Bossavit Computational Electromagnetism, 998. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL -and JanSciences 20 University 9 / 29

Discrete derham Diagrams We now have a basis for finite dimensional subspaces of the derham Diagram: H d 0 H(curl ) grad {λ i } D 0 {η ij } (grad) d H(div ) curl D {η ijk } (curl) d 2 L 2 div D 2 {χ T } (div) These are called the primal cochain spaces in Discrete Exterior Calculus: C 0 D 0 (grad) C D (curl) C 2 D 2 C 3 (div) Supposing for a moment we can construct conforming interpolation functions on the dual mesh, we also have a sequence of dual cochain spaces: C 3 D T 0 (div) C 2 DT (curl) C DT 2 C 0 (grad) DESBRUN, HIRANI, LEOK, MARSDEN Discrete Exterior Calculus, arxiv:math/050834v2 [math.dg], 2005 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences0 University / 29

Discrete Exterior Derivative The discrete exterior derivative D is the transpose of the boundary operator. 2 4 7 2 4 7 D 0 5 4 matrix with entries 0, ± The discrete exterior derivative on the dual mesh is D T 3 2 3 5 3 2 5 3 4 6 2 8 +3 5+2 D T 3 2+3 3 5 4 5 matrix 3 with entries 3 0, ± 2 5 3 2 5 3 These cochain vectors and derivative matrices are the building blocks for equation discretization. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences University / 29

Discrete Magnetostatics - Primal Returning to the magnetostatics problem, we can discretize the equations in two ways: Continuous Equations: div b = 0, b = h, curl h = j Primal Discrete Equations, with b as a primal 2-cochain: D 2 B = 0, M 2 B = H, D T H = J. Dual Discrete Equations, with b as a dual 2-cochain: D T 0 B = 0, M B = H, D H = J. The discrete Hodge Star M transfers information between complementary dimensions on dual meshes. In this example, we use the identity matrix for M. 3 2 5 3 3 2 3 5 M 5 5 matrix 3 2 3 5 3 2 5 3 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences3 University / 29

Discrete Hodge Stars Discretization of the Hodge star operator is non-canonical. Existing inverse discrete Hodge stars are either too full or too empty for use in discretizations on dual meshes We present a novel dual discrete Hodge star for this purpose using polyhedral vector interpolation functions primal mesh simplex σ k dual mesh cell σ k type reference definition M k M k DIAGONAL [Desbrun et al.] (M Diag k ) ij := σk i δ σj k ij diagonal diagonal WHITNEY [Dodziuk],[Bell] (M Whit k ) ij := η σ k η i σ k j T sparse (full) DUAL [G, Bajaj] ((M Dual k ) ) ij := η σ k η i σ k (full) sparse j T Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences4 University / 29

Condition Number of Discrete Hodge Stars Theorem [G, Bajaj] The condition number of ( M Dual k condition number of M Diag k ) is governed by different mesh criteria than the and M Whit k. v = (0, 0) v 3 = (P, ) 2 v 2 = (0, ) v 4 = ( P, ) 2 Condition numbers as functions of P: P M Diag M Whit ( M Dual ) 2 6.3 3.2.5 5 7.2 9.9.3 0 34.6 2.6.4 order O(P) O(P) O() Dual Formulations of Mixed Finite Element Methods. Submitted, 200. omputational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences5 University / 29

Dual-based Linear Systems Independence of primal and dual discrete Hodge stars implies accuracy vs. speed tradeoffs are possible between primal and dual methods. Ex: Fewer elements in dual mesh smaller system faster. Ex: Better condition number in dual system more accurate. Primal Linear System, with b as a primal 2-cochain: D 2 B = 0, M 2 B = H, D T H = J. ( ) ( ) ( ) M2 D T 2 B H0 =. D 2 0 P 0 Here, H 0 C satisfies D T H 0 = J and H is defined by H := H 0 + D T 2 P. Thus D T H = DT (H 0 + D T 2 P) = J is assured. Dual Linear System, with b as a dual 2-cochain: M D T 0 B = 0, B = H, D H = J. ( ) ( ) ( ) M D 0 B H0 D T =. 0 0 P 0 Here, H 0 C satisfies D H 0 = J and H is defined by H := M B. Thus D H = D (H 0 + D 0 P) = J is assured. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences6 University / 29

Dual-based Linear Systems The duality of the systems is easily visualized via the cochain sequences: C 0 (grad) C (curl) C 2 (div) C 3 (M ) H 0 B J (D 0 ) T D J B (D ) T H M 2 D 2 0 C 3 C 2 (div) (curl) C (grad) C 0 Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences7 University / 29

The DEC-deRham Diagram for R 3 We combine the Discrete Exterior Calculus maps with the L 2 derham sequence. derham: H grad I 0 P 0 H(curl ) P curl H(div ) primal: C 0 D 0 C D C 2 D 2 M 0 dual: C 3 M 0 (D 0 ) T I M C 2 M (D ) T I 2 M 2 C P 2 M 2 div (D 2 ) T L 2 I 3 M 3 C 3 C 0 P 3 M 3 The combined diagram can be used to formulate dual-based discretizations for many problems including electromagnetics, Darcy flow, and electrodiffusion. The question remains: How do we construct polyhedral vector interpolation functions? Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences8 University / 29

Scalar Interpolation: Generalized Barycentric Functions Let Ω be a convex polygon in R 2 with vertices v,..., v n. Functions λ i : Ω R, i =,..., n are called barycentric coordinates on Ω if they satisfy two properties: Non-negative: λ i 0 on Ω. 2 Linear Completeness: For any linear function L : Ω R, L = n L(v i )λ i. It can be shown that any set of barycentric coordinates under this definition also satisfy: n 3 Partition of unity: λ i. y 4 Linear precision: i= n v i λ i (x) = x. i= 5 Interpolation: λ i (v j ) = δ ij. x y i= x Theorem [Warren, 2003] If the λ i are rational functions of degree n 2, then they are unique. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences20 University / 29

Triangulation Coordinates Let T be a triangulation of Ω formed by adding edges between the v j in some fashion. Define : Ω R λ Tri i,t to be the barycentric function associated to v i on triangles in T containing v i and identically 0 otherwise. Trivially, these are barycentric coordinates on Ω. Theorem [Floater, Hormann, Kós, 2006] For a fixed i, let T m denote any triangulation with an edge between v i and v i+. Let T M denote the triangulation formed by connecting v i to all the other v j. Any barycentric coordinate function λ i satisfies the bounds 0 λ Tri i,t m (x) λ i (x) λ Tri i,t M (x), x Ω. () Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences2 University / 29

Wachspress Coordinates Let x Ω and define A i (x) and B i as the areas shown. Define the Wachspress weight function as w Wach i (x) = B i j i,i A j (x). The Wachspress coordinates are then given by the rational functions λ Wach i (x) = wi Wach n j= w Wach j (x) (x) (2) Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences22 University / 29

Wachspress Coordinates Example Let x = v. Note A (x) = area of (v, v, v 2 ) = 0. Similarly A 5(x) = 0 λ Wach w Wach (x) = B A 2 (x)a 3 (x)a 4 (x) = W w Wach 2 (x) = B 2 A 3 (x)a 4 (x)a 5(x) = 0 w Wach 3 (x) = B 3 A 4 (x)a 5(x)A (x) = 0 w Wach 4 (x) = B 4 A 5(x)A (x)a 2 (x) = 0 w Wach 5 (x) = B 5A (x)a 2 (x)a 3 (x) = 0 (x) = w Wach (x) w Wach i (x) = W W = 2 (x) = w 2 Wach (x) w Wach (x) = 0 λ Wach Similarly λ Wach 3 (x) = λ Wach 4 (x) = λ Wach 5 (x) = 0. This is an illustration of the property λ Wach i (v j ) = δ ij Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences23 University / 29

Sibson (Natural Neighbor) Coordinates Let P denote the set of vertices {v i } and define P = P {x}. C i := V P (v i ) = {y Ω : y v i < y v j, j i} = area of cell for v i in Voronoi diagram on the points of P, D(x) := V P (x) = {y Ω : y x < y v i, i} = area of cell for x in Voronoi diagram on the points of P. By a slight abuse of notation, we also define The Sibson coordinates are defined to be λ Sibs i (x) := D(x) C i D(x) D(x) C i := V P (x) V P (v i ). or, equivalently, λ Sibs i (x) = D(x) C i n j= D. j(x) C j omputational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences24 University / 29

Optimal Coordinates Let g i : Ω R be the piecewise linear function satisfying g i (v j ) = δ ij, g i linear on each edge of Ω. The optimal coordinate function λ Opt i equations with g i as boundary data, { ( λ Opt i is defined to be the solution of Laplace s ) = 0, on Ω, λ Opt i = g i. on Ω. These coordinates are optimal in the sense that they minimize the norm of the gradient over all functions satisfying the boundary conditions, { } λ Opt i = argmin λ H (Ω) : λ = g i on Ω. (3) Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences25 University / 29

Polyhedral H(Curl) Vector Interpolantion Let {λ i } denote a set of generalized barycentric coordinates for a polygon (2D) or polyhedra (3D). Define for each edge v i v j :: Theorem [G,Bajaj] η ij := λ i λ j λ j λ i Constructing Whitney-like -forms analogously to the triangular case produces globally H(curl )-conforming vector fields. v k PROOF: Consider edge v i v j and λ k associated to a different vertex v k. Then the edge is part of the zero level set of λ k. Hence λ k must be perpendicular to the edge at all points along it and any summand λ l λ k has no tangential component on the edge. Therefore, the tangential components only depend on λ i and λ j. Hence the H(curl ) conformity constraints are satisfied. To decide which definition of {λ i } is suitable, we need error estimates. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences26 University / 29

Error Estimates: 2D Scalar Case The optimal convergence estimate for a finite element method bounds the interpolation error in H -norm of an unknown function u by a constant multiple of the mesh size times the H 2 semi-norm of u: u I 0 u H (Ω) C diam(ω) u H 2 (Ω), u H2 (Ω). (4) (Note that I 0 u assumes u is known or computed at vertices of the dual mesh.) Theorem [G, Rand, Bajaj] Assume certain standard geometric quality conditions on the dual mesh can be guaranteed. Then a dual formulation of a finite element method using any of the coordinate systems has the optimal convergence estimate on the mesh. Error Estimates for Generalized Barycentric Interpolation. Submitted, 200. Example showing necessity of geometric criteria for Wachspress coordinates. Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences27 University / 29

Future Work Efficient computation of λ i basis functions Error estimates for polyhedral vector functions H(div )-conforming vector elements for polyhedral meshes Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences28 University / 29

Questions? Thank you for inviting me to visit. Slides and pre-prints available at http://www.math.utexas.edu/users/agillette Computational Visualization Center, I C E S (Department The University of Mathematics of TexasInstitute at Austinof Computational Engineering LANL - Jand 20 Sciences29 University / 29