Polygonal, Polyhedral, and Serendipity Finite Element Methods

Transcription

1 Polygonal, Polyhedral, and Serendipity Finite Element Methods Andrew Gillette Department of Mathematics University of Arizona ASU Computational and Applied Math Seminar Slides and more info at: agillette/ Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

2 What are a priori FEM error estimates? Poisson s equation in R n : Given a domain D R n and f : D R, find u such that strong form u = f u H 2 (D) weak form u φ = f φ φ H 1 (D) discrete form D D u h φ h = Typical finite element method: D D f φ h φ h V h finite dim. H 1 (D) Mesh D by polytopes {Ω} with vertices {v i }; define h := max diam(ω). Fix basis functions λ i with local piecewise support, e.g. barycentric functions. Define u h such that it uses the λ i to approximate u, e.g. u h := i u(v i)λ i A linear system for u h can then be derived, admitting an a priori error estimate: u u h H 1 (Ω) C h p u H }{{} p+1 (Ω), u H p+1 (Ω), }{{} approximation error optimal error bound provided that the λ i span all degree p polynomials on each polytope Ω. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

3 Two trends in contemporary finite element research 1 Polygonal / polyhedral domain meshes 2 Serendipity higher order methods Greater geometric flexibility can alleviate known difficulties with simplicial and cubical elements. TALISCHI, PAULINO, PEREIRA, MENEZES, PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Structural and Multidisciplinary Optimization, SUKUMAR Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons. Computer Methods in Applied Mechanics and Engineering, Long observed but only recently formalized theory ensuring order p function approximation with many fewer basis functions than expected. ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, DA VEIGA, BREZZI, CANGIANI, MANZINI, RUSSO Basic Principles of Virtual Element Methods, M3AS, u u h H 1 (Ω) C h p u H p+1 (Ω), u H p+1 (Ω) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

4 Table of Contents 1 Linear elements on polygons 2 Quadratic serendipity elements on polygons 3 Order r serendipity elements on n-cubes 4 Future directions Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

5 Outline 1 Linear elements on polygons 2 Quadratic serendipity elements on polygons 3 Order r serendipity elements on n-cubes 4 Future directions Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

6 The generalized barycentric coordinate approach Let Ω R 2 be a convex polygon with vertex set V. We say that a set of functions λ v : Ω R are generalized barycentric coordinates (GBCs) on Ω if they satisfy λ v 0 on Ω and L = v V L(v v)λ v, L : Ω R linear. Familiar properties are implied by this definition: λ v 1 vλ v(x) = x v V }{{} partition of unity v V } {{ } linear precision λ vi (v j ) = δ ij }{{} interpolation traditional FEM family of GBC reference elements Bilinear Map Unit Diameter Affine Map T Reference Element Physical Element T Ω Ω Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

7 Many generalizations to choose from... Triangulation FLOATER, HORMANN, KÓS, A general construction of barycentric coordinates over convex polygons, λ Tm i (x) λ i (x) λ T M i (x) 1 Wachspress WACHSPRESS, A Rational Finite Element Basis, WARREN, Barycentric coordinates for convex polytopes, Sibson / Laplace SIBSON, A vector identity for the Dirichlet tessellation, HIYOSHI, SUGIHARA, Voronoi-based interpolation with higher continuity, Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

8 Many generalizations to choose from... x α i ri α i 1 v i+1 v i Mean value FLOATER, Mean value coordinates, FLOATER, KÓS, REIMERS, Mean value coordinates in 3D, v i 1 u = 0 Harmonic WARREN, SCHAEFER, HIRANI, DESBRUN, Barycentric coordinates for convex sets, CHRISTIANSEN, A construction of spaces of compatible differential forms on cellular complexes, Many more papers could be cited (maximum entropy coordinates, moving least squares coordinates, surface barycentric coordinates, etc...) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

9 Comparison via eyeball norm Triangulated Triangulated Wachspress Mean Value Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

10 Comparison via eyeball norm Wachspress Sibson Mean Value Discrete Harmonic Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

11 Optimal convergence estimates on polygons Let Ω be a convex polygon with vertex set V. For linear elements, an optimal convergence estimate has the form u u(v)λ v C diam(ω) u H 2 (Ω), u H 2 (Ω). (1) v V H }{{ 1(Ω) }{{}} optimal error bound approximation error The Bramble-Hilbert lemma in this context says that any u H 2 (Ω) is close to a first order polynomial in H 1 norm. VERFÜRTH, A note on polynomial approximation in Sobolev spaces, Math. Mod. Num. An., DEKEL, LEVIATAN, The Bramble-Hilbert lemma for convex domains, SIAM J. Math. An., For (1), it suffices to prove an H 1 -interpolant estimate over domains of diameter one: u(v)λ v C I u H 2 (Ω), u H1 (Ω). (2) v V H 1 (Ω) For (2), it suffices to bound the gradients of the {λ v}, i.e. prove C λ R such that λ v L 2 (Ω) C λ. (3) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

12 Geometric criteria for convergence estimates v1 v1 To bound the gradients of the coordinates, we need control of the element geometry. Let ρ(ω) denote the radius of the largest inscribed circle. The aspect ratio γ is defined by γ = diam(ω) ρ(ω) (2, ) Three possible geometric conditions on a polygonal mesh: G1. BOUNDED ASPECT RATIO: γ < such that γ < γ G2. MINIMUM EDGE LENGTH: d > 0 such that v i v i 1 > d G3. MAXIMUM INTERIOR ANGLE: β < π such that β i < β Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

13 Polygonal Finite Element Optimal Convergence Theorem In the table, any necessary geometric criteria to achieve the a priori linear error estimate are denoted by N. The set of geometric criteria denoted by S in each row taken together are sufficient to guarantee the estimate. G1 (aspect ratio) G2 (min edge length) G3 (max interior angle) Triangulated λ Tri - - S,N Wachspress λ Wach S S S,N Sibson λ Sibs S S - Mean Value λ MV S S - Harmonic λ Har S - - G, RAND, BAJAJ Error Estimates for Generalized Barycentric Interpolation Advances in Computational Mathematics, 37:3, , 2012 RAND, G, BAJAJ Interpolation Error Estimates for Mean Value Coordinates, Advances in Computational Mathematics, 39:2, , Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

14 A geometric criterion for polyhedra Observe that on triangles of fixed diameter: λ large large interior angle small altitude at a vertex For Wachspress coordinates, we generalize to polygons: λ Wach large small altitude at a vertex and then to polyhedra: Given a simple, convex d-dimensional polytope P R d, let h := minimum distance from a vertex to a hyper-plane of a non-incident face. Then λ Wach large h small Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

15 Upper and lower bounds on polytopes On a polytope P R n, define Λ := sup x P v λ Wach v (x). simple convex polytope in R n 1 h Λ n-simplex in R n 1 h Λ 2n h n + 1 h hyper-rectangle in R n 1 Λ n + n h h regular k-gon in R 2 2(1 + cos(π/k)) h Λ 4 h Note that lim 2(1 + cos(π/k)) = 4, so the bound is sharp in R 2. k FLOATER, G, SUKUMAR Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numerical Analysis, Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

16 Outline 1 Linear elements on polygons 2 Quadratic serendipity elements on polygons 3 Order r serendipity elements on n-cubes 4 Future directions Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

17 From linear to quadratic elements A naïve quadratic element is formed by products of linear GBCs: {λ i } pairwise products {λ aλ b } Why is this naïve? For a k-gon, this construction gives k + ( k 2) basis functions λaλ b The space of quadratic polynomials is only dimension 6: {1, x, y, xy, x 2, y 2 } Conforming to a linear function on the boundary requires 2 degrees of freedom per edge only 2k functions needed! Problem Statement Construct 2k basis functions associated to the vertices and edge midpoints of an arbitrary k-gon such that a quadratic convergence estimate is obtained. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

18 Polygonal Quadratic Serendipity Elements We define matrices A and B to reduce the naïve quadratic basis. filled dot = Lagrangian domain point = all functions in the set evaluate to 0 except the associated function which evaluates to 1 open dot = non-lagrangian domain point = partition of unity satisfied, but not Lagrange property {λ i } k pairwise products {λ aλ b } k + ( ) k 2 A {ξ ij } 2k B {ψ ij } Linear Quadratic Serendipity Lagrange 2k Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

19 From quadratic to serendipity The bases are ordered as follows: ξ ii and λ aλ a = basis functions associated with vertices ξ i(i+1) and λ aλ a+1 = basis functions associated with edge midpoints λ aλ b = basis functions associated with interior diagonals, i.e. b / {a 1, a, a + 1} Serendipity basis functions ξ ij are a linear combination of pairwise products λ aλ b : ξ ii. ξ i(i+1) = A λ aλ a. λ aλ a+1. λ aλ b = c11 11 cab 11 c 11 (n 2)n c ij 11 c ij ab c ij (n 2)n c n(n+1) 11 c n(n+1) ab c n(n+1) (n 2)n λ aλ a. λ aλ a+1. λ aλ b Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

20 From quadratic to serendipity We require the serendipity basis to have quadratic approximation power: Constant precision: 1 = ξ ii + 2ξ i(i+1) i Linear precision: x = v i ξ ii + 2v i(i+1) ξ i(i+1) i Quadratic precision: xx T = v i v T i ξ ii + (v i v T i+1 + v i+1 v T i )ξ i(i+1) i ξ b(b+1) ξ bb ξ b(b 1) Theorem Constants {cij ab } exist for any convex polygon such that the resulting basis {ξ ij } satisfies constant, linear, and quadratic precision requirements. ξ a(a 1) ξ aa ξ a(a+1) λ a λ b Proof: We produce a coefficient matrix A with the structure A := [ I A ] where A has only six non-zero entries per column and show that the resulting functions satisfy the six precision equations. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

21 Pairwise products vs. Lagrange basis Even in 1D, pairwise products of barycentric functions do not form a Lagrange basis at interior degrees of freedom: 1 λ 0 λ 0 λ 1 λ 1 1 ψ 00 ψ 11 ψ01 λ 0 λ 1 Pairwise products 1 Lagrange basis 1 Translation between these two bases is straightforward and generalizes to the higher dimensional case. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

22 From serendipity to Lagrange {ξ ij } B {ψ ij } [ψ ij ] = ψ 11 ψ ψ nn ψ 12 ψ ψ n1 = ξ 11 ξ ξ nn ξ 12 ξ ξ n1 = B[ξ ij ]. Andrew Gillette - U. Arizona ( ) Polytope and Serendipity FEM ASU Seminar - Apr / 41

23 Serendipity Theorem {λ i } pairwise products {λ aλ b } A {ξ ij } B {ψ ij } Theorem Given bounds on polygon aspect ratio (G1), minimum edge length (G2), and maximum interior angles (G3): A is uniformly bounded, B is uniformly bounded, and span{ψ ij } P 2 (R 2 ) = quadratic polynomials in x and y We obtain the quadratic a priori error estimate: u u h H 1 (Ω) C h 2 u H 3 (Ω) RAND, G, BAJAJ Quadratic Serendipity Finite Element on Polygons Using Generalized Barycentric Coordinates, Math. Comp., 2011 Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

24 Special case of a square v14 v4 v34 v3 v23 Bilinear functions are barycentric coordinates: λ 1 = (1 x)(1 y) λ 2 = x(1 y) λ 3 = xy λ 4 = (1 x)y v1 ξ 11 ξ 22 ξ 33 ξ 44 ξ 12 ξ 23 ξ 34 ξ 14 v12 = v2 Compute [ξ ij ] := [ ] I A [λaλ b ] λ 1 λ 1 λ 2 λ 2 λ 3 λ 3 λ 4 λ 4 λ 1 λ 2 = λ 2 λ 3 λ 3 λ 4 λ 1 λ /2 1/ /2 1/ /2 1/ /2 1/2 (1 x)(1 y)(1 x y) x(1 y)(x y) xy( 1 + x + y) (1 x)y(y x) (1 x)x(1 y) x(1 y)y (1 x)xy (1 x)(1 y)y span { ξ ii, ξ i(i+1) } = span {1, x, y, x 2, y 2, xy, x 2 y, xy 2} =: S 2 (I 2 ) Hence, this provides a computational basis for the serendipity space S 2 (I 2 ) defined in ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

25 Numerical evidence for non-affine image of a square Instead of mapping, use quadratic serendipity GBC interpolation with mean value coordinates: n ( vi + v ) i+1 u h = I qu := u(v i )ψ ii + u ψ i(i+1) 2 n = 2 n = 4 i=1 Non-affine bilinear mapping u u h L 2 (u u h ) L 2 n error rate error rate 2 5.0e-2 6.2e e e e e e e e e e e ARNOLD, BOFFI, FALK, Math. Comp., 2002 Quadratic serendipity GBC method u u h L 2 (u u h ) L 2 n error rate error rate e e e e e e e e e e e e e e e e Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

26 Outline 1 Linear elements on polygons 2 Quadratic serendipity elements on polygons 3 Order r serendipity elements on n-cubes 4 Future directions Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

27 Serendipity elements on squares O(h 2 ) O(h 3 ) O(h 4 ) O(h 5 ) O(h 6 ) For r 4: O(h r ) tensor product: r 2 + 2r + 1 degrees of freedom O(h r 1 ) serendipity: (r 2 + 3r + 6) degrees of freedom 2 u u h H 1 (Ω) C h r u H }{{} r+1 (Ω), u H r+1 (Ω). }{{} approximation error optimal error bound Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

28 Mathematical challenges Basis functions must be constructed to implement serendipity elements. Current constructions lack key mathematical properties, limiting their broader usage Goal: Construct basis functions for serendipity elements satisfying the following: Symmetry: Accommodate interior degrees of freedom that grow according to triangular numbers on square-shaped elements. Tensor product structure: Write as linear combinations of standard tensor product functions. Hierarchical: Generalize to methods on n-cubes for any n 2, allowing restrictions to lower-dimensional faces. O(h 2 ) O(h 3 ) O(h 4 ) O(h 5 ) O(h 6 ) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

29 Which monomials do we need? O(h 3 ) serendipity element: { total degree at most cubic (req. for O(h 3 ) approximation) }} { { 1, x, y, x 2, y 2, xy, x 3, y 3, x 2 y, xy 2, x 3 y, xy 3, x 2 y 2, x 3 y 2, x 2 y 3, x 3 y 3 } }{{} at most cubic in each variable (used in O(h 3 ) tensor product methods) We need an intermediate set of 12 monomials! The superlinear degree of a polynomial ignores linearly-appearing variables. Example: sldeg(xy 3 ) = 3, even though deg(xy 3 ) = 4 ( n ) Definition: sldeg(x e 1 1 x e 2 2 x n en ) := e i # {e i : e i = 1} {1, x, y, x 2, y 2, xy, x 3, y 3, x 2 y, xy 2, x 3 y, xy 3, x 2 y 2, x 3 y 2, x 2 y 3, x 3 y 3 } }{{} superlinear degree at most 3 (dim=12) ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41 i=1

30 Superlinear polynomials form a lower set Given a monomial x α := x α 1 1 x α d d, associate the multi-index of d non-negative integers α = (α 1, α 2,..., α d ) N d 0. Define the superlinear norm of α as α sprlin := so that the superlinear multi indices are S r = d α j, j=1 α j 2 { α N d 0 : α sprlin r Observe that S r has a partial ordering µ α means µ i α i. }. Thus S r is a lower set, meaning We can thus apply the following recent result. α S r, µ α = µ S r Theorem (Dyn and Floater, 2013) Fix a lower set L N d 0 and points z α R d for all α L. For any sufficiently smooth d-variate real function f, there is a unique polynomial p span{x α : α L} that interpolates f at the points z α, with partial derivative interpolation for repeated z α values. DYN AND FLOATER Multivariate polynomial interpolation on lower sets, J. Approx. Th., to appear. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

31 Partitioning and reordering the multi-indices By a judicious choice of the interpolation points z α = (x i, y j ), we recover the dimensionality associations of the degrees of freedom of serendipity elements. y 5 y 1 y 4 y 5 y 3 y 4 y 2 y 3 y 1 y 2 The order 5 serendipity element, with degrees of freedom color-coded by dimensionality. y 0 x 0 x 1 x 2 x 3 x 4 x 5 The lower set S 5, with equivalent color coding. y 0 x 0 x 2 x 3 x 4 x 5 x 1 The lower set S 5, with domain points z α reordered. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

32 Symmetrizing the multi-indices By collecting the re-ordered interpolation points z α = (x i, y j ), at midpoints of the associated face, we recover the dimensionality associations of the degrees of freedom of serendipity elements. y 1 y 1 y 5 y 4 y 3 y 2, y 3, y 4, y 5 y 2 y 0 x 0 x 2 x 3 x 4 x 5 x 1 The lower set S 5, with domain points z α reordered. y 0 x 0 x 2, x 3, x 4, x 5 x 1 A symmetric reordering, with multiplicity. The associated interpolant recovers values at dots, three partial derivatives at edge midpoints, and two partial derivatives at the face midpoint. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

33 2D symmetric serendipity elements Symmetry: Accommodate interior degrees of freedom that grow according to triangular numbers on square-shaped elements. O(h 2 ) O(h 3 ) O(h 4 ) O(h 5 ) O(h 6 ) O(h 7 ) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

34 Tensor product structure The Dyn-Floater interpolation scheme is expressed in terms of tensor product interpolation over maximal blocks in the set using an inclusion-exclusion formula coefficient calculator +1 1 Put differently, the linear combination is the sum over all blocks within the lower set with coefficients determined as follows: Place the coefficient calculator at the extremal block corner. Add up all values appearing in the lower set. The coefficient for the block is the value of the sum. Hence: black dots +1; white dots -1; others 0. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

35 Linear combination of tensor products Tensor product structure: Write basis functions as linear combinations of standard tensor product functions = Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

36 3D elements Hierarchical: Generalize to methods on n-cubes for any n 2, allowing restrictions to lower-dimensional faces. Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

37 3D coefficient computation Lower sets for superlinear polynomials in 3 variables: Decomposition into a linear combination of tensor product interpolants works the same as in 2D, using the 3D coefficient calculator at left. (Blue +1; Orange -1). FLOATER, G, Nodal bases for the serendipity family of finite elements, Submitted, Available as arxiv: Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

38 Outline 1 Linear elements on polygons 2 Quadratic serendipity elements on polygons 3 Order r serendipity elements on n-cubes 4 Future directions Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

39 Future Directions Expand serendipity results from cubes to polyhedra. Incorporate elements into finite element software packages. Analyze speed vs. accuracy trade-offs. n = r 2n dim Q r r 2 + 2r + 1 dim S r n = (r 2 + 3r + 6) dim Q r r 3 + 3r 2 + 3r + 1 dim S r (r 3 + 6r r + 24) Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41

40 Acknowledgments Chandrajit Bajaj UT Austin Alexander Rand UT Austin / CD-adapco Michael Holst UC San Diego National Biomedical UC San Diego Computation Resource Michael Floater University of Oslo N. Sukumar UC Davis Thanks for the invitation to speak! Slides and pre-prints: Andrew Gillette - U. Arizona Polytope ( ) and Serendipity FEM ASU Seminar - Apr / 41