Serendipity Finite Element Methods for Cardiac Electrophysiology Models

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1 Serendipity Finite Element Methods for Cardiac Electrophysiology Models Andrew Gillette Department of Mathematics University of California, San Diego agillette/ Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

2 Monodomain Equations for Electrophysiology Numerical approximation of solutions to the monodomain equations allows the simulation of electrical propagation on patient-specific cardiac geometries. v = transmembrane potential u = extracellular potential s = state of ionic currents F(v, s) = voltage-dependent state update I(v, s) = ionic current across membrane Find u, v : Ω R such that ds = F(s, v) dt in Ω λ 1 + λ (M i v) = v t + I(v, s) in Ω λ (M i v) = (M i u) in Ω (M i v) ˆn = 0 on Ω (M i u) ˆn = 0 on Ω M i, M e = intra-, extra-cellular conductivity tensors, satisfying M e = λm i Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Application Motivations and Challenges Scientific and clinical motivations: Provide models of electrical activity at spatial and temporal resolutions beyond MRI and EKG data.

3 Application Motivations and Challenges Scientific and clinical motivations: Provide models of electrical activity at spatial and temporal resolutions beyond MRI and EKG data. Allow patient-specific modeling for real-time simulations during surgery. Analyze spiral waves that occur during atrial fibrillation: [ show videos ] Competing theories of the biophysical nature of this phenomenon have been proposed; computational simulations and mathematical analysis can provide evidence for or against treatment strategies (e.g. where to cauterize heart cells) Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

4 Application Motivations and Challenges Computational challenges: Survey of various codes on a benchmark problem revealed large discrepancies in the estimation of activation time, a basic measure of accuracy. In the application context, Péclet number = k h2 6 D = (reaction rate) (mesh element length)2 6 (diffusivity constant) which can lead to Gibbs type phenomena of spurious oscillations. This can be overcome either by using many mesh elements (small h values) or higher order methods, both of which can be computationally expensive. Using different functions for geometry and function approximation is expensive; unified methods are needed. Serendipity finite element methods help address these challenges. > 1 Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

5 Table of Contents 1 Geometric Decompositions for Serendipity FEM 2 The Cubic Tensor Product Case 3 The Cubic Serendipity Space Case 4 Future Directions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

6 Outline 1 Geometric Decompositions for Serendipity FEM 2 The Cubic Tensor Product Case 3 The Cubic Serendipity Space Case 4 Future Directions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

7 What is a serendipity finite element method? Goal: Efficient, accurate approximation of the solution to a PDE over Ω R n. Standard O(h r ) tensor product finite element method in R n : Mesh Ω by n-dimensional cubes of side length h. Set up a linear system involving (r + 1) n degrees of freedom (DoFs) per cube. For unknown continuous solution u and computed discrete approximation u h : u u h H 1 (Ω) C h r u H }{{} r+1 (Ω), u H r+1 (Ω). }{{} approximation error optimal error bound A O(h r ) serendipity FEM converges at the same rate with fewer DoFs per element: O(h) O(h 2 ) O(h 3 ) O(h) O(h 2 ) O(h 3 ) tensor product elements serendipity elements Example: For O(h 3 ), d = 3, 50% fewer DoFs 50% smaller linear system Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

21, ϑ 31, ϑ 24, ϑ 34 } 14 24 34 44 13 43 12 42 11 21 31 41 monomials basis functions domain points Previously known basis functions employ Legendre polynomials These functions bear no symmetrical

8 What is a geometric decomposition? A geometric decomposition for a finite element space is an explicit correspondence: {1, x, y, x 2, y 2, xy, x 3, y 3, x 2 y, xy 2, x 3 y, xy 3 } {ϑ 11, ϑ 14, ϑ 41, ϑ 44, ϑ 12, ϑ 13, ϑ 42, ϑ 43, ϑ 21, ϑ 31, ϑ 24, ϑ 34 } monomials basis functions domain points Previously known basis functions employ Legendre polynomials These functions bear no symmetrical correspondence to the domain points 1 (x 1)(y 1) ( x 2 1 ) (y 1) 1 5 x ( x 2 1 ) (y 1) 4 2 vertex edge (quadratic) edge (cubic) SZABÓ AND I. BABUŠKA Finite element analysis, Wiley Interscience, A canonical geometric decomposition can aid in higher order generalizations. Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Motivations and Related Topics Goal: Construct geometric decompositions of serendipity spaces using linear combinations of standard tensor product functions. Focus: Cubic Hermites.

9 Motivations and Related Topics Goal: Construct geometric decompositions of serendipity spaces using linear combinations of standard tensor product functions. Focus: Cubic Hermites. Isogeometric analysis: Finding basis functions suitable for both domain description and PDE approximation avoids the expensive computational bottleneck of re-meshing. COTTRELL, HUGHES, BAZILEVS Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Modern mathematics: Finite Element Exterior Calculus, Discrete Exterior Calculus, Virtual Element Methods... ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, DA VEIGA, BREZZI, CANGIANI, MANZINI, RUSSO Basic Principles of Virtual Element Methods, M3AS, Flexible Domain Meshing: Serendipity type elements for Voronoi meshes provide computational benefits without need of tensor product structure. RAND, G., BAJAJ Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates, Mathematics of Computation, in press. Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

10 Outline 1 Geometric Decompositions for Serendipity FEM 2 The Cubic Tensor Product Case 3 The Cubic Serendipity Space Case 4 Future Directions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

11 Cubic Hermite Geometric Decomposition: 1D {1, x, x 2, x 3 } {ψ 1, ψ 2, ψ 3, ψ 4 } approxim. geometry monomials basis functions domain points Cubic Hermite Basis on [0, 1] Approximation: x r = ψ 1 ψ 2 ψ 3 := ψ 4 1 3x 2 + 2x 3 x 2x 2 + x 3 x 2 x 3 3x 2 2x ε r,i ψ i, for r = 0, 1, 2, 3, where [ε r,i ] = i= Geometry: u = u(0)ψ 1 + u (0)ψ 2 u (1)ψ 3 + u(1)ψ 4, u P 3 ([0, 1]) }{{} cubic polynomials Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

12 Cubic Hermite Geometric Decomposition: 2D { x r y s } } 0 r, s 3 {{ } Q 3 ([0, 1] 2 ) { ψi (x)ψ j (y) 1 i, j 4 } monomials basis functions domain points Approximation: x r y s = 4 4 ε r,i ε s,j ψ ij, for 0 r, s 3, ε r,i as in 1D. i=1 j=1 Geometry: ψ 11 ψ 21 ψ 22 u = u (0,0) ψ 11 + xu (0,0) ψ 21 + yu (0,0) ψ 12 + x yu (0,0) ψ 22 +, u Q 3 ([0, 1] 2 ) Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Cubic Hermite Geometric Decomposition: 3D { x r y s z t } } 0 r, s, t 3 {{ } Q 3 ([0, 1] 3 ) ψ

444 124 114 214 314 414 123 113 213 313 413 122 112 212 312 412 121 111 211 311 411 monomials

13 Cubic Hermite Geometric Decomposition: 3D { x r y s z t } } 0 r, s, t 3 {{ } Q 3 ([0, 1] 3 ) ψ i(x)ψ j (y)ψ k (z) 1 i, j, k monomials basis functions domain points Approximation: x r y s z t = ε r,i ε s,j ε t,k ψ ijk, for 0 r, s, t 3, ε r,i as in 1D. i=1 j=1 k=1 Geometry: Contours of level sets of the basis functions: ψ 111 ψ 112 ψ 212 ψ 222 Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

14 Two families of finite elements on cubical meshes Q r Λ k ([0, 1] n ) tensor product spaces early work: RAVIART, THOMAS 1976, NEDELEC 1980 more recently: ARNOLD, BOFFI, BONIZZONI arxiv: , 2012 ( degree r in each variable) S r Λ k ([0, 1] n ) serendipity finite element spaces (superlinear degree r) early work: STRANG, FIX An analysis of the finite element method 1973 more recently: ARNOLD, AWANOU FoCM 11:3, 2011, and arxiv: , The superlinear degree of a polynomial ignores linearly-appearing variables. Q 3 Λ 0 ([0,1] 2 ) (dim=16) {}}{ n = 2 : { 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 } }{{} S 3 Λ 0 ([0,1] 2 ) (dim=12) Q 3 Λ 0 ([0,1] 3 ) (dim=64) {}}{ n = 3 : { 1,..., xyz, x 3 y, x 3 z, y 3 z,..., x 3 yz, xy 3 z, xyz 3, x 3 y 2,..., x 3 y 3 z 3 } }{{} S 3 Λ 0 ([0,1] 3 ) (dim=32) Q r Λ k and S r Λ k and have the same key mathematical properties needed for FEEC (degree, inclusion, trace, subcomplex, unisolvence, commuting projections) but for fixed k 0, r, n 2 the serendipity spaces have fewer degrees of freedom Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

15 Outline 1 Geometric Decompositions for Serendipity FEM 2 The Cubic Tensor Product Case 3 The Cubic Serendipity Space Case 4 Future Directions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

16 Cubic Hermite Serendipity Geom. Decomp: 2D Theorem [G, 2012]: A Hermite-like geometric decomposition of S 3 ([0, 1] 2 ) exists. { x r y s } } sldeg 3 {{ } S 3 ([0, 1] 2 ) { ϑ lm (to be defined) } monomials basis functions domain points Approximation: x r y s = lm ε r,i ε s,j ϑ lm, for superlineardegree(x r y s ) 3 Geometry: ϑ 11 ϑ 21 u = u (0,0) ϑ 11 + xu (0,0) ϑ 21 + yu (0,0) ϑ 12 + u S 3 ([0, 1] 2 ) Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

17 Cubic Hermite Serendipity Geom. Decomp: 2D Proof Overview: 1 Fix index sets and basis orderings based on domain points: V = vertices (11, 14,...) E = edges (12, 13,...) D = interior (22, 23,...) [ϑ lm ] := [ ϑ 11, ϑ 14, ϑ 41, ϑ 44, ϑ 12, ϑ 13, ϑ 42, ϑ 43, ϑ 21, ϑ 31, ϑ 24, ϑ 34 ], [ψ ij ] := [ ψ 11, ψ 14, ψ 41, ψ 44, ψ 12, ψ 13, ψ 42, ψ 43, ψ 21, ψ 31, ψ 24, ψ 34, ψ 22, ψ 23, ψ 32, ψ 33 ] }{{}}{{}}{{} indices in V indices in E indices in D 2 Define a matrix H with entries h lm ij so that lm V E, ij V E D. 3 Define the serendipity basis functions ϑ lm via [ϑ lm ] := H[ψ ij ] and show that the approximation and geometry properties hold. Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

18 Hermite Style Serendipity Functions (2D) ϑ 11 (x 1)(y 1)( 2 + x + x 2 + y + y 2 ) ϑ 14 (x 1)(y + 1)( 2 + x + x 2 y + y 2 ) ϑ 41 (x + 1)(y 1)( 2 x + x 2 + y + y 2 ) ϑ 44 (x + 1)(y + 1)( 2 x + x 2 y + y 2 ) ϑ 12 (x 1)(y 1) 2 (y + 1) ϑ [ϑ lm ] = 13 = (x 1)(y 1)(y + 1) 2 ϑ 42 (x + 1)(y 1) 2 (y + 1) 1 8 ϑ 43 (x + 1)(y 1)(y + 1) 2 ϑ 21 ϑ (x 1) 2 (x + 1)(y 1) 31 ϑ (x 1)(x + 1) 2 (y 1) 24 (x 1) 2 (x + 1)(y + 1) ϑ 34 (x 1)(x + 1) 2 (y + 1) ϑ 11 ϑ 21 ϑ 31 Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

19 Cubic Hermite Serendipity Geom. Decomp: 3D Theorem [G, 2012]: A Hermite-like geometric decomposition of S 3 ([0, 1] 3 ) exists. { x r y s z t } } sldeg 3 {{ } S 3 ([0, 1] 3 ) { ϑ lmn (to be defined) } monomials basis functions domain points Approximation: x r y s z t = lmn ε r,i ε s,j ε t,k ϑ lmn, for superlineardegree(x r y s z t ) 3 Geometry: u = u (0,0,0) ϑ xu (0,0,0) ϑ yu (0,0,0) ϑ zu (0,0,0) ϑ u S 3 ([0, 1] 3 ) ϑ 111 ϑ 112 Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

20 Cubic Hermite Serendipity Geom. Decomp: 3D Proof Overview: 1 Fix index sets and basis orderings based on domain points. [ϑ lmn ] := [ ϑ 111,..., ϑ 444, ϑ 112,..., ϑ 443 ], [ψ ijk ] := [ ψ 111,..., ψ 444, ψ 112,..., ψ 443, ψ 122,..., ψ 433, ψ 222,..., ψ 333 ] }{{}}{{}}{{}}{{} indices in V indices in E indices in F indices in M 2 Define a matrix W with entries h lmn ijk (where lmn V E) 3 Define the serendipity basis functions ϑ lmn via [ϑ lmn ] := W[ψ ijk ] and show that the approximation and geometry properties hold. [ϑ lmn ] = ϑ 111 ϑ 114. ϑ 442 ϑ 443 (x 1)(y 1)(z 1)( 2 + x + x 2 + y + y 2 + z + z 2 ) (x 1)(y 1)(z + 1)( 2 + x + x 2 + y + y 2 z + z 2 ) =. (x + 1)(y + 1)(z 1) 2 (z + 1) (x + 1)(y + 1)(z 1)(z + 1) Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

21 Cubic Bernstein Serendipity Geom. Decomp: 2D, 3D Cubic Bernstein Basis on [0, 1] ψ 1 (1 x) 3 ψ 2 ψ 3 := (1 x) 2 x (1 x)x 2 ψ 4 x Theorem [G, 2012]: Bernstein-like geometric decompositions of S 3 ([0, 1] 2 ) and S 3 ([0, 1] 3 ) exist. ξ 11 (x 1)(y 1)( 2 2x + x 2 2y + y 2 ) ξ 14 (x 1)(y + 1)( 2 2x + x 2 + 2y + y 2 ) [ξ lm ] = 1. =. 16 ξ 24 (x 1) 2 (x + 1)(y + 1) ξ 34 (x 1)(x + 1) 2 (y + 1) ξ 111 (x 1)(y 1)(z 1)( 5 2x + x 2 2y + y 2 2z + z 2 ) ξ 114 (x 1)(y 1)(z + 1)( 5 2x + x 2 2y + y 2 + 2z + z 2 ) [ξ lmn ] = 1. =. 32 ξ 442 (x + 1)(y + 1)(z 1) 2 (z + 1) ξ 443 (x + 1)(y + 1)(z 1)(z + 1) 2 Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

22 Bernstein Style Serendipity Functions (2D) β 11 β 21 β 31 ξ 11 ξ 21 ξ Bicubic Bernstein functions (top) and Bernstein-style serendipity functions (bottom). Note boundary agreement with Bernstein functions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Computational Evidence Serendipity cubic Hermite-like method implemented in Matlab to solve Poisson s equation with exact solution u(x, y) = sin(x) e y u u h L 2 (u u h ) L 2 n error rate error rate

23 Computational Evidence Serendipity cubic Hermite-like method implemented in Matlab to solve Poisson s equation with exact solution u(x, y) = sin(x) e y 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 n = 2 Confirms the expected cubic order a priori error estimate n = 32 u u h H 1 (Ω) C h 3 u H }{{} 4 (Ω), u H 4 (Ω), }{{} approximation error optimal error bound Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

24 Outline 1 Geometric Decompositions for Serendipity FEM 2 The Cubic Tensor Product Case 3 The Cubic Serendipity Space Case 4 Future Directions Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

25 Serendipity spaces for large r n = 2 Let p(x, y) := (1 + x)(1 x)(1 y)(1 + y). Observe p P 4 S 4, but p 0 on ([0, 1] 2 ). When r > 3, there are interior domain points for the serendipity spaces as characterized in ARNOLD, AWANOU The serendipity family of finite elements, Found. Comp. Math, 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) Thus, when r 2n, the expected computational savings by serendipity methods is 50% in 2D and 83% in 3D. I am currently pursuing related theoretical results with Snorre Christiansen and Michael Floater (both from U. Oslo). Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Simula for organizing this workshop! Slides and pre-prints: http://ccom.ucsd.

26 Acknowledgments Matt Gonzalez Kevin Vincent Poorya Mirkhosravi Andrew McCulloch Michael Holst National Biomedical Computation Resource Thanks to Simula for organizing this workshop! Slides and pre-prints: Andrew Gillette - UCSD ( )Serendipity FEM for EP Simula - May / 26

Geometric Decomposition of Serendipity Finite Element Spaces

Geometric Decomposition of Serendipity Finite Element Spaces Andrew Gillette Department of Mathematics University of California, San Diego http://ccom.ucsd.edu/ agillette/ Joint Mathematics Meetings 2013