Adding a new finite element to FreeFem++:

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1 Adding a new finite element to FreeFem++: the example of edge elements Marcella Bonazzoli 1 2, Victorita Dolean 1,3, Frédéric Hecht 2, Francesca Rapetti 1, Pierre-Henri Tournier 2 1 LJAD, University of Nice Sophia Antipolis, Nice, France 2 LJLL, Pierre and Marie Curie University, Paris, France - Inria Alpines 3 University of Strathclyde, Glasgow, UK December 15, 2017 Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

Definition of a finite element A finite element (FE) is a triple (K, P, Σ): K - geometrical element of the mesh T h over Ω ex: triangle in 2d, tetrahedron in 3d, P - finite-dimensional space of

2 Definition of a finite element A finite element (FE) is a triple (K, P, Σ): K - geometrical element of the mesh T h over Ω ex: triangle in 2d, tetrahedron in 3d, P - finite-dimensional space of functions on K, Σ - set of linear functionals ξ i acting on P (degrees of freedom). Example of P 1 Lagrange finite element (in 2d): K - triangle, P - space of polynomials of degree 1 on a triangle basis functions: barycentric coordinates λ n1,λ n2,λ n3 (hat functions) Σ - functionals giving the values at mesh nodes: ξ ni : w w(x ni ) Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

3 Definition of a finite element P 1 Lagrange finite element: basis functions: barycentric coordinates λ n1,λ n2,λ n3 (hat functions) degrees of freedom: values at mesh nodes ξ ni : w w(x ni ) Property (duality) ξ ni (λ nj ) = λ nj (x ni ) = δ ij = { 1 if i = j 0 if i j Consequence: the coefficients of the interpolation operator Π h are the values at mesh nodes Π h : H 1 (T ) V h (T ), u u h = m c j λ nj, with c j = ξ nj (u) = u(x nj ) j=1 Indeed if ξ ni (λ nj ) = δ ij, then ξ ni (u h ) = m j=1 c j ξ ni (λ nj ) = c i Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

4 Adding a new finite element to FreeFem++ To add a new finite element to FreeFem++, write a C++ plugin that defines a C++ class. Main ingredients the basis functions and their derivatives (class function FB): defined locally in a triangle/tetrahedron K, the interpolation operator (class constructor): define the computation of the degrees of freedom (quadrature formulas for more complicated FE) It requires degrees of freedom ξ i in duality with the basis functions! Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

5 Edge finite elements Edge finite elements are well suited for the approximation of the electric field in Maxwell s equations: conformal discretization of H(curl, Ω) = {v L 2 (Ω) 3, v L 2 (Ω) 3 } In FreeFem++ edge finite elements of degree 1 in 3d: keyword Edge03d, we added the edge elements of degree 2, 3 in 3d to use them: load "Element_Mixte3d"; keywords Edge13d, Edge23d. [Bonazzoli, Dolean, Hecht, Rapetti, An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell s equations. Accepted for publication in CAMWA. Preprint HAL <hal >] Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

6 Edge finite elements Tetrahedral mesh T h of Ω, V h H(curl, Ω) Low order edge finite elements (degree r = 1, Nédélec) Given a tetrahedron T T h, the local basis functions are associated with the oriented edges e = {n i, n j } of T : w e = λ ni λ nj λ nj λ ni, (the λ nl are the barycentric coordinates). oriented edges, 4 they are vector functions, they ensure the continuity of the tangential component across inter-element interfaces, degrees of freedom: ξ e : w 1 e e w t e, 2 e 5 e 1 e 3 e 6 e 2 1 e 4 3 duality: ξ e (w e ) = δ ee Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

7 Edge finite elements Visualization of basis functions in 2d: ξ e : w 1 { 1 if e = e w t e, ξ e (w e ) = δ ee = e e 0 if e e Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

8 High order edge finite elements The basis functions Generators of degree r = k + 1 Given T T h, for all oriented edges e of T and for all multi-indices k = (k 1, k 2, k 3, k 4 ) of weight k = k 1 + k 2 + k 3 + k 4, define: w {k,e} = λ k w e, where λ k = (λ n1 ) k1 (λ n2 ) k2 (λ n3 ) k3 (λ n4 ) k4. Only barycentric coordinates! Still V h H(curl, Ω) 4 E.g. degree r = 2 k = 1 λ 1 w e1, λ 2 w e1, λ 3 w e1, λ 4 w e1, λ 1 w e2, λ 2 w e2, λ 3 w e2, λ 4 w e2, e 5 e 3 e 6 λ 1 w e3, λ 2 w e3, λ 3 w e3, λ 4 w e3, λ 1 w e4, λ 2 w e4, λ 3 w e4, λ 4 w e4, e 1 1 e 2 3 λ 1 w e5, λ 2 w e5, λ 3 w e5, λ 4 w e5, λ 1 w e6, λ 2 w e6, λ 3 w e6, λ 4 w e6. 2 e 4 Select linearly independent basis functions! (dim = 20) [Rapetti, Bossavit, Whitney forms of higher degree, SIAM J. Num. Anal., 47(3), 2009] Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

9 High order edge finite elements The basis functions 4 e 5 e 3 e 6 e 1 = {1, 2}, e 2 = {1, 3}, e 3 = {1, 4}, e 4 = {2, 3}, e 5 = {2, 4}, e 6 = {3, 4}, f 1 = {2, 3, 4}, f 2 = {1, 3, 4}, f 3 = {1, 2, 4}, f 4 = {1, 2, 3}. e 1 1 e 2 3 degree r = 2 : 2 e 4 Edge-type basis functions: w 1 = λ 1 w e1, w 2 = λ 2 w e1, w 3 = λ 1 w e2, w 4 = λ 3 w e2, w 5 = λ 1 w e3, w 6 = λ 4 w e3, w 7 = λ 2 w e4, w 8 = λ 3 w e4, w 9 = λ 2 w e5, w 10 = λ 4 w e5, w 11 = λ 3 w e6, w 12 = λ 4 w e6, Face-type basis functions: w 13 = λ 4 w e4, w 14 = λ 3 w e5, w 15 = λ 4 w e2, w 16 = λ 3 w e3, w 17 = λ 4 w e1, w 18 = λ 2 w e3, w 19 = λ 3 w e1, w 20 = λ 2 w e2. Choice using the global numbers of the vertices! Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

10 High order edge finite elements The degrees of freedom (dofs) Revisitation of classical dofs Define the dofs on T T h for degree r 1 as the functionals: ξ e : w 1 (w t e ) q, q P r 1 (e), e E(T ), e e ξ f : w 1 (w t f,i ) q, q P r 2 (f ), f F(T ), f f t f,i two sides of f, i = 1, 2, ξ T : w 1 (w t T,i ) q, q P r 3 (T ), T T t T,i three sides of T, i = 1, 2, 3. As polynomials q, use products of barycentric coordinates [Bonazzoli, Rapetti, High order finite elements in numerical electromagnetism: dofs and generators in duality, NUMA 2017] Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

11 High order edge finite elements The degrees of freedom (dofs) 4 e 3 e 6 e 1 = {1, 2}, e 2 = {1, 3}, e 3 = {1, 4}, e 4 = {2, 3}, e 5 = {2, 4}, e 6 = {3, 4}, f 1 = {2, 3, 4}, f 2 = {1, 3, 4}, f 3 = {1, 2, 4}, f 4 = {1, 2, 3}. e 5 2 e 1 1 e 4 e 2 3 degree r = 2: for e = {n i, n j }, P 1 (e) = span(λ ni, λ nj ); P 0 (f ) = span(1); no volume dofs Edge-type dofs: ξ 1 : w 1 e 1 e 1 (w t e1 ) λ 1, ξ 2 : w 1 e 1 ξ 11 : w 1 e 6 e 6 (w t e6 ) λ 3, ξ 12 : w 1 e 6 e 1 (w t e1 ) λ 2,... e 6 (w t e6 ) λ 4, Face-type dofs: ξ 13 : w 1 f 1 f 1 (w t e4 ), ξ 14 : w 1 f 1 f 1 (w t e5 ),... ξ 19 : w 1 f 4 f 4 (w t e1 ), ξ 20 : w 1 f 4 f 4 (w t e2 ). Same choice as for generators Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

12 High order edge finite elements Restoring duality (for r > 1) To restore duality between basis functions ~w j and dofs ξ i : Ṽ ij = ξ i (~w j ) = δ ij take linear combinations of the old basis functions w j with coefficients given by the entries of V 1. Properties of the matrix V ij = ξ i (w j ): V does not depend on the metrics of the tetrahedron T (but pay attention to orientation!), V is blockwise lower triangular, V 1 entries are integer numbers. [Bonazzoli, Dolean, Hecht, Rapetti, An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell s equations. Accepted for publication in CAMWA. Preprint HAL <hal >] Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

13 High order edge finite elements Restoring duality degree r = 2, basis functions and dofs listed as before (ordering and choice!): V 1 = Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

Numerical results h as various optoelectronic rchitectural and automotive TIR.

cylindrical chamber with 5 rings of 32 antennas (rectangular waveguides) hs of Figures 8.4.3 8.4.5. uce the results and graphs of e-index slab given by Eqs.

You will need to implement e results and graphs of Figures produce the results and graphs Waveguides are used to transfer

Some common guiding structures are shown in the figure be These include the typical coaxial cable, the two-wire and

In practice, the choice of structure is dictated by: (a) the desired operating frequ band, (b) the amount of power to be

14 Numerical results h as various optoelectronic rchitectural and automotive TIR. Modeling a brain imaging system (ANR project MEDIMAX) Microwave brain imaging system prototype (EMTensor, Austria): Waveguid cylindrical chamber with 5 rings of 32 antennas (rectangular waveguides) hs of Figures uce the results and graphs of e-index slab given by Eqs. (8.6.1) of the slab given in (8.6.8). the perfect lens property to the d to the presence of losses by 4. You will need to implement e results and graphs of Figures produce the results and graphs Waveguides are used to transfer electromagnetic power efficiently from one poin space to another. Some common guiding structures are shown in the figure be These include the typical coaxial cable, the two-wire and mictrostrip transmission l hollow conducting waveguides, and optical fibers. In practice, the choice of structure is dictated by: (a) the desired operating frequ band, (b) the amount of power to be transferred, and (c) the amount of transmis losses that can be tolerated. over, derive the Brewster angle f possible Brewster angles and produce the results and graphs Γ i Γ w Fig Typical waveguiding structures. Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

15 Numerical results Variational formulation in FreeFem++ Ω = [ ] ( E) ( v) γ 2 E v i=1 Γ i iβ(e n) (v n) Γ j g j v, v V h V, V = {v H(curl, Ω), v n = 0 on Γ w }. load "Element_Mixte3d" fespace Vh(Th,Edge13d); // Edge03d, Edge13d, Edge23d Vh<complex> [Ex,Ey,Ez], [vx,vy,vz]; macro Curl(ux,uy,uz) [dy(uz)-dz(uy),dz(ux)-dx(uz),dx(uy)-dy(ux)] // EOM macro Nvec(ux,uy,uz) [uy*n.z-uz*n.y,uz*n.x-ux*n.z,ux*n.y-uy*n.x] // EOM varf medimax([ex,ey,ez], [vx,vy,vz]) = int3d(th)(curl(ex,ey,ez) *Curl(vx,vy,vz) - gamma^2*[ex,ey,ez] *[vx,vy,vz]) + int2d(th,ports)(1i*beta*nvec(ex,ey,ez) *Nvec(vx,vy,vz)) + on(guide,ex=0,ey=0,ez=0); varf medimaxrhs([ex,ey,ez], [vx,vy,vz]) = int2d(th,portj)([vx,vy,vz] *[Gjx,Gjy,Gjz]) + on(guide,ex=0,ey=0,ez=0); Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

Numerical results (FreeFem++ with HPDDM) Plastic-filled cylinder (non dissipative!

One transmitting antenna in the second ring from the top.

16 Numerical results (FreeFem++ with HPDDM) Plastic-filled cylinder (non dissipative!) immersed in the matching solution inside the imaging chamber. One transmitting antenna in the second ring from the top. Norm of the real part of the solution field: [Bonazzoli, Dolean, Rapetti, Tournier. Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging. 2017] Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

17 Numerical results (FreeFem++ with HPDDM) Solve 32 linear systems (same matrix, different right-hand sides) Pseudo-block GMRES with a Domain Decomposition preconditioner 1024 cores on Curie supercomputer Reference solution with 114 million (complex-valued) unknowns. Edge finite elements of degree 1, 2 for different mesh sizes: relative error M 2.4M 5.2M 8.5M 13M 21M 27M time to solution (s) 43M 46M Degree 1 Degree 2 74M Relative error Err = j,i S ij S ref ij 2 j,i Sref ij 2 (the S ij are the measurable quantities, calculated from the solution E) For Err 0.1 degree 1: 21 M unks, 130 s degree 2: 5 M unks, 62 s Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

18 Conclusion The user can add new finite elements to FreeFem++: define basis functions and interpolation operator (need duality!). FreeFem++ is based on a natural transcription of the variational formulation of the considered boundary value problem. The high order finite elements make it possible to attain a given accuracy with much fewer unknowns and much less computing time than the degree 1 approximation. The parallel implementation in HPDDM of the domain decomposition preconditioner is essential to be able to solve the considered large scale linear systems. Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

19 Conclusion Thank you for your attention! Marcella Bonazzoli (LJLL, Inria) Adding a new finite element 15/12/ / 19

Efficient domain decomposition methods for the time-harmonic Maxwell equations

Efficient domain decomposition methods for the time-harmonic Maxwell equations Marcella Bonazzoli 1, Victorita Dolean 2, Ivan G. Graham 3, Euan A. Spence 3, Pierre-Henri Tournier 4 1 Inria Saclay (Defi