hp-adaptive Finite Elements in the C++ library Concepts

Transcription

1 hp-adaptive Finite Elements in the C++ library Concepts Kersten Schmidt Group POEMS, INRIA, France, Seminar for Applied Mathematics, ETH Zurich, Switzerland 8th July 9, NanoOptics Workshop, Zürich

2 Outline Space discretisation by h-, p- and hp-fem Maxwell s equations Variational formulation Linear system of equations Mesh refinement or increasing the (polynomial) order Resolving material interfaces essential p-fem gives exponential convergence for smooth interfaces hp-fem = suitable combination of h- and p-refinement exponential convergence for interface corners Applications Implementation in Concepts Overview Basis functions with local support Hierarchical basis based on Jacobi polynomials on reference cell Numerical quadrature on reference cell for curved physical cells General Algorithm for Matrix Assembling FEM, DG-FEM and BEM K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

3 Maxwell s equations Electromagnetic wave phenomena are described by Maxwell source problem in time domain permeability µ(x),permittivity ε(x, t), incident E inc (x, t) and initial field E init (x) µ (x) E(x, t) + ε(x, t) t E(x, t) = E inc (x, t) E(x, ) = E init (x) Looking for E(x, t) Maxwell source problem in frequency domain permeability µ(x),permittivity ε(x, ω), incident E inc (x), frequency ω µ (x) E(x) ε(x, ω)ω E(x) = E inc (x) Looking for E(x) Maxwell eigenvalue problem permeability µ(x),permittivity ε(x, ω) Looking for modes (E(x), ω) µ (x) E(x) ε(x, ω)ω E(x) = K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

4 Space discretisation Ansatz with finite functions to approximate exact solution E(x) E N (x) = NX i= c i E i (x) i.e. E E N < ɛ(n) N Finite Differences (FDM) Finite Elements (here p = ) Mesh refinement Expect algebraic convergence ɛ(n) N α Plane Wave Expansion (PWM) Finite Elements (fixed mesh) Increasing Order Expect exponential convergence ɛ(n) exp( βn α ) K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

5 Space discretisation of Maxwell s equation E.g. Maxwell eigenvalue problem µ (x) E(x) ε(x, ω) ω E(x) = ( ) Ansatz with finite functions to approximate exact solution E(x) E N (x) = X N i= c i E i (x) ( ) Finite Differences (FDM) Taylor expansion of E(x)around each grid point to derive stencils for the differential operators. Standard FDTD: Yee algorithm equivalence with variational form, where ansatz functions are Z-splines and test functions are Dirac distributions, J.T. Becerra Sagredo, SAM Report 3-, ETH Zurich. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

6 Space discretisation of Maxwell s equation E.g. Maxwell eigenvalue problem µ (x) E(x) ε(x, ω) ω E(x) = ( ) Ansatz with finite functions to approximate exact solution E(x) E N (x) = X N i= c i E i (x) ( ) Finite Differences (FDM) Taylor expansion of E(x)around each grid point to derive stencils for the differential operators. Standard FDTD: Yee algorithm equivalence with variational form, where ansatz functions are Z-splines and test functions are Dirac distributions, J.T. Becerra Sagredo, SAM Report 3-, ETH Zurich. Galerkin method for Finite Elements and Plane Wave Expansion multiply ( ) with test function F (x) integrate over the domain Ω, use integration by parts formula and insert boundary conditions insert ansatz ( ) and each basis function E i (x) as test function Z µ (x) E N (x) E i (x) ε(x, ω) ω E N (x) E i (x)dx = Ω Same procedure for Maxwell source problem in time or frequency domain. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

7 Space discretisation of Maxwell s equation E.g. Maxwell eigenvalue problem µ (x) E(x) ε(x, ω) ω E(x) = ( ) Ansatz with finite functions to approximate exact solution E(x) E N (x) = X N i= c i E i (x) ( ) Finite Differences (FDM) Taylor expansion of E(x)around each grid point to derive stencils for the differential operators. Standard FDTD: Yee algorithm equivalence with variational form, where ansatz functions are Z-splines and test functions are Dirac distributions, J.T. Becerra Sagredo, SAM Report 3-, ETH Zurich. Galerkin method for Finite Elements and Plane Wave Expansion multiply ( ) with test function F (x) integrate over the domain Ω, use integration by parts formula and insert boundary conditions insert ansatz ( ) and each basis function E i (x) as test function Z µ (x) E N (x) E i (x) ε(x, ω) ω E N (x) E i (x)dx = Ω Same procedure for Maxwell source problem in time or frequency domain. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

8 Space discretisation of Maxwell s equation E.g. Maxwell eigenvalue problem µ (x) E(x) ε(x, ω) ω E(x) = ( ) Ansatz with finite functions to approximate exact solution E(x) E N (x) = X N Galerkin method for Finite Elements and Plane Wave Expansion multiply ( ) with test function F (x) i= c i E i (x) ( ) integrate over the domain Ω, use integration by parts formula and insert boundary conditions insert ansatz ( ) and each basis function E i (x) as test function Z µ (x) E N (x) E i (x) ε(x, ω) ω E N (x)e i (x)dx = Ω Same procedure for Maxwell source problem in time or frequency domain. Strong theory of variational problems, i.e. stability, consistency and convergence coming from functional analysis and mathematical physics for the continuous and numerical analysis for the discrete spaces. Plane wave expansion originately used by mathematical physicists (Schrödinger equation), where Finite Elements come from civil engineering. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

9 Eigenvalue problem in D 6 TE mode at k =, for ε = 9 in [, ] 4 (x) ε(x) h = ω h(x) in [, ] with first non-zero eigenvalue ω = arccos(/4).5.5 ε(x) p = p = p = 3 8 x h = h = / h = / K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

10 Eigenvalue problem in D TE mode at k =, for ε = 9 in [, ] (x) ε(x) h = ω h(x) in [, ] with first non-zero eigenvalue ω = arccos(/4) ωh ω /ω h-fem p = h-fem p = h-fem p = 3 p = p = p = 3 dof h = h = / h = / K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

11 Eigenvalue problem in D TE mode at k =, for ε = 9 in [, ] (x) ε(x) h = ω h(x) in [, ] with first non-zero eigenvalue ω = arccos(/4) p = p = p = 3 ωh ω /ω h-fem p = h-fem p = h-fem p = 3 p-fem dof h = = p-fem with exponential convergence h = / h = / K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

12 Eigenvalue problem in D TE mode at k =, for ε = 9 in [, ] (x) ε(x) h = ω h(x) in [, ] with first non-zero eigenvalue ω = arccos(/4) p = p = p = 3 ωh ω /ω h-fem p = h-fem p = h-fem p = 3 p-fem dof h = = p-fem with exponential convergence h = / h = / ωh ω /ω 3 5 h-fem p = h-fem p = h-fem p = 3 7 p-fem dof K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

13 Eigenvalue problem in D Smooth, periodic dielectricity Plane Wave expansion ε(x).5.5 ωh ω /ω Jumping dielectricity ε(x) x dof 8 4 ε(x) 6 4 ωh ω /ω x dof Global plane wave expansions exponential convergence for smooth material coefficients convergence only algebraic for jumping material coefficients R. Norton, PhD thesis, University of Bath, Sept. 8. Need to resolve material interfaces spectral methods like p-fem conserves exponential convergence K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 6 / 35

14 Eigenvalue problem in D Smooth, periodic dielectricity Plane Wave expansion ε(x).5.5 ωh ω /ω Jumping dielectricity ε(x) x dof Plane Wave expansion versus h- and p-fem 8 4 ε(x) x Global plane wave expansions ωh ω /ω 8 6 PWM h-fem p = h-fem p = h-fem p = 3 p-fem exponential convergence for smooth material coefficients convergence only algebraic for jumping material coefficients R. Norton, PhD thesis, University of Bath, Sept. 8. Need to resolve material interfaces spectral methods like p-fem conserves exponential convergence K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 6 / 35 dof

15 Resolving material interfaces in D FDM / PWM Level Level Level Level 3 h-fem Level Level Level Level 3 Resolving by mesh refinement p-fem p = p = p = 3 p = 4 Resolving by curved elements K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 7 / 35

16 Eigenvalue problems in D Photonic crystal bandstructure dispersive material Photonic crystal composed of cylinders of a polymer with embedded quantum dots in Silicon D. Hermann et. al., Phys. Rev. B 77, 8. Comparison of different FEM and quadratic EWP in k FreeFEM++ with p = without curved cells Code of M. Richter (Univ. Karlsruhe) with p = and curved cells Concepts p-fem with curved cells 3 εp ηα(ω) ε (ω) = ε p + να(ω) α(ω) = ε d (ω) ε p ε d (ω)+ ε p ε d (ω) = + ω p ω ω iγω Relative error (log ) (p ref.) p-fem h-fem h-fem (curved) N (log ) C. Engström, C. Hanfer and K. Schmidt, CTN 6, p , 9 K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 8 / 35

17 Eigenvalue problems in D Photonic crystal wave-guide Supercell approach in a finite cell around a Photonic Cyrstal W-waveguide Joint work with Roman Kappeler (IfE, ETH) S.Soussi, SIAM Journal on Numerical Analysis 43, p. 75, 5. a a a Floquet-transformation in periodicity direction Periodic boundary conditions on horizontal boundaries Eigenvalue problems linear in λ = ω or quadratic in k. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 9 / 35

18 Eigenvalue problems in D Photonic crystal wave-guide Comparison with other codes MPB MIT Photonic-Bands (Group of St.G.Johnson, MIT) Expansion in plane waves, mostly used code for band structure computation. St.G.Johnson and J.D.Joannopoulos, Optics Express 8, p. 73 9,. Comsol FE solver with broad class of applications Linear eigenvalue problem for ω(k) 4 4 error error Concepts mesh for W wave-guide with 5 rows of holes (r =.3a) and ε(x) = degrees of freedom time [s] uniformly sampled on D Brillouin zone (5 points, 3 bands) Sun Fire 46, Dual Core AMD Opteron.53 GHz, Linux K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

19 Eigenvalue problems in D Photonic crystal bandstructure different pattern Contrast ε =, Diameter of cylinder or length of square.9 a, Convergence of p-fem at k = (π/, )/a ωh ω /ω 4 6 ωh ω /ω cylindrical holes dielectric veins curved dielectric rods ( dof) /3 8 cylindrical holes dielectric veins curved dielectric rods ( dof) /3 K.Schmidt, P.Kauf, CMAME 98, p , Mar 9. Weaker convergence behaviour for p-fem in case of material corners Theory : we can expect in general only algebraic convergence K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

20 hp-adaptive refinement in D hp-adaptive FEM Combination of cell refinement and polynomial order enrichment Geometric mesh Exponential convergence also with sharp corners expected λ λ N C exp( βn /3 ). Layers of constant polynomial order around interface corner Polynomial order is raised away from interface corner Cells are refined close to interface corner Occurrence of hanging nodes, possibly multiple hanging nodes per edge for different refinement in two materials K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

21 hp-adaptive refinement in D Maxwell eigenvalue problem BENCHMAX, M. Dauge Benchmark computations for Maxwell equations for the approximation of highly singular solutions TE transmission problem in a square approximated with hp-fem Z Z curl D e(x) curl D e (x)dx = ω ε(x)e(x) e (x)dx Ω Ω Geometry DomE Mesh Convergence of third eigenvalue ε ε ε ε.5.5 ε =, ε =.5.5 relative error - - weighted reg., s = edge elements equivalent problem ndof Ph. Frauenfelder, PhD thesis, ETH Zurich, 4. Approximation of highly singular eigenfunctions with exponential convergence K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

22 hp-adaptive refinement in D Photonic crystal bandstructure Dielectric veins of thickness. a with ε = 8.9 and air holes (ε = ) Coarse Mesh with 9 cells p-fem with algebraic convergence, as expected hp-fem with exponential convergence, as expected relative errors TE mode p-fem N relative errors 5 TM mode N K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

23 hp-adaptive refinement in D Photonic crystal bandstructure Dielectric veins of thickness. a with ε = 8.9 and air holes (ε = ) Coarse Mesh with 9 cells p-fem with algebraic convergence, as expected hp-fem with exponential convergence, as expected relative errors TE mode p-fem hp-fem start with p= N relative errors 5 TM mode N K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

24 Space discretisation by h-, p- and hp-fem Approximation of exact solution E(x) by a finite number of functions P N i= c i E i (x) h-fem, Finite Differences,... mesh refinement p-fem, Global Plane Wave Expansion,... increasing order hp-fem contains both, each cell can be refined or function order can be increased take best choice (a-priori or a-posteriori) in D and 3D material interfaces has to be resolved curved cells hp-fem is able to compute with asymptotically exponential convergence K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

25 Space discretisation by h-, p- and hp-fem Approximation of exact solution E(x) by a finite number of functions P N i= c i E i (x) h-fem, Finite Differences,... mesh refinement p-fem, Global Plane Wave Expansion,... increasing order hp-fem contains both, each cell can be refined or function order can be increased take best choice (a-priori or a-posteriori) in D and 3D material interfaces has to be resolved curved cells hp-fem is able to compute with asymptotically exponential convergence Wave propagation Waves have to be resolved ( 3 dof per wavelength) before we have convergence towards right solution (independent of discretisation scheme). Then, p-fem is more effective (better increase p = 64 7 on a one-cell mesh than refinement of mesh with 64 cells (p = ) three times ( 5 cells). 3 3 K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 6 / 35

26 Space discretisation by h-, p- and hp-fem Approximation of exact solution E(x) by a finite number of functions P N i= c i E i (x) h-fem, Finite Differences,... mesh refinement p-fem, Global Plane Wave Expansion,... increasing order hp-fem contains both, each cell can be refined or function order can be increased take best choice (a-priori or a-posteriori) in D and 3D material interfaces has to be resolved curved cells hp-fem is able to compute with asymptotically exponential convergence Resolving material interfaces Geometries with small details small cells? smallness in or directions anisotropic cells with anisotropic function order domain decomposition and parallel computing Small periodic structures (finite photonic crystals) basis functions with micro-structure (generalised FEM) Yesterday s talk of H. Brandsmeier, joint project with Prof. Ch. Schwab. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 7 / 35

27 Space discretisation by h-, p- and hp-fem Approximation of exact solution E(x) by a finite number of functions P N i= c i E i (x) h-fem, Finite Differences,... mesh refinement p-fem, Global Plane Wave Expansion,... increasing order hp-fem contains both, each cell can be refined or function order can be increased take best choice (a-priori or a-posteriori) in D and 3D material interfaces has to be resolved curved cells hp-fem is able to compute with asymptotically exponential convergence Resolving material interfaces Geometries with small details small cells? smallness in or directions anisotropic cells with anisotropic function order domain decomposition and parallel computing Small periodic structures (finite photonic crystals) basis functions with micro-structure (generalised FEM) Yesterday s talk of H. Brandsmeier, joint project with Prof. Ch. Schwab. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 7 / 35

28 Classification of FEM Finite Element Methods Continuous FEM BEM DG-FEM R Ω µ E E ε ω E E dx = weak solution E H(curl, Ω) FE space is conforming V N H(curl, Ω) (continuous) FEM of boundary integral eq. meshing of surfaces smaller matrices double integral with kernel matrices are dense Fast BEM singular integrals due to singular kernel non-conforming, discontinuous spaces discontinuity over facettes vanishes for N larger matrices, less entries for large p additional terms over facettes or Lagrange multiplicator Edge elements Existing h-, p- and hp-version of Continuous FEM, BEM and DG-FEM Generalised FEM use of problem-adapted basis functions, e.g. local plane wave, singular functions near corners, functions with micro-structure K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 8 / 35

29 Implemenation in Concepts Numerical C++ class library Concepts Started in 995 and further developed at Seminar for Applied Mathematics of ETH Zurich, mainly by Ch. Lage, Ph. Frauenfelder, A.-M. Matache, G. Schmidlin, K. Schmidt, P. Kauf, H. Brandsmeier and severals students. Modular programming Use of interfaces defined in base classes. Re-usability in the development, e.g. various bilinearforms on curved cells. Generality, e.g. meshes with curved cells of mixed type. Concept Oriented Design using mathematical principles. Ph.Frauenfelder and Ch.Lage, Math. Model. Numer. Anal. 36, p ,. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 9 / 35

30 Implemenation in Concepts Numerical C++ class library Concepts hp-adaptive FEM on tensor product elements (quadrilateral, hexahedron) with anisotropic polynomial degree and arbitrary number of hanging nodes per facette for scalar problems in H (Ω) in D, D and 3D, for vectorial problems in (H (Ω)) d in D and 3D, for vectorial problems in H(curl, Ω) in D, for mixed problems in H(curl, Ω) H (Ω) in D. Fast BEM (panel clustering and wavelets) in 3D on triangular surface meshes. Fast writeable sparse matrix format based on hash tables. Indirect linear solver CG, GMRES (intern). Direct linear solver SuperLU, Umfpack, Pardiso (interfaces to libraries). Eigenvalue solver ARPACK, Trilinos (interfaces to libraries). Various time integration schemes, e.g. Euler, Newmark, Runge-Kutta. Solution output or function of the solution, like its gradient, to Matlab, Tecplot, Gnuplot. Post-processing of the solution, e.g. ohmic power loss by integrating Im(ε)E. Class documentation in Doxygen. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

31 Implemenation in Concepts Mesh Import in D and 3D Coord.dat Elm.dat Attrib.dat Radia.dat Concepts Input Generator in D Joint work with R. Kappeler, S. Belfanti, B. Jin (IfE, ETH) Python library for scripting of geometry input Graphical User Interface with Qt Pre-defined pattern, like this K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

32 Implemenation in Concepts Input files HexaWguide-n5-r3-withAirL4.cig string inputpath "../cig/hexa_w_guides" string coordfile "Hexa_W_guide_n5_r3_withAirL4_Coord.dat" string elemfile "Hexa_W_guide_n5_r3_withAirL4_Elms.dat" string attribfile "Hexa_W_guide_n5_r3_withAirL4_Attr.dat" string radiafile "Hexa_W_guide_n5_r3_withAirL4_EdgRadia.dat" string edgecorrfile "Hexa_W_guide_n5_r3_withAirL4_EdgCorr.dat" array double epsilon { } string mode "TE" string refinement "p3, p in 3" string nonrefl_bdattrib "9" double epsarpack e- bool graphics true int graphicpoints 7 Concepts mesh for super-cell of W wave-guide Call for PC bandstructure calculation: PCEVsolve -f HexaWguide-n5-r3-withAirL4.cig K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. / 35

33 Implemenation in Concepts Main C++ file. Reading the input file. Importing the mesh M 3. Defining the FE space on M and perform (adaptive) refinements in h and p 4. Defining the material formula, e.g. for µ(x) and ε(x) 5. Defining the involved bilinearforms with their material or the linearforms for sources. 6. Assembling of the system matrices and vector of the rhs. 7. Solving Time integration scheme with possibly several time solving of linear systems for Maxwell source problem in time domain. Solving a linear system for Maxwell source problem in time domain. Solving an matrix eigenvalue problem for Maxwell eigenvalue problem. 8. Post-processing Graphical output of the solution or function of the solution. Computation of physical quantities like Ohms losses or transmission coefficients. Flexibility by writing problem adapted main files. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

34 Implemenation in Concepts Scalar continuous basis functions (subspace of H (Ω)) vertex function edge function inner or bubble function Representation by local shape function on reference cell [, ] mapped to physical cell.5.5 F K Φ i (x) K = φ k,l (F K x), Continuity is assured by T- or connectivity matrix for each element K Φ i (x) K = X [T K ] kl,i φ kl (F K x), K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

35 Implemenation in Concepts Scalar continuous basis functions (subspace of H (Ω)) vertex function edge function inner or bubble function Representation by local shape function on reference cell [, ] mapped to physical cell.5.5 F K Φ i (x) K = ±φ k,l (F K x), Continuity is assured by T- or connectivity matrix for each element K Φ i (x) K = X [T K ] kl,i φ kl (F K x), K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

36 Implemenation in Concepts Scalar continuous basis functions (subspace of H (Ω)) vertex function edge function inner or bubble function Representation by local shape function on reference cell [, ] mapped to physical cell.5.5 F K Φ i (x) K = φ k,l (F K x), Continuity is assured by T- or connectivity matrix for each element K Φ i (x) K = X [T K ] kl,i φ kl (F K x), K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 4 / 35

37 Implemenation in Concepts Hierarchical basis of H (Ω) We use Ainsworth Coyle s basis with anisotropic polynomial degrees on quadrilateral or hexahedral elements Definition on the reference element η N i,j (ξ, η) = P i (ξ)p j (η) i =,..., p +, j =,..., p + ˆK - ξ P (ξ) = ξ, P (ξ) = +ξ, P i (ξ) = ξ +ξ P, i (ξ) P, i (ξ)... Jacobi polynomials # shape functions = (p + )(p + ) = (p + ) for p = p Vertex function Occuring of P ( ) or P ( ) in both directions K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

38 Implemenation in Concepts Hierarchical basis of H (Ω) We use Ainsworth Coyle s basis with anisotropic polynomial degrees on quadrilateral or hexahedral elements Definition on the reference element η N i,j (ξ, η) = P i (ξ)p j (η) i =,..., p +, j =,..., p + ˆK - ξ P (ξ) = ξ, P (ξ) = +ξ, P i (ξ) = ξ +ξ P, i (ξ) P, i (ξ)... Jacobi polynomials # shape functions = (p + )(p + ) = (p + ) for p = p Edge function Occuring of only one of P ( ) or P ( ) in one directions K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

39 Implemenation in Concepts Hierarchical basis of H (Ω) We use Ainsworth Coyle s basis with anisotropic polynomial degrees on quadrilateral or hexahedral elements Definition on the reference element η N i,j (ξ, η) = P i (ξ)p j (η) i =,..., p +, j =,..., p + ˆK - ξ P (ξ) = ξ, P (ξ) = +ξ, P i (ξ) = ξ +ξ P, i (ξ) P, i (ξ)... Jacobi polynomials # shape functions = (p + )(p + ) = (p + ) for p = p Inner or bubble function Occuring of P i ( ) with i > K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 5 / 35

40 Implemenation in Concepts Hierarchical basis of H(curl, Ω) We use Ainsworth Coyle s basis with anisotropic polynomial degrees on quadrilateral elements Definition on the reference element η ˆK N i,j, (ξ, η) = P i+ (ξ)p (η)` i =,..., p, j j =,..., p + N i,j, (ξ, η) = P i (ξ)p j+ (η)` i =,..., p +, j =,..., p - ξ P (ξ) = ξ, P (ξ) = +ξ, P i (ξ) = ξ +ξ P, i (ξ) P, i (ξ)... Jacobi polynomials # shape functions = (p + )(p + ) + (p + )(p + ) = (p + )(p + ) for p = p Edge function Occuring of P ( ) or P ( ) K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 6 / 35

41 Implemenation in Concepts Hierarchical basis of H(curl, Ω) We use Ainsworth Coyle s basis with anisotropic polynomial degrees on quadrilateral elements Definition on the reference element η ˆK N i,j, (ξ, η) = P i+ (ξ)p (η)` i =,..., p, j j =,..., p + N i,j, (ξ, η) = P i (ξ)p j+ (η)` i =,..., p +, j =,..., p - ξ P (ξ) = ξ, P (ξ) = +ξ, P i (ξ) = ξ +ξ P, i (ξ) P, i (ξ)... Jacobi polynomials # shape functions = (p + )(p + ) + (p + )(p + ) = (p + )(p + ) for p = p Inner or bubble function Occuring of P i ( ) with i > K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 6 / 35

42 Implemenation in Concepts Evaluation on the physical element Nodal element Edge element N i,j (F K x) J N i,j,k (F K x) Element mass matrix for nodal elements Z Z N i,j (F K x)n i,j (FK x) dx = N i,j (ξ, η)n i K ˆK,j (ξ, η) det J dξdη Element stiffness matrix for nodal elements Z Z N i,j (F K x) N i,j (FK x) dx = ( ˆ N i,j (ξ, η)) J J ˆ N i K ˆK,j (ξ, η) det J dξdη Element mass matrix for edge elements Z Z N i,j,k (F K x) J J N i,j,k (FK x) dx = K ˆK N i,j,k (ξ, η) J J N i,j,k (ξ, η) det J dξdη Element stiffness matrix for edge elements Z ( J N i,j,k (F K x)) J N i,j,k (FK x) dx K Z = det J ˆ N i,j,k (ξ, η) det J ˆ N i ˆK,j,k (ξ, η) det J dξdη Numerical quadrature on the reference cell ˆK K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 7 / 35

43 Implemenation in Concepts Evaluation on the physical element Nodal element Edge element N i,j (F K x) J N i,j,k (F K x) Element mass matrix for nodal elements Z Z N i,j (F K x)n i,j (FK x) dx = N i,j (ξ, η)n i K ˆK,j (ξ, η) det J dξdη Element stiffness matrix for nodal elements Z Z N i,j (F K x) N i,j (FK x) dx = ( ˆ N i,j (ξ, η)) J J ˆ N i K ˆK,j (ξ, η) det J dξdη Element mass matrix for edge elements Z Z N i,j,k (F K x) J J N i,j,k (FK x) dx = K ˆK N i,j,k (ξ, η) J J N i,j,k (ξ, η) det J dξdη Element stiffness matrix for edge elements Z ( J N i,j,k (F K x)) J N i,j,k (FK x) dx K Z = ˆ N i,j,k (ξ, η) ˆ N i ˆK,j,k (ξ, η) det J dξdη Numerical quadrature on the reference cell ˆK K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 7 / 35

44 Implemenation in Concepts Sparsity pattern for a rectangle curl D -curl D element matrix identity element matrix K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 8 / 35

45 Implemenation in Concepts Sparsity pattern for a parallelogramme curl D -curl D element matrix identity element matrix K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 9 / 35

46 Implemenation in Concepts Sparsity pattern for a trapezoid curl D -curl D element matrix identity element matrix Sum factorisation reduces computational effort from O(p 6 ) to O(p 5 ) K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

47 Implemenation in Concepts Condition number of A + M in dependence of polynomial degree 8 condition number shape functions p A... element stiffness matrix M... element mass matrix K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

48 Implemenation in Concepts Curved cells η x 3 (ξ) p ˆK ξ F K p 3 x 4 (η) p x (ξ) p x (η) Blending Mapping as linear extension of edge parametrisation into the interior F K ξ = ( η) x (ξ) + ( + ξ) x (η) + ( + η) x 3 (ξ) + ( ξ) x 4 (η) 4 ( ξ)( η) p 4 ( + ξ)( η) p 4 ( + ξ)( + η) p... Jacobian is linear combination of derivates of edge parametrisations and itself «x x J = = «( η) x ξ η (ξ) + x (η) +... x (η) + ( + ξ) x (η) +... Blending technique needs formula for edge parametrisation and derivates Currently implemented: Circular, Ellipsoid, Parabel, Parallel of other one K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 3 / 35

49 Implemenation in Concepts General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) [B] ij = b(φ j, Φ i ) = X X b KK (Φ j, Φ K i K ) K K FEM : K = K DG-FEM : K, K are neighbours BEM : all combination of K and K K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 33 / 35

50 Implemenation in Concepts General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) [B] ij = b(φ j, Φ i ) = X X b KK (Φ j, Φ K i K ) K K = X dim(k) dim(k X X ) X b [T K ] m,j φ K m, [T K ] n,i φ K A n K K m= n= FEM : K = K DG-FEM : K, K are neighbours BEM : all combination of K and K K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 33 / 35

51 Implemenation in Concepts General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) [B] ij = b(φ j, Φ i ) = X X b KK (Φ j, Φ K i K ) K K FEM : K = K = X K X K DG-FEM : K, K are neighbours dim(k) X m= dim(k ) X [T K ] m,j [T K ] n,i b KK n= BEM : all combination of K and K φ K m, φk n K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 33 / 35

52 Implemenation in Concepts General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) [B] ij = b(φ j, Φ i ) = X X b KK (Φ j, Φ K i K ) K K FEM : K = K = X K X K dim(k) X m= DG-FEM : K, K are neighbours dim(k ) X n= BEM : all combination of K and K [T K ] m,j [T K ] n,i {z } connectivity local to global [B KK ] nm, {z } element matrix K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 33 / 35

53 Implemenation in Concepts General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) [B] ij = b(φ j, Φ i ) = FEM : K = K = X K X K dim(k) X m= DG-FEM : K, K are neighbours dim(k ) X n= BEM : all combination of K and K [T K ] m,j [T K ] n,i {z } connectivity local to global [B KK ] nm, {z } element matrix,, over of over of Get by calling with, Multiply with, and copy result into K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 33 / 35

54 Implemenation in Concepts Main C++ file. Reading the input file. Importing the mesh M 3. Defining the FE space on M and perform (adaptive) refinements in h and p 4. Defining the material formula, e.g. for µ(x) and ε(x) 5. Defining the involved bilinearforms with their material or the linearforms for sources. 6. Assembling of the system matrices and vector of the rhs. 7. Solving Time integration scheme with possibly several time solving of linear systems for Maxwell source problem in time domain. Solving a linear system for Maxwell source problem in time domain. Solving an matrix eigenvalue problem for Maxwell eigenvalue problem. 8. Post-processing Graphical output of the solution or function of the solution. Computation of physical quantities like Ohms losses or transmission coefficients. Flexibility by writing problem adapted main files. K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 34 / 35

55 Conclusion Space discretisation by h-, p- and hp-fem Mesh refinement or increasing the (polynomial) order Variational formulation Linear system of equations Resolving material interfaces essential p-fem gives exponential convergence for smooth interfaces hp-fem = suitable combination of h- and p-refinement expon. convergence for interface corners Resolution of number of wave lengthes after coarse resolution an increasing p is more efficient Geometries with small details anisotropic cells, domain decomposition, gfem with micro-structure Applications Bandstructure of disperse photonic crystal (with C. Engström, C. Hafner) Modes in Photonic Crystal Waveguides (with R. Kappeler) Benchmark for highly singular Maxwell solutions (with P. Frauenfelder) PC Bandstructure for polygonial interfaces (with P. Kauf) Implementation in Concepts Modular Programming, Concepts oriented design hp-fem in D and 3D with hanging nodes (edge elements in D) Basis functions with local support Hierarchical basis based on Jacobi polynomials on reference cell Numerical quadrature on reference cell considering shape of curved cell (Blending technique) General Algorithm for Matrix Assembling FEM, DG-FEM and BEM (hp, hanging nodes) K.Schmidt 8th July 9, NanoOptics Workshop, Zürich p. 35 / 35