ngsxfem: Geometrically unfitted discretizations with Netgen/NGSolve (

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1 ngsxfem: Geometrically unfitted discretizations with Netgen/NGSolve ( Christoph Lehrenfeld (with contributions from F. Heimann, J. Preuß, M. Hochsteger,...) NGSolve user meeting, Vienna, June 15, 2017 Institute for Numerical and Applied Mathematics, University of Göttingen

2 Table of contents Example problems Requirements for geometrically unfitted FEM Numerical integration on cut elements Jupyter demo Conclusion & Outlook

3 Geometry description via level sets Level set function φ = 0, x Γ(t), φ(x, t) < 0, x Ω 1(t), > 0, x Ω 2(t). Problems: PDEs on interior domain Ω 1 Ω 2 Ω 1 Γ interface problems involving Ω 1, Ω 2 and Γ PDEs on the interface Γ description only implicit Properties geometry evolution with PDE for φ meshes are unfitted C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

4 Example problem 1: Stationary fictitious domain problem Stationary geometrically unfitted domain (Γ(t) = Γ): Features: α u = f in Ω = {φ < 0}, α u n = g N on Γ N {φ = 0}, u = g D on Γ D {φ = 0}. CutFEM : Finite element space V Γ h = V h Ω Imposition of Dirichlet values = Lagrange multiplier / Nitsche techniques Weak formulations require integrals on Ω dx, Γ dx: Numerical integration on cut elements Small cuts introduce stability and conditioning problems (stabilizations?!) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

5 Example problem 2: Stationary interface problem Stationary geometrically unfitted domain (Γ(t) = Γ): Features: α u = f in Ω 1 Ω 2, [[ α u n]] = 0 on Γ, [[u]] = 0 on Γ, u = 0 on Ω. Solution has kinks across Γ: CutFEM : Vh Γ = V h Ω1 V h Ω2 or XFEM : Vh Γ = V h Vh x Imposition of interface conditions = Lagrange multiplier / Nitsche techniques Numerical integration on cut elements Small cuts (stability / conditioning) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) 3 20 %

6 Example problem 3: Stationary surface PDE problem (Laplace Beltrami) Stationary geometrically unfitted domain (Γ(t) = Γ): Features: Γ u = f on Γ. Solution defined only on Γ Trace/CutFEM : Vh Γ = V h Γ Numerical integration on cut elements Approximation of normal/tangential direction Ambiuigity in normal directions (near) singular matrices (conditioning) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

7 Example problem 4: Unsteady Poisson problem Geometrically unfitted moving domain (Γ(t) = {φ(x, t) = 0}): tu u = f in Ω(t) = {φ(x, t) < 0}, α u n = g N on Γ N (t) {φ(x, t) = 0}, u = g D on Γ D (t) {φ(x, t) = 0}, u = u 0 in Ω(0) = {φ(x, 0) < 0}. t Features: Time-dependent finite element spaces Space-Time: Space-time finite element spaces Numerical integration on space-time cut elements Characteristic FEM: Characteristic back-tracking integrator + features of stationary unf. problems... Γ x C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

8 Remarks Remarks on Examples 1-4 Appear in two-phase flows Vector-valued versions relevant (Stokes interface, mean curvature calculation,..) Background discretization can be conforming or nonconforming C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) 6 40 %

9 Existing features Related existing features in NGSolve: Background finite element spaces (scalar/vector, continuous/discontinuous) Convenient integral forms (VOL/BND/BBND) Handling of additional dofs: UNUSED DOFs Easy set up of preconditioners Curved elements (isoparametric FE) Restriction of integration on some elements... bfi = SymbolicBFI( u*v, BND) bfi.setdefinedonelements(..) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

10 The new features of ngsxfem New features in ngsxfem: CutInfo (level set cut topology): Domain type per (VOL/BND) element: NEG: completely in {φ < 0} POS: completely in {φ > 0} IF: interseceted by {φ = 0} Cut ratio T Ω / T (detect small cuts) GetDofsOfElements(Vh,neg) dof-handling for unfitted FE spaces, V h = V h Ω Extended FE spaces, V h = V h V x h Ghost penalty stabilization: Marking of relevant facets (as BitArray) Differential operator for [[ l nu]] across facets deformed mesh Θh(Th) quadrilaterals triangles Numerical integration on cut elements undeformed mesh Th deformation order: 2 Θh (Vh 2)2 integration order: 4 C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

11 Numerical integration on cut simplices Tesselation Approximation by piecewise linear level set function = piecewise linear interface Γ lin + explicit domain approx. + robust 2nd order x 3 c 3 x 4 c 1 c 2 x 1 x 2 x 3 c 3 x 1 x 2 c 3 c 2 x 1 x 2 x 1 c 3 x 4 c 1 c 2 c 3 c 1 c 2 T 1 = T a T b T c T 2 = T d C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) 9 60 %

12 Numerical integration on cut simplices Tesselation Approximation by piecewise linear level set function = piecewise linear interface Γ lin + explicit domain approx. + robust 2nd order x 3 x 4 c 3 c 3 c 3 c 2 c 1 x 2 x 1 c 2 x 3 x 2 x 2 x 1 x 1 x 1 c 3 x 4 c 3 c 2 c 1 c 2 c 1 T 1 = T a T b T c T 2 = T d fdx or uvdx in ngsxfem: Ω lin Ω lin Integrate(levelset domain={"levelset":lsetp1,"domain type":neg},cf=f) SymbolicBFI(levelset domain={"levelset":lsetp1,"domain type":neg},u*v) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) 9 60 %

13 Parametric mapping for higher order geometrical accuracy + higher order, implicit low order, explicit Basic concept of isoparametric unfitted FEM Start from (multi-)linear level set I h φ (h) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

14 Parametric mapping for higher order geometrical accuracy + Θ h higher order, implicit low order, explicit Basic concept of isoparametric unfitted FEM higher order, explicit Start from (multi-)linear level set I h φ (h) Construct a mapping of the underlying mesh s.t. I h φ φ Θ h C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

15 Parametric mapping for higher order geometrical accuracy + Θ h higher order, implicit low order, explicit Basic concept of isoparametric unfitted FEM higher order, explicit Start from (multi-)linear level set I h φ (h) Construct a mapping of the underlying mesh s.t. I h φ φ Θ h lsetmeshadap=levelsetmeshadaptation(mesh,order=order) deformation=lsetmeshadap.calcdeformation(levelset) Θ h mesh.setdeformation(deformation) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

16 Examples C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

17 Numerical integration in ngsxfem Numerical integration on cut elements with φ lin + mesh transformation Θ h Integration on linear level set domains on simplices: Geometrical decomposition into simplices Integration by transformation to reference domain (Ω lin ) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

18 Numerical integration in ngsxfem Numerical integration on cut elements with φ lin + mesh transformation Θ h Integration on linear level set domains on simplices: Geometrical decomposition into simplices Integration by transformation to reference domain (Ω lin ) Transformation factors (det(dθ h ),...) are automatically considered due to NGSolves mesh deformation handling (ALETransformation) Changes in quadrature (Ω lin i Ω i,h = Θ h(ω lin i )) Quadrature after mapping, dist( Ω i, (Ω i,h)) O(h k+1 ), ω i > 0: f dx f dx = f dx ω i det(dθ h(x i)) f (Θ h(x i)) Ω i Ω i,h Θ h (Ω lin ) i T T h i C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

19 Numerical integration in ngsxfem Numerical integration on cut elements with φ lin + mesh transformation Θ h Integration on linear level set domains on simplices: Geometrical decomposition into simplices Integration by transformation to reference domain (Ω lin ) Transformation factors (det(dθ h ),...) are automatically considered due to NGSolves mesh deformation handling (ALETransformation) Changes in quadrature (Ω lin i Ω i,h = Θ h(ω lin i )) Quadrature after mapping, dist( Ω i, (Ω i,h)) O(h k+1 ), ω i > 0: f dx f dx = f dx ω i det(dθ h(x i)) f (Θ h(x i)) Ω i Ω i,h Θ h (Ω lin ) i accuracy depends on Θ h, but cut topology of Ω lin i T T h Consequences unchanged Guaranteed stability of quadrature: Ω i,h uv dx I h(ω i,h; uv), (positiveness of quadrature weights ω i det(dθ h(x i)) ) i C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

20 Mapping of quadrature points (order = 1) quadrilaterals triangles undeformed mesh Th deformed mesh Θh(Th) deformation order: 1 Θ h (V 1 h )2 integration order: 2 C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

21 Mapping of quadrature points (order = 2) quadrilaterals triangles undeformed mesh Th deformed mesh Θh(Th) deformation order: 2 Θ h (V 2 h )2 integration order: 4 C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

22 Mapping of quadrature points (order = 3) quadrilaterals triangles undeformed mesh Th deformed mesh Θh(Th) deformation order: 3 Θ h (V 3 h )2 integration order: 6 C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

23 Mapping of quadrature points (order = 4) quadrilaterals triangles undeformed mesh Th deformed mesh Θh(Th) deformation order: 4 Θ h (V 4 h )2 integration order: 8 C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

24 Jupyter demos! C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

25 Conclusion Θ Ω h boundary approximation Ω h Ω h Demands from Ω h unfitted FEM Φ Ω h id Φ Ω h Handling cut information (cut elements...) Φ Ω h =id Numerical integration in cut elements Stabilization on parts of the mesh ch interface approximation Ω h Ω Ω 2 Ω 1 Ω Γ Θ Γ h Ω 1,h Ω 2,h Γ h Φ Γ h =id Tools in ngsxfem Ω 1,h Ω ch 1 CutInfo Φ Γ new FESpaces h Φ Γ h id (XFESpace, SpaceTimeFESpace,...) Levelset-domain integration Γ h for SymbolicBFI/LFI and Integrate Ω Γ explicit 2,h high Ω 2 Φ Γ h =id order geometry approximation with trafo Θ h Possibilities High order accurate unfitted FEM Simplex and quad meshes Fictitious domain / interface problems / surface PDEs Scalar and vector-valued problems Time dependent problem:... C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

26 Outlook: Numerical integration in space-time in ngsxfem (poster) Space time integration Space-time FE for φ lin ( order k t in time, order 1 in space) Iterated integrals with space-time prism (in regions with same cut topology) Combine with space-time finite element for Θ h : Ω [t n 1, t n ] Ω T +(t n ) t 3 = t n T (tn ) Θ h Θ h(t +(t n )) t 3 = t n Θ h(t (t n )) t 2 t 1 t 2 t 1 tfe = ScalarTimeFE(k_t) t 0 = t n 1 t 0 = t n 1 fes = H1(mesh, order=1) Q T st_fes = SpaceTimeFESpace(fes,tfe) lsetp1 = GridFunction(st_fes)... SymbolicBFI(levelset_domain={"levelset":lsetp1,..},u*v,time_order=4) Θ h (Q T ) C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %

27 Thank you for your attention! Questions / Comments? C. Lehrenfeld, ngsxfem ( June 15, 2017, Vienna (NGSolve User Meeting 2017) %