Nitsche-type Mortaring for Maxwell s Equations

Transcription

1 Nitsche-type Mortaring for Maxwell s Equations Joachim Schöberl Karl Hollaus, Daniel Feldengut Computational Mathematics in Engineering at the Institute for Analysis and Scientific Computing Vienna University of Technology Dzevat Omeragic, Schlumberger-Doll Research Markus Wabro, CST AG Joachim Schöberl Page 1

2 Problem setup Domain decomposition on non-matching meshes: Contents: Method for scalar model problem Nitsche-method for Maxwell s equation Numerical results Joachim Schöberl Page 2

3 A model problem Poisson equation: u = f on Ω 1 Ω 2 u = 0 at (Ω 1 Ω 2 ) Interface conditions on Γ: [u] := u 1 u 2 = 0 n1 u 1 + n2 u 2 = 0 Joachim Schöberl Page 3

4 Mortar method Pose the constraint as additional equation. Find u H 1 0,D (Ω 1) H 1 0,D (Ω 1) and λ H 1/2 (Γ) such that Ω 1 Ω 2 u v + Γ [v]λ = fv v Γ [u]µ = 0 µ The Lagrange parameter λ is the normal flux n u. Requires stability condition for finite element spaces (LBB). Leads to an indefinite system matrix. Joachim Schöberl Page 4

5 Nitsche / Discontinuous Galerkin method Allows discontinuous approximation by keeping extra boundary terms: Find u H 1 0,D (Ω 1) H 1 0,D (Ω 1) such that Ω 1 Ω 2 u v Γ n u [v] n v [u] + α Γ Γ [u] [v] = fv v with soft penalty term α p 2 /h sufficiently large. No extra stability condition is required. Leads to a symmetric positive definite stiffness matrix. Joachim Schöberl Page 5

6 Integration of boundary terms Both methods require to compute integrals of finite element functions from different meshes: Mortar method: Γ u 2µ Nitsche method: Γ v 1 n u 2 Requires the calculation of an intersection mesh Complicated implementation, in particular on curved interfaces in 3D Joachim Schöberl Page 6

7 Hybrid Nitsche method - derivation Introduce a new variable for the primal unknown on the interface: λ := u Γ Multiply by test-functions, and integrate by parts: u v n u v = Ω i Γ i fv v H 1 (Ω 1 ) H 1 (Ω 2 ) Use continuity of n u and introduce single-valued test function µ on interface: u v n u (v µ) = fv v µ i Γ Use u = λ on Γ to symmetrize and stabilize with α p 2 /h. u v n u (v µ) n v (u λ) + α i Γ Γ Γ (u λ)(v µ) = fv v µ For α chosen right, the discrete formulation is stable independent of the choice of fe spaces for u and λ. Joachim Schöberl Page 7

8 Discretizing and numerical integration In general, numerical integration is still difficult. We propose to use smooth B-spline functions for discretizing the hybrid variable λ. This allows evaluation in global coordinates efficient numerical integration by Gauss-rules on the surface elements u u λ Joachim Schöberl Page 8

9 Numerical experiments in 2D f = x in circle, else f = 0. Solution u: Solution u/ x: Finite element order p = 5. Joachim Schöberl Page 9

10 Numerical experiments in 3D f = xz in cylinder, else f = 0. Solution u: Solution u/ x: Finite element order p = 4. Joachim Schöberl Page 10

11 Maxwell s Equations Time harmonic Maxwell s equations curl µ 1 curl u + κu = j in Ω i with κ = iωσ ω 2 ɛ, and Transmission conditions E = iωu, H = µ 1 curl u. u 1 n 1 + u 2 n 2 = 0, µ 1 1 curl u 1 n 1 + µ 1 2 curl u 2 n 2 = 0. Joachim Schöberl Page 11

12 Hybrid Nitsche formulation proceed as in the scalar case: Ω i {µ 1 curl u curl v + κu v} + Ω i µ 1 curl u (v n) = Ω i j v add symmetry and penalty terms: find (u, λ) such that { 2 i=1 Ω i µ 1 {curl u curl v + κu v} + + Ω i µ 1 curl u [(v µ) n] Ωi µ 1 curl v [(u λ) n] + αp2 µh Ω i [(u λ) n] [(v µ) n] } = Ω j v, where u, v H(curl, Ω 1 ) H(curl, Ω 2 ), and λ, µ are tangential vector valued fields on the interface. Joachim Schöberl Page 12

13 Overpenalization of gradient fields The natural energy norm is For gradient fields u = φ, this norms scales as u 2 = µ 1 curl u 2 L 2 + κ u 2 L 2 φ 2 = O(κ) This is small for small frequencies/conductivities. The norm for the Nitsche method is (u, λ) 2 = 2 i=1 { } µ 1 curl u 2 Ω i + κ u 2 Ω i + αµ 1 (u λ) n 2 Γ But, the last term of this norm does not scale with κ for gradient fields. Thus, the penalty term u λ leads to an overpenalization of the jump for gradient fields. Joachim Schöberl Page 13

14 Scalar potential at the boundary Goal: Want to replace the continuity condition (u i λ) n i = 0 i = 1, 2 by (u i φ i λ) n i = 0 φ i φ Γ = 0 with arbitrary scalar fields φ 1 = φ 2 = φ Γ on the boundary. This allows to scale the penalty terms for gradients and rotations differently. Joachim Schöberl Page 14

15 Variational formulation { 2 i=1 Ω i {µ 1 curl u curl v + κuv}+ µ 1 curl u [(v µ) n] + µ 1 curl v [(u φ λ) n]+ Ω i Ω i α µ 1 [(u φ λ) n][(v ψ µ) n]+ Ω i } α κ(φ φ Γ )(ψ ψ Γ ) Ω i = Ω jv Joachim Schöberl Page 15

16 A boundary identity Testing the weak form with v = ψ gives κu ψ + µ 1 curl u ( ψ n) = Ω i Ω i Ω j ψ Taking the divergence in the strong form, and integrating by parts leads to div(κu) ψ = div j ψ Ω i Ω i κu ψ + κu n ψ = j ψ + j n ψ Ω i Ω i Ω i Ω i Adding up leads to the boundary relation Ω i µ 1 curl u ( ψ n) + Ω i κu n ψ = Ω i j n ψ Joachim Schöberl Page 16

17 { 2 i=1 Final variational formulation {µ 1 curl u curl v + κuv} + µ 1 curl u [(v ψ µ) n] Ω i Ω i + µ 1 curl v [(u φ λ) n] + α µ 1 [(u φ λ) n][(v ψ µ) n] Ω i Ω i } κu n (ψ ψ Γ ) κv n (φ φ Γ ) + α κ(φ φ Γ )(ψ ψ Γ ) = Ω i Ω i Ω i { 2 i=1 jv j n ψ Ω i Ω i } u, v... H(curl) conforming element basis functions on Ω i φ, ψ... H 1 conforming element basis functions on Ω i Γ λ, µ... tangential vector valued spline functions on Γ φ Γ, ψ Γ... scalar spline functions on Γ Joachim Schöberl Page 17

18 Magnetostatics Permanent magnet (red) with two domains (green and blue) Joachim Schöberl Page 18

19 Magnetic flux magnetic flux x-component of magnetic flux Finite element order p = 4. Joachim Schöberl Page 19

20 LWD-Tool borehole with soil tool with antennas Joachim Schöberl Page 20

21 LWD-Tool B-field B-field, field-lines: Joachim Schöberl Page 21

22 LWD-Tool Numerical results for first order elements: frequency [khz] standard, rec volt [nv] Nitsche, rec volt [nv] dofs dofs i i i i i i i i 5255 Numerical results for second order elements: frequency [khz] standard, rec volt [nv] Nitsche, rec volt [nv] dofs dofs i i i i i i i i 5255 Joachim Schöberl Page 22

23 Conclusions and Ongoing Work We have Hybrid Nitsche-type mortaring for scalar equation Stable transmission conditions for low frequency Maxwell s equations Interface fields discretized by smooth B-spline spaces for simple numerical integration We work on Error estimators and adaptivity for spline space and numerical integration Iterative solvers, in particular BDDC domain decomposition methods The methods are implemented within Netgen/NGSolve software [K. Hollaus, D. Feldengut, J. Schberl, M. Wabro, D. Omeragic: Nitsche-type Mortaring for Maxwell s Equations PIERS Proceedings, , July 5-8, Cambridge, USA 2010] Joachim Schöberl Page 23