Spectral Performance of Nitsches Method Isaac Harari, Uri Albocher - PowerPoint PPT Presentation

Spectral Performance of Nitsche’s Method Isaac Harari, Uri Albocher ∗ Tel Aviv University December 2018 ∗ also, Afeka, Tel Aviv Academic College of Engineering

A great deal of the ICASE research was conducted. . . on one of the many oversize blackboards. . . Saul Abarbanel was a prominent practitioner of this art. . . [Salas, 2018] Eigenvalues, stability of IBVPs: [Carpenter-Gottlieb-A., 1993] [A.-Chertock, 2000] 1

Background Interface problems: • Composites with complex microstructure • Multiphase flow • Biofilm growth • Interaction Challenge: • Stationary interface with complex geometry • Interface with evolving geometry Meshing strategies: + R • Compatible with remeshing • Incompatible with special treatment R - Embedded features: Finite element mesh non-conforming with interface. 2

Approaches (related to DG & IP): • X-FEM • Immersed FEM • CutFEM [Hansbo, Burman, Kreiss] • Finite Cell • Universal mesh • Shifted boundary Dirichlet boundary conditions ( ∼ interface constraints): • Conventional approach: conforming mesh + interpolation • Weak enforcement (fluids [Bazilevs-Hughes, 2007]) – Hybrid formulation, Lagrange multipliers (mortar, FETI) – Penalty methods – Nitsche’s method 3

Nitsche’s method Relaxation of essential boundary conditions [Courant, 1943]. Lagrange multipliers for Dirichlet constraints [Jones, 1964], [Babuˇ ska, 1973]. inf-sup condition [Pitk¨ aranta, 1979–81]. Least-squares stabilization to circumvent inf-sup [Barbosa-Hughes, 1991]. Bubble stab. [Mourad-Dolbow-H., 2007], RFB [Dolbow-Franca, 2008]. Penalty method + variational consistency [Nitsche, 1971]. Rediscovered and connected to stabilized methods [Stenberg, 1995]. Nitsche coeff. ≡ stab. parameter. Used: domain decomposition, contact, discont. Galerkin, meshless... 4

Basic idea −∇ · ( κ ∇ u ) = f in Ω u = g on Γ Weak form, essential bc’s: u = g , v = 0 on Γ � � ∇ v · κ ∇ u d Ω = vf d Ω Ω Ω u, v ∈ H 1 (Ω) Hybrid approach: Lagrange multiplier λ (& µ ), � � � ∇ v · κ ∇ u d Ω − vλ d Γ = vf d Ω Ω Γ Ω � � − µu d Γ = − µg d Γ Γ Γ 5

Euler-Lagrange equation, flux weakly imposes Dirichlet bc’s λ = κ ∇ u · n on Γ Stabilized formulation, parameter α > 0 λ = κ ∇ u · n + α ( g − u ) on Γ Nitsche method (reduces to weak form when u, v satisfy bc’s) � � � � ∇ v · κ ∇ u d Ω − vκ ∇ u · n d Γ − κ ∇ v · n u d Γ + vαu d Γ = Ω Γ Γ Γ � � � vf d Ω − κ ∇ v · n g d Γ + vαg d Γ Ω Γ Γ “Penalty” method + variational consistency (easily verified). 6

α = ? Strategies: • Ad hoc 1. Unstabilized 2. Empirical • Discrete trace inequalities 1. Bound [H.-Hughes, ’92], [Warburton-Hesthaven, 2003], [Evans-Hughes, 2013] 2. Compute 7

α = ? Discrete trace inequality, config. dep. const., C > 0 � κ ∇ v h · n � 2 Γ ≤ C � v h � 2 κ [Barbosa-Hughes, 1991], [Stenberg, 1995]. α > C = ⇒ coercivity Find C from global gen. eigenvalue prob. [Griebel-Schweitzer, 2003]. Elem.-level inequality, estimate C (only for elements on boundary). E.g. linear triangle L = meas(Γ e ∩ Γ) C ≥ κL/A, α e = 2 C . In practice, for good numerical performance 8

Spectral behavior Weak enforcement = ⇒ + dof’s = ⇒ + sol’ns (indef.) Stabilization = ⇒ coercivity Approx. of exact spectrum? Spectral investigations: • Characterize operator. • Insight to BVP. • Nitsche’s method for eigenvalue problems. 9

Upshot Main goal: ≈ spectrum of std. discrete formulation. Challenge: Complementary discrete sol’ns. Main result: Standard 350 Reduced Nitsche Reduced Nitsche spectrum 300 ≈ spectrum of 250 std. discrete formulation. 200 L 2 150 Favorable implications: BVP (condition) 100 eigenvalue problems 50 explicit dynamics. 0 0 5 10 15 20 25 30 35 modal index 10

Terminology Compatible discretization: Cont. of unknown field + Dirichlet bc’s/interface cond’s. Conforming mesh: Elements are fitted (matched, uncut, untrimmed, aligned?). + boundary/interface. Nitsche’s method enforces surface constraints weakly: Framework for incompatible discretization. Accommodates non-conforming meshes with cut (trimmed) elements. 11

Spectral behavior of Nitsche’s method Elliptic eigenvalue problem ∆ u + λu = 0 in Ω u = 0 on Γ u, w ∈ H 1 (Ω) Nitsche formulation (bc’s) ( ∇ w, ∇ u ) − ( w, u ,n ) Γ − ( w ,n , u ) Γ + α ( w, u ) Γ − λ ( w, u ) = 0 � �� � a ( w, u ) For u, w ∈ H 1 0 (Ω) , reduces to std. formulation. 12

Parameter sensitivity Specific eigenpair { λ r , u r } , r = 1 , 2 , . . . a ( u r , u r ) − λ r ( u r , u r ) = 0 Rayleigh quotient R ( v ) = a ( v, v ) such that λ r = R ( u r ) � v � 2 Boundary quotient B ( v ) = � v � 2 dλ r Γ show that dα = B ( u r ) � v � 2 Note (stablization ∼ stiffen.?) dλ dα ≥ 0 13

Complementary discrete solutions Eigenvalues ( ∈ R + ): Weak form 0 < λ 1 ≤ λ 2 ≤ . . . Eigenfunctions u r ∈ H 1 0 (Ω) , L 2 (Ω) -ortho. u h ∈ V h 0 ⊂ H 1 dim V h Std. formulation 0 (Ω) , 0 = N { λ h r , u h N eigenpairs, r } ≈ { λ r , u r } λ h u h r ≥ λ r , r retain L 2 (Ω) -ortho. � � u h ∈ V h ⊂ H 1 (Ω) , dim V h = N + dim V h / V h Nitsche 0 N eigenpairs + ?, retain L 2 (Ω) -ortho. V h = V h � � ⊥ V h 0 ⊕ 0 � ⊥ = dim � � � V h V h / V h dim 0 0 14

Example: cut bilinear quad dim V h = 4 Aligned Unaligned 4 3 N 1 ,N 4 N 2 ,N 3 Γ 1 2 � � � � V h / V h V h / V h dim = 2 dim =? 0 0 15

Illustrative construction Consider 0 = span { u r } N V h r =1 Decompose u h = u 0 + u ⊥ , u 0 = Pu h ∈ V h 0 , . . . Nitsche decouples ( ∇ w 0 , ∇ u 0 ) − λ ( w 0 , u 0 ) = 0 a ( w ⊥ , u ⊥ ) − λ ( w ⊥ , u ⊥ ) = 0 N eigenpairs (exact), { u r , R ( u r ) } , st B ( u r ) = 0 � � : projections of u h along V h � � ⊥ V h / V h V h Complementary pairs dim 0 onto 0 0 � � N u h , u r � u h − u ⊥ = � u r � 2 u r & R ( u ⊥ ) ( > 0 for α > C ) r =1 � u ⊥ � → 0 as N → ∞ B ( u ⊥ ) > 0 16

Simple example � � V h = span { 1 , cos( πx/L ) , sin( rπx/L ) } V h / V h Ω =]0 , L [ , dim = 2 , 0 � � ( rπ/L ) 2 , sin( rπx/L ) N eigenpairs, , B ( u r ) = 0 2 complementary solutions u = 1 u = cos ( π x / L ) 1.0 1.0 N = 10 N = 10 0.8 100 100 0.5 0.6 0.0 0.4 u ⊥ u ⊥ 0.2 - 0.5 0.0 - 1.0 - 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x L x L Edge “modes”: support ≈ boundary strips of width L/N . B ( u ⊥ ) ∼ π 2 N/ (2 L ) , R ( u ⊥ ) ∼ B ( u ⊥ )( α ∓ 2 N/L ) , C = 2 N/L 17

In gen. Nitsche, u h ∈ V h ⊂ H 1 (Ω) , doesn’t decouple, but still useful V h = V h � � ⊥ V h 0 ⊕ 0 Eigenpairs, { λ h r , u h r } approx. { λ r , u r } , r = 1 , 2 , . . . , dim V h 0 ∂λ h ∂α = B ( u h r r ) ≈ 0 � ⊥ , � � � V h V h / V h Complementary sol’ns, approx’s of ft’ns in # = dim 0 0 support ≈ strip O ( h ) along boundary. B > 0 = ⇒ R ’s (indef.) increase w/ α ( → > 0) . Artifact of discretization (mesh- & α -dep.) Mechanism to enforce constraint. 18

Numerical studies Consider entire spectrum. Cut elements (parallel to mesh lines, vol. fraction η ): • Highlight essential features (cf. std. formulations). • Structured meshes of bilinear elem’s, C = 1 / ( ηh ) . “Sliver” cut can lead to poor discretization [de Prenter et al., 2018] Std. discrete formulation: Finite # of discrete eigenpairs approx. lower exact eigenpairs. Eigenvalues ( ∈ R + ): approx. ≥ exact. Nitsche formulation: + complementary sol’ns. Wish to separate two types of sol’ns. Eigenvalues are positive ( ≥ exact?). R ’s of complementary sol’ns are indefinite ( α < C ). 19

Rectangular domain L × 2 L Dirichlet bc’s top & bottom. n 2 + m 2 / 4 π 2 � � λL 2 = � nπx � � mπ y � u = cos sin L 2 L 4 × 8 elements, Std./Nitsche on top. η = 1 , unif. squares Uni = Nitsche = Std η < 1 , Uni unchanged Nitsche = unif. rect’s stretched vertically Std = Nitsche, but compatible Nitsche: +5 complementary sol’ns. (Similar for L-shaped domain w/re-entrant corner.) 20

α = 0 , η = 0 . 5 Eigenfunctions (first 8) ≈ 0 on Γ = ⇒ B ( u ) ≈ 0 . Complementary Edge “modes”: support ≈ cut elements. Idealize: support = cut elements = ⇒ B ( u ) ≈ 3 / ( ηh ) . 21

α = 0 , vary η η = 1 0 . 5 0 . 2 0 . 1 0 . 01 Eigenft’n #3 Complem. #2 − 3 R ( u ) ≈ ( ηh ) 2 22

Eigenpairs vs. complementary sol’ns α = 0 B ( u ) separates solutions 800 = 1 Uni Std = 0.5 700 3.0 Nitsche = 0.1 = 0.1 600 = 0.01 2.5 500 2.0 h ( u ) 400 L 2 1.5 300 200 = 0.5 1.0 100 0.5 0 0.0 100 0 5 10 15 20 25 30 35 modal index As η decreases: Prediction B ( u ) ≈ 3 / ( ηh ) for complementary improves. Nitsche e-values ≈ Std, split from Uni for higher modes. R ’s of complementary sol’ns become more negative. 23

Dependence on α dλ Recall dα = B ( u ) Idealized � dλ 0 , eigenvalues dα ≈ 3 / ( ηh ) , complementary (in fact 3 . 3 / ( ηh ) ) η = 0 . 5 η = 0 . 1 800 800 600 600 400 400 L 2 L 2 200 200 0 0 200 200 0 1 2 3 0 1 2 3 / C / C Veering present, region more restricted as η decreases. 24