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:

• Compatible with remeshing
• Incompatible with special treatment

Embedded features: Finite element mesh non-conforming with interface.

R- R

+

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

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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...

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Basic idea

−∇ · (κ∇u) = f in Ω u = g

• n Γ

Weak form, essential bc’s: u = g, v = 0

• n Γ
• Ω

∇v · κ∇u dΩ =

• Ω

vf dΩ Hybrid approach: Lagrange multiplier λ (& µ), u, v ∈ H1(Ω)

• Ω

∇v · κ∇u dΩ −

• Γ

vλ dΓ =

• Ω

vf dΩ −

• Γ

µu dΓ = −

• Γ

µg dΓ

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Euler-Lagrange equation, flux weakly imposes Dirichlet bc’s λ = κ∇u · n

• n Γ

Stabilized formulation, parameter α > 0 λ = κ∇u · n + α(g − u)

• n Γ

Nitsche method (reduces to weak form when u, v satisfy bc’s)

• Ω

∇v · κ∇u dΩ −

• Γ

vκ∇u · n dΓ −

• Γ

κ∇v · nu dΓ +

• Γ

vαu dΓ =

• Ω

vf dΩ −

• Γ

κ∇v · ng dΓ +

• Γ

vαg dΓ “Penalty” method + variational consistency (easily verified).

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α = ? Strategies:

• Ad hoc
• 1. Unstabilized
• 2. Empirical
• Discrete trace inequalities
• 1. Bound

[H.-Hughes, ’92], [Warburton-Hesthaven, 2003], [Evans-Hughes, 2013]

• 2. Compute

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α =? Discrete trace inequality, config. dep. const., C > 0 κ∇vh · n2

Γ ≤ Cvh2 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 C ≥ κL/A, L = meas(Γe ∩ Γ) In practice, for good numerical performance αe = 2C.

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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.

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Upshot

Main goal: ≈ spectrum of std. discrete formulation. Challenge: Complementary discrete sol’ns. Main result: Reduced Nitsche spectrum ≈ spectrum of

• std. discrete formulation.

Favorable implications: BVP (condition) eigenvalue problems explicit dynamics.

5 10 15 20 25 30 35

modal index

50 100 150 200 250 300 350

L2

Standard Reduced Nitsche

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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.

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Spectral behavior of Nitsche’s method

Elliptic eigenvalue problem ∆u + λu = in Ω u =

• n Γ

Nitsche formulation (bc’s) u, w ∈ H1(Ω) (∇w, ∇u) − (w, u,n)Γ − (w,n, u)Γ + α(w, u)Γ

• a(w, u)

−λ(w, u) = 0 For u, w ∈ H1

0(Ω), reduces to std. formulation.

12

Parameter sensitivity

Specific eigenpair {λr, ur}, r = 1, 2, . . . a(ur, ur) − λr(ur, ur) = 0 Rayleigh quotient R(v) = a(v, v) v2 such that λr = R(ur) Boundary quotient B(v) = v2

Γ

v2 show that dλr dα = B(ur) Note (stablization ∼ stiffen.?) dλ dα ≥ 0

13

Complementary discrete solutions

Weak form Eigenvalues (∈ R+): 0 < λ1 ≤ λ2 ≤ . . . Eigenfunctions ur ∈ H1

0(Ω), L2(Ω)-ortho.

• Std. formulation

uh ∈ Vh

0 ⊂ H1 0(Ω),

dim Vh

0 = N

N eigenpairs, {λh

r, uh r} ≈ {λr, ur}

λh

r ≥ λr,

uh

r retain L2(Ω)-ortho.

Nitsche uh ∈ Vh ⊂ H1(Ω), dim Vh = N + dim

• Vh/Vh
• N eigenpairs + ?,

retain L2(Ω)-ortho. Vh = Vh

0 ⊕

• Vh

⊥ dim

• Vh

⊥ = dim

• Vh/Vh
• 14

Example: cut bilinear quad

dim Vh = 4

N1,N4 Unaligned

Γ

1 2 3 4 Aligned N2,N3

dim

• Vh/Vh
• = 2

dim

• Vh/Vh
• =?

15

Illustrative construction

Consider Vh

0 = span {ur}N r=1

Decompose uh = u0 + u⊥, u0 = Puh ∈ Vh

0 , . . .

Nitsche decouples (∇w0, ∇u0) − λ(w0, u0) = a(w⊥, u⊥) − λ(w⊥, u⊥) = N eigenpairs (exact), {ur, R(ur)} , st B(ur) = 0 Complementary pairs dim

• Vh/Vh
• : projections of uh along Vh

0 onto

• Vh

⊥ u⊥ = uh −

N

• r=1
• uh, ur
• ur2 ur

& R(u⊥) (> 0 for α > C) u⊥ → as N → ∞ B(u⊥) > 0

16

Simple example

Ω =]0, L[, dim

• Vh/Vh
• = 2,

Vh = span {1, cos(πx/L), sin(rπx/L)} N eigenpairs,

• (rπ/L)2, sin(rπx/L)
• ,

B(ur) = 0 2 complementary solutions

N = 10 100

0.0 0.2 0.4 0.6 0.8 1.0

• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

x L

u⊥

u = 1 N = 10 100

0.0 0.2 0.4 0.6 0.8 1.0

• 1.0
• 0.5

0.0 0.5 1.0

x L

u⊥

u = cos(πx/L)

Edge “modes”: support ≈ boundary strips of width L/N. B(u⊥) ∼ π2N/(2L), R(u⊥) ∼ B(u⊥)(α ∓ 2N/L), C = 2N/L

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In gen. Nitsche, uh ∈ Vh ⊂ H1(Ω), doesn’t decouple, but still useful Vh = Vh

0 ⊕

• Vh

⊥ Eigenpairs, {λh

r, uh r} approx. {λr, ur}, r = 1, 2, . . . , dim Vh

∂λh

r

∂α = B(uh

r) ≈ 0

Complementary sol’ns, approx’s of ft’ns in

• Vh

⊥ , # = dim

• Vh/Vh
• 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 × 2L Dirichlet bc’s top & bottom. λL2 = π2 n2 + m2/4

• u

= cos

• nπx

L

• sin
• mπ y

2L

• 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.)

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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

Eigenft’n #3 Complem. #2 R(u) ≈ −3 (ηh)2 η = 1 0.5 0.2 0.1 0.01

22

Eigenpairs vs. complementary sol’ns α = 0

B(u) separates solutions

0.0 0.5 1.0 1.5 2.0 2.5 3.0

h (u)

= 1 = 0.5 = 0.1 = 0.01 5 10 15 20 25 30 35

modal index

100 100 200 300 400 500 600 700 800

L2

= 0.1 = 0.5 Uni Std Nitsche

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

Recall dλ dα = B(u) Idealized dλ dα ≈

• 0,

eigenvalues 3/(ηh), complementary (in fact 3.3/(ηh)) η = 0.5 η = 0.1

1 2 3

/C

200 200 400 600 800

L2

1 2 3

/C

200 200 400 600 800

L2

Veering present, region more restricted as η decreases.

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Eigenvalue veering

Occurs in models with parameters, continuous as well as discrete. Known in diverse disciplines (under different names). The rule in one-parameter, self-adjoint systems. “adjacent eigenvalues. . . appear to be on a collision course; yet at the last minute they turn aside” [Lax] Eyecatching, but ultimately immaterial.

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Reduced form

Recall, B(u) separates solutions, but needs entire spectrum. In practice, interested in lower eigenpairs. Reduce syst. by eliminating Nitsche dof’s (# = dim

• Vh/Vh
• ).

Precludes complementary solutions. E.g. Irons-Guyan reduction (diag. scal.) preserves matrix structure. Can solve alg. EVP as usual. In contrast to original formulation dλr dα = B(ur) in fact dλ dα ≥ 0

26

Original vs. reduced form

Coarse mesh 1 × 2 (η = 1) 7 6 λL2 =

• 2αL ± A

14 + 2αL ± A

• A =
• (2αL)2 − 14αL + 14
• Reduced

λL2 =      3 15 5.2 + 2αL 5 + 2αL

• dλ

dα < 0

2 4 6 8 10

/C

5 5 10 15 20 25 30

L2

Stablization ∼ stiffen.

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No veering, improved cond., little dep. on α. η = 0.5 η = 0.1

1 2 3 4 5 6 7 8 9 10

/C

50 100 150 200 250 300 350

L2

1 2 3 4 5 6 7 8 9 10

/C

100 200 300 400 500 600 700

L2

Expect no repercussions from veering on sol’n of BVP.

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α = 2C

Reduced Nitsche spectrum ≈ spectrum of std. discrete formulation Useful for explicit dynamics (λmax).

5 10 15 20 25 30 35

modal index

100 100 200 300 400 500 600 700 800

L2

= 0.1 = 0.5 Std Reduced Nit. 1 5 10 15 20 25 30 35

modal index

0.0 0.1 0.2 0.3 0.4 0.5 0.6

(u)

= 1 = 0.5 = 0.1

Eigenfunctions: B indicates degree of satisfaction of BC’s

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Pythagorean eigenvalue identity

[Strang-Fix,1973]. Errors in eigenvalues and eigenfunctions for each mode. Extend to Nitsche: conventional form (uh

r = 1)

λh

r − λr

λr +

• uh

r − ur

• 2 = a
• uh

r − ur, uh r − ur

• λr

, r = 1, 2, . . . , N W/o scaling of eigenfunctions

• λh

r − λr

• uh

r2 + λr

• uh

r − ur

• 2 = a
• uh

r − ur, uh r − ur

• Indep. of scaling of ur, except for sign. Set sign st

(uh

r, ur) > 0

In practice ur = uh

r = 1

30

η = 0.5

mode 3 20 35

1 2 3 4 5 6 7 8 9 10

/C

1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.200

||uh

r

ur||2

red

||uh

r

ur||2

std

(

h r r)red

(

h r r)std

a(uh

r

ur, uh

r

ur)red (||uh

r

ur||2

E)std

1 2 3 4 5 6 7 8 9 10

/C

0.99 1.00 1.01 1.02 1.03 1.04 1.05 1 2 3 4 5 6 7 8 9 10

/C

0.99 1.00 1.01 1.02 1.03 1.04 1.05

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η = 0.1

mode 3 20 35

1 2 3 4 5 6 7 8 9 10

/C

0.9990 0.9995 1.0000 1.0005 1.0010 1.0015 1.0020

||uh

r

ur||2

red

||uh

r

ur||2

std

(

h r r)red

(

h r r)std

a(uh

r

ur, uh

r

ur)red (||uh

r

ur||2

E)std

1 2 3 4 5 6 7 8 9 10

/C

0.99900 0.99925 0.99950 0.99975 1.00000 1.00025 1.00050 1.00075 1.00100 1 2 3 4 5 6 7 8 9 10

/C

1.000 1.002 1.004 1.006 1.008 1.010

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Summary

Nitsche formulation admits mesh- & α-dep. complementary discrete solutions associated with enforcing constraint ( # = dim

• Vh/Vh
• ).

Boundary quotient separates complementary solutions from eigenpairs. Eigenpairs exhibit little dependence on stabilization, B ≈ 0.

• Compl. values increase roughly linearly w/α (→ coercivity), B > 0.

Reduced form is free of complementary solutions. BVP: Static condensation for conditioning. No sensitivity to veering. Dynamic reduction (Irons-Guyan) for eigenvalue problems and explicit dynamics.

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