Coupling of BEM and FEM in the Time Domain for Fluid-Structure - - PowerPoint PPT Presentation

Coupling of BEM and FEM in the Time Domain for Fluid-Structure Interaction

Ernst P . Stephan1 (joint with Heiko Gimperlein2)

1: Leibniz Universität Hannover, Germany 2: Heriot–Watt University and Maxwell Institute, Edinburgh, UK

RICAM Workshop Linz, 11. Nov. 2016

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 1 / 23

Outline

Fluid-structure interaction: Set-up & well-posedness FEM-BEM in time domain: Reduction to boundary, variational formulation, discretization A priori and a posteriori error analysis Some numerical experiments

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 2 / 23

Fluid-structure interaction

Elastic body Ω ⊂ Rd, d = 2, 3 submersed in a fluid: u : R+ × Ω → Rd ̺1∂2

t u − divσ(u) = f in R+ × Ω ,

u = ∂tu = 0 for t = 0 Wave equation in Ωc = Rd \ Ω: p : R+ × Ω → R ∂2

t p − ∆p = 0 in R+ × Ωc ,

p = ∂tp = 0 for t = 0 Transmission conditions on Γ = ∂Ω: n exterior unit normal, data: ̺1, ̺2 densities, f load, pI incident wave σ(u)n + (∂tp + ∂tpI)n = 0 and ̺2∂tu · n + ∂np + ∂npI = 0

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 3 / 23

Space–time anisotropic Sobolev spaces

Hr

σ(R+, Hs(Rd)) defined using Fourier–Laplace transform F for σ > 0:

{ψ : supp ψ ⊂ R+×Rd,

• R+iσ dω
• Rd dξ|ω|2r(|ω|2 + |ξ|2)s|Fψ(ω, ξ)|2 < ∞}
• Hr

σ(R+, Hs(Ω)), Hr σ(R+, Hs(Γ)) with norms · r,s,Ω, · r,s,Γ.

(σ − ∂2

t )r/2 (σ − ∂2 t − ∆)s/2ψ(t, x) ∈ L2(R + × Rd)

Fourier–Laplace transform: t → ω = η + iσ, x → ξ Ha-Duong, Bamberger 1986, . . .

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 4 / 23

Well-posedness of fluid-structure interaction

Theorem (Felipe 1994 + ε, cf. also Hsiao-Sayas-Weinacht 2015)

Assume s ≥ 0 f ∈ H1+s

σ

(R+, H−1(Ω))d pI ∈ H3+s

σ

(R+, H

1 2 (Γ)), ∂npI ∈ H3+s

σ

(R+, H− 1

2 (Γ))

Then there exists a unique solution (u, p) ∈ H1+s

σ

(R+, H1(Ω))d × Hs

σ(R+, H1(Ωc)) ,

which depends continuously on the data.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 5 / 23

Green’s functions G / Huygens potential

G(t − s, x, y) = H(t − s − |x − y|) 2π

• (t − s)2 + |x − y|2

(2d) G(t − s, x, y) = δ(t − s − |x − y|) 4π|x − y| (3d) satisfy wave equation for point source in Rt × Rd

x:

∂2

t G(t, x, x′) − ∆xG(t, x, x′) = δ(t, x − x′)

Single layer ansatz for wave equation / Huygens potential

p(t, x) = Sq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy, x ∈ Rd \ Γ is continuous and solves wave equation on Rd \ Γ: ∂2

t p − ∆p = 0.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 6 / 23

Huygens potential / integral operators

Single layer ansatz for wave equation / Huygens potential

p(t, x) = Sq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy , x ∈ Rd \ Γ , is continuous and solves wave equation on Rd \ Γ: ∂2

t p − ∆p = 0.

Single layer operator Vq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy , x ∈ Γ . S and V: there are no differences – it’s just where x is.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 7 / 23

Huygens potential / integral operators

Single layer ansatz for wave equation / Huygens potential

p(t, x) = Sq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy , x ∈ Rd \ Γ , is continuous and solves wave equation on Rd \ Γ: ∂2

t p − ∆p = 0.

Single layer operator Vq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy , x ∈ Γ . Adjoint of double layer operator K′ϕ(t, x) =

• R+×Γ

∂G ∂nx (t − τ, x, y)ϕ(τ, y) dτ dsy , x ∈ Γ .

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 7 / 23

Fluid-structure interaction: reduce wave equation to Γ

Elastic body Ω ⊂ Rd submersed in a fluid: ̺1∂2

t u − divσ(u) = f in R+ × Ω ,

u = ∂tu = 0 for t = 0 Wave equation in Ωc = Rd \ Ω: p(t, x) = Sq(t, x) =

• R+×Γ

G(t − τ, x, y)q(τ, y) dτ dsy , x ∈ Rd \ Γ Transmission conditions on Γ = ∂Ω: σ(u)n + (∂tp + ∂tpI)n = 0 , ̺2∂tu · n + ∂np + ∂npI = 0 σ(u)n + (∂tVq + ∂tpI)n = 0 , ̺2∂tu · n + (− 1

2 + K′)q + ∂npI = 0

Solve the coupled red system with FEM / BEM in the time domain.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 8 / 23

Weak formulation in space

̺1̺2

• Ω

∂2

t u˙

v dx + ̺2

• Ω

σ(u) : ǫ(˙ v) dx − ̺2

• Γ

σ(u)n · ˙ v ds = ̺2

• Ω

f ˙ v dx ̺2

• Γ

σ(u)n · ˙ v ds + ̺2

• Γ

V∂tq˙ vn ds = −̺2

• Γ

∂tpI ˙ vn ds −̺2

• Γ

˙ un

• V∂tq′

ds −

• Γ
• (−1/2 + K′)q
• V∂tq′ ds =
• Γ

∂npI V∂tq′ ds

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 9 / 23

Space-time variational formulation

Bilinear form, with ε(v) = 1

2(∇v + (∇v)T) and σ > 0:

• B((u, q), (v, q′)) =
• R+ e−2σt

̺1̺2

• Ω

(∂2

t u) · (∂tv) dx + ̺2

• Ω

σ(u) : ∂tε(v) dx + ̺2

• Γ

(V∂tq)(∂tv · n) ds − ̺2

• Γ

(∂tu · n)(V∂tq′) ds −

• Γ

((− 1

2 + K′)q)(V∂tq′) ds

• dt

defined on (H1

σ(R+, H1(Ω))d × D(V))2, where

D(V) = {q ∈ H1

σ(R+, H− 1

2 (Γ)) : Vq ∈ H1

σ(R+, H− 1

2 (Γ))} .

Note: V : H1

σ(R+, H− 1

2 (Γ)) → H0

σ(R+, H

1 2 (Γ)) Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 10 / 23

Space-time variational formulation

Bilinear form, with ε(v) = 1

2(∇v + (∇v)T) and σ > 0:

• B((u, q), (v, q′)) =
• R+ e−2σt

̺1̺2

• Ω

(∂2

t u) · (∂tv) dx + ̺2

• Ω

σ(u) : ∂tε(v) dx + ̺2

• Γ

(V∂tq)(∂tv · n) ds − ̺2

• Γ

(∂tu · n)(V∂tq′) ds −

• Γ

((− 1

2 + K′)q)(V∂tq′) ds

• dt

Linear functional: F(v, q) =

• R+ e−2σt

−̺2

• Γ

(∂tpI)(∂tv·n)+ ∂pI

∂n (V∂tq) ds+ρ2

• Ω f ˙

v dx

• dt .

Variational formulation: Find (u, q) ∈ H1

σ(R+, H1(Ω))d × D(V) s.t.

• B((u, q), (v, q′)) = F(v, q′)

for all (v, q′) ∈ H1

σ(R+, H1(Ω))d × D(V).

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 10 / 23

Coercivity / continuity of the bilinear form

• B((u, q), (v, q′)) =
• R+ e−2σt

̺1̺2

• Ω

(∂2

t u) · (∂tv) dx + ̺2

• Ω

σ(u) : ∂tε(v) dx + ̺2

• Γ

(V∂tq)(∂tv · n) ds − ̺2

• Γ

(∂tu · n)(V∂tq′) ds −

• Γ

((− 1

2 + K′)q)(V∂tq′) ds

• dt

Theorem (Felipe 1994)

u2

1,1,Ω+q2 1,− 1

2,Γ+Vq2

1,− 1

2 ,Γ

B((u, q), (u, q)) σ u2

0,1,Ω+q2 0,− 1

2 ,Γ

Difference of time derivatives between upper and lower bounds: Wave equation is hyperbolic. V∂t coercive with loss Parseval: Im ω = σ > 0 1 2π

• R+iσ

ˆ f(ω)ˆ g(ω) dω =

• R+ e−2σtf(t)g(t) dt

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 11 / 23

Discretization

Ω = ∪N

i=1Ti, Γ = ∪M i=1Γi regular triangulations

each Γi is a face of one Tj Wp

h piecewise polynomial functions of degree p on Ω = ∪N i=1Ti

(continuous if p ≥ 1) Vp

h restrictions of functions in Wp h to Γ

[0, T) = ∪L

n=1[tn−1, tn), tn = n(∆t)

Vq

∆t piecewise polynomial functions of degree q in time

(continuous and vanishing at t = 0 if q ≥ 1) tensor products in space-time: Wp,q

h,∆t = Wp h ⊗ Vq ∆t, Vp,q h,∆t = Vp h ⊗ Vq ∆t

Abboud, Joly, Rodriguez, Terrasse 2011: DGFEM Sayas, Hsiao et al. 2015–: convolution quadrature

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 12 / 23

Galerkin FEM-BEM in space-time

Variational formulation: Find (u, q) ∈ H1

σ(R+, H1(Ω))d × D(V) s.t.

• B((u, q), (v, q′)) = F(v, q′)

for all (v, q′) ∈ H1

σ(R+, H1(Ω))d × D(V).

Conforming Galerkin discretization: Find (˜ u, ˜ q) ∈ (Wp1,q1

h,∆t )d × Vp2,q2 k,∆t s.t.

• B((˜

u, ˜ q), (˜ v, ˜ q′)) = Fk,∆t(˜ v, ˜ q′) for all (˜ v, ˜ q′) ∈ (Wp1,q1

h,∆t )d × Vp2,q2 k,∆t .

Fk,∆t(v, q) =

• R+ e−2σt

− ̺2

• Γ(∂tpI

k,∆t)(∂tvn) ds +

• Γ
• ∂pI

∂n

• k,∆t(V∂tq) ds

+ ρ2

• Ω f ˙

v dx

• dt .

Corollary

The variational formulation and its Galerkin discretization admit unique solutions.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 13 / 23

A priori error estimate

Coercivity of bilinear form & inverse estimate in time imply:

Theorem

(u, q) ∈ H1

σ(R+, H1(Ω))d × D(V) solution of continuous problem

(˜ u, ˜ q) ∈ (Wp1,q1

h,∆t )d × Vp2,q2 k,∆t Galerkin solution

Assume (u, q) is sufficiently smooth. Then ˜ u − u2

0,1,Ω + ˜

q − q2

0,− 1

2,Γ pI

k,∆t − pI2 2,− 1

2 ,Γ +

• ∂pI

∂n

• k,∆t − ∂pI

∂n 2 2,− 1

2,Γ

+ (1 + (∆t)−2)u − ˜ w2

0,1,Ω + u − ˜

w2

3,− 1

2 ,Γ

+ q − ˜ r2

3,− 1

2,Γ

for all (˜ w,˜ r) ∈ (Wp1,q1

h,∆t )d × Vp2,q2 k,∆t .

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 14 / 23

A posteriori error estimate

Theorem

(u, q) ∈ H1

σ(R+, H1(Ω))d × D(V) solution of continuous problem

(˜ u, ˜ q) ∈ (Wp1,q1

h,∆t )d × Vp2,q2 k,∆t Galerkin solution

Then ˜ u − u2

0,1,Ω + ˜

q − q2

0,− 1

2,Γ η2

1 + η2 2 + η2 3 + η2 4,

where η2

1 = ̺1∂2 t ˜

u − divσ(˜ u) − f2

0,0,Ω ,

η2

2 =

• Γi∩Γ=∅

max{h, ∆t}[nσ(˜ u)]2

1,1/2,Γi ,

η2

3 = max{k, ∆t}nσ(˜

u) + V∂t˜ qn + ∂tpIn2

1,1/2,Γ ,

η2

4 = ̺2∂t˜

u · n + (− 1

2 + K′)˜

q + ∂pI

∂n 2 2,− 1

2,Γ . Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 15 / 23

A posteriori estimate for Dirichlet bvp: V ˙ φ = ˙ f

elliptic: Carstensen–Stephan ’95 Does elliptic a posteriori analysis generalize to wave equation? ˙ φh,∆tpw polynom. degr. p in space, q in time.

Theorem 3 (reliability)

Assume R = ˙ f − V ˙ φh,∆t ∈ L2([0, T], H1(Γ)) = ⇒ φ − φh,∆t2

0,− 1

2,Γ RL2([0,

T],H1(Γ))

• ∆t∂tRL2([0,

T],L2(Γ))

+ h · ∇RL2([0,

T],L2(Γ))).

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 16 / 23

Proof of upper bound

b(φ, ψ) = ∞ e−2σt

Γ V ˙

φ(t, x)ψ(t, x) dΓx dt φ − φh,∆t2

0,− 1

2,Γ

• ∞

dt e−2σt

• Γ

dΓ V( ˙ φ − ˙ φh,∆t)(φ − φh,∆t) = ∞ dt e−2σt

• Γ

dΓ (˙ f − V ˙ φh,∆t)(φ − ψh,∆t) R0, 1

2,Γ φ − ψh,∆t0,− 1 2 ,Γ .

interpolation inequality: R2

0, 1

2,Γ R0,1,ΓR0,0,Γ .

residual orthogonal: R ⊥ ψh,∆t in H0

σ(R+, H0(Γ)) .

interpolation h, ∆t R0,0,Γ ≤ inf R − ψh,∆tL2([0,

T],L2(Γ)),

ψh,∆t ∈ Vp,q

h,∆t

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 17 / 23

Adaptivity: time-averaged error indicator

1

Start with coarse space-time grid: (∆t)i ≃ (∆x)i ≃ h0 ∀∆i

2

Solve discretization of b(φ, ψ) = ˙ f, ψ.

3

Compute time-integrated error indicator η(∆i)

4

• i η(∆i) < ε

= ⇒ STOP

5

η(∆i) > θηmax = ⇒ ∆i → ∆/4, (∆t)i → (∆t)i

2

6

GO TO 2.

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 18 / 23

Adaptive Alg. Solve → Estimate → Mark → Refine

η2

∆ = Nt 1 η∆(In)2 error indicator residual

V ˙ φ = ˙ f, ˙ f =

• 2,

x1 > 0 0, x1 ≤ 0. η∆(In)2 = tn

tn−1

• ∆[h∇Γ(˙

f − V ˙ φh,∆t)]2 ˙ φh,∆t pw. constant Refine if η∆i > θηmax Hierarchical estimator: test with new basis functions ψnew on smaller triangles: ηnew =

1 Nold

4

i=1˙

f − V ˙ ϕh,∆t, ψi

new,

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 19 / 23

residual hierarchical 4th level

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 20 / 23

residual error indicator, CFL≈ 0.48

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 21 / 23

hierarchical error indicator, CFL≈ 0.48

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 22 / 23

Conclusions & References

Fluid-structure interaction: FEM-BEM coupling in time-domain well-posedness, a priori and residual a posteriori error estimates loss of derivatives compared to elliptic analysis → non-optimal convergence rates adaptivity based on time-averaged indicator

• H. Gimperlein, M. Maischak, E. P

. Stephan, Adaptive time-domain boundary element methods and engineering applications, survey 2016, to appear.

• H. Gimperlein, C. Oezdemir, E. P

. Stephan, Analysis of a time-dependent FEM/BEM coupling method for fluid-structure interation.

Thank you

Ernst P . Stephan (Hannover) Fluid-Structure Interaction in Time-Domain RICAM 2016 23 / 23