applications of integral equation methods to a class of fundamental problems … · 2017-12-23 ·...

Outline

Applications of Integral Equation Methods to aClass of Fundamental Problems in Mechanics

and Mathematical Physics

George C. Hsiao

Department of Mathematical SciencesUniversity of DelawareNewark, Delaware USA

www.math.udel.edu/ ˜ hsiao

International Conference”Continuum Mechanics and Related Problems of Analysis”

September 9-14 , 2011, Tbilisi, Georgia

Outline

Outline

1 Some Historical Remarks

2 Exterior Boundary Value Problems

3 Transmission Problems

4 Concluding Remarks

Some Historical Remarks Exterior Boundary Value Problems Transmission Problems Concluding Remarks

Some Historical Remarks


Muskhelishvili’s monograph (1958 in English)


Simple layer potentials

Chapter 7. The Dirichlet Problem


Fichera, G., Linear elliptic equations of higher order in twoindependent variables and singular integral equations, withapplications to anisotropic inhomogeneous elasticity. InPartial Differential Equations and Continuum Mechanics,Langer, R. E. (ed). The University of Wisconsin Press:Wisconsin, 1961; 55-80.

Fichera’s method:It is a general method in terms of simple–layer potential fortreating Dirichlet problems for a large class of elliptic equationsof higher order with variable coefficients in the plane.


A model problem in R2

−∆u = 0 in Ω ( or Ωc := R2 \ Ω)

u|Γ = f on Γ := ∂Ω

(+Cond. at∞)

u(x) =

∫Γ

E(x , y)σ(y) dsy , x ∈ R2 \ Γ

Vσ :=

∫Γ

E(x , y)σ(y) dsy = f on Γ

E(x , y) = − 12π

log |x − y |,

−∆ E(x , y) = δ(|x − y |)


A modified Fichera’s method

∫Γ

∂E∂sx

(x , y)σ(y) dsy =∂

∂sxf (x) on Γ

u(x) =

∫Γ

E(x , y)σ(y)dsy + ω , x ∈ R2\Γ

Vσ + ω = f on Γ∫Γ σ ds = A

(MBIE)

Theorem [Hsiao and MacCamy 1973]

Given (f ,A) ∈ C1,α(Γ)× R, (MBIE) has a unique solution pairs(σ, ω) ∈ C0,α(Γ)× R.


A special class of BVPs

∆mu − s∆m−1u = 0 in Ω (or Ωc),

∂m−1u∂xm−k

1 ∂xk−12

= fk on Γ,

(+ Conds at∞), 1 ≤ k ≤ m, m = 1 or 2.

Hsiao, G. C. and R. C. MacCamy, Solution of boundaryvalue problems by integral equations of the first kind. SIAMRev. 1973; 15 : 687-705.Hsiao, G. C. and W. F. Wendland, A finite element methodfor some integral equations of the first kind. J. Math. Anal.Appl. 1977; 58 : 449-481.


A fundamental result (m = 1, s = 0)

Theorem [Hsiao and Wendland 1977].Under the assumption:

maxx , y ∈ Γ

|x − y | < 1,

the simple-layer boundary integral operator

Vσ :=

∫Γ

E(x , y)σ(y) dsy , x ∈ Γ

satisfies the inequalities

γ1 ||σ||2−1/2 6 γ2 ||Vσ||21/2 6 〈σ,Vσ〉 6 γ3 ||σ||2−1/2

for all σ ∈ H−1/2(Γ), where γi ’s are constants.


A weak solution

Given f ∈ H1/2(Γ), σ ∈ H−1/2(Γ) is said to be a weak solutionof the boundary integral equation

Vσ = f on Γ,if it satisfies the variational form

〈χ,Vσ〉 = 〈χ, f 〉 ∀ χ ∈ H−1/2(Γ)

Here H−1/2(Γ) = dual of H1/2(Γ) is the energy space for theoperator V .


Remarks for the general Γ

Vσ(x) := − 12 π

∫Γ

log|x − y | σ(y)dsy

= − 12 π

∫Γ

log

(|x − y |

2d

)σ(y)dsy

− c∫

Γσ(y)dsy

with c = 12π log(2d) and d = maxx , y∈Γ |x − y |.


Garding’s inequalities

Vσ(x) :=

∫Γ

E(x , y)σ(y)dsy , x ∈ Γ

〈σ,Vσ〉 = c0 ||σ||2H−1/2(Γ)− c1 ||σ||2H−(1/2+ε)(Γ)

⟨(σω

),B(σω

)⟩≥ c0

||σ||2H−1/2(Γ)

+ |ω|2

− c1

||σ||2H−(1/2+ε)(Γ)

+ |ω|2

for all (σ, ω) ∈ H−1/2(Γ)× R; ε > 0, a constant.


Boundary operator B for the modified system

B(σω

):=

( V 1∫Γ· ds 0

)(σω

)=

(Vσ + ω∫

Γ σds

)=

(fA

)

⟨(χκ

),B(σω

)⟩= 〈χ,Vσ〉+ 〈χ,1〉ω + κ〈σ, 1〉

for (σ, ω) ∈ H−1/2(Γ)× R and for all (χ, κ) ∈ H−1/2(Γ)× R


Theorem [Hsiao and Wendland 1977]

Given (f ,A) ∈ H1/2(Γ)× R, there exists a unique solution pair(σ, ω) ∈ H−1/2(Γ)× R of the system

〈χ,Vσ〉+ 〈χ,1〉ω = 〈χ, f 〉,

κ〈σ, 1〉 = κA

for all (χ, κ) ∈ H−1/2(Γ)× R.


A closely related consequence

For σ ∈ H−1/2(Γ), let u(x) :=∫

Γ E(x , y)σ(y)dsy , x ∈ R2 \ Γ.Then u ∈ H1

loc(R2) satisfies

−∆u = 0, x ∈ R2 \ Γ, [u]|Γ = 0, [∂u∂n

]|Γ = σ

u − A log|x | = o(1) as |x | → ∞, A = − 12π

∫Γσds.

([v ]|Γ := v− − v+)

ΓRΓ

Ω

ΩcR

n

n

1

〈σ,Vσ〉 =

∫Ω|∇u|2dx +

∫Ωc

R

|∇u|2dx +

∫ΓR

∂u∂n

u ds

= aΩ(u,u) + aΩcR

(u,u) + · · ·


ΓRΓ

Ω

ΩcR

n

n

1

〈σ,Vσ〉 =

∫Ω|∇u|2dx +

∫Ωc

R

|∇u|2dx +

∫ΓR

∂u∂n

u ds

= aΩ(u,u) + aΩcR

(u,u) + · · ·

Garding’s inequality for the bilinear 〈σ,Vσ〉 for the boundaryintegral operator V on the boundary Γ is a consequence of thecorresponding Gariding’s inequality of the bilinear formassociated with a related transmission problem for the partialdifferential operator −∆ in the domain Ω ∪ Ωc


Generalization

Costabel and Wendland (1986) gave a systematic approach fora general class of boundary integral operators associated withstrongly elliptic boundary value problems. For such class ofBIOs, Garding’s inequality is a consequence of strong ellipticityof the corresponding BVP for the PDE.

Costabel, M. and Wendland, W. L.: Strongly ellipticity ofboundary integral operators. J. Reine Angew. Mathematik372 (1986) 34-63.Hsiao, G. C., and Wendland, W. L., Boundary IntegralEquations, Applied Mathematical Series, 164,Springer-Verlag, 2008


Exterior Boundary ValueProblems


Viscous flow past an obstacle


Γ

Ω

R2 \Ω

u∞

1

u :=Q|Q∞|

, p :=DPµ|Q∞|

σ :=DΣ

µ|Q∞|= −pI +

(∇u +∇uT

)

Re :=D|Q∞|µ/ρ

<< 1


Solution of the modified Stokes problem


For given (q,A) ∈ (C1+α(Γ))2 × R2 with∫

Γ q · n ds = 0, thereexists a unique “solution” pair (u,p) of the modified Stokesproblem which can be represented by the simple-layer of theform

u(x) =

∫Γ

E(x , y) · σ(y)dsy + ω, p(x) =

∫Γε(x , y) · σ(y)dsy

for x ∈ Ωc , and (σ,ω) ∈ (Cα(Γ))2 ×R2 is the unique solution ofthe system

Vσ + ω = q on Γ and − 14π

∫Γσ ds = A

subject to the constrain:∫

Γ σ · n ds = 0.


The simply-layer BIO V :

Vσ(x) :=

∫Γ

E(x , y) · σ(y)dsy , x ∈ Γ.

The fundamental solution (E, ε) of the Stokes equationsdefined by

−∆E(x , y) +∇ε(x , y) = δ(|x − y |)Idiv E(x , y) = 0

can be computed explicitly :

E(x , y) = − 14πlog |x − y |+ (1 + γo − log 4)I +

(x − y)(x − y)

|x − y |2t

,

ε(x , y) =1

2π∇y log |x − y | with the Euler’s constant γo.


Main results


Let D be any compact subset of Ωc . Then the classicalsolution (u,p) of the viscous flow problem admits theasymptotic expansions:

u(x) =2∑

k=1

uk (x)

(log Re)k + O((log Re)−3),

p(x) =2∑

k=1

pk (x)

(log Re)k + O((log Re)−3)

as Re→ 0+, uniformly on D.


The force f exerted on the obstacle by the fluid admits theasymptotic expansion

f = 4π

A1

(log Re)+

A2

(log Re)2

+O((log Re)−3)

as Re→ 0+.

Here (uk ,pk ) are solutions of the modified Stokes problem withq = 0 and A = Ak for k = 1,2. The constant vectors A1 and A2are coincided with those obtained by the singular perturbationprocedure developed by Hsiao and MacCamy in [1973].Moreover the constant vector A1 is independent of the shape ofthe obstacle. In particular,

A1 = −e1,

if u∞ = e1 ( the unit vector in the x1 -direction).


Potential flow past an airfoil

The NACA airfoils are airfoil shapes for aircraft wings developed by the National

Advisory Committee for Aeronautics (NACC). The shape of the NACA airfoils is

described using a 4 -digit series following the word NACA. The formula for the shape of

a NACA 00xx foil, with xx being replaced by the percentage of thickness to chord.


Formulation (velocity q = (q1,q2)′ )

∇× q :=∂q2

∂x1− ∂q1

∂x2= 0 in Ωc

∇ · q :=∂q1

∂x1+∂q2

∂x2= 0 in Ωc

q · n|Γ = 0 on Γ

q− q∞ = o(1) as |x | → ∞lim

Ωc3x→TE|q(x)| = | q |TE exists at TE

+ (Kutta-Joukowski cond.)


Circulationκ :=

∫Γ

q · dx = −∫

Γ

∂u∂n

ds

The Kutta-Joukowski condition states that the circulationaround the airfoil should be such that the perturbedvelocity q− q∞ should be finite and continuous at thetrailing edge TE.


Formulation (Stream Function q = (∇ψ)⊥)

u := ψ − ψ∞, ψ∞ := −q∞⊥ · x

−∆u = 0 in Ωc ,

u|Γ = −ψ∞ on Γ,

u +κ

2πlog|x | = ω + O(|x |−1) as |x | → ∞

+ (Kutta-Joukowski cond.)

Remark: A subsonic flow past a given profile Γ is uniquelydetermined by its free stream velocity q∞ and its circulation κ(or equivalently the Kutta -Joukowski condition at TE).


Theorem

Given ψ∞ := −q⊥∞ · x , the problem of potential flow past anairfoil has a unique solution pair (κ,u) ∈ R× H2+ε

loc (Ωc).

Green’s Representation for u (µ := u|Γ, σ := ∂u/∂n|Γ):

u(x) =

∫Γµ(x)

∂E∂ny

(x , y)dsy −∫

ΓE(x , y)σ(y)dsy + ω, x ∈ Ωc

∼ 12π

(

∫Γσ ds)log|x |+ ω + O(|x |−1)

=⇒∫

Γσds = −κ.


Boundary Integral Equations

Vσ − ω = f ,∫

Γσds = −κ

+(Kutta-Joukowski cond.)

f := (−1 + θx/2π)I + K(−ψ∞)


Circulation and flow pattern


Solution Procedure: σ = σ0 − κσ1, ω = ω0 − κω1,

Vσ0 − ω0 = f , Vσ1 − ω1 = 0,∫Γσ0ds = 0,

∫Γσ1ds = 1.

κ := limρ→0

ρβσ0

ρβσ1, β > 0 (to be determined)

σi = ciσs + regular term, i = 0,1

σs = O(ρπ

2π−απ−1) = O(ρ−1−α2−α ), β := 1−α

2−α

=⇒ σ0 − κσ1 = (c0 − κc1)σs + regular term

κ = c0/c1

(see e.g., Hsiao (1990), Hsiao, Marcozzi & Zhang (1993))


Transmission Problems


A nonlinear interface problem

Let Ω ⊂ R3 be a bounded, simply connected domain withboundary Γ. We assume that Ω is decomposed into twosubdomains: Ωe and Ωp :

n

nΓ

Γ0

Ωp

Ωe

1

Ωe ⊂ Ω, an inner region with linear elastic materialΩp = Ω \ Ωe, an annual region with elasto-plastic material .


The interface problem can be formulated as follows:

For given body force f in Ωp, find the displacement field usatisfying

−L(u) := − divσ(u) = 0 in Ωe

−N(u) := −divσp(u) = f in Ωp

u|Γ = 0 on Γ,

u− = u+, Tp(u)− = T(u)+ on Γ0,

Here the traction operators Tp and T are defined by

Tp(u)− = σp(u)n|Γ0 and T(u)+ = σ(u)n|Γ0 ,

where n denotes the outward unit normal to Γ0.


Linear elastic material

For the linear elastic material in Ωe, we assume the standardHooke’s law for the isotropic material,

σ(u)(x) = λ div u(x)I + 2µε(u)(x)

for all x ∈ Ωe, where λ and µ are the Lame constants such thatµ > 0, and 3λ+ 2µ > 0. Hence, the partial differential equationin Ωe is linear, namely the Lame equation,

L(u) := divσ(u) = 0


The elasto-plastic material

For the elasto-plastic material in Ωp, we assume theHencky-Mises stress-strain relation:

σp(u)(x) =

[κ(x)− 2

3µ(x ,G(u)(x)

]div u(x)I

+ 2µ(x ,G(u)(x))ε(u)(x)

where κ : Ωp → R is the bulk modulus, µ : Ωp × R+ → R theLame function, and

G(u) := ε∗(u) : ε∗(u) with ε∗(u) := ε(u)− 13

(div u)I

being the deviator of the small strain tensor.


Here the functions κ and µ are supposed to be continuous, andµ(x , ·) to be continuously differentiable in R+ such that thefollowing estimates hold:

0 < κ0 < κ(x) ≤ κ1

for all x ∈ Ωp, while

0 < µ0 ≤ µ(x , η) ≤ 32κ(x), and

0 < µ1 ≤ µ(x , η) + 2(∂

∂ηµ(x , η)

)η ≤ µ2

for all (x , η) ∈ Ωp × R+.


The interface problem can be formulated as follows:

For given body force f in Ωp, find the displacement field usatisfying

−L(u) := − divσ(u) = 0 in Ωe

−N(u) := −divσp(u) = f in Ωp

u|Γ = 0 on Γ,

u− = u+, Tp(u)− = T(u)+ on Γ0,

Here the traction operators Tp and T are defined by

Tp(u)− = σp(u)n|Γ0 and T(u)+ = σ(u)n|Γ0 ,

where n denotes the outward unit normal to Γ0.


Reduction to non-local problem

From the Betti formula, we represent u in Ωe in the form

u(x) =

∫Γ0

E(x , y)T(u)−(y)dsy −∫

Γ0

(Ty (E(x , y)))tu−(y)dsy

= S(T(u)−)−D(u−), x ∈ Ωe,

from which, we now arrive at the system of BIOs on Γ0:

u− = V (T(u)−) +

(12

I − K)

u−,

T(u)− =

(12

I + K ′)

(T(u)−) + Wu−

where V ,K ,K ′ and W are the four basic boundary integraloperators:


Four basic boundary integral operators

Vσ(x) :=

∫Γ0

E(x , y) · σ(y)dsy (simple-layer),

Kµ(x) :=

∫Γ0

(TyE(x , y)

)t· µ(y)dsy (double-layer)

K ′σ(x) :=

∫Γ0

TxE(x , y) · σ(y)dsy (transpose of K ),

Wµ(x) := −Tx

∫Γ0

(TyE(x , y)

)t· µ(y)dsy (hypersingular).

E(x , y) =λ+ 3µ

8πµ(λ+ 2µ)

1

|x − y | I +

(λ+ µ

λ+ 3µ

)(x − y)(x − y)t

|x − y |3

is the fundamental tensor for the Lame equation .


u− = V (T(u)−) +

(12

I − K)

u−,

T(u)− =

(12

I + K ′)

T(u)− + Wu−

Now we use the interface conditions:

u− = u+, Tp(u)+ = T(u)− := σ

.

Vσ −(

12

I + K)

u+ = 0 on Γ0

Tp(u)+ = (12

I + K ′)σ + Wu+ on Γ0


Nonlocal boundary problem

Nonlocal boundary problem for the unknowns u and σ:

−N(u) = f in Ωp

u|Γ = 0 on Γ,

Tp(u)+ = (12

I + K ′)σ + Wu+ on Γ0

together with the nonlocal boundary condition

Vσ −(

12

I + K)

u+ = 0 on Γ0.


Variational formulation

(H1Γ (Ωp))3 = v ∈ (H1(Ωp))3 : v|Γ = 0.

Given f ∈ (L2(Ωp))3, find (u,σ) ∈ (H1Γ (Ωp))3 × (H−1/2(Γ0))3

such that

AΩp (u,v) + BΓ0((u,σ), (v,λ)) =

∫Ωp

f · v dx .

for all (v,λ) ∈ (H1Γ (Ωp))3 × (H−1/2(Γ0))3.

In the formulation, BΓ0 is the bilinear form on Γ0,

BΓ0((u,σ), (v,λ)) := 〈(

12

I + K ′)σ,v+〉Γ0 + 〈Wu+,v+〉Γ0

+〈λ,Vσ〉Γ0 − 〈λ,(

12

I + K)

u+〉Γ0


An operator equation

Let H∗ denote the dual of H := (H1Γ (Ωp))3 × (H−1/2(Γ0))3.

Define a nonlinear operator A : H → H∗ on H by putting

[A(u,σ), (v,λ)] := AΩp (u,v) + BΓ0((u,σ), (v,λ)).

Then the nonlocal boundary problem can be reformulated inthe form of an operator equation for the unknowns (u,σ):

A(u,σ) = F ,

where F ∈ H∗ is defined by

[F , (v,λ)] :=

∫Ωp

f · v dx .


Main results

TheoremUnder the assumptions on the bulk modulus κ and on the Lamefunction µ, we have the following:

The operator A is both strongly monotone and Lipschitzcontinuous on H.Given f ∈ (L2(Ωp))3, there exists a unique solution(u,σ) ∈ H of the operator equation:

A(u,σ) = F .

(see, e.g., Gatica and Hsiao (1989, 1990))


Remarks:

Linear-Nonlinear Transmission Problems

Strongly monotone type:Polizzoto (1987), Gatica & Hsiao (1989, 1990, 1992,1995), Stephan & Costabel (1990), Carstensen (1992),Gatica & Wendland (1994)Non-monotone type:Gatica & Hsiao (1992), Berger, Warneke, & Wendland(1993, 1994), Feistauer, Hsiao, Kleinman & Tezaur (1994,1995).


Fluid-solid interaction in Rd ,d = 2,3

Elastic Body

Scattered Field

Incident Acoustic WaveCompressible Fluid

Γ

Ω

Ωc = Rd \Ω

1


Find u ∈ (H1(Ω))d in Ω and ps ∈ H1loc(Ωc) in Ωc such that

(E)

∆∗u + ρω2u = 0 in Ω (elastic body)

∆ps + k2ps = 0 in Ωc (fluid)

(B)

T(u)− = −(ps+ + pinc)n

u− · n =1

ρfω2

(∂ps+

∂n+∂pinc

∂n) on Γ

(C)∂ps

∂r− ikps = o(r−(d−1)/2) as r = |x | −→ ∞.


Basic idea

The basic idea is agin to transform the original transmissionproblem to an equivalent nonlocal boundary problem for u inthe bounded domain Ω together with an appropriate boundaryintegral equation for ps+ on the interface Γ in terms of thetransmission conditions on the interface. For instance, if werepresent ps in Ωc ,

ps(x) =

∫Γ

∂E∂ny

(x , y)ps+(y)dsy −∫

ΓE(x , y)

∂ps+(y)

∂nydsy

= D(ps+)(x)− S(∂ps+

∂n)(x),

then we arrive at a non-local boundary problem of the form


Nonlocal boundary problem

Given pinc ∈ H1/2(Γ), find (u,ps+) ∈ (H1(Ω))3 × H1/2(Γ)satisfying

∆∗u + ρω2u = 0 in Ω

T(u) = −(ps+ + pinc)n on Γ

Wps+ + ρfω2(

12

I + K ′)(u− · n) = (12

+ K ′)∂pinc

∂non Γ

TheoremAssume that the homogeneous BVP of (E), (B), (C) has notraction free solution and that k is not an exceptional value.Then the corresponding non-local boundary problem hasunique solution (u,ps+) in (H1(Ω))3 × H1/2(Γ).

(see, e.g., Hsiao (1989), Hsiao, Kleinman & Roach (2000) )


Representations of ps in Ωc

Method Rep. of ps in Ωc TractionT(u) on Γ

1 −(ps+ + pinc)n

2(a) ps = Dps+ − Sσ (same as above)

2(b) ps+ ∈ H1/2(Γ), σ ∈ H−1/2(Γ) (same as above)

3 σ := ρfω2(u− · n)− ∂pinc

∂n −((12 I + K )ps+ − Vσ + pinc)n

4 −(ps+ + pinc)n

5 ps = Dµ, µ ∈ H1/2(Γ) −((12 I + K )µ+ pinc)n

6(a) ps = −Sφ −(−Vφ+ pinc)n

6(b) φ ∈ H−1/2(Γ) (same as above)

7 ps = Dµ+ iηSµ, µ ∈ H1/2(Γ) −((12 I + K )µ+ iηVµ+ pinc)n


Nonlocal BC’s

Method NLBC on Γ Weak Form

1 Wps+ + (12 I + K ′)σ = 0 < ·,q >0= 0, q ∈ H1/2(Γ)

2(a) (12 I − K )ps+ + Vσ = 0 < ·,q >1/2= 0, q ∈ H1/2(Γ)

2(b) (same as above) < ·,Wq >0= 0, q ∈ H1/2(Γ)

3 Wps+ + (12 I + K ′)σ = 0 < ·,q >0= 0, q ∈ H1/2(Γ)

4 Wps+ + (12 I + K ′)σ < ·,q >0= 0, q ∈ H1/2(Γ)

+iη((12 I − K )p+ + Vσ) = 0

5 Wµ+ σ = 0 < ·, ν >0= 0 , ν ∈ H1/2(Γ)

6(a) (12 I − K ′)φ− σ = 0 < ·, ψ >−1/2= 0, ψ ∈ H−1/2(Γ)

6(b) (same as above) < ·,Vψ >0= 0, ψ ∈ H1/2(Γ)

7 Wµ+ iη(12 I − K ′)µ+ σ = 0 < ·, ν >0= 0, ν ∈ H1/2(Γ)


Concluding Remarks


Contact, crack and screen problems

2011: Gachechiladze, A., Gachechiladze, Gwinner, J., Natroshvili, D., Contactproblems with friction for hemitropic solids: boundary variational inequalityapproach, Appl. Anal. 90 279-303

2004: Natroshvili, D. and Wendland, W. L., Boundary variational inequalities inthe theory of interface cracks: In Operator Theory: Advances and Appl. 147387-402

1991: Hsiao, G.C., Stephan, E. P. and Wendland, W. L., On the Dirichlet problemin elasticity for a domain exterior to an arc, ıJ. Comp. Appl. Math. 34 1-19

1990: Han, H., A direct boundary element method for Signorini problems,Mathematics of Computation, 55 115-128.

1988: Han, H. and Hsiao, G.C., The boundary element method for a contactproblem: In Theory and Applications of Boundary Elements, Q. Du and M.Tanaka eds., Tsinghua University Press, Beijing, 33-38

1986: Stephan, E. P., Boundary integral equations for magnetic screens in R3,Proc. Royal Soc. Edinburgh 102A 189-210


Numerical experiments

Viscous Flow Pattern

Flow Flow PatternPattern

Viscous Flow Past Viscous Flow Past an Obstacle an Obstacle

& Absolute Error & Absolute Error EstimateEstimate

Hsiao, G. C., Kopp, P., and Hsiao, G. C., Kopp, P., and WendlandWendland, W. L.,, W. L.,Some applications of a Some applications of a GalerkinGalerkin –– Collocation method for Collocation method for boundary integral equations of the first kind, boundary integral equations of the first kind, Math. Method. Math. Method. ApplAppl. . SciSci. . 6 6 (1984) 280(1984) 280----325.325.

Hsiao, G.C., Kopp, P., and Wendland, W.L., Some applications of a Galerkin-collocation

method for boundary integral equations of the first kind, Math. Meth. Appl. Sci. 6

(1984) 280-325


High Stress Gradients

High Stress GradientsHigh Stress GradientsG.C. Hsiao, E. Schnack, and W.L.Wendland, Hybride coupled finite-boundary element methods for Elliptic systems of second order,Comput. Methods Appl. Mech. Engrg. 190 (2000) 431-485

Hsiao, G. C., Schnack, and Wendland, W. L., Hybrid coupled finite-boundary element

methods for elliptic system of second order, Comput. Meth. Mech. Engrg. 190 (2000)

431-485


Radar Cross Section

Hsiao, G. C., Kleinman, R. E., and Wang, D. Q., Applications of boundary integral

methods in 3-D electromagnetic scattering, J. Comp. Appl. Math., 104 (1999) 89-110


Fluid-Solid interaction

Time-harmonic behaviour of the pressure

Acoustic source: Plane incident wave from leftka = 1.0,R = 0.95λ


ïxï

ïyï

Real part of pressure

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï1

ï0.5

0

0.5

1

ïxï

ïyï

Imag.part of pressure

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï1.5

ï1

ï0.5

0

0.5

ïxï

ïyï

Real part of xïcoord.displacement

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï10

ï8

ï6

ï4

ï2

0

2

4

6

ïxï

ïyï

Imag.part of xïcoord.displacement

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï5

0

5

10

15


Displacement Fields

ïxï

ïyï

Real part of xïcoord.displacement

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï10

ï8

ï6

ï4

ï2

0

2

4

6

ïxïïyï

Imag.part of xïcoord.displacement

ï6 ï4 ï2 0 2 4 6ï6

ï4

ï2

0

2

4

6

ï5

0

5

10

15

Elschner, J., Hsiao, G.C., and Rathsfeld, A., Reconstruction of elastic obstacles from

the far-field data of scattered acoustic waves, Memoirs on Differential Equations and

Mathematical Physics, 53 (2011) .


Further references

1985: Gatica, N. G., and Hsiao, G. C., Boundary-field equation Methods for aClass of Nonlinear Problems, Pitman Research Notes in Mathematics Series331, Longman

2004: Stephan, E. P., Coupling of Boundary Element Methods and FiniteElement methods Chapter 13 in Encyclopedia of Computational Mechanics,Edited by Erwin Stein, Rene de Borst and Thomas J.R. Hughes. Volume 1:Fundamentals, pp.375-412. c© 2004 John Wiley & Sons, Ltd. ISBN:0–470–84699–2

2004: Hsiao, G. C., and Wendland, W. L., Boundary Element Methods:Foundations and Error Analysis Chapter 12 in Encyclopedia of ComputationalMechanics, Edited by Erwin Stein, Rene de Borst and Thomas J.R. Hughes.Volume 1: Fundamentals, pp.339–373. c© 2004 John Wiley & Sons, Ltd. ISBN:0–470–84699–2

2008: Hsiao, G. C., and Wendland, W. L., Boundary Integral Equations, AppliedMathematical Series, 164, Springer-Verlag.


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