Applications of Integral Equation Methods to a Class of Fundamental Problems in Mechanics and Mathematical Physics

Transcription

1 Outline Applications of Integral Equation Methods to a Class of Fundamental Problems in Mechanics and Mathematical Physics George C. Hsiao Department of Mathematical Sciences University of Delaware Newark, Delaware USA hsiao International Conference Continuum Mechanics and Related Problems of Analysis September 9-14, 2011, Tbilisi, Georgia

2 Outline Outline 1 Some Historical Remarks 2 Exterior Boundary Value Problems 3 Transmission Problems 4 Concluding Remarks

3 Some Historical Remarks Exterior Boundary Value Problems Transmission Problems Concluding Remarks Some Historical Remarks

4 Muskhelishvili s monograph (1958 in English)

5 Simple layer potentials Chapter 7. The Dirichlet Problem

6 Fichera, G., Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anisotropic inhomogeneous elasticity. In Partial Differential Equations and Continuum Mechanics, Langer, R. E. (ed). The University of Wisconsin Press: Wisconsin, 1961; Fichera s method: It is a general method in terms of simple layer potential for treating Dirichlet problems for a large class of elliptic equations of higher order with variable coefficients in the plane.

7 A model problem in R 2 u = 0 in Ω ( or Ω c := R 2 \ Ω) u Γ = f on Γ := Ω (+Cond. at ) u(x) = E(x, y)σ(y) ds y, x R 2 \ Γ Γ V σ := Γ E(x, y)σ(y) ds y = f on Γ E(x, y) = 1 log x y, 2π E(x, y) = δ( x y )

8 A modified Fichera s method Γ E s x (x, y)σ(y) ds y = u(x) = Γ s x f (x) on Γ E(x, y)σ(y)ds y + ω, x R 2 \Γ V σ + ω = f on Γ Γ σ ds = A (MBIE) Theorem [Hsiao and MacCamy 1973] Given (f, A) C 1,α (Γ) R, (MBIE) has a unique solution pairs (σ, ω) C 0,α (Γ) R.

9 A special class of BVPs m u s m 1 u = 0 in Ω (or Ω c ), m 1 u x m k 1 x k 1 = f k on Γ, 2 (+ Conds at ), 1 k m, m = 1 or 2. Hsiao, G. C. and R. C. MacCamy, Solution of boundary value problems by integral equations of the first kind. SIAM Rev. 1973; 15 : Hsiao, G. C. and W. F. Wendland, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 1977; 58 :

10 A fundamental result (m = 1, s = 0) Theorem [Hsiao and Wendland 1977]. Under the assumption: max x, y Γ x y < 1, the simple-layer boundary integral operator V σ := E(x, y)σ(y) ds y, x Γ satisfies the inequalities Γ γ 1 σ 2 1/2 γ 2 V σ 2 1/2 σ, V σ γ 3 σ 2 1/2 for all σ H 1/2 (Γ), where γ i s are constants.

11 A weak solution Given f H 1/2 (Γ), σ H 1/2 (Γ) is said to be a weak solution of the boundary integral equation V σ = f on Γ, if it satisfies the variational form χ, V σ = χ, f χ H 1/2 (Γ) Here H 1/2 (Γ) = dual of H 1/2 (Γ) is the energy space for the operator V.

12 Remarks for the general Γ V σ(x) := 1 log x y σ(y)ds y 2 π Γ = 1 ( ) x y log σ(y)ds y 2 π Γ 2d c σ(y)ds y Γ with c = 1 2π log(2d) and d = max x, y Γ x y.

13 Gårding s inequalities V σ(x) := Γ E(x, y)σ(y)ds y, x Γ σ, V σ c 0 σ 2 H 1/2 (Γ) c 1 σ 2 H (1/2+ɛ) (Γ) (σ ), B ω ( ) } σ c ω 0 { σ 2H 1/2(Γ) + ω 2 { } c 1 σ 2 H (1/2+ɛ) (Γ) + ω 2 for all (σ, ω) H 1/2 (Γ) R; ɛ > 0, a constant.

14 Boundary operator B for the modified system B ( ) σ := ω ( V 1 ds 0 Γ ) (σ ) ( ) V σ + ω = ω Γ σds = ( ) f A (χ ), B κ ( ) σ = χ, V σ + χ, 1 ω + κ σ, 1 ω for (σ, ω) H 1/2 (Γ) R and for all (χ, κ) H 1/2 (Γ) R

15 Theorem [Hsiao and Wendland 1977] Given (f, A) H 1/2 (Γ) R, there exists a unique solution pair (σ, ω) H 1/2 (Γ) R of the system χ, V σ + χ, 1 ω = χ, f, κ σ, 1 = κa for all (χ, κ) H 1/2 (Γ) R.

16 A closely related consequence For σ H 1/2 (Γ), let u(x) := Γ E(x, y)σ(y)ds y, x R 2 \ Γ. Then u H 1 loc (R2 ) satisfies u = 0, x R 2 \ Γ, [u] Γ = 0, [ u n ] Γ = σ u A log x = o(1) as x, A = 1 σds. 2π ([v] Γ := v v + ) Γ n n Ω Γ Γ R Ω c R σ, V σ = u 2 dx + u 2 u dx + Ω Ω c Γ R R n u ds = a Ω (u, u) + a Ω c R (u, u) +

17 Ω Ω c R Γ n n Γ R σ, V σ = u 2 dx + u 2 u dx + Ω Ω c Γ R R n u ds = a Ω (u, u) + a Ω c R (u, u) + 1 Gårding s inequality for the bilinear σ, V σ for the boundary integral operator V on the boundary Γ is a consequence of the corresponding Gåriding s inequality of the bilinear form associated with a related transmission problem for the partial differential operator in the domain Ω Ω c

18 Generalization Costabel and Wendland (1986) gave a systematic approach for a general class of boundary integral operators associated with strongly elliptic boundary value problems. For such class of BIOs, Gårding s inequality is a consequence of strong ellipticity of the corresponding BVP for the PDE. Costabel, M. and Wendland, W. L.: Strongly ellipticity of boundary integral operators. J. Reine Angew. Mathematik 372 (1986) Hsiao, G. C., and Wendland, W. L., Boundary Integral Equations, Applied Mathematical Series, 164, Springer-Verlag, 2008

19 Some Historical Remarks Exterior Boundary Value Problems Transmission Problems Concluding Remarks Exterior Boundary Value Problems

20 Viscous flow past an obstacle

21 Γ u Ω R 2 \ Ω u := Q DP, p := Q µ Q σ := DΣ ( ) µ Q = pi + u + u T Re := D Q µ/ρ << 1

22 Dimensionless BVP (P Re ) u + Re(u u) + p = 0; u = 0 in Ω c, u Γ = 0 on Γ, u u, p 0 as x. The force exerted on the obstacle by the fluid is f := F µ Q = σ[ˆn] ds. Stokes Paradox The reduced problem (P 0 ) has no solution. Γ

23 A modified Stokes problem u 0 + p 0 = 0; u 0 = 0 in Ω c, u 0 Γ = q on Γ, u A log x = O(1), p 0 = o(1) as x, where A is any given vector in R 2 and q is a prescribed function satisfying q ˆn ds = 0. Γ

24 Solution of the modified Stokes problem Theorem [Hsiao and MacCamy 1973] For given (q, A) (C 1+α (Γ)) 2 R 2 with Γ q ˆn ds = 0, there exists a unique solution pair (u, p) of the modified Stokes problem which can be represented by the simple-layer of the form u(x) = E(x, y) σ(y)ds y + ω, p(x) = ε(x, y) σ(y)ds y Γ for x Ω c, and (σ, ω) (C α (Γ)) 2 R 2 is the unique solution of the system V σ + ω = q on Γ and 1 σ ds = A 4π subject to the constrain: Γ σ ˆn ds = 0. Γ Γ

25 The simply-layer BIO V : V σ(x) := E(x, y) σ(y)ds y, x Γ. Γ The fundamental solution (E, ɛ) of the Stokes equations defined by E(x, y) + ɛ(x, y) = δ( x y )I can be computed explicitly : div E(x, y) = 0 E(x, y) = 1 4π {log x y + (1 + γ o log 4)}I + ε(x, y) = 1 2π y log x y with the Euler s constant γ o. t (x y)(x y) x y 2,

26 Main results Theorem [Hsiao and MacCamy 1982] Let D be any compact subset of Ω c. Then the classical solution (u, p) of the viscous flow problem admits the asymptotic expansions: u(x) = p(x) = 2 k=1 2 k=1 u k (x) (log Re) k + O((log Re) 3 ), p k (x) (log Re) k + O((log Re) 3 ) as Re 0 +, uniformly on D.

27 The force f exerted on the obstacle by the fluid admits the asymptotic expansion { } A1 f = 4π (log Re) + A 2 (log Re) 2 +O((log Re) 3 ) as Re 0 +. Here (u k, p k ) are solutions of the modified Stokes problem with q = 0 and A = A k for k = 1, 2. The constant vectors A 1 and A 2 are coincided with those obtained by the singular perturbation procedure developed by Hsiao and MacCamy in [1973]. Moreover the constant vector A 1 is independent of the shape of the obstacle. In particular, A 1 = ê 1, if u = ê 1 ( the unit vector in the x 1 -direction).

28 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.

29 Formulation (velocity q = (q 1, q 2 ) ) q := q 2 x 1 q 1 x 2 = 0 in Ω c q := q 1 x 1 + q 2 x 2 = 0 in Ω c lim Ω c x TE q ˆn Γ = 0 on Γ q q = o(1) as x q(x) = q TE + (Kutta-Joukowski cond.) exists at TE

30 Circulation κ := Γ u q dx = Γ n ds The Kutta-Joukowski condition states that the circulation around the airfoil should be such that the perturbed velocity q q should be finite and continuous at the trailing edge TE.

31 Formulation (Stream Function q = ( ψ) ) u := ψ ψ, ψ := q x u = 0 in Ω c, u Γ = ψ on Γ, u + κ 2π log x = ω + O( x 1 ) as x + (Kutta-Joukowski cond.) Remark: A subsonic flow past a given profile Γ is uniquely determined by its free stream velocity q and its circulation κ (or equivalently the Kutta -Joukowski condition at TE).

32 Theorem Given ψ := q x, the problem of potential flow past an airfoil has a unique solution pair (κ, u) R H 2+ɛ loc (Ωc ). Green s Representation for u (µ := u Γ, σ := u/ n Γ ): u(x) = µ(x) E (x, y)ds y E(x, y)σ(y)ds y + ω, x Ω c n y Γ 1 2π ( σ ds)log x + ω + O( x 1 ) Γ = σds = κ. Γ Γ

33 Boundary Integral Equations V σ ω = f, σds = κ +(Kutta-Joukowski cond.) f := {( 1 + θ x /2π)I + K }( ψ ) Γ

34 Circulation and flow pattern

35 Solution Procedure: σ = σ 0 κσ 1, ω = ω 0 κω 1, V σ 0 ω 0 = f, V σ 1 ω 1 = 0, σ 0 ds = 0, σ 1 ds = 1. Γ Γ κ := lim ρ 0 ρ β σ 0 ρ β σ 1, β > 0 (to be determined) σ i = c i σ s + regular term, i = 0, 1 σ s = O(ρ π 2π απ 1 ) = O(ρ 1 α 2 α ), β := 1 α 2 α = σ 0 κσ 1 = (c 0 κc 1 )σ s + regular term κ = c 0 /c 1 (see e.g., Hsiao (1990), Hsiao, Marcozzi & Zhang (1993))

36 Some Historical Remarks Exterior Boundary Value Problems Transmission Problems Transmission Problems Concluding Remarks

37 A nonlinear interface problem Let Ω R 3 be a bounded, simply connected domain with boundary Γ. We assume that Ω is decomposed into two subdomains: Ω e and Ω p : Γ n Ωp Γ Ωe n Ω e Ω, an inner region with linear elastic material Ω p = Ω \ Ω e, an annual region with elasto-plastic material.

38 The interface problem can be formulated as follows: For given body force f in Ω p, find the displacement field u satisfying L(u) := div σ(u) = 0 in Ω e N(u) := divσ p (u) = f in Ω p u Γ = 0 on Γ, u = u +, T p (u) = T(u) + on Γ 0, Here the traction operators T p and T are defined by T p (u) = σ p (u)ˆn Γ0 and T(u) + = σ(u)ˆn Γ0, where ˆn denotes the outward unit normal to Γ 0.

39 Linear elastic material For the linear elastic material in Ω e, we assume the standard Hooke s law for the isotropic material, σ(u)(x) = λ div u(x)i + 2µε(u)(x) for all x Ω e, where λ and µ are the Lamé constants such that µ > 0, and 3λ + 2µ > 0. Hence, the partial differential equation in Ω e is linear, namely the Lamé equation, L(u) := div σ(u) = 0

40 The elasto-plastic material For the elasto-plastic material in Ω p, we assume the Hencky-Mises stress-strain relation: σ p (u)(x) = [κ(x) 23 ] µ(x, G(u)(x) div u(x)i + 2 µ(x, G(u)(x))ε(u)(x) where κ : Ω p R is the bulk modulus, µ : Ω p R + R the Lamé function, and G(u) := ε (u) : ε (u) with ɛ (u) := ε(u) 1 (div u)i 3 being the deviator of the small strain tensor.

41 Here the functions κ and µ are supposed to be continuous, and µ(x, ) to be continuously differentiable in R + such that the following estimates hold: for all x Ω p, while 0 < κ 0 < κ(x) κ 1 0 < µ 0 µ(x, η) 3 κ(x), and ( 2 ) 0 < µ 1 µ(x, η) + 2 µ(x, η) η µ 2 η for all (x, η) Ω p R +.

42 The interface problem can be formulated as follows: For given body force f in Ω p, find the displacement field u satisfying L(u) := div σ(u) = 0 in Ω e N(u) := divσ p (u) = f in Ω p u Γ = 0 on Γ, u = u +, T p (u) = T(u) + on Γ 0, Here the traction operators T p and T are defined by T p (u) = σ p (u)ˆn Γ0 and T(u) + = σ(u)ˆn Γ0, where ˆn denotes the outward unit normal to Γ 0.

43 Reduction to non-local problem From the Betti formula, we represent u in Ω e in the form u(x) = E(x, y)t(u) (y)ds y Γ 0 (T y (E(x, y))) t u (y)ds y Γ 0 = S(T(u) ) D(u ), x Ω e, from which, we now arrive at the system of BIOs on Γ 0 : ( 1 u = V (T(u) ) + ( ) 1 T(u) = 2 I + K 2 I K ) u, (T(u) ) + W u where V, K, K and W are the four basic boundary integral operators:

44 Four basic boundary integral operators V σ(x) := E(x, y) σ(y)ds y (simple-layer), Γ 0 ( t K µ(x) := T y E(x, y)) µ(y)dsy (double-layer) Γ 0 K σ(x) := T x E(x, y) σ(y)ds y (transpose of K ), Γ 0 ( t W µ(x) := T x T y E(x, y)) µ(y)dsy (hypersingular). E(x, y) = Γ 0 { ( ) λ + 3µ 1 λ + µ (x y)(x y) t 8πµ(λ + 2µ) x y I + λ + 3µ x y 3 is the fundamental tensor for the Lamé equation. }

45 ( 1 u = V (T(u) ) + ( ) 1 T(u) = 2 I + K Now we use the interface conditions: 2 I K ) u, T(u) + W u u = u +, T p (u) + = T(u) := σ. ( ) 1 V σ 2 I + K u + = 0 on Γ 0 T p (u) + = ( 1 2 I + K )σ + W u + on Γ 0

46 Nonlocal boundary problem Nonlocal boundary problem for the unknowns u and σ: N(u) = f in Ω p u Γ = 0 on Γ, T p (u) + = ( 1 2 I + K )σ + W u + on Γ 0 together with the nonlocal boundary condition ( ) 1 V σ 2 I + K u + = 0 on Γ 0.

47 Variational formulation (H 1 Γ (Ω p)) 3 = {v (H 1 (Ω p )) 3 : v Γ = 0}. Given f (L 2 (Ω p )) 3, find (u, σ) (HΓ 1(Ω p)) 3 (H 1/2 (Γ 0 )) 3 such that A Ωp (u, v) + B Γ0 ((u, σ), (v, λ)) = f v dx. Ω p for all (v, λ) (H 1 Γ (Ω p)) 3 (H 1/2 (Γ 0 )) 3. In the formulation, B Γ0 is the bilinear form on Γ 0, ( ) 1 B Γ0 ((u, σ), (v, λ)) := 2 I + K σ, v + Γ0 + W u +, v + Γ0 ( ) 1 + λ, V σ Γ0 λ, 2 I + K u + Γ0

48 An operator equation Let H denote the dual of H := (H 1 Γ (Ω p)) 3 (H 1/2 (Γ 0 )) 3. Define a nonlinear operator A : H H on H by putting [A(u, σ), (v, λ)] := A Ωp (u, v) + B Γ0 ((u, σ), (v, λ)). Then the nonlocal boundary problem can be reformulated in the form of an operator equation for the unknowns (u, σ): A(u, σ) = F, where F H is defined by [F, (v, λ)] := f v dx. Ω p

49 Main results Theorem Under the assumptions on the bulk modulus κ and on the Lamé function µ, we have the following: The operator A is both strongly monotone and Lipschitz continuous on H. Given f (L 2 (Ω p )) 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))

50 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).

51 Fluid-solid interaction in R d, d = 2, 3 Elastic Body Γ Scattered Field Ω Incident Acoustic Wave Ω c = R d \ Ω Compressible Fluid

52 Find u (H 1 (Ω)) d in Ω and p s H 1 loc (Ωc ) in Ω c such that (E) { u + ρω 2 u = 0 in Ω (elastic body) p s + k 2 p s = 0 in Ω c (fluid) (B) { T(u) = (p s+ + p inc )ˆn u ˆn = 1 ( p s+ ρ f ω 2 n + pinc n ) } on Γ (C) p s r ikp s = o(r (d 1)/2 ) as r = x.

53 Basic idea The basic idea is agin to transform the original transmission problem to an equivalent nonlocal boundary problem for u in the bounded domain Ω together with an appropriate boundary integral equation for p s+ on the interface Γ in terms of the transmission conditions on the interface. For instance, if we represent p s in Ω c, p s (x) = Γ E (x, y)p s+ (y)ds y n y Γ = D(p s+ )(x) S( ps+ n )(x), E(x, y) ps+ (y) ds y n y then we arrive at a non-local boundary problem of the form

54 Nonlocal boundary problem Given p inc satisfying H 1/2 (Γ), find (u, p s+ ) (H 1 (Ω)) 3 H 1/2 (Γ) u + ρω 2 u = 0 in Ω T(u) = (p s+ + p inc )ˆn on Γ Wp s+ + ρ f ω 2 ( 1 2 I + K )(u ˆn) = ( K ) pinc n on Γ Theorem Assume that the homogeneous BVP of (E), (B), (C) has no traction free solution and that k is not an exceptional value. Then the corresponding non-local boundary problem has unique solution (u, p s+ ) in (H 1 (Ω)) 3 H 1/2 (Γ). (see, e.g., Hsiao (1989), Hsiao, Kleinman & Roach (2000) )

55 Representations of p s in Ω c Method Rep. of p s in Ω c TractionT(u) on Γ 1 (p s+ + p inc )n 2(a) p s = Dp s+ Sσ (same as above) 2(b) p s+ H 1/2 (Γ), σ H 1/2 (Γ) (same as above) 3 σ := ρ f ω 2 (u ˆn) pinc n (( 1 2 I + K )ps+ V σ + p inc )ˆn 4 (p s+ + p inc )ˆn 5 p s = Dµ, µ H 1/2 (Γ) (( 1 2 I + K )µ + pinc )ˆn 6(a) p s = Sφ ( V φ + p inc )ˆn 6(b) φ H 1/2 (Γ) (same as above) 7 p s = Dµ + iηsµ, µ H 1/2 (Γ) (( 1 2 I + K )µ + iηv µ + pinc )ˆn

56 Nonlocal BC s Method NLBC on Γ Weak Form 1 Wp s+ + ( 1 2 I + K )σ = 0 <, q > 0 = 0, q H 1/2 (Γ) 2(a) ( 1 2 I K )ps+ + V σ = 0 <, q > 1/2 = 0, q H 1/2 (Γ) 2(b) (same as above) <, Wq > 0 = 0, q H 1/2 (Γ) 3 Wp s+ + ( 1 2 I + K )σ = 0 <, q > 0 = 0, q H 1/2 (Γ) 4 Wp s+ + ( 1 2 I + K )σ <, q > 0 = 0, q H 1/2 (Γ) +iη(( 1 2 I K )p+ + V σ) = 0 5 W µ + σ = 0 <, ν > 0 = 0, ν H 1/2 (Γ) 6(a) ( 1 2 I K )φ σ = 0 <, ψ > 1/2 = 0, ψ H 1/2 (Γ) 6(b) (same as above) <, V ψ > 0 = 0, ψ H 1/2 (Γ) 7 W µ + iη( 1 2 I K )µ + σ = 0 <, ν > 0 = 0, ν H 1/2 (Γ)

57 Some Historical Remarks Exterior Boundary Value Problems Transmission Problems Concluding Remarks Concluding Remarks

58 Contact, crack and screen problems 2011: Gachechiladze, A., Gachechiladze, Gwinner, J., Natroshvili, D., Contact problems with friction for hemitropic solids: boundary variational inequality approach, Appl. Anal : Natroshvili, D. and Wendland, W. L., Boundary variational inequalities in the theory of interface cracks: In Operator Theory: Advances and Appl : Hsiao, G.C., Stephan, E. P. and Wendland, W. L., On the Dirichlet problem in elasticity for a domain exterior to an arc, ıj. Comp. Appl. Math : Han, H., A direct boundary element method for Signorini problems, Mathematics of Computation, : Han, H. and Hsiao, G.C., The boundary element method for a contact problem: In Theory and Applications of Boundary Elements, Q. Du and M. Tanaka eds., Tsinghua University Press, Beijing, : Stephan, E. P., Boundary integral equations for magnetic screens in R 3, Proc. Royal Soc. Edinburgh 102A

59 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. Method. Appl. Viscous Sci. 6 (1984) Flow Pattern Numerical experiments Viscous Flow Past an Obstacle & Absolute Error Estimate 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)

60 Hybride coupled finite-boundary element methods for Elliptic systems of second order, Comput. Methods Appl. High Stress Mech. Gradients Engrg. 190 (2000) 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)

61 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)

62 Fluid-Solid interaction Time-harmonic behaviour of the pressure Acoustic source: Plane incident wave from left ka = 1.0, R = 0.95λ

63 Real part of pressure Imag.part of pressure y 0 0 y x x 6 Real part of x coord.displacement 6 6 Imag.part of x coord.displacement y 0 2 y x x 5

64 Displacement Fields 6 Real part of x coord.displacement 6 6 Imag.part of x coord.displacement y 0 2 y x x 5 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).

65 Further references 1985: Gatica, N. G., and Hsiao, G. C., Boundary-field equation Methods for a Class of Nonlinear Problems, Pitman Research Notes in Mathematics Series 331, Longman 2004: Stephan, E. P., Coupling of Boundary Element Methods and Finite Element methods Chapter 13 in Encyclopedia of Computational Mechanics, Edited by Erwin Stein, René de Borst and Thomas J.R. Hughes. Volume 1: Fundamentals, pp c 2004 John Wiley & Sons, Ltd. ISBN: : Hsiao, G. C., and Wendland, W. L., Boundary Element Methods: Foundations and Error Analysis Chapter 12 in Encyclopedia of Computational Mechanics, Edited by Erwin Stein, René de Borst and Thomas J.R. Hughes. Volume 1: Fundamentals, pp c 2004 John Wiley & Sons, Ltd. ISBN: : Hsiao, G. C., and Wendland, W. L., Boundary Integral Equations, Applied Mathematical Series, 164, Springer-Verlag.

66 We ve-come-a-long-way to 2008!