FINITE ELEMENT SOLUTION OF SCATTERING IN COUPLED FLUID-SOLID SYSTEMS by Mirela O. Popa B.S., Harvey Mudd College, Claremont CA, 1995 M.S.

Transcription

1 FINITE ELEMENT SOLUTION OF SCATTERING IN COUPLED FLUID-SOLID SYSTEMS by Mirela O. Popa B.S., Harvey Mudd College, Claremont CA, 995 M.S., University of Colorado at Denver, 997 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 2002

2 This thesis for the Doctor of Philosophy degree by Mirela O. Popa has been approved by Jan Mandel Leopoldo P. Franca William L. Briggs Lynn S. Bennethum Charbel Farhat Date

3 Popa, Mirela O. (Ph.D., Applied Mathematics) Finite element solution of scattering in coupled fluid-solid systems Thesis directed by Professor Jan Mandel ABSTRACT In this thesis we investigate the mathematical theory of wave scattering by an obstacle. The obstacle considered is a bounded elastic body in a fluid domain. We analyse finite element methods and show existence and uniqueness of the solution for the coupled fluid-solid interaction problem in more than one dimension. We study the stability and regularity properties of acoustic wave scattering by introducing interpolation of spaces and scaled norms. This type of analysis, to the author s knowledge, has not been investigated. Then we consider a multigrid method for the coupled fluid-solid interaction model problem in higher dimension. We use the Gårding Inequality to obtain coercivity, then we use the Fredholm Alternative to analyze spectral properties and show uniqueness and existence of solutions. We need only weak regularity assumptions, and give a rigorous treatment of scale of spaces with constants independent of wave number k. A new approach is used to show stability of the coupled problem by using intermediate spaces and norms. iii

4 Finally, we present a multigrid algorithm to solve a coupled solidfluid interface problem. To the author s knowledge, multigrid methods for the coupled acoustic-elastic problem have not been investigated. In this thesis, we formulate such a method and present numerical experiments from a prototype implementation in MATLAB. This abstract accurately represents the content of the candidate s thesis. I recommend its publication. Signed Jan Mandel iv

5 DEDICATION To my family.

6 ACKNOWLEDGMENTS I would like to thank my advisor, Prof. Jan Mandel, for his support, guidance, and generosity. This research was supported by the National Science Foundation under grants ECS and DMS , and by the Office of Naval Research under grant N

7 CONTENTS Figures ix Tables xi. Introduction Existing Methods Theoretical Preliminaries Sobolev Spaces Finite Element Approximation Gårding Inequality Generalized Korn Inequality Fredholm Alternative Trace Theorem Riesz Representation Theorem Lax-Milgram Theorem Hilbert Interpolation Spaces Statement of Coupled Problem Derivation of the Coupled Problem Elastic Waves Acoustic Waves Boundary Conditions Solid-Fluid Interface Conditions vii

8 4.2 Variational Form of Coupled Problem Weak Formulation Hilbert scale Analysis Gårding Inequality for the Coupled Problem Existence of Solution Intermediate Spaces Intermediate norms Regularity of Solution for Coupled Problem Discretization and Error Bound Multigrid Method Multigrid for the Coupled Problem GMRES BICG-STAB Gauss-Seidel Numerical Results Numerical Verification of the Discretization Computational Results with Multigrid Conclusion and Future Work References viii

9 FIGURES Figure 4. Problem setup Configuration of sample points for Table 7.. Sample points,2,3,4 are fluid pressure values, and sample points 5 and 6 are u x and u y displacements, respectively Log log plot of difference of solution values x h at sample points for mesh size 0 0 to and extrapolated exact solution x, k= Solution for a mesh, k = 5, right-hand side modified for the Dirichlet boundary condition p = p 0 on Γ d Exact solution for a mesh, k = 5, right-hand side is modified for the Dirichlet boundary condition p = p 0 on Γ d Three multigrid iterations, 20 smoothing steps, smoother GM- RES, 2 levels to solve a mesh, k = 5, right-hand side is modified for the Dirichlet boundary condition p = p 0 on Γ d Contour of an obstacle (0.4 m) and (0.2 m) in the x and the y direction, respectively. The gap is on the y axis. The size of the gap is 0.5 or 50% in the x direction and 0.4 or 40% in the y direction ix

10 7.7 Contour of an obstacle (0.2 m) and (0.4 m) in the x and the y direction, respectively. The gap is on x axis. The size of the gap is 0.4 or 40% in the x and 0.5 or 50% in the y direction Solution for a mesh with ushaped obstacle (0.2 m) by (0.4 m). The obstacle has a gap of size 0.4 by 0.5 on the y axis. The wave number is k = 5, and the right-hand side is modified for the Dirichlet boundary condition p = p 0 on Γ d Solution for a mesh ushaped obstacle (0.2 m) by (0.4 m). The obstacle has a gap of 0.4 by 0.5 on the x axis. The wave number is k = 5, and the right-hand side is modified for the Dirichlet boundary condition p = p 0 on Γ d Residual reduction as a function of adding coarse meshes, decreasing mesh size h while keeping k 3 h 2 constant Residual reduction as a function of adding coarse meshes, decreasing mesh size h while keeping k 3 h 2 constant Residual reduction as a function of adding smoothing steps, decreasing mesh size h while keeping k 3 h 2 constant Residual reduction as a function of adding smoothing steps, decreasing mesh size h while keeping k 3 h 2 constant Relative residual varies as mesh size h is decreased for the case in which k 3 h 2 is constant and in the case of resonance Solution for a mesh, k = 4π, and right-hand side modified for the Dirichlet boundary condition p = p 0 on Γ d. A gap of half wavelength in x and y direction x

11 TABLES Table 7. Values of the solution at the 6 sample points as explained in (Fig.7.), mesh size h is halved at each run and wave number is kept constant k = Exact solution is x and solution at mesh size h is x h h = step size, k = wave number, sm = number of smoothing steps, lv = number of levels, mth = iterative method used for the smoothing algorithm, it = number of multigrid iterations, res red = residual reduction, rel rs = relative residual, res fl = residual in the fluid part, and res el = residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRES preconditioned by D (as explained in (7.5)), GA is Gauss-Seidel h = step size, k = wave number, sm = number of smoothing steps, lv = number of levels, mth = iterative method used for the smoothing algorithm, it = number of multigrid iterations, res red = residual reduction, rel rs = relative residual, res fl = residual in the fluid part, and res el = residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRES preconditioned by D (as explained in (7.5)), GA is Gauss-Seidel xi

12 7.5 Iteration counts for multigrid V cycle to achieve a relative residual of order e-6, for smoothers GMS=GMRES; BGS=BICG- STAB; GMD=GMRES preconditioned by the inverse of the lower triangular part of A h = step size, k = wave number, sm = number of smoothing steps, lv = number of levels, mth = iterative method used for the smoothing algorithm, it = number of multigrid iterations, res red = residual reduction, rel rs = relative residual, res fl = residual in the fluid part, and res el = residual in the elastic part. BC is BICG-STAB, GM is GMRES, GD is GMRES preconditioned by D (as explained in (7.5)) h = step size, k = wave number, sm = number of smoothing steps, lv = number of levels, it = number of multigrid iterations, res red = residual reduction, rel rs = relative residual, res fl = residual in the fluid part, and res el = residual in the elastic part h = step size, k = wave number, sm = number of smoothing steps, lv = number of levels, mth = iterative method used for the smoothing algorithm, it = number of multigrid iterations, res red = residual reduction, rel rs = relative residual, res fl = residual in the fluid part, and res el = residual in the elastic part. BC is BICG-STAB, GM is GMRES, GT is GMRES preconditioned by the inverse of the lower triangular part of A, GA=Gauss-Seidel. 04 xii

13 7.9 h = the step size, k = the wave number, sm = the number of smoothing steps, lv = the number of levels, sm =smoother, mth = the iterative method used for the smoothing algorithm, it = the number of multigrid iterations, res red = the residual reduction, rl res = the relative residual, rs fl = the residual in the fluid part, and rs el = the residual in the elastic part. BC is BICG-STAB, GT is GMRES preconditioned by the inverse of the lower triangular part of A h = the step size, k = the wave number, sm = the number of smoothing steps, lv = the number of levels, mth = the iterative method used for the smoothing algorithm, it = the number of multigrid iterations, res red = the residual reduction, rl res = the relative residual, rs fl = the residual in the fluid part, and rs el = the residual in the elastic part. BC is BICG-STAB, GT is GMRES preconditioned by the inverse of the lower triangular part of A h = the step size, x is size of obstacle in x direction, y size of obstacle in y direction, gap x is size of the gap on the x axis as a percentage, gap y is size of gap on the y axis, k = the wave number, rs red = the residual reduction, rl rs = the relative residual, rs fl = the residual in the fluid part, and rs el = the residual in the elastic part xiii

14 7.2 h = the step size, x is size of obstacle in x direction, y size of obstacle in y direction, gp x is size of the gap on the x axis as a percentage, gp y is size of gap on the y axis, k = the wave number, rs red = the residual reduction, rl rs = the relative residual, rs fl = the residual in the fluid part, and rs el = the residual in the elastic part h = the step size, k = the wave number, rel res = the relative residual, rel res fl = the relative residual in the fluid part, and rel res el = the relative residual in the elastic part GMRES preconditioned by ILU h = the step size, k = the wave number, rel res = the relative residual, res fl = the residual in the fluid part, and res el = the residual in the elastic part h = the step size, k = the wave number, conv fctr = convergence factor or the residual reduction, rel res = the relative residual, res fl = the residual in the fluid part, and res el = the residual in the elastic part h = the step size, k = the wave number, conv fctr = convergence factor or the residual reduction, rel res = the relative residual, res fl = the residual in the fluid part, and res el = the residual in the elastic part xiv

15 7.7 Residual Reduction per unit of work for one Multigrid cycle. h = the step size, x is size of obstacle in x direction, y size of obstacle in y direction, gp x is size of the gap on the x axis as a percentage, gp y is size of gap on the y axis, k = the wave number, rs red = the residual reduction, rl rs = the relative residual, rs fl = the residual in the fluid part, and rs el = the residual in the elastic part xv

16 . Introduction Time harmonic acoustics of coupled fluid-solid systems in fluid pressure and solid displacement formulation has many wide ranging applications. For this reason it has been extensively studied in recent years in an effort to predict the dynamic response of elastic objects immersed in fluids. Wave propagation in fluid, assuming time-harmonic behavior, is described by the Helmholtz equation, p + k 2 p = 0, (.) where the the critical parameter is the wave number k. In elastic media, waves propagate in the form of oscillations of the stress field, and are described by the elastodynamic Helmholtz equation, τ + ω 2 ρu = 0, (.2) where u is displacement, ρ is density of the elastic medium, and τ is the stress tensor. Wave propagation in a composite medium, consisting of a fluid part and an elastic part, is described by (.) and (.2), in the acoustic and in the solid part, respectively, together with boundary conditions imposed on the fluid solid interface. Important applications of elastic wave scattering are found in geophysical exploration, seismology, the response of structures to seismic waves,

17 acoustic response of elastic objects immersed in fluids, response of aircraft parts to dynamic loads, and applications of ultrasound to biological systems for diagnostic of therapeutic purposes [64, 69]. A significant growth in the literature employing scattering problems can be seen in recent years; [4, 48, 50, 34, 60, 59, 2]. The question of existence and uniqueness of solutions of Helmholtz problems was addressed by the end of the 950s; [46, 3, 58]. In this thesis, we analyze a time-harmonic solution of coupled fluidsolid interaction model problem in more than one dimension. We analyse finite element methods, show existence and uniqueness of the solution, and study its stability and regularity properties by introducing interpolation of spaces and scaled norms. We give a rigorous treatment of scale of spaces with constants independent of the wave number k. We present a multigrid method for the solution of linear systems arising from the finite element discretization of the coupled fluid-solid system. The obstacle considered is a bounded elastic body embedded in fluid. It is known, that numerical solutions to the Helmholtz equation deteriorate for increasing wave numbers k, [37, 36], which is called the pollution effect. The effect of pollution is that the wave number of the finite element solution is different from the wave number of the exact solution, and pollution means that the local error has a global effect, [4, 20]. In one dimension the pollution effect has been extensively studied, and different approaches have been proposed that lead to solutions that do not suffer the pollution effect, 2

18 [3, 4]. In two and three dimensions it has been shown that pollution cannot be avoided, [4]. In this thesis we will not be concerned with the pollution effect or dispersion, and we will keep in mind that for large wave numbers, numerical pollution in the error dominates the error of interpolation. Completely different methods are required to address the pollution effect. Iterative methods consisting of alternating solution in the fluid and the solid region are known [4, 50]. Numerical solution of elliptic partial differential equations, in two or three dimensions, is a typical application for iterative solvers based on the multilevel paradigm. In contrast to other methods, multigrid does not depend on the separability of the equations; thus using multigrid methods for solving the Helmholtz equation in modeling acoustic scattering is classical [25, 3, 63, 62]; for more recent developments, see [22, 44, 45, 62]. The remainder of the thesis is organized as follows. In Chapter 2, we give an overview of some existing methods and approaches to solving the problem of scattering by an elastic obstacle in an acoustic fluid. In particular, we describe the analysis of a fluid-solid interaction problem in one dimension studied by Makridakis et. al. [48]. In Chapter 3, we present some theoretical preliminaries, and describe notation we use later in this thesis. In Chapter 4, we present the relations of linear wave physics, derive the differential equations with an emphasis on the boundary conditions that describe the solid-fluid interaction, and then we define the spaces necessary to derive the weak formulation of the coupled problem. Chapter 5 is concerned with analyzing the existence 3

19 and uniqueness of the solution of scattering by an elastic obstacle in an acoustic fluid in a bounded region. We use the Gårding Inequality to obtain coercivity, then we use the Fredholm Alternative to analyze the spectral properties and show existence of solution. We show that if a solution exists, then the solution is unique. By using intermediate spaces, we show stability of the solution. In Chapter 6, we present a multigrid algorithm to solve a coupled solid-fluid interface problem. In Chapter 7, we present numerical experiments from a prototype implementation in MATLAB. 4

20 2. Existing Methods The scattering and propagation of waves is a classical area, which has engaged the interest of several generations of mathematicians, physicists, and engineers. Extensive research in the area of acoustics, electro-magnetics, and elastodynamics has resulted in many mathematical and computational techniques [5, 8, 47, 64, 68, 69, 70]. Bielak, MacCamy, and Zeng [8] studied acoustic scattering in an unbounded domain, representing the solution in the unbounded domain by a combination of potentials. Demkowicz [5, 6] considered elastic scattering in unbounded domains for spherical shells submerged in fluid, and showed that an inf-sup condition holds for the reduced problem on the boundary of the scatterer. In [2], Djellouli, Farhat, Tezaur, and Macedo apply a finite element method coupled with the Bayliss-Turkel-like non reflecting boundary condition to solve direct acoustic problems. Domain decomposition approaches to solving the Helmholtz equation are in papers by Hetmaniuk and Farhat [34], and Tezaur, Macedo and Farhat [59]. Analysis and domain decomposition methods for the time dependent coupled fluid-solid interaction problem are found in papers by Cummings and Feng [4, 24]. Mandel in [50] presents a domain decomposition method for solving the time harmonic coupled fluid-solid interaction 5

21 problem. Ohayon and Valid [55] present symmetric variational formulations for the problem of transient and modal analysis of bounded coupled fluid-structure linear systems, taking into account gravity and compressibility effects. Generalized added mass operators, independent of time and circular frequency, are introduced. Numerical results are presented for an incompressible hydroelastic modal analysis of a liquid propelled launch vehicle, elasto-acoustic modal analysis of an incompressible structure containing a compressible gas, and an elastic cylinder partially filled with liquid under gravity effects. In [22], Elman et. al. present a multigrid method enhanced by Krylov subspace iterations for the discrete Helmholtz equation. In this thesis, we extend the approach of [22] to the coupled problem. Makridakis et. al. [48] proved existence and regularity of the solution of elastic scattering in bounded domains in one dimension, using the Gårding inequality and an inf-sup condition, which leads to asymptotically optimal estimates. Our results extend [48] to more than one dimension; however, some tools which work in one dimension are not available here, so we do not obtain asymptotically optimal estimates. More details on the approaches outlined above follow. Bielak, MacCamy, and Zeng [8] discuss a scattering problem and its solution by a coupling method. Existence, uniqueness, convergence, as well as accuracy of the numerical approximations is also presented. This is accomplished by using potential theory and three different representations of the pressure p to solve the coupled fluid-solid problem. The domain is unbounded, 6

22 and it is separated into a bounded region Ω in R 3, with boundary Γ, and exterior Ω +. The domain Ω represents an inhomogeneous elastic obstacle and Ω + is a compressible, non viscous, homogeneous fluid. Three different coupling problems are presented, of which, the first two both fail for different sets of frequencies. The third coupling method is based on potential theory, and is shown to be stable. Another approach to the problem of scattering by an elastic obstacle in an acoustic fluid was presented by Demkowicz [5, 6]. The main difference between what Demkowicz did and this thesis is that Demkowicz investigates rigid scattering and vibrations of an elastic submerged shell. From the spectral decomposition of the operator for an elastic spherical shell in fluid, he computes the LBB constant as a function of the wave number k, and shows its effects on convergence. This work is motivated by the earlier paper [7, 8], where asymptotic convergence was studied for both finite and boundary element methods on acoustic problems, and a -D model acoustic interaction problem is presented. Elastic scattering problems investigated by Demkowicz in [5, 6] assume an elastic spherical shell freely floating in fluid. The spectral decomposition for this operator is computed, and finding the LBB constant reduces to solving a saddle point problem. Pointwise infimum of the spectral decomposition show dependence of the LBB constant on the wave number k. Demkowicz is able to prove that the magnitude of the LBB constant depends upon the distance from the nearest resonant frequency, and that without a 7

23 strict control of the discrete LBB constant during the solution process, the results may be unreliable. Cummings and Feng [4] present a domain decomposition method for the time dependent system of coupled acoustic and elastic interaction problems. Two classes of iterative methods are proposed for decoupling the domain problem into fluid and solid subdomains, and replacing the physical interface conditions with equivalent relaxation conditions as the transmission conditions. The nonoverlapping domain decomposition methods developed are regarded as Jacobi and Gauss-Seidel type algorithms, and they use convex combinations of the original physical interface conditions to transmit information between the subdomains. Strong convergence in the energy norm for the fluid-solid interaction problem is shown for the iterative methods, and their findings are supported by numerical test results. Feng [24] analyzes some finite Galerkin approximations for the time dependent fluid-solid interaction model, and presents a domain decomposition method for the time dependent system of coupled elastic acoustic problem. An optimal order apriori error estimates in L (H )-norm and in L (L 2 )-norm for the semi-discrete and fully discrete Galerkin approximation to the solution of the model is established. Higher order time derivatives of the errors are used for the error estimates of the interface conditions describing the interaction between the fluid and the solid. To handle the terms involving the interface conditions, the boundary duality argument due to Douglas and Dupont [38] 8

24 is used. The error estimates for the fully discrete methods are obtained by averaging error equations at different time steps and choosing some nonstandard test functions. Feng defines a second order discrete-time Galerkin method, and presents a generalized parallelizable nonoverlapping domain decomposition method for solving the coupled fluid-solid interaction problem. Another domain decomposition method for time harmonic coupled fluid-solid systems that decomposes the fluid and solid domains into nonoverlapping subdomains is presented by Mandel in [49]. The continuity of the solution uses Lagrange multipliers to ensure that the values of the degrees of freedom coincide on the interface between the subdomains. This method is known as the FETI-H domain decomposition method originally proposed by Farhat and Roux [23]. Mandel finds that the division into subdomains does not need to match across the wet interface. The system is augmented by duplicating the degrees of freedom on the wet interface, and the original degrees of freedom are eliminated. The intersubdomain Lagrange multipliers and the duplicates of the degrees of freedom on the wet interface are retained and form the reduced problem. The resulting system is solved by iterations preconditioned by a coarse space correction. Numerical results are presented for a bounded 2D region. Elman et. al. [22] study the exterior Helmholtz problem in an unbounded domain. The unbounded domain is truncated to a finite domain 9

25 by introducing an artificial boundary on which the radiation boundary condition approximates the outgoing Sommerfeld radiation condition. In this paper multigrid methods are used to solve the discretized Helmholtz equation. The authors identify difficulties arising in a standard multigrid iteration for the Helmholtz equation, and analyze and test techniques designed to address these difficulties. Some difficulties encountered are with the smoothing and coarse grid corrections. Standard smoothers such as Jacobi and Gauss-Seidel relaxation become unstable for indefinite problems, since there are error components that are amplified by these smoothers. The difficulties with the coarse grid corrections are due to the poor approximation of the Helmholtz operator on very coarse meshes. The approach used in [22] for smoothing is to use standard damped Jacobi relaxation when it works reasonably well, on fine enough grids, and then, to replace it with a Krylov subspace iteration when standard damped Jacobi fails as smoother. For the coarse grid correction, the number of eigenvalues that are handled poorly during the correction is identified, and an acceleration for multigrid is introduced by using multigrid as a preconditioner for an outer Krylov subspace iteration. The authors observe that multigrid does a poor job of eliminating some modes from the error, so it converges slowly or even diverges in some cases, and an outer Krylov subspace iteration is needed for the method to be robust. GMRES is used as the Krylov subspace method. In this thesis we extended the approach of [22] to the coupled problem. 0

26 The analysis for a fluid-solid interaction problem in one dimension done by Makridakis, Ihlenburg, and Babuška [48], is the closest to the analysis presented in this thesis. The approach taken in [48] focuses on the stability of the continuous problem (2.) and on the stability and convergence of the discrete problem (2.9) with respect to the wave number k. The problem is a D layered fluid-solid-fluid medium with configuration Ω = Ω Ω 2 Ω 3 = [0, L], L > 0, p xx + k 2 p = g, in Ω (au x ) x + k 2 gu = f, in Ω 2 (2.) p xx + k 2 p = g 2, in Ω 3 with radiation boundary conditions at the boundary of Ω, p x (0) + ikp(0) = 0, p x (L) ik p ( L) = 0, transmission conditions on the interface boundary at x, p x (x ) k 2 u(x ) = 0, p(x ) + au x (x ) = 0, and at x 2, p x (x 2 ) k 2 u(x 2 ) = 0, p(x 2 ) + au x (x 2 ) = 0.

27 The space considered is H = H (Ω ) H (Ω 2 ) H (Ω 3 ). With the weak formulation, we find U = (p, u, p) H such that B(U, V ) = (F, V ) 0,H V H (2.2) where V = (q, v, q) H, and B(U, V ) = p x q x dx k 2 p qdx k 2 u(x ) q(x ) ikp(0) q(0) Ω Ω +k 2 au x v x dx k 4 gu vdx k 2 p(x ) v(x ) + k 2 p(x 2 ) v(x 2 ) (2.3) Ω 2 Ω 2 + p qdx + k 2 u(x 2 ) q(x2 ) ik p(l) q(l) Ω 3 p x qx dx k 2 Ω 3 (F, V ) 0,H = (g, q) + k 2 (f, v) + (g 2, q). (2.4) The standard definition for the inner product is used, (u, v) = Ω i u v dx. With a suitable choice of norms, (U, V ) 0,H = (p, q) + k 2 (u, v) + ( p, q), (2.5) (U, V ),H = (U x, V x ) 0,H + k 2 (U, V ) 0,H, (2.6) it was shown using variational techniques, that if the right-hand side F L 2, then the solution satisfies a regularity estimate of the form U,H C F 0,H, (2.7) where C is a constant independent of k. The estimate (2.7) establishes the uniqueness of the solution of (2.2), and is needed in the analysis of the discrete 2

28 problem. Uniqueness combined with the fact that the variational form satisfies a Gårding type inequality, yields the existence of the solution U. Using an estimate of the form (2.7) for a properly chosen auxiliary problem, the Babuška- Brezzi condition for the bilinear form B(, ) is proved: ReB(U, V ) sup γ V H v,h k U,H, U H. (2.8) This approach follows earlier Babuška work relying on the LBB constant. LBB condition (2.8) is equivalent to a bound on the norm of the solution operator, which depends on k linearly. Using (2.7) and (2.8), other similar regularity estimates are derived. These estimates are useful in the convergence analysis of the numerical methods. If S h is a suitable finite element subspace of H consisting of piecewise polynomial functions, approximation U h S h to U is defined as the solution of B(U h, φ) = (F, φ) 0,H, φ S h. (2.9) In [48], the authors showed the existence of a unique solution of (2.9), provided the quantity h d k2 is small enough, where h is the maximum mesh size and d is the polynomial degree of the functions of S h. It is also shown that the discrete analog of the LBB condition (2.8) is satisfied on S h. The authors used (2.7) to show that an optimal estimate of the form U U h,h C inf φ S h U φ,h, (2.0) 3

29 holds, where C is a positive constant independent of h and k. The approximation properties of S h are known, thus (2.0) implies convergence of optimal order of the finite element approximations in the,h norm. In D, the final estimate has no dependence on wave number k. In more than D, the equivalent of (2.8) does not hold; instead we use estimates and FEM error bounds that rely on the Gårding inequality. We have dependence on wave number k, because the estimate relies on the boundedness of (I βk 2 G), where β is a constant independent of k and G is a compact operator with finite but unknown norm. 4

30 3. Theoretical Preliminaries In this chapter, we describe notation and summarize some concepts and algorithms used throughout this thesis. Let Ω be a bounded open connected domain in R n, n, 2, 3, with Lipschitz-continuous boundary Γ. Let Ω = Γ d Γ n with Γ d, Γ n disjoint, and meas(γ d ) > 0. We will denote by Γ d, Γ n the parts of the boundary with Dirichlet and Neumann boundary conditions, respectively. 3. Sobolev Spaces Let W q p (Ω) be the usual Sobolev space of complex valued functions W q p (Ω) = { f L loc : f W q p (Ω) < }, where q is a non-negative integer, and f W q p (Ω) = ( α q Dwf α p /p L (Ω)). p We use the multi-index notation α for denoting partial derivatives by an n- tuple with non-negative integer components, α = (α,..., α n ), with the length of α given by α = n i= α i. For p = 2, the space Wp q (Ω) is a Hilbert space, and we denote W q 2 (Ω) by H q (Ω). For q =, we obtain the Hilbert space H equipped with the norm f H = ( f 2 L 2 (Ω) + /2 f 2 L (Ω)). In addition to 2 integer-order Sobolev spaces, there are fractional-order Sobolev spaces. For 5

31 s R, 0 < s <, p <, and q 0, we recall u p := Wp q+s (Ω) u p Wp q (Ω) + with the seminorm [54] α =q Ω Ω u (α) (x) u (α) (y) p x y n+sp dxdy, u p := Wp q+s (Ω) α =q Ω Ω u (α) (x) u (α) (y) p x y n+sp dxdy, where Ω R n. For more details, see [2, 0, 54]. 3.2 Finite Element Approximation The finite element method (FEM) is a general technique for the numerical solution of partial differential equations in structural engineering. From the engineering point of view, the method was thought of as a generalization of earlier methods in structural engineering for beams, frames, and plates, where the structure was subdivided into small parts, called finite elements, with known simple behavior. The presentation here follows [0], where more details can be found. To fix ideas, as a simple example, consider the second order elliptic equation with Dirichlet boundary condition, Au = f in Ω (3.) u = 0 on Ω, where Ω is a Lipschitz domain in R 2 or R 3, and n Au(x) := i,j= x j ( a ij (x) u ) (x) x i + n k= b k (x) u x k (x) + b 0 (x)u(x) (3.2) 6

32 with the matrix [a(i, j)] symmetric, positive definite, and bounded on Ω. Multiplying by a test function v V (Ω), where the Sobolev space V (Ω) = H (Ω), is a given set of admissible functions, and integrating by parts, the model problem (3.) can be written in the variational form, Find u V (Ω) such that a(u, v) = F (v) for all v V (Ω), (3.3) where F : V R is a functional with F (v) = (f, v), and the bilinear form a(, ) : V V R given by a(u, v) := Ω n i,j= a ij u x i v x j + n k= b k u x k v + b 0 uv dx (3.4) is defined for all u and v in the Sobolev space V (Ω) = H (Ω). The functions v V (Ω) usually represent a continuously varying quantity, such as a displacement in an elastic body, and F (v) is the total energy associated with v. In general, the functions in V (Ω) cannot be described by a finite number of parameters, and so the problem cannot be solved directly. The standard Galerkin approximation approach is to look for an approximate solution of (3.3) in a finite dimensional subspace V h (Ω) of the space V (Ω), in which the weak form is posed. The space V h (Ω) consists of simple functions only depending on finitely many parameters, usually chosen to be piecewise polynomials. This leads to a finite-dimensional problem. The Galerkin approximation is the solution to the following problem: Find u h V h (Ω) such that a(u h, v h ) = F (v h ) for all v h V h (Ω). (3.5) 7

33 When a basis is chosen for V h, the Galerkin approximation leads to a system of equations. Let {φ i } N i= be a basis for V h (Ω). Assuming that u h = equation (3.5) becomes N u i φ i, i= N a(φ i, φ j )u i = (f, φ j ), j =,..., N. i= The left-hand side matrix K = [a(φ i, φ j )] is called the stiffness matrix, the right-hand side vector [(f, φ j )] is called the load vector, and the vector of unknowns [u i ] is referred to as the vector of degrees of freedom. For a wide class of approximation spaces V h (Ω), u h is a good approximation for u. The choice of the finite dimensional subspace V h (Ω) is influenced by the variational formulation, accuracy requirements, and regularity properties of the exact solution. The construction of suitable spaces V h uses triangulation, which splits the domain Ω into small disjoint simple geometries. In R 2, triangular and rectangular shapes are considered, and for R 3, tetrahedrons and hexahedrons are used. By imposing certain assumptions on the triangulation, the finer the triangulation, the closer a finite element Galerkin solution is to the exact solution. As in the example above, functions in V h arise from a polynomial interpolation on the elements of the triangulation, and are generated by basis functions that are usually polynomial on each element of the triangulation of the domain. The supports of basis functions have only small overlaps. Every 8

34 polynomial defined on a given region is uniquely determined by its values and perhaps the values of its derivatives at some nodal points. Thus, each function in V h is determined by a set of values at nodal points, so called degrees of freedom. Simple examples of finite element spaces include spaces formed by continuous linear or triangle regions in R 2, or tetrahedrons in R 3. These spaces are referred to as P and Q respectively. In this thesis we use standard Q finite element spaces. For finite element spaces, the standard basis functions chosen are the continuous piecewise linear functions that take the value at one node point and the value 0 at other node points. Thus, the unknowns in the linear system arising from the discretization are the degrees of freedom of the Galerkin approximation. Since the overlaps of the supports of the basis functions are small, the stiffness matrix is sparse. 3.3 Gårding Inequality A bilinear form a(, ) on a normed linear space H, is said to be bounded (or continuous), if c < such that a(u, v) c u H v H u, v H, and coercive on V H if c 2 > 0 such that a(v, v) c 2 v 2 H v V. (3.6) There can be well-posed elliptic problems (3.) for which the corresponding variational problem (3.3) is not coercive, although a suitably large additive 9

35 constant can always make it coercive as follows. By Gårding inequality there is a constant, κ <, such that a(v, v) + κ v 2 L 2 (Ω) α v 2 H (Ω), (3.7) for some α > 0, where a(, ) is defined in equation (3.4). 3.4 Generalized Korn Inequality Korn s inequality plays an important role in establishing the existence and uniqueness of a solution in linearized elasticity, and it is used to establish coercivity of the operator. Let Ω R n be a domain with Lipschitz boundary. There exists a constant C k > 0, such that for all u (H (Ω)) n, [40, 53] e(u) 2 (L 2 (Ω)) + n n u 2 (L 2 (Ω)) C k u 2 n (H (Ω)) n, (3.8) where e(u) = 2 ( u + ( u)t ) is the strain tensor (see section 4.). 3.5 Fredholm Alternative Compactness of a linear operator is essential in Fredholm s theory. For X and Y normed spaces, an operator T : X Y is called a compact linear operator if T is linear, and if for every bounded subset M of X, the closure T (M) is compact, i.e. every sequence in T (M) has a convergent subsequence whose limit is an element of T (M). A bounded linear operator A : X X on a normed space X is said to satisfy the Fredholm alternative [42] if A is such that either (I) or (II) holds: (I) The nonhomogeneous equations Ax = y, A f = g, have solutions x and 20

36 f, respectively, for every given y X and g X the dual space of X, the solutions being unique. (II) The homogeneous equations Ax = 0, A f = 0, have the same number of linearly independent solutions x,..., x n and f,..., f n, respectively. The nonhomogeneous equations Ax = y, A f = g, have a solution if and only if y and g are such that f k (y) = 0, g(x k ) = 0 (k =,..., n), respectively. The particular important case is that in which both Ax = 0 and Af = 0 admit only the trivial solution; then there is a solution of Ax = y for any y Y. Compact operators and the Fredholm alternative theory are related by the following result. Let T : X Y be a compact linear operator on a normed space X, and let λ 0. Then T λ = T λi satisfies the Fredholm alternative. 3.6 Trace Theorem Let Ω be a bounded open set of R n with a Lipshitz boundary Ω and Γ a subset of the boundary. Consider the Sobolev space H m (Ω); the trace γu = u Γ of a function u H m (Ω) on Γ is a bounded linear operator, and is defined as a restriction of the function u to the boundary γ : H m (Ω) H m /2 (Γ), m > [27]. There is a constant C 2 Γ(m), such that u Γ H m /2 (Γ) C Γ u H m (Ω). (3.9) 2

37 3.7 Riesz Representation Theorem Any continuous linear functional F on a Hilbert space H, can be represented uniquely in terms of the inner product, F (x) = x, z, where z depends on F, is uniquely determined by f, and has norm z H = F H. In general, let H, H 2 be Hilbert spaces and, h : H H 2 K a bounded sesquilinear form. Then h has a representation (x, y) = Sx, y, where S : H H 2 is bounded linear operator, uniquely determined by h, and has norm S = h. 3.8 Lax-Milgram Theorem Given a Hilbert space (V, (, )), a continuous, coercive bilinear form a(, ) and a continuous linear functional F V, there exists a unique u V such that a(u, v) = F (v), v V. Since the bilinear form a(, ) is coercive, we have the estimate u V c 2 F V, where c 2 is the coercivity constant (3.6). 3.9 Hilbert Interpolation Spaces There are many ways in which one can define an interpolation space, with the most common methods being the real and the complex interpolation methods. We will describe the complex interpolation method, and we will follow [4], for the real interpolation method [0, 6]. Here and in section 5.2, we use the notation X a X b to denote the continuous embedding of X a in X b (i.e. convergence in X a strongly implies convergence in X b strongly), [58]. Similarly, we let X a c X b denote compact 22

38 embedding of X a in X b (i.e. the embedding operator is compact). We are interested in the interpolation theory of Hilbert spaces. Given two Hilbert spaces X 0 and X X 0 with inner products (, ) X0, (, ) X and norms X0, X, respectively, we will define Hilbert spaces that interpolate between them. The spaces X θ = [X 0, X ] θ, 0 < θ < are called interpolation spaces between X 0 and X. Let an unbounded positive definite self-adjoint operator A exist in X 0, with dense domain D(A) = X, such that u X = Au X0, u D(A). The existence of operator A follows from the Riesz representation theorem. For 0 < θ <, the set [X 0, X ] θ = X θ := { u X 0 : u [X0,X ] θ < } (3.0) forms a Hilbert scale of spaces (X θ, Xθ ) given by X θ = D(A θ ), with norm u [X0,X ] θ = A θ u X0. The following theorem is known as the Convexity Theorem, and will be used later. Theorem 3. [0] Suppose that X i and Y i (i = 0, ) are two pairs of Banach spaces, and that A is a linear operator that maps X i to Y i. Then A maps X θ to Y θ for 0 < θ <. Moreover, A Xθ Y θ A θ X 0 Y 0 A θ X Y. (3.) 23

39 4. Statement of Coupled Problem 4. Derivation of the Coupled Problem In this section, we summarize some of the basic relations of linear wave physics, starting with elastic waves, proceeding to acoustic waves, and then fluid-solid interaction. Derivation of the coupled problem is standard, and this section is intended for completeness only. We will derive the differential equations and the boundary conditions that describe the fluid-solid interaction. We are interested in the time-harmonic case, and we assume that all waves are steady-state with circular frequency ω. An acoustic wave that is incident onto an elastic obstacle is not totally reflected, and part of the incident energy is transmitted in the form of elastic vibrations. Basic ideas regarding the nature of the elastic field were established long before the application of elasticity theory [64]. We will use the word field to refer to any physical quantities that vary in space and time. For time harmonic fields, the scalar and vector potentials satisfy the scalar and vector Helmholtz equations. 4.. Elastic Waves In an elastic medium, waves propagate in the form of small oscillations of the stress field. In a solid elastic material, the physical quantities that are of interest are the displacement, the stress, the 24

40 strain tensors and, if any are present, the body forces. The speed of propagation is usually denoted by c. The governing equations for the elastic medium are obtained from the basic relations of continuum mechanics. Consider the momentum equation (ρv) t + (ρvv) τ = f. Assuming no external forces, the momentum equation becomes (ρv) t + (ρvv) τ = 0. Assume that the density, ρ(x, t), and the velocity, v(x, t), undergo small fluctuations about equilibrium values, ρ 0 and v 0, respectively. The domain as a whole is assumed to be at rest (i.e. v 0 = 0). We linearize the momentum equation by writing v = ṽ + v 0 ρ = ρ + ρ 0 where ṽ, ρ represent small perturbations about the equilibrium values. The term ρvv can be linearized as ρvv = ρ(ṽ + v 0 )(ṽ + v 0 ) = ρ(ṽṽ + ṽv 0 + v 0 ṽ + v 0 v 0 ) 0, 25

41 since ṽṽ is negligible and v 0 = 0. Similarly, ρv = ( ρ + ρ 0 )(ṽ + v 0 ) = ρṽ + ρv 0 + ρ 0 ṽ + ρ 0 v 0 ρ 0 ṽ, since ρṽ is also assumed negligible. The momentum equation then becomes (ρ 0 v) t τ = 0, (4.) where we have dropped the tilde notation for simplicity. Since ρ 0 is the constant equilibrium value, equation (4.) becomes ρ 0 v t τ = 0. (4.2) To linearize equation (4.2) and write it in terms of displacement we assume small oscillations. The partial derivative,, is related to the total derivative t d dt by the nonlinear expression [43] d dt = t + v. If u(r, t) is the vector field of particle displacement at position r, then in order to formulate equation (4.2) in terms of displacement, we write v(r, t) = du (r, t) dt = u t + v u u t, 26

42 where u is assumed to be of the same order as v so that v u is negligible. So, v t = 2 u t, (4.3) 2 and we obtain the linearized form of the momentum equation in terms of displacement 2 u ρ 0 τ = 0. (4.4) t2 For wave propagation, it is generally assumed that a time-dependent scalar field F (x, t) can be separated as F (x, t) = f(x)e iωt, where f is a stationary amplitude function. A similar convention holds for vector fields. Assuming a time harmonic solution to equation (4.4) and letting u(x, t) = û(x)e iωt, we obtain ω 2 ρû + τ = 0, (4.5) where τ is the stress tensor. The symmetric stress tensor τ (x) is also known as the Cauchy stress tensor at the point x Ω e. With the assumption of small deformations, the strains are related to the displacements by the linearized equations, and are defined by e(u) = 2 ( u + ( u)t ). (4.6) 27

43 By Generalized Hooke s Law, stress is a linear function of strain, where the strain assumes small displacements. We then have τ ij = C ijkl e kl, (4.7) where C ijkl is a fourth order elastic stiffness tensor which, for an isotropic medium, is invariant under rotations and reflections, so that it takes on the form C ijkl = λδ ij δ kl + µδ jl δ jk (4.8) where λ, µ are the Lamé coefficients. Substituting equation (4.8) into (4.7) we get τ ij = λδ ij e kk + 2µe ij = λδ ij k u k + µ( i u j + j u i ). Thus we obtain the equations governing the elastic medium ω 2 ρu + τ = 0 τ = λ I( u) + 2µe(u), where τ is the stress tensor, e(u) is the strain tensor, u is displacement, ρ is density of the elastic medium, and λ, µ are the Lamé coefficients of the elastic medium Acoustic Waves Acoustic waves (sound) are small oscillations of pressure in a compressible ideal fluid (acoustic medium) [35], and are 28

44 associated with local motions of the particles of the fluid and not with bodily motion of the fluid itself. The difference between fluids and elastic solids is that fluids cannot support shear stresses in the absence of internal friction. In viscous fluids, frictional forces are generated when gradients in velocity are present. Here we consider inviscid media, so the fluid cannot support any shear forces. We assume that the fluid is compressible, i.e. the density of the fluid changes as a consequence of flow processes. The equations governing the acoustic medium are obtained from fundamental laws for compressible fluids. In this section, the differential equations governing acoustic wave propagation in a liquid or gaseous medium are described following the derivation in [65]. As with the derivation of the differential equations governing the elastic medium, we assume that the density, velocity, and pressure, undergo small fluctuations about their respective mean variables: ρ 0, v 0, and p 0. We also assume that the fluid as a whole is at rest, i.e. v 0 = 0. The simplest constitutive equations encountered in continuum mechanics are those for an ideal fluid, where the pressure field is isotropic and depends only on density and temperature; then the stress field τ is represented by τ = p(ρ, T )I, (4.9) where ρ is the density of the fluid and T is the absolute temperature. Substituting equation (4.9) into the linearized momentum equation (4.2) we 29

45 obtain ρ 0 v t Taking the curl of equation (4.0), we see [65] ( pi) = 0. (4.0) v = 0 v = Φ, (4.) where Φ is a scalar field called the velocity potential. Equation (4.) holds because of the assumption of a nonviscous fluid. Using equation (4.) in equation (4.0), we obtain p = ρ 0 Φ t. (4.2) From thermodynamics, we can write the pressure as a function of entropy and density. We assume that the acoustic wave propagation is an adiabatic process at constant entropy, and that the changes in density are small. Then using the Taylor series representation, we have p = p 0 + ( ) p dρ p = ρ S ( ) p dρ, (4.3) ρ S where p = p 0 + p. We define the adiabatic compressibility κ S via ( ) p = c 2 =, (4.4) ρ κ S S ρ 0 where c is the speed of the acoustic waves and depends on material properties. Using equation (4.3), (4.4), and (4.2) in equation (4.0), we obtain the time dependent wave equation for the velocity potential 2 Φ 2 Φ = 0. (4.5) c 2 t2 30

46 With the assumption of time-harmonic waves of frequency ω, the wave equation (4.5) can be transformed to the scalar Helmholtz equation 2 Ψ c 2 i2 ω 2 Ψ = 0, or 2 Ψ + k 2 Ψ = 0 (4.6) where Ψ is amplitude, i = is the imaginary unit, and k = ω c. (4.7) The physical parameter k is called the wave number. The physical interpretation of the parameter k is the number of waves per 2π units. Hence, k characterizes the oscillatory behavior of the exact solution. The larger the value of k, the greater the spacial frequency of the waves Boundary Conditions In order to set up a well-posed problem and to model the physics, the equations need to be complemented by boundary conditions. The most common boundary conditions imposed on a model problem are the Dirichlet and Neumann boundary conditions. Holding the pressure constant on the boundary is an essential, or Dirichlet, boundary condition of the form p = p 0. For a rigid surface, Neumann boundary conditions are imposed, p n = 0, where n is the outer unit normal to Ω. This condition models a free boundary, i.e., no external forces are acting at the boundary [58]. The physical requirement that all radiated waves are outgoing leads to the Sommerfeld radiation condition, which can be interpreted as a boundary condition at infinity. The Sommerfeld condition is usually replaced by its 3