Computer solution of fluid-structure vibrations on unbounded domains^{∗}

Tomás Barrios and Rodolfo Rodríguez

Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Abstract. The aim of this research is to obtain an efficient code to compute the steady state vibrations of an elastic object submerged into a compressible fluid, in case that the fluid domain is unbounded. An example of application is the computation of scattered acoustic waves typically used for underwater detection by means of sonar (see for instance [1]). The submerged object is assumed to be linearly elastic and is described by means of its displacement field, discretized by piecewise linear continuous elements. The fluid is supposed inviscid, barotropic and compressible; it is described by means of a scalar displacement potential discretized with the same type of elements. The radiation at infinity, as given by Sommerset condition, is taken into account by means of a DtN (Dirichlet-to-Neumann) operator. To do this, an artificial circular boundary containing the whole elastic object is introduced and the DtN operator is approximated by using a truncated Fourier expansion in the eigenmodes of the exterior problem (see for instance [2]). Numerical results validating the developed code are presented.

Keywords— Fluid-structure interaction, Helmholtz problem, coupled DtN/FEM methods.

References

[ 1] M. Lenoir M. Vullierme-Ledard and C. Hazard, Variational formulation for the determination of resonant states in scattering problems SIAM Journal on Mathematical Analysis 23, 579-608, (1992).

[ 2] D. Givoli, Numerical Methods for Problems in Infinite Domains. Studies in Applied Mathematics 33, Elsevier, 1992.

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through Program A on Numerical Analysis.

A parallel domain decomposition scheme for Fredholm integral equations^{∗}

Manuel C. Campos and Carlos E. Pérez

Departamento de Ingeniería Matemática, Universidad de Concepción Casilla 160-C, Concepción, Chile.

Abstract. We use a domain decomposition technique to show that the solution of some Fredholm integral equations of the second kind can be obtained by solving a system of equations, of the same type, but defined on subdomains. In order to solve the system we use a parallelizable iterative scheme and we give a suficient condition for its convergence. The system is discretized by Nyström method and convergence of the numerical procedure is also considered. Finally, we apply the corresponding parallel scheme to some model problems and show the performance results obtained on a parallel computer Origin 2000 with 6 processors. Keywords— Parallel Processing, Fredholm Integral Equations, Nyström Methods, Domain Decomposition Methods.

References

[ 1] R. Aggarwal, D. Dellwo and M. Friedman, Parallel solution of Fredholm integral equations of the second kind by accelerated projection methods. Parallel Computing 19, 1005-1115, (1993).

[ 2] Ph. Anselone, Collectively Compact Operator Approximation Theory and

Applications to Integral Equations. Prentice Hall, Englewood Cliffs, (1971). [ 3] K. Atkinson, The Numerical Solution of Integral Equations of the Second

Kind. Cambridge University Press, (1997). [ 4] R. Babolian and L. Delves, Parallel solution of Fredholm integral equations.

Parallel Computing 12, 95-106, (1989). [ 5] M. Campos and C.E. Pérez, Tratamiento numérico de ecuaciones integrales

mediante técnicas de separación de dominios. Technical Report 97-15, Departamento de Ingeniería Matemática, Universidad de Concepción, (1997).

[ 6] M. Campos and C.E. Pérez, A domain decomposition scheme for Fredholm integral equations using Nyström discretization. Submitted to Parallel Computing, (1998).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis and by the Dirección de Investigación de la Universidad de Concepción through the research project 94.13.04-1.

Nyström and Galerkin separation techniques for Fredholm integral equations of the second kind^{∗}

Manuel C. Campos and Carlos E. Pérez

Departamento de Ingeniería Matemática, Universidad de Concepción. Casilla 160-C, Concepción, Chile.

Abstract. We present the Domain Separation technique as a procedure that permits to work on subsets of the original nodes set in the discretized integral equation. In order to achieve this, it is necessary to define projection and restriction operators between the originally discretized problem and the new lower-size subproblems. The scope of this approach is to get always lower-size problems and parallelism for the resolution of the integral equations. Also, we show numerical results with Nyström quadrature and some projection operators arising in the Galerkin method for integral equations. Keywords— Integral Equations, Nyström Method, Galerkin Method, Multigrid Techniques, Domain Reduction Method.

References

[ 1] M. Campos and C. Pérez, Paralelización de técnicas para ecuaciones integrales de Fredholm de segunda especie por métodos de cuadratura y multimallas.Technical Report 97-09, Departamento de Ingeniería Matemática, Universidad de Concepción, (1997).

[ 2] M. Campos and C. Pérez, Tratamiento numérico de ecuaciones integrales mediante técnicas de Separación de Dominios. Technical Report 97-15, Departamento de Ingeniería Matemática, Universidad de Concepción, (1997).

[ 3] M. Campos and C. Pérez, A Domain Decomposition scheme for Fredholm integral equations using Nyström discretization. Submitted to Parallel Computing (1998).

[ 4] J. Xu, Theory of Multilevel Methods. Ph. D. Thesis, Cornell University,

(1989). [ 5] C. Douglas and W. Miranker, Constructive interference in parallel algorithms.

SIAM Journal on Numerical Analysis 25, 376-398, (1988).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis and by the Dirección de Investigación de la Universidad de Concepción through the research project 94.13.04-1.

On stationary model of magneto-micropolar fluid in exterior domains^{∗}

Mario Durán

Facultad de Matemáticas, Pontificia Universidad Católica, Casilla 306, Santiago 22, Chile

Abstract. As in many non-bounded physical situations, the study of the dynamics of the magnetomicropolar fluid model considered on an unbounded domain plays an important and useful role. We often find solid-fluid structures in which a bounded obstacle stops or impedes the flow of the surrounding fluid and the spatial volume of the external environment of the body, namely exterior domain, is extensively much larger than the obstacle. ¿From the modeling point of view, this case may be regarded as a compact domain located in all of IR3. Let K denote this compact subset, and let Ω denote its complement in IR3, that is, Ω = Kc. We deal with the existence of a weak solutions for equations that describe the motion of a viscous incompressible magneto-micropolar fluid in the exterior domain Ω (see [1]). Such mathematical model (e.g., see [2]) reads: Find the three-dimensional fields (u, w, h) : Ω → IR9 and the scalar functions (p, q) : Ω → IR2 which satisfy the system of equations:

−(μ + χ)Δu + (u · ∇)u + ∇(p + \frac{r}{2}|h|^{2}) = χ rot w + r (h · ∇)h + f in Ω , (1.1)
−γΔw − (α + β)∇ div w + j (u · ∇)w + 2χw = χ rot u + g in Ω ,  (1.2)
−νΔh + (u · ∇)h − (h · ∇)u + ∇q = 0 in Ω ,  (1.3)
div u = 0, div h = 0 in Ω ,  (1.4)
u(x) = w(x) = h(x) = 0, on ∂Ω .  (1.5)

Here u(x), w(x), h(x) ∈ IR3 denote, respectively, the velocity, the microrotational velocity, and the magnetic field of the fluid at point x ∈ Ω, and p(x), q(x) ∈ IR denote the hydrostatic and magnetic pressures at the same place. The values μ, χ, r, α, β, γ, j and ν are constants associated to properties of the material and f(x), g(x) ∈ IR3 are given external fields. To complete the system of equations, we establish the behaviour of the solution at infinity. We consider the classical homogeneous decay

lim_{|x|→∞} u(x) = lim_{|x|→∞} w(x) = lim_{|x|→∞} h(x) = 0 . (1.6)

It is important to remark that our case involves solutions in the space J0(Ω), defined as

J_{0}(Ω) = Completion of C_{0,σ}^{∞}(Ω) in norm ∥∇ϕ∥_{L^{2}(Ω)} ,

hence no weighted spaces are required.

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by the Chilean Grants: # 1971056 (Fondecyt), # 97/04E (Dipuc) and # C-13222/7 (Fundación Andes).

Magneto-Micropolar Fluids

Equation (1.1) has the familiar form of the Navier-Stokes equations though coupled with (1.2) and (1.3). Equation (1.2) describes the motion inside the macro-volumes as they undergo micro rotational effects, which are represented by the micro rotational velocity vector w. For fluids with no micro structure, this velocity vanishes and we deal with a magnetohydrodynamics system. For Newtonian fluids, where χ = 0, equation (1.1) decouples from equation (1.2). Equation (1.3), which is the equation for h, is none other than the Maxwell system in which the electrical field is determined in a posteriori way. It is also important to note that if h = 0, we consider the well known stationary asymmetric fluid model. In this lecture, to prove existence of weak solutions we use “the extending domain method”as in Ladyzhenskaya[3] and Heywood [4]. We also discuss the uniqueness of solutions. Thus we study weak solutions the same way as in the classical study of the Navier-Stokes equations. Finally, we propose a scheme to carry out the numerical experience in the bounded domain case (see [5]).

Keywords— Magneto-Micropolar Fluids, Exterior Domains, Non-linear EDP.

References

[ 1] M. Durán, E. Ortega-Torres and M. Rojas-Medar, Stationary Solution of MAgneto-Micropolar Fluid Equations in Exterior Domains. Submitted to Communication on Applied Nonlinear Analysis, (1998).

[2] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions. Int. J. Enging Sci. 12, 657-663, (1974).

[ 3] O. A. Ladyszhenskaya, The mathematical theory ofviscous incompressible flow.

Second edition. Gordon and Breach, New York, (1969). [ 4] J. G. Heywood, The Navier-Stokes equations: on the existence, regularity and

decay of solutions. Indiana Univ. Math. J. 29, 639-681, (1980). [ 5] M. Durán, E. Ortega-Torres and M. Rojas-Medar, Stationary asymmetric

fluids: Numerical analysis. Submitted to Applied Numerical Mathematics, (1998).

Domain decomposition methods for exterior boundary value problems in potential theory^{∗}

G.N. Gatica†, G.C. Hsiao‡ and M.E. Mellado†

†Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

‡Department of Mathematical Sciences, University of Delaware Newark, Delaware 19716, USA.

Abstract. In this paper we present new domain decomposition schemes for a class of linear and nonlinear boundary value problems from potential theory. We first use the boundary integral equation method to derive a Dirichlet-to-Neumann mapping to transform the exterior problem into an equivalent problem on a bounded domain. This bounded domain is decomposed into a finite number of annular subdomains and different types of interface unknowns are considered. The Steklov-Poincaré operators are expressed either by the finite element method or a combination between finite element and boundary element methods. The corresponding interface problems are solved by using Robin-type preconditioners which yields iteration-by-subdomain algorithms well suited for parallel processing. Also, we present contractivity results and finite dimensional approximations. Finally, some numerical experiments on a parallel computer are also provided.

Keywords— Dirichlet-to-Neumann Mapping, Steklov-Poincaré Operator, Boundary Element Method, Finite Element Method, Domain Decomposition, Preconditioned Conjugate Gradient Method.

References

[ 1] G.N. Gatica, E.C. Hernández and M.E. Mellado, A domain decomposition method for linear exterior boundary value problems. Applied Mathematics Letters 11, 6, 1-9, (1998).

[ 2] G.N. Gatica and G.C. Hsiao, The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 189, 442-461, (1995).

[ 3] G.N. Gatica and M.E. Mellado, Nonoverlapping domain decomposition methods for linear and nonlinear exterior boundary value problems. In: Proceedings of the Fourth World Congress on Computational Mechanics, Buenos

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis and by the Universidad de Concepción through the Dirección de Investigación and Facultad de Ciencias Físicas y Matemáticas.

Domain Decomposition Method

Aires, Argentina, june 29 - july 2, 1998, S. Idelsohn and E. Onate (Editors), CIMNE, Barcelona, (1998).

[ 4] G.C. Hsiao, B.N. Khoromskij and W.L. Wendland, Boundary integral operators and domain decomposition. Preprint 94-11, Mathematisches Institut A, Universität Stuttgart, (1994).

[ 5] G.C. Hsiao and W.L. Wendland, Domain decomposition via boundary element methods. Technical Report 92-10, Department of Mathematical Sciences, University of Delaware, (1992).

Primal and dual formulations for the coupling of mixed-FEM and BEM^{∗}

Gabriel N. Gatica†, Salim Meddahi‡ and Jean-Marie Thomas§

†Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

‡Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo, 33007 Oviedo, España.

§Laboratoire de Mathématiques Appliquées, UPRES A 5033, Université de Pau et CNRS, Avenue de l’Université, 64000 Pau, France.

Abstract. This paper deals with the coupling of mixed finite element and boundary element methods. The procedure, which may also use Dirichlet-to-Neumann mappings, is applied to several transmission problems in physics and engineering sciences, particularly potential theory and elastostatics. Our analysis includes both primal and dual mixed methods, and consider linear and nonlinear boundary value problems. We study the different types of variational formulations arising from the coupling procedure and derive results of existence and uniqueness of solution for the continuous and discrete schemes. A-priori and a-posteriori estimates for the corresponding error are also provided. Keywords— Mixed Finite Elements, Boundary Integral Equation Method, Dirichlet-to-Neumann Mappings.

References

[ 1] G.R. Barrenechea, G.N. Gatica and J.-M. Thomas, Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems. Numerical Functional Analysis and Optimization 19, 7-32, (1998).

[ 2] G.N. Gatica, Combination of mixed finite element and Dirichlet-to-Neumann methods in nonlinear plane elasticity. Applied Mathematics Letters 10, 29-35, (1997).

[ 3] G.N. Gatica and S. Meddahi, An a-posteriori error estimate for the coupling of BEM and mixed-FEM. Technical Report 98-16, Departamento de Ingeniería Matemática, Universidad de Concepción, (1998).

[ 4] G.N. Gatica and W.L. Wendland, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Applicable Analysis 63, 39-75, (1996).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis, by the CNRS-Conicyt Program, and by Fondecyt-Conicyt through the research project 1980122.

Mixed-FEM and BEM

[ 5] G.N. Gatica and W.L. Wendland, Coupling of mixed finite elements and boundary elements for a hyperelastic interface problem. SIAM Journal on Numerical Analysis 34, 2335-2356, (1997).

[ 6] S. Meddahi, J. Valdés, O. Menéndez, and P. Pérez, On the coupling of boundary integral and mixed finite element methods. Journal of Computational and Applied Mathematics 69, 113-124, (1996).

Quality improvement of Delaunay triangulations for finite element and finite volume discretization methods^{∗}

Nancy Hitschfeld and Maria C. Rivara

Departamento de Ciencias de la Computación, Universidad de Chile Blanco Encalada 2120, Santiago, Chile.

Abstract. In this paper we discuss the application of a new longest-edge based algorithm capable of improving the quality of Delaunay triangulations for finite element and finite volume discretization methods. The resulting triangulations are Delaunay triangulations, whose boundary triangles (triangles with at least one edge on the boundary or on a material interface) do not have obtuse angles opposite to any boundary or interface edges. In addition, the algorithm guarantees that, for non geometry restricted triangles, the minimum and maximum angles of the triangles are bounded (minimum angle greater than or equal to 30∘ and maximum angle less than or equal to 120∘).

Keywords— Longest-Edge, Delaunay Triangulations, FEM, Finite Volume Methods.

References

[ 1] M.C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrids techniques. International Journal for Numerical Methods in Engineering 20, 745-756, (1984).

[ 2] N. Hitschfeld, P. Conti and W. Fichtner, Mixed elements trees: A generalization of modified oc trees for the generation of meshes for the simulation of complex 3-D semiconductor devices. IEEE Transactions on CAD/ICAS 12, 1714–1725, (1993).

[ 3] M.C. Rivara, New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations. International Journal for Numerical Methods in Engineering 40, 3313-3324, (1997).

[ 4] N. Hitschfeld and M.C. Rivara, Automatic construction of quality nonobtuse boundary Delaunay triangulations. Submitted to Mathematics of Computation, (1998).

[ 5] N. Hitschfeld, M.C. Rivara and M. Palma, Improving the quality of Delaunay triangulations for the control volume discretization method. Simulation of semiconductor processes and devices. Springer-Verlag, pp. 189-192. 1998.

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis.

Numerical comparison between relaxation and nonlinear equilibrium models. Application to chemical engineering^{∗}

Francois James†, Marie Postel‡ and Mauricio Sepúlveda§

†Mathématiques, Applications et Physique Mathématique d’Orléans, UMR CNRS 6628, Université d’Orléans, F-45067 Orléans Cedex 2, France.

‡Laboratoire de’Analyse Numérique, UMR CNRS 7598, BC 187, 4 Place Jussieu, 75252 Paris Cedex 05, France.

§Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

Abstract. A model to take into account the finite exchange kinetic for the equilibrium between phases in the chromatography experiment and the distillation is studied. The resulting hyperbolic system with a nonlinear relaxation term is then formally treated with a Chapman-Enskog type expansion. A first order correction to the classical quasilinear hyperbolic model is derived which consists in a nonlinear diffusion term. Numerical schemes for both models– relaxed and parabolic – are then tested and compared for different initial and boundary values. A range of validity for the first order approximation is obtained through numerical simulations where its computational performance indicates that it should be advantageously used.

References

[ 1] G. Guiochon, S. Golshan Shirazi, and A. Katti, Fundamentals of Preparative

and Nonlinear Chromatography. Academic Press, Boston, 1994. [ 2] F. James, Convergence results for conservation laws with a reflux boundary

condition and a relaxation term. SIAM Journal on Mathematical Analysis 29, 5, 1200-1223, (1998).

[ 3] S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimension. Communications on Pure and Applied Mathematics 48, 235-277, (1995).

[ 4] C-W Shu, TVB boundary treatment for numerical solutions of conservation laws. Mathematics of Computation 49, 179, 123-134, (1987).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis, by the CNRS-Conicyt Program, and by Fondecyt-Conicyt through the research project 1980123.

Nonlinear equilibrium models

[ 5] F. James, M. Postel, and M. Sepulveda, Numerical comparison between relaxation and nonlinear equilibrium models. Application to chemical engineering. Technical Report 98-18, Departamento de Ingeniería Matemática, Universidad de Concepción, (1998).

The aliasing error in wavelet boundary element methods^{∗}

Freddy W. Paiva and Cristián G. Pérez

Departamento de Ingeniería Matemática, Universidad de Concepción Casilla 160-C, Concepción, Chile.

Abstract. We first observe that the wavelet boundary element method (WBEM), as presented in [ 1], consists in considering a wavelet transform in which the coefficients are obtained by sampling instead of numerical quadrature. In this paper we use the Sampling Theorem for Wavelet subspaces from [ 2], to determine the aliasing error in the multiscale approximation. In other words, we find an upper bound for the L2−error between a function f ∈ L2[0, L], which is L-periodic, Hölder continuous in (0, L) and unbounded at x = 0, and the interpolation function constructed in a subspace of fixed resolution. In addition, we show that this interpolation function reduces to a linear combination of the translates of the scale function of that given subspace. The corresponding coefficients are obtained multiplying a circulant matrix by a vector containing the sampling values of the function, to which the singularity sustraction method has been previously applied. Some numerical examples are also provided. Keywords— Wavelets, Reproducing Kernel, Riesz Basis, Zak Transform, Circulant Matrix, Aliasing Error, Integral Equation, Boundary Element.

References

[ 1] D.L. Donoho, Interpolating wavelet transforms. Technical Report, Depart-

ment of Statistics, Stanford University, october (1992). [ 2] Y. Meyer, Ondelettes et Opérateurs I: Ondelettes. Hermann, Paris, (1990). [ 3] J.G. Ren, Wavelet methods for boundary integral equations. Communications

on Numerical Methods in Enginnering 13, 373-385, (1997). [ 4] G.G. Walter, A sampling theorem for wavelet subspace. IEEE Transactions

on Information Theory 38, 2, 881-884, (1992).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis and by the Dirección de Investigación de la Universidad de Concepción through the research project 97.013.008-1.2.

Longest-edge refinement algorithms in 3-dimensions^{∗}

Maria C. Rivara† and Angel Plaza‡

†Departamento de Ciencias de la Computación, Universidad de Chile Blanco Encalada 2120, Santiago, Chile.

‡ Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, España.

Abstract. Longest-edge algorithms are well known techniques for the refinement of triangular 2-dimensional meshes, which guarantee strong non degeneracy properties over the refined meshes: they produce triangles whose minimum angle is bounded as a function of the quality of the initial triangulation, they produce a finite number of similarly distinct triangles, a fractal behavior around each vertex holds, and both the percentage of good-quality triangles and the area covered by these triangles increase as the refinement proceeds. In this paper, some mesh non-degeneracy properties for longest-edge 3-dimensional algorithms are discussed. Empirical experimentation showing analogous behavior to the 2-dimensional case is also presented.

Keywords— Longest-Edge Algorithms, Mesh Refinements, Non-Degeneracy.

References

[ 1] M.C. Rivara, Mesh refinement processes based on the generalized bisection of

simplices. SIAM Journal on Numerical Analysis 21, 604-613, (1984). [ 2] M.C. Rivara, Algorithms for refining triangular grids suitable for adaptive

and multigrids techniques. International Journal for Numerical Methods in Engineering 20, 745-756, (1984).

[ 3] M.C. Rivara and G. Iribarren, The 4-triangles longest-side partition of triangles and linear refinement algorithms. Mathematics of Computation 65, 1485-1501, (1996).

[ 4] M.C. Rivara and A. Plaza, Mesh Refinement / Derefinement based on the 8-Tetrahedra Longest-Edge Partition. Department of Computer Science, University of Chile, October 1998.

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis.

Modeling and numerical solution of elastoacoustic vibrations with interface damping^{∗}

Rodolfo Rodríguez

Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Abstract. This paper deals with the damped transmission of small amplitude vibrations between two media: a linear elastic structure and an inviscid, compressible, barotropic fluid, both separated by a thin layer of viscoelastic material with a given frequency dependent impedance. Much attention has been recently paid to this kind of problems, mostly related to the goal of decreasing the level of noise in aircrafts and cars (see for instance [ 4]). We consider a displacement formulation for both, the fluid and the solid, which has been thoroughly analized in the case of an undamped interaction in some previous papers ([ 1], [ 2], [ 3]). Standard finite elements for the structure combined with Raviart-Thomas elements for the fluid has been shown therein to provide a reliable method to compute the vibration modes of the coupled system. In particular, the method was proved to be free of spurious non-zero frequency modes. In this paper we present a simplified mathematical model for the damped transmission conditions which allows us to introduce an efficient numerical method to compute the vibrations of the coupled system. It consists of relaxing the kinematic constrain: the classical slipping condition on the interface is substituted by

p = iωZ(ω)(u^{S} · n − u^{F} · n),

where p is the pressure on the interface, ω the excitation frequence, Z(ω) the (complex) frequency dependent acoustic impedance of the viscoelastic material and uS · n and uF · n the normal displacements of the solid and the fluid, respectively. The method is used to compute the vibrations transmitted to the fluid from an external source acting on the structure and the (complex) resonance frequencies of the coupled damped system. The relation between these and the response peaks of the system is also analized.

Keywords— Damped Acoustic Vibrations, Quadratic Eigenvalue Problems.

References

[ 1] A. Bermúdez and R. Rodríguez, Finite element computation of the vibration modes of a fluid-solid system. Computer Methods in Applied Mechanics and Engineering 119, 355-370, (1994).

∗ To be delivered at the Fifth French-Latinamerican Congress on Applied Mathematics

to be held at the Universidad de Chile, Santiago, during the period december 15-18, 1998. This research was partially supported by FONDAP-Conicyt through the Program A on Numerical Analysis, and by Fondecyt-Conicyt through the research project 1960615.

Elastoacoustic vibrations

[ 2] A. Bermúdez, R. Durán, M.A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM Journal on Numerical Analysis 32, 1280-1295, (1995).

[ 3] A. Bermúdez, L. Hervella-Nieto and R. Rodríguez, Numerical Solution of Three Dimensional Elastoacustic Problems. In Numerical Methods in Engineering ECCOMAS ‘96, J.A. Désidéri et al., eds., J. Wiley & Sons, 1996, 874-880.

[ 4] V. Kehr-Candille and R. Ohayon, Elasto-acoustic damped vibrations. Finite element and modal reduction methods. In New Advances in Computational Structure Mechanics, O.C. Zienkiewicz and P. Ladevèze, eds., Elsevier, 1992.