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The Partition of Unity Finite Element Method (PUFEM) has been widely used for the numerical simulation of the Helmholtz equation in different physical settings. In fact, it is a numerical pollution-free alternative method to the classical piecewise polynomial-based finite element methods. Taking into account a plane wave enrichment of the piecewise linear finite element method, the main goal of this work is focused on the derivation of the numerical dispersion relation and the robustness analysis of the PUFEM discretization when a spurious perturbation is presented in the wave number value used in the enrichment definition. From the one-dimensional Helmholtz equation, the discrete wave number is estimated based on a Bloch’s wave analysis and a priori error estimates are computed explicitly in terms of the mesh size, the wave number, and the perturbation value.

[Abstract] During the last decade, several authors have addressed that the Perfectly Matched Layers (PML) technique can be used not only for the computation of the near-field in time-dependent and time-harmonic scattering problems, but also to compute numerically the resonances in open cavities. Despite such complex resonances are not natural eigen-frequencies of the physical system, the numerical determination of this kind of eigenvalues provides information about the model, what can be used in further applications. The present work will be focused on two main specific goals—firstly, the mathematical analysis of the frequency-dependent highly non-linear eigenvalue problem associated to the computation of resonances with the standard PML technique. Second, the implementation of a robust numerical method to approximate resonances in open cavities. This work has been supported by MTM project PID2019-108584RB-I00 (MICINN), accreditations ED431C 2018/33 -M2NICA (Xunta de Galicia & ERDF) and ED431G 2019/01 -CITIC (Xunta de Galicia & ERDF) Xunta de Galicia; ED431C 2018/33 Xunta de Galicia; ED431G 2019/01

We compare the Helmholtz equation in pressure and displacement formulation.We are not aware of any similar investigations and results.We use Lagrangian elements resp. Raviart-Thomas elements for the discretization.We show that Raviart-Thomas elements account for a reasonable alternative to Lagrangian elements in some cases. In this article, we compare the finite element solution of the pressure and the displacement formulations for the acoustic problem in the frequency domain. The Helmholtz differential equation is solved for both, the spectral and the source problems, and each of these is formulated in terms of pressure and displacement. The pressure formulation allows to use standard Lagrangian elements whereas Raviart-Thomas elements are applied for the displacement formulation. Both formulations are tested in several examples. It is shown that Raviart-Thomas elements account for a reasonable alternative to Lagrangian elements in some cases.

This paper presents the problem of modeling poroelastic material on a moving structure. It presents the ambiguity of using estimated or measured surface impedance on a rigid structure as opposed to transfer impedance, which takes into account the movement of the structure to which a porous material is fixed. Biot's model for prediction of surface impedance is presented, and the transfer impedance is defined on the basis of this model. A case where several layers of porous material are fixed to a moving wall is also studied. Finally, it presents the level of agreement with experimental measurements in a coupled fluid–structure problem when an impedance considering a fixed structure and an impedance considering a moving structure are used.

The main goal of this work consists in the numerical simulation of a multichannel passive–active noise control system based on devices involving a particular kind of cell. Each cell consists of a parallelepipedic box with all their faces rigid, except one of them, where a porous veil or a rigid micro-perforated plate (MPP) is placed. Firstly, the frequency response of a single passive cell is computed, when it is surrounded by an unbounded air domain (an anechoic room) and harmonic excitations are imposed. For the numerical solution of this three-dimensional problem, the original unbounded domain is truncated by using exact perfectly matched layers (PML) and the resulting partial differential equation (PDE) is discretized with a standard finite element method. Secondly, the passive cells are transformed into active by assuming that the opposite face to the passive one may vibrate like a piston in order to reduce noise. The corresponding multichannel active control problem is stated and analyzed in the framework of the optimal control theory. A numerical method is proposed to assess and compare different control configurations.

This work deals with a coupled acoustic problem: a fluid contained in a cavity whose walls are all rigid except for the top, where a flexible plate is placed. The fluid is described by means of its pressure, meanwhile for the plate the Reissner-Mindlin model is used. A finite element method to solve this coupled problem, based on Lagrange hexahedral elements in the fluid and four node MITC elements in the plate is presented. A comparison between numerical and experimental results is presented. Then, the behaviour of an absorbing layer attached to the plate is described by using a wall impedance condition and also by means of two different fluid equivalent models. Again, a comparison between numerical and experimental results is presented for the covered problem.

This article is a bibliographical revision concerning acoustic absorbing materials, also known as poroelastics. These absorbing materials are a passive medium use extensively in the industry to reduce noise. This review presents the fundamental parameters that define each of the parts comprising these materials, as well as current experimental methods used to measure said parameters. Further along, we will analyze the principle models of characterization in order to study the behaviour of poroelastic materials. Given the lack of accuracy of the standing wave method three absorbing materials are characterized using said principle models. A comparison between measurements with the standing wave method and the predicted surface impedance with the models is shown.

We introduce an optimal bounded perfectly matched layer (PML) technique by choosing a particular absorbing function with unbounded integral. With this choice, spurious reflections are avoided, even though the thickness of the layer is finite. We show that such choice is easy to implement in a finite element method and overcomes the dependency of parameters for the discrete problem. Finally, its efficiency and accuracy are illustrated with some numerical tests.

The aim of this paper is to study the time-harmonic scattering problem in a coupled fluid-porous medium system. We consider two different models for the treatment of porous materials: the Allard–Champoux equations and an approximate model based on a wall impedance condition. Both models are compared by computing analytically their respective solutions for unbounded planar obstacles, considering successively plane and spherical waves. A numerical method combining an optimal bounded PML and finite elements is also introduced to compute the solutions of both problems for more general axisymmetric geometries. This method is used to compare the solutions for a spherical absorber.

We study the Helmholtz equation with a Sommerfeld radiation condition in an unbounded domain. We provethe existence of an exact bounded perfectly matched layer (PML) for this problem, in the sense that we recover theexact solution in the physical domain by choosing a singular PML function in a bounded domain. We approximatethe solution for the PML problem using a standard nite element method and assess its performance throughnumerical tests.R esum eNous etudions l’ equation de Helmholtz avec une condition de radiation de Sommerfeld dans un domaine nonborn e. Pour ce probl eme nous prouvons l’existence d’une couche born ee parfaitement adapt ee et exacte, au sensque nous retrouvons la solution exacte dans le domaine physique. Nous approchons la solution du probl eme PMLavec une m etho de standard d’ el ements nis et nous montrons ses bonnes propriet es sur des exemples test. Version fran˘caise abr eg eeDans ce travail, nous etudions le probl eme de Helmholtz avec une condition de radiation de Sommerfelddans un domaine non born e (voir (1)). Dans cette equation nous utilisons la notation k := !=c pour lenum ero d’onde, avec ! la fr equence et c la vitesse de propagation. La fonction u

In this paper we analyze the non symmetric pressure/displacement formulation of the elastoacoustic vibration problem and show its equivalence with the (symmetric) stiffness coupling formulation. We introduce discretizations for these problems based on Lagrangian finite elements. We show that both formulations are also equivalent at discrete level and prove optimal error estimates for eigenfunctions and eigenvalues. Both formulations are rewritten such as to be solved with a standard Matlab eigensolver. We report numerical results comparing the efficiency of the methods over some test examples.

This paper deals with the numerical approximation of the bending of a plate modeled by Reissner--Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is based on the family of elements called MITC (mixed interpolation of tensorial components). We consider two lowest-order methods of this family on quadrilateral meshes. Under mild assumptions we obtain optimal H1 and L2 error estimates for both methods. These estimates are valid with constants independent of the plate thickness. We also obtain error estimates for the approximation of the plate vibration problem. Finally, we report some numerical experiments showing the very good behavior of the methods, even in some cases not covered by our theory.

This paper deals with a finite element method to compute the vibrations of a coupled fluid–solid system subject to an external harmonic excitation. The system consists of an acoustic fluid and a plate, with a thin layer of a noise damping viscoelastic material separating both media. The fluid is described by displacement variables whereas the plate is modeled by Reissner–Mindlin equations. Face elements are used for the fluid and MITC3 elements for the bending of the plate. The effect of the damping material is taken into account by adequately relaxing the kinematic constraint on the fluid–solid interface. The nonlinear eigenvalue problem arising from the free vibrations of the damped coupled system is also considered. The dispersion equation is deduced for the simpler case of a fluid in a hexahedral rigid cavity with an absorbing wall. This allows computing analytically its eigenvalues and eigenmodes and comparing them with the finite element solution. The numerical results show that the coupled finite element method neither produces spurious modes nor locks when the thickness of the plate becomes small. Finally the computed resonance frequencies are compared with those of the undamped problem and with the complex eigenvalues of the above nonlinear spectral problem.

We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness $t>0$ , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when $t\to 0$ . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method.

In this paper, the interior elastoacoustic problem in a 3D domain is solved. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation, a non-standard discretization is used, consisting of classical linear tetrahedral finite element for the solid and Raviart–Thomas elements of lowest order for the fluid. A new unknown is introduced on the interface between solid and fluid to impose the transmission conditions.

In this paper a finite element approximation of the free vibration of folded plates is studied. Naghdi model, including bending, shear and membrane terms for the plate, is considered. Quadrilateral low order MITC (Mixed Interpolation Tensorial Component) elements are used for the bending and shear effect, coupled with standard quadratic elements enriched with a drilling degree of freedom for the membrane term. Convergence properties and optimal order error estimates are proved. Numerical examples, showing the good behavior of the method, are presented for one and two folded plates with different thickness and crank angles.

This paper deals with the approximation of the vibration modes of plates and shells using the MITC4 finite element method. We use the classical Naghdi model over a reference domain. We assess the performance of this approach for both structures by means of numerical experiments.

Publisher Summary This chapter discusses a modal synthesis method to solve the elastoacoustic vibration problem. The chapter determines the small amplitude vibration modes of a coupled fluid–solid system: an inviscid barotropic compressible fluid contained in an elastic structure. The solid is described by displacement variables, whereas displacement potential is used for the fluid. A modal reduction method is also introduced that leads to a matrix symmetric eigenvalue problem to compute approximate solutions of the spectral problem. Numerical experiments exhibiting the good performance of the method are also provided in the chapter. The governing equations for free harmonic small amplitude motions for the coupled system are also provided. The analysis is restricted to the two-dimensional case, although the method could also be applied to three-dimensional problems.

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family. Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding L 2 -estimate for a load problem which is proven here. This optimal order L 2 -estimate is of interest in itself. Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory., Facultad de Ciencias Exactas, Consejo Nacional de Investigaciones Científicas y Técnicas