Mathematical Problems in Engineering

Volume 2019, Article ID 4834521, 18 pages

https://doi.org/10.1155/2019/4834521

## Adaptive Analysis of Acoustic-Elastodynamic Interacting Models Considering Frequency Domain MFS-FEM Coupled Formulations

^{1}Structural Engineering Department, Federal University of Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil^{2}ISISE, Department of Civil Engineering, University of Coimbra, 3030-788 Coimbra, Portugal

Correspondence should be addressed to D. Soares Jr.; rb.ude.fjfu@seraos.mifled

Received 2 November 2018; Accepted 26 December 2018; Published 6 January 2019

Academic Editor: Mijia Yang

Copyright © 2019 D. Soares Jr. and L. Godinho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work discusses adaptive iterative coupling strategies for the frequency domain analysis of interacting acoustic-elastodynamic models. The method of fundamental solutions (MFS) is used to analyze acoustic fluids, whereas the finite element method (FEM) is employed to discretize elastodynamic solids. Flexible and optimized iterative MFS-FEM coupling procedures are considered, allowing independent discretizations to be adopted for each subdomain. In this context, it is easy to implement adaptive refinements and enable enhanced analyses. Two adaptive coupling approaches are discussed, based on multiple and single iterative algorithms. Numerical results are presented to illustrate the performance of the proposed techniques.

#### 1. Introduction

The analysis of coupled acoustic-elastodynamic systems is a complex and demanding task, which involves different physical phenomena and governing equations. It can be more efficiently handled by making use of different numerical methods for each part of the problem. A number of the research works that have been published on this topic have looked at BEM-FEM coupling algorithms [1–8]. Indeed, the BEM (boundary element method) is quite suitable to handle infinite or semi-infinite media, while the FEM (finite element method) has been widely studied and developed to handle structural systems; a combination of both methods thus seems like a natural choice.

Many of the above references account for the full-coupling between the BEM and the FEM directly by establishing a fully coupled system matrix [1–5]; however, this option can be quite inefficient. This is because the symmetric and sparse/banded character of the FEM matrix is significantly affected by the presence of the BEM components, and optimized solvers, usually used by the FEM, cannot be employed anymore. Additionally, the different properties of fluid and solid media may lead to badly conditioned matrices, which affect the accuracy of the methodology. Finally, standard direct coupling methodologies require compatible discretizations (with matching nodes along the interfaces) and this greatly affects the flexibility and versatility of the technique. Although alternatives can be used to allow incompatible discretizations (such as interpolation along the interface), this usually increases the complexities when setting up the problem.

In order to overcome these difficulties, iterative coupling procedures have been presented, considering both time and frequency domain acoustic-elastodynamic interaction analyses, and taking into account not only BEM-FEM coupling, but also considering other possibilities [6–10]. Studies have shown that iterative coupling approaches allow BEM and FEM subdomains to be analyzed separately, leading to smaller and better-conditioned systems of equations (different solvers, suitable for each subdomain, may be employed). Furthermore, a small number of iterations are usually required for the algorithm to converge and the matrices related to the smaller governing systems of equations do not need to be treated (inverted, triangularized, etc.) at each iterative step, thus providing an efficient methodology. Different methods have been tested for frequency domain analysis. For the elastodynamic subdomain, the FEM [8, 10] and the newer meshless methods, including the Kansa’s method [9, 10] and the meshless local Petrov-Galerkin method [10], have been employed with success, while for the acoustic fluid, both the BEM [8, 10] and the method of fundamental solutions (MFS) [9, 10] have been used in iterative coupling algorithms. Previous works [9–12] have also shown that the MFS can be particularly well suited for the analysis of the acoustic subdomain, even surpassing the BEM in terms of efficiency and accuracy. Thus, iteratively coupling the MFS with the FEM can be a robust and efficient alternative strategy to handle acoustic-elastodynamic problems in the frequency domain. It is worth noting that although the MFS is quite well suited for this type of analysis, it requires previously defining the position of a set of virtual sources, located outside the domain of analysis, which are used to construct the approximation of the acoustic pressure field. The definition of such positions is still an open problem, which has been widely discussed in the literature, and some methods have been proposed for their optimal determination, such as in [13]; a number of alternative methods have also been proposed to overcome this difficulty, such as those described in [14–16].

Differently from earlier studies, the present paper reports on the analysis of acoustic-elastodynamic systems using an efficient adaptive iterative coupling between the MFS and the FEM. In fact, adaptive remeshing can be employed in association with the analyses of different physical problems. For the case of elastoplasticity, remeshing has been applied in the context of the BEM-FEM coupled analysis by Elleithy et al. [17, 18], who updated the subdomains modelled by the FEM and the BEM according to the evolution of the plastic zones of the model. Regarding dynamic problems, although contributions can be found in the literature on elastodynamics (such as [19–21]) and acoustics (such as [22, 23]), not many studies report on adaptive methods for coupled acoustic-elastodynamic interaction problems. Of the few there are, we would mention García et al. [24], who applied an adaptive FEM algorithm to solve solid-fluid interaction problems governed by the Navier-Stokes equation and Demkowicz and Oden [25, 26], who applied an hp-adaptive coupled BEM-FEM model to study problems of elastic scattering, assuming the fluid to be governed by the Helmholtz equation.

In these works, the BEM-FEM coupling is performed in a direct form, requiring the construction of a fully coupled matrix. Here, a very different approach is developed: the regions modelled by the MFS do not change during the analysis, and thus the fully populated matrices of the MFS are computed only once, which increases efficiency; as for the FEM, a coarse discretization is used to start the algorithm, and this is adaptively enriched as the solution evolves. In this context, and since refining only takes place within the FEM meshes, nonmatching nodes must be allowed at the MFS-FEM interfaces; otherwise changing the MFS point distributions would require the MFS matrices to be recomputed, with a consequent loss of efficiency. The use of an iterative coupling algorithm is thus highly desirable because of its superior performance and because it is very flexible.

In this work, two different iterative coupling strategies are proposed and discussed, both offering mesh adaptivity. The first is a more standard approach in which the coupling between the two domains is computed iteratively, with the FEM mesh then being analyzed and refined according to the final coupled solution. The problem is then reanalyzed using the new mesh. The second approach, on the other hand, incorporates the mesh adaptivity within the coupling iterative process, thus producing a much more efficient procedure with significant computational savings while still providing the same accuracy level. For both cases, the use of different methods to address the acoustic and elastodynamic problems (MFS and FEM, respectively), which are handled separately, is a significant advantage of the iteratively coupled approaches. Indeed, the MFS allows directly accounting for the infinite character of the problem, while the FEM is quite adequate to model a finite solid subdomain, which may contain complex details and spatially variable properties. Thus, their joint use in iterative coupling strategies, without the requirement of matching discretizations in the contact interfaces, leads to a powerful numerical tool, which incorporates what can be called the “best of two worlds”.

The paper is organized as follows: first, we present the governing equations of the acoustic and elastodynamic models and briefly discuss the basic aspects of the MFS and FEM. We then describe the proposed adaptive iterative MFS-FEM coupling algorithms and look at some numerical applications, illustrating the performance and potentialities of the proposed techniques.

#### 2. Numerical Modelling

This section sets out the basic governing equations of the model and the main aspects of the numerical techniques focused upon. Here, acoustic fluids are handled by the MFS and elastodynamic solids by the FEM.

##### 2.1. Acoustic Medium Discretization

The Helmholtz wave equation can be written aswhere stands for the acoustic pressure distribution, represents body source terms, stands for the wavenumber, represents the wave propagation velocity, and and represent the frequency and the spatial domain of analysis, respectively. Subscript commas indicate partial space derivatives (index notation is adopted). The boundary conditions of the model are given bywhere the prescribed values are indicated by overbars and represents the normal velocity of the acoustic fluid along the boundary, whose unit outward normal vector is represented by ( stands for the mass density). The boundary of the model is denoted by () and the domain by . Equation (2a) stands for essential (or Dirichlet) boundary conditions and (2b) stands for natural (or Neumann) boundary conditions.

In the MFS, the solution is approximated by a linear combination of fundamental solutions centred on virtual sources (), placed outside the domain of interest to avoid singularities in the response:In (3), stands for the Green’s function of the model, is related to domain source terms, and represents coefficients to be determined. By applying approximation (3) to the model, a system of algebraic equations can be obtained, as indicated below:where the entries of matrix and vector are given byand for 2D analysis the Green’s function expression can be given for an infinite medium by = = , where stands for the second type Hankel’s function of order 0, and the term represents the distance between the collocation () and the virtual source () points.

Once the system of (4) is solved (i.e., vector is computed), the approximate solution at any point of interest can be obtained using definition (3). For more details considering the numerical modelling by the method of fundamental solutions, see [11, 27].

##### 2.2. Solid Medium Discretization

The elastic wave equation is given bywhere and stand for displacement and body force distribution components, respectively. The notation for space derivatives employed in (1) is once again adopted. In (7), is the dilatational wave velocity and is the shear wave velocity; they are given by and , where is the mass density and and are the Lamé constants. Equation (7) can be obtained from the combination of the following basic mechanical equations:where and are stress and strain tensor components, respectively, and is the Kronecker delta ( = 1, for and = 0, for ). Equation (8a) is the momentum equilibrium equation; (8b) represents the constitutive law of the elastic model; and (8c) stands for kinematic relations. The boundary conditions of the model are given bywhere, once again, the prescribed values are indicated by overbars and denotes the traction vector along the boundary.

In the FEM, the solution is approximated by a local (i.e., at element level) interpolation: In (10), stands for local interpolating functions, describes the number of nodes in the element, and represents nodal values to be determined. Applying approximations (10), a system of algebraic equations is obtained once the FEM is implemented, as indicated below:where matrices and and vector are given byand** N**,** B,** and** D** stand for the interpolation, the strain, and the constitutive matrices of element , respectively.

Once the system of (11) is solved (i.e., vector is computed), the approximate solution at any point of interest can be obtained using definition (10). For more details considering the numerical modelling by the finite element method, see [28–30].

#### 3. Coupling Formulation

For the coupled analysis in question, the following continuity and equilibrium equations must hold at the interfaces between the acoustic fluid and solid subdomains:where and stand for normal (normal to the common interface) displacements and tractions, respectively.

The coupling between the acoustic fluid (MFS) and the elastic solid (FEM) subdomains of the model is enabled by implementing an iterative procedure that performs a successive renewal of the relevant variables at the acoustic-elastodynamic interface. The proposed approach is based on the imposition of prescribed velocities and tractions at the fluid and solid common interfaces, respectively, following relations (14a) and (14b). Since the subdomains are analyzed separately, independent discretizations can easily be considered at each subdomain without requiring matching nodes on common interfaces, thereby enhancing the flexibility of the technique. This is especially important when remeshing is considered and adaptive refinement is implemented, as it is the case here. To ensure and/or to speed up convergence, a relaxation parameter *λ* is introduced in the iterative coupling algorithm. The effectiveness of the iterative process depends to a great extent on the selection of this relaxation parameter, since an inappropriate choice for *λ* can significantly increase the number of iterations in the analysis or, even worse, make convergence unfeasible.

In the subsections that follow, first, the basic steps of the iterative coupling procedure are described. Afterwards, we give an expression for an optimal relaxation parameter and then discuss the introduction of the adaptive refinement.

##### 3.1. Iterative Coupling

In the iterative step of the MFS-FEM acoustic-elastodynamic coupling, the MFS subdomain is analyzed and vector is computed, as described in Section 2.1 (the superscript indicates the iterative step of the analysis). In this case, the MFS analysis takes into account prescribed velocities at the collocation points on the common interfaces, which are given by the previous iterative step (in the first iterative step, null prescribed velocities are assumed). Once is computed, the fluid pressures at the FEM nodes of the common interfaces are evaluated, using (3). These pressures are then employed to compute the prescribed tractions at the FEM interface, following (14b) (i.e., ):Here, an interpolation approach, analogous to that described by (10), is used to describe the pressure distribution along the FEM interface, as indicated by the r.h.s. of (15), where describes the number of nodes at the interface of the element.

Once the prescribed values at the common interfaces of the FEM are known, the FEM subdomains can be analyzed so that the displacement vector can be computed as described in Section 2.2. Once is computed, the velocities at the MFS collocation points of the common interface can be evaluated, following approximation (10) and (14a) (i.e., ):

As previously explained, we have used relaxation parameters to ensure and/or to speed up the convergence of the iterative process. Thus, the prescribed velocities that are employed by the MFS subdomains in the next iterative step are computed as follows:where stands for the relaxation parameter. Once the prescribed values at the common interfaces of the MFS are known, the algorithm goes on to the next iterative step, repeating all the procedures described above until convergence is achieved.

##### 3.2. Optimal Relaxation Parameter

In order to evaluate an optimal relaxation parameter, the following square error functional is minimized here: where stands for the MFS prescribed values at the solid-fluid interfaces.

Taking into account the relaxation of the prescribed values for the () and () iterations, (19a) and (19b) may be written, based on the definition in (17):

Substituting (19a) and (19b) into (18) yieldswhere the inner product definition is employed (e.g., ) and new variables, as defined in (21), are considered.

To find the optimal that minimizes the functional , (20) is differentiated with respect to *λ* and the result is set to zero, as described below:

Rearranging the terms in (22) yieldswhich is an easy to implement expression that provides an optimal value for the relaxation parameter , at each iterative step. It should be noted that other alternatives for the calculation of the relaxation parameter can be found in the literature, such as in [31–33]; however, this expression leads to lower computational costs than some of these alternatives (see, for instance, [33]).

Additionally, one should keep in mind that the computed relaxation parameter is a complex number since the problem is formulated in the frequency domain. This complex number computation could be ranged (e.g., by imposing ), but we have observed that faster convergence is usually achieved in the iterative process if a nonrestricted relaxation parameter selection, provided by (23), is considered. Moreover, although we found that the iterative process is relatively insensitive to the value of the relaxation parameter used for the first step, a real value of is assumed in all the cases discussed here.

##### 3.3. Adaptive Discretization

This work uses triangular finite elements since discretizations considering this type of element are easier to adaptively refine. Adaptive discretizations are widely used nowadays and a typical loop for an adaptive FEM through local refinement basically involves 4 steps, namely: (i) solve; (ii) estimate; (iii) mark; (iv) refine/coarsen. Thus, the FEM subdomains are analyzed and the solution in the current triangular mesh obtained (first step). The error is then estimated using the computed solution (second step) and it is used to mark (third step) a set of triangles that are to be refined, which is done while keeping the triangulation shape regularity and conformity (fourth step). The adaptive procedure implemented here is based on the package provided by Chen and Zhang [34].

We implemented two iterative approaches with respect to the adaptive coupling formulation. The first approach used a multiple iterative algorithm. In this case, the entire coupling iterative loop is carried out within each iterative step of the adaptive iterative loop. This approach is expected to be very computationally demanding because engaged iterative procedures are employed. To avoid this computational cost, we propose a second approach, which is expected to be more efficient. In this alternative formulation, a unified single iterative loop is considered and the coupling and adaptive analyses are carried out together in the same iterative step. As illustrated in the next section, this seems to be a good approach because the number of iterative steps required by the iterative coupling procedure is not significantly increased by handling the two iterating procedures together in the same iterative loop.

Figure 1 shows sketches for the multiple and the single iterative algorithms, taking into account the application being studied.