In this work, coupled numerical analysis of interacting acoustic and dynamic models is focused. In this context, several numerical methods, such as the finite difference method, the finite element method, the boundary element method, meshless methods, and so forth, are considered to model each subdomain of the coupled model, and multidomain decomposition techniques are applied to deal with the coupling relations. Two basic coupling algorithms are discussed here, namely the explicit direct coupling approach and the implicit iterative coupling approach, which are formulated based on explicit/implicit time-marching techniques. Completely independent spatial and temporal discretizations among the interacting subdomains are permitted, allowing optimal discretization for each sub-domain of the model to be considered. At the end of the paper, numerical results are presented, illustrating the performance and potentialities of the discussed methodologies.

Usually, an engineer is faced with the analysis of a problem where two or more different physical systems interact with each other, so that the independent solution of any one system is impossible without simultaneous solution of the others. Such systems are known as coupled, and the intensity of such coupling is dependent on the degree of interaction [

In the present work, several numerical methods are considered to discretize the different subdomains of the global model, taking into account interface coupled analyses. Although nowadays there are several powerful numerical techniques available, none of them can be considered most appropriate for all kinds of analysis, and, usually, the coupling of different numerical methodologies is necessary to analyze complex problems more effectively. In this context, the coupling of different numerical methods is recommended, in order to profit from their respective advantages and to evade their disadvantages. Two basic coupling algorithms are discussed here, considering multidomain decomposition techniques. In the first algorithm, explicit time-marching procedures are employed for wave propagation analysis at some subdomains of the model. Since explicit algorithms allow the computation of the current time-step response as function of only previous time-steps information; those subdomains can be independently analyzed directly, at each time step, allowing the development of an explicit direct coupling approach (ExDCA). On the other hand, when implicit time-marching procedures are considered, the computation of the current time-step response depends on the current time-step information, and interacting subdomains modeled by these techniques cannot be independently analyzed directly, being an iterative procedure necessary to analyze these coupled subdomains, once multidomain decomposition techniques are regarded. For this case, a second coupling algorithm is discussed here, referred to as implicit iterative coupling approach (ImICA).

Taking into account an explicit direct or an implicit iterative multidomain decomposition technique, the coupling of several numerical procedures is carried out here. In this work, the coupling of the finite difference method (FDM), finite element method (FEM), boundary element method (BEM), and meshless local Petrov-Galerkin method (MLPG) is focused. In the last decades, these methodologies have been intensively applied to model acoustic-dynamic coupled models, taking into account different coupling strategies and time- and frequency-domain analyses. Considering the FDM, Vireaux [

When time-domain acoustic-dynamic coupled analyses are focused, the coupling of media with different properties (high properties contrast) and/or the coupling of numerical procedures with different spatial/temporal behavior may lead to inaccurate results or, even worse, instabilities. Thus, it is important to develop robust discretization techniques that not only are able to provide accurate and stable analyses, but also are computationally efficient. In this work, a multilevel time-step procedure is presented, as well as nonmatching interface nodes techniques are referred, allowing each subdomain of the model to be independently and optimally discretized, efficiently improving the accuracy and the stability of the analyses.

The paper is organized as follows: first, basic equations concerning acoustic and dynamic models are presented, as well as interface interacting relations; in the sequence, numerical modeling of the acoustic/dynamic subdomains is briefly addressed taking into account domain- and boundary-discretization techniques. In Section

In this section, acoustic and elastic wave equations are briefly presented. Each one of these wave propagation models is used to mathematically describe different subdomains of the global problem. At the end of the section, basic equations concerning the coupling of acoustic and dynamic subdomains are described.

The scalar wave equation is given by

(i) boundary conditions (

(ii) initial conditions (

The elastic wave equation for homogenous media is given by

(i) boundary conditions (

(ii) initial conditions (

On the acoustic-dynamic interface boundaries, the dynamic subdomain normal (normal to the interface) accelerations (

Several numerical methods can be applied to discretize each subdomain of the coupled acoustic-dynamic model, according to their properties and advantages/disadvantages. In the following sub-sections, some numerical methods are briefly discussed, addressing their basic characteristics.

In the numerical methods based on domain discretization, the whole domain of the model is discretized into basic structures (elements, cells, points, etc.), and the spatial treatment of the governing equations is carried out considering these basic structures. In this case, matrix system of equations, as indicated in (

In (

The FDM was one of the first methods developed to analyze complex problems governed by differential equations [

Once the spatial treatment of the governing equations is carried out by a domain-discretization technique and (

In (

In boundary-discretization methods, just the boundary of the model is discretized, taking into account once again some basic structure, such as elements and point distributions. In this case, transient fundamental solutions are employed, and mixed approaches are focused, rendering numerical procedures based on more than one field incognita. The matrix system of equations that arises considering this kind of discretization can be written as

In the present work, the boundary element method (BEM) is focused as a boundary-discretization technique [

There are also some “hybrid” formulations that are difficult to classify as a domain-or a boundary-discretization technique. This is the case, for instance, for some meshless techniques that are based on local boundary discretization (see, e.g., the LBIE—local boundary integral equation method [

In this work, the global model is divided in different subdomains, and each subdomain is analysed independently (as an uncoupled model), taking into account the numerical discretization techniques discussed in Section

In the first procedure (i.e., the ExDCA), explicit time-marching schemes (e.g., the central difference method, the Green-Newmark method, etc.) are employed in some of the subdomains that are analyzed by domain-discretization methods. In the second procedure (ImICA), implicit time-marching schemes are considered within the subdomains. Since the ImICA is based on implicit algorithms (

For both explicit direct and implicit iterative coupling procedures, it is appropriate to consider different temporal discretizations within each subdomain. This is the case since optimal time steps are usually quite different taking into account dynamic and acoustic models, as well as different discretization techniques (especially taking into account some time-marching schemes that are conditionally stable). For instance, as it has been extensively reported in the literature, for small time steps, the time-domain BEM may become unstable, whereas, for large time-steps, excessive numerical damping may occur [

In order to consider different time steps in each subdomain, interpolation/extrapolation procedures along time are performed. Here, several schemes are considered for this temporal data manipulation, according to the discretization techniques involved. For instance, when the BEM is considered discretizing an interacting subdomain, temporal interpolation and extrapolation procedures are carried out based on the BEM time interpolation functions. In this case, time extrapolation procedures can be applied with confidence since they are consistent with the time-domain BEM formulation. Once time interpolation and extrapolation techniques are being employed, coupled implicit subdomains can be easily independently analysed (ImICA) taking into account different time steps. If explicit subdomains are considered (ExDCA), a subdomain solution can be computed independently of the current time step. As a consequence, just time interpolation procedures, associated with subcycling techniques, may be necessary if different time steps are required. Using these temporal data manipulations, optimal modelling in each subdomain may be achieved, which is very important regarding flexibility, efficiency, accuracy, and stability.

In the explicit direct coupling (as well as in the implicit iterative coupling), natural boundary conditions are prescribed at the acoustic and at the dynamic subdomains common interfaces. Two explicit direct coupling approaches are discussed here, the first one considering acoustic explicit subdomains and the second one considering dynamic explicit subdomains. For both approaches, the acoustic subdomain time steps are considered larger than the dynamic subdomain time steps (when different time-steps are regarded), since the wave propagation velocities in solids are usually higher than in acoustic fluids.

In the first explicit direct coupling algorithm discussed here, the pressures related to the acoustic subdomains are computed directly, since their evaluation only takes into account results corresponding to previous time steps (

The detailed algorithm for this first ExDCA is presented in Table

ExDCA-1 algorithm.

Time-step loop (based on | |

(1) Acoustic subdomains analyses: evaluation of | |

(2) Subcycling (until | |

(2.1) pressure temporal interpolation: | |

(2.2) force-pressure compatibility (spatial interpolation): | |

(2.3) dynamic subdomains analyses: evaluation of | |

(2.4) evaluation of time derivatives of | |

(3) Flux-acceleration compatibility (spatial interpolation): | |

(4) Evaluation of time derivatives of |

In this work, this first algorithm is employed associated to FEM-FEM coupled procedures in which the acoustic subdomains are modelled considering the Green-Newmark method (explicit technique), and the dynamic subdomains are modelled considering the Newmark method (implicit technique), as well as to FEM-FEM, and FEM-FDM coupled procedures in which all subdomains are modelled considering the central difference method (explicit technique).

In the second explicit direct coupling algorithm focused here, the displacements related to the dynamic subdomains are computed directly, since their evaluation only takes into account results corresponding to previous time steps (

The detailed algorithm for this second ExDCA is presented in Table

ExDCA-2 algorithm.

Time-step loop (based on | |

(1) Dynamic subdomains analyses: evaluation of | |

(2) Evaluation of time derivatives of | |

(3) Acceleration temporal extrapolation: | |

(4) Flux-acceleration compatibility (spatial interpolation): | |

(5) Acoustic subdomains analyses: evaluation of | |

(6) Pressure temporal interpolation: | |

(7) Force-pressure compatibility (spatial interpolation): | |

(8) Evaluation of time derivatives of | |

(9) Evaluation of time derivatives of |

In the implicit iterative approach, each subdomain of the model is analysed independently (as in the ExDCA), and a successive renewal of the variables at the common interfaces is performed, until convergence is achieved. In order to maximize the efficiency and robustness of the iterative coupling algorithm, the evaluation of an optimised relaxation parameter is introduced, taking into account the minimisation of a square error functional.

Initially, in the ImICA, the dynamic subdomains are analysed and the displacements at the common interfaces are evaluated, as well as its time derivatives. A relaxation parameter

The algorithm representing the ImICA is presented in Table

ImICA algorithm.

Time-step loop (based on | |

(1) Iterative analysis (until convergence): | |

(1.1) dynamic subdomains analyses: evaluation of | |

(1.2 or 1.3) evaluation of time derivatives of | |

(1.3 or 1.2) adoption of a relaxation parameter: | |

(1.4) acceleration temporal extrapolation: | |

(1.5) flux-acceleration compatibility (spatial interpolation): | |

(1.6) acoustic subdomains analyses: evaluation of | |

(1.7) pressure temporal interpolation: | |

(1.8) force-pressure compatibility (spatial interpolation): | |

(2) Evaluation of time derivatives of | |

(3) Evaluation of time derivatives of |

The effectiveness of the iterative coupling methodology is intimately related to the relaxation parameter selection; an inappropriate selection for

Substituting (

To find the optimal

In the following sub-sections, some numerical applications are presented, illustrating the performance and potentialities of the discussed coupling methodologies. In the first application, a multidomain column is analyzed, considering several geometrical and physical configurations, as well as coupling procedures. In this case, acoustic-acoustic, acoustic-dynamic, and dynamic-dynamic coupled models are discussed, taking into account axisymmetric, two-dimensional, and three-dimensional configurations. In the second application, a dam-reservoir system is analyzed, considering once more several coupling techniques. In this case, a two-dimensional model is focused, and some advanced analyses are carried out, such as the modeling of nonlinear behavior and infinite media. In the last application, a tube of steel, submerged in water, is analyzed. In this case, axisymmetric models are focused, and, once again, several geometric and numeric configurations are considered. Along the applications discussed here, a large scope of coupling procedures is presented, namely: (i) for the ExDCA—FEM-FEM, FEM-BEM and FEM-FDM coupling procedures; (ii) for the ImICA—FEM-BEM, DBEM-BEM (which is referred to here as BEM-BEM 1), BEM-BEM (which is referred to here as BEM-BEM 2) and MLPG-MLPG coupling procedures. In this way, the reader can compare and better visualize some benefits and drawbacks of each methodology, considering an ample range of configurations.

The first example is that of a prismatic body behaving like a one-dimensional column. Initially, the column is analysed as an acoustic model [

Column model: (a) FEM-BEM acoustic-acoustic two-dimensional model; (b) MLPG-MLPG and FEM-FEM acoustic-dynamic two-dimensional model; (c) FEM-BEM acoustic-acoustic axisymmetric model; (d) FEM-FEM and FEM-FDM acoustic-acoustic/acoustic-dynamic/dynamic-dynamic three-dimensional model; (e) sketch of the three-dimensional model spatial discretization.

In Figure

Time-history results at points A and B taking into account FEM-BEM coupling procedures and different temporal discretizations for each subdomain: (a) explicit direct coupling analysis; (b) implicit iterative coupling analysis.

ExDCA

ImICA

In a second approach for the column model, the acoustic-dynamic coupled problem is focused (fluid-solid column [

Two spatial-temporal MLPG discretizations are considered to analyse the model, namely: (i) discretization 1—153 nodes are employed to spatially discretize each subdomain, and the time step adopted is

Time-history results at points A, B, and C taking into account MLPG-MLPG coupling procedures (ImICA) and different refinement levels: (a) discretization 1; (b) discretization 2.

The same fluid-solid column is analysed considering FEM-FEM coupled procedures based on the Green-Newmark method and on the Newmark method (ExDCA). In this case, 200 square finite elements are employed to discretize each subdomain of the model, and the time discretization is specified by

Time-history results for the solid-fluid column at points A, B, and C taking into account FEM-FEM coupling procedures (ExDCA) and different physical models: (a) homogeneous wave propagation velocities; (b) heterogeneous wave propagation velocities.

In a third approach for the column model, the propagation of acoustic waves through a prismatic circular column is analysed (axisymmetric model [

Time-history results for acoustic pressures taking into account FEM-BEM coupling procedures (ImICA) applied to a heterogeneous axisymmetric model: (a) results at point A; (b) results at point B.

In a fourth approach for the column model, three-dimensional analyses are considered, taking into account FEM-FEM and FEM-FDM explicit coupling approaches (ExDCA) based on the central difference time-marching method [

Three different numerical models are considered to simulate this problem, taking into account different coupling procedure combinations. A sketch of the three models adopted is presented in Figure

Two numerical analyses are considered, namely: (i) homogeneous analysis, where the entire column is considered composed by

Time-history results (displacement/pressure x time) at the interface of the three-dimensional column taking into account FEM-FEM and FEM-FDM coupling procedures (ExDCA): (a) homogeneous analysis; (b) heterogeneous analysis.

Considering this first example, the advantages of the discussed multidomain decomposition procedures may be highlighted under several aspects: different time steps are easily adopted for each subdomain and, as a consequence, the algorithm becomes quite robust even when considering media with high properties contrast; moreover, less systems of equations need to be solved along the time-marching process; not all subdomains need to be considered at initial time steps, the activation/initialisation of different subdomains may be controlled based on the properties of the model (wave propagation velocities, etc.), saving most of the computational effort of the first time steps, and so forth.

In this second example, a dam-reservoir system, as depicted in Figure ^{2}; ^{3}. The adjacent water is characterized by a mass density ^{3} and a wave velocity

Sketch of the dam-reservoir system.

Several ImICA and ExDCA are employed to analyze the dam-reservoir system. Taking into account the ImICA, the following discretizations are considered: (i) FEM-BEM: in this case, 93 quadrilateral finite elements are employed to discretize the dam, and the fluid is discretized by constant-length boundary elements (

Time-history results for the dam-reservoir system considering the ImICA: (a)

Taking into account the ExDCA, the following discretizations are considered: (i) FEM-BEM—same as in the ImICA. The time-steps adopted are

Time-history results for the dam-reservoir system considering the ExDCA: (a)

In this example, the advantages of employing different discretization procedures to analyze different subdomains of the global model can be explored. For instance, for the semi-infinite fluid domain, the BEM can be regarded as an appropriate discretization technique (infinite domain analysis), whereas domain-discretization methods can be applied to model the dam and consider some eventual more complicate behavior. In Figure

Time-history results considering linear and nonlinear material behavior and FEM-BEM implicit iterative coupling analyses.

The results presented so far are obtained taking into account a closed-domain dam, (null displacements are prescribed at the base of the dam and null fluxes are prescribed at the base of the storage lake). As is well known, boundary element formulations are an extremely elegant tool to model infinite media. As a consequence, in the present BEM-BEM 2 coupling context, analyses considering an opened-domain dam (acoustic-dynamic coupling also being carried out at the base of the storage lake) can be provided very easily. For the opened-domain dam case, time-history results are depicted in Figure

Time-history results for the opened-domain dam-reservoir system considering a BEM-BEM implicit iterative coupling analysis.

Scaled displacement results for the dam (

In this application, two analyses of a tube of steel submerged in water (axisymmetric models) are carried out. A sketch of the first model is depicted in Figure

Sketch of the tube submerged in water: (a) first case of analysis; (b) second case of analysis; (c) detail of the FEM mesh adopted to model the neighbourhood of the spherical source.

Time-history results for the tube of steel submerged in water (first case of analysis): (a) pressures at point A; (b) pressures at point B; (c) horizontal displacements at point C.

A sketch of the second model is depicted in Figure

The results achieved for the hydrodynamic pressure at the receiver (hydrophone) are depicted in Figures

Numerical results (pressure x time) at the receiver (second case of analysis) considering FEM-FDM explicit direct coupling procedures and (a) punctual and (b) spherical sources. (c) Experimental results at the receiver.

Photos of the experiment: (a) tube installation through the water tank input gate; (b) tank facilities; (c) tube inside the tank (view through the gate); (d) acoustic transductor ITC 1032 (source/receptor).

Pressure distribution for the punctual source case (FDM mesh) at three different moments: (a) begin of propagation; (b) wave fronts due to the faster propagation through the tube wall (head waves); (c) reinforcement of amplitude.

The present paper discusses multidomain decomposition techniques to model the propagation of interacting acoustic-elastic waves considering several coupling procedures. Two basic algorithms are presented here, namely, the ExDCA (explicit direct coupling approach) and the ImICA (implicit iterative coupling approach), which are based on explicit and implicit time-marching schemes, respectively, and multidomain decomposition coupling procedures. Within the context of these two basic algorithms, several coupled numerical methods are presented along the paper, such as FEM-FEM, FEM-FDM, FEM-BEM, BEM-BEM, DBEM-BEM, and MLPG-MLPG. Independent temporal and spatial (i.e., no matching nodes in common interfaces) discretizations within interacting subdomains are also discussed in the paper, being several applications of the discussed multilevel time-step algorithm presented along Section

The financial support by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais), and PETROBRAS (Project no. 0050.0011058.05.3) is greatly acknowledged.