Coupled FE–BE method for eigenvalue analysis of elastic structures submerged in an infinite fluid domain

For thin elastic structures submerged in heavy fluid, e.g., water, a strong interaction between the structural domain and the fluid domain occurs and significantly alters the eigenfrequencies. Therefore, the eigenanalysis of the fluid–structure interaction system is necessary. In this paper, a coupled finite element and boundary element (FE–BE) method is developed for the numerical eigenanalysis of the fluid–structure interaction problems. The structure is modeled by the finite element method. The compressibility of the fluid is taken into consideration, and hence the Helmholtz equation is employed as the governing equation and solved by the boundary element method (BEM). The resulting nonlinear eigenvalue problem is converted into a small linear one by applying a contour integral method. Adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenvalues of interest. The Burton–Miller formulation is applied to tackle the fictitious eigenfrequency problem of the BEM, and the optimal choice of its coupling parameter is investigated for the coupled FE–BE method. Numerical examples are given and discussed to demonstrate the effectiveness and accuracy of the developed FE–BE method. Copyright © 2016 John Wiley & Sons, Ltd.     

FE/FMBE coupling to model fluid–structure interaction

In this paper, a finite element (FE)/fast multipole boundary element (FMBE)‐coupling method is presented for modeling fluid–structure interaction problems numerically. Vibrating structures are assumed to consist of elastic or sound absorbing materials. An FE method (FEM) is used for this part of the solution. This structural sub‐domain is embedded in a homogeneous fluid. The case where the boundary of the structural sub‐domain has a very complex geometry is of special interest. In this case, the BE method (BEM) is a more suitable numerical tool than FEM to account for the sound propagation in the homogeneous fluid. The efficiency of the BEM is increased by using FMBEM. The BE‐surface mesh required is directly generated by the FE‐mesh used to discretize the structural sub‐domain and the absorbing material. This FE/FMBE‐coupling method makes it possible to predict the effects of arbitrarily shaped absorbing materials and vibrating structures on the sound field in the surrounding fluid numerically. The coupling method proposed is used to study the acoustic behavior of the lining of an anechoic chamber and that of an entire anechoic chamber in the low‐frequency range. The numerical results obtained are compared with the experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

A fast BE‐FE coupling scheme for partly immersed bodies

Fluid–structure coupled problems are investigated to predict the vibro‐acoustic behavior of submerged bodies. The finite element method is applied for the structural part, whereas the boundary element method is used for the fluid domain. The focus of this paper is on partly immersed bodies. The fluid problem is favorably modeled by a half‐space formulation. This way, the Dirichlet boundary condition on the free fluid surface is incorporated by a half‐space fundamental solution. A fast multipole implementation is presented for the half‐space problem. In case of a high density of the fluid, the forces due to the acoustic pressure, which act on the structure, cannot be neglected. Thus, a strong coupling scheme is applied. An iterative solver is used to handle the coupled system. The efficiency of the proposed approach is discussed using a realistic model problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction

To predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well‐known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton–Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated. Copyright © 2008 John Wiley & Sons, Ltd.     

A new acoustic interface element for fluid-structure interaction problems

A new comprehensive acoustic 2-D interface element capable of coupling the boundary element (BE) and finite element (FE) discretizations has been formulated for fluid–structure interaction problems. The Helmholtz equation governing the acoustic pressure in a fluid is discretized using the BE method and coupled to the FE discretization of a vibrating structure that is in contact with the fluid. Since the BE method naturally maps the infinite fluid domain into finite node points on the fluid–structure interface, the formulation is especially useful for problems where the fluid domain extends to infinity. Details of the BE matrix computation process adapted to FE code architecture are included for easy incorporation of the interface element in FE codes. The interface element has been used to solve a few simple fluid–structure problems to demonstrate the validity of the formulation. Also, the vibration response of a submerged cylindrical shell has been computed and compared with the results from an entirely finite element formulation.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

BOUNDARY ELEMENT–FINITE ELEMENT COUPLED EIGENANALYSIS OF FLUID–STRUCTURE SYSTEMS

The eigenanalysis of acoustical cavities with flexible structure boundaries, such as a fluid-filled container or an automobile cabin enclosure, is considered. An algebraic eigenvalue problem formulation for the fluid–structure problem is presented by combining the acoustic fluid boundary element eigenvalue analysis method and the structural finite elements. For many practical eigenproblems, use of finite elements to discretize the fluid domain leads to large stiffness and mass matrices. Since the acoustic boundary element discretization requires putting nodes only on the wetted surface of the structure, the size of the eigenproblem is reduced considerably, thus reducing the eigenvalue extraction effort. Futhermore, unlike in ordinary cases, the finite element discretization of pressure–displacement based fluid–structure problem gives rise to unsymmetric matrices. Therefore, the fact that the boundary element formulation produces unsymmetric matrices does not introduce additional difficulties here compared to the finite element case in the choice of an eigenvalue extraction procedure. Examples are included to demonstrate the fluid–structure eigenanalysis using boundary elements for the fluid domain and finite elements for the structure.     

Fast BEM–FEM mortar coupling for acoustic–structure interaction

A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure–acoustic field interaction. Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non‐matching grids. The resulting saddle‐point problem is solved by an approximate Uzawa‐type scheme in which the matrix–vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method. The algorithm is demonstrated on the example of a cavity‐backed elastic panel. Copyright © 2005 John Wiley & Sons, Ltd.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

Coupled adjoint‐based sensitivities in large‐displacement fluid‐structure interaction using algorithmic differentiation

A methodology for the calculation of gradients with respect to design parameters in general fluid‐structure interaction problems is presented. It is based on fixed‐point iterations on the adjoint variables of the coupled system using algorithmic differentiation. This removes the need for the construction of the analytic Jacobian for the coupled physical problem, which is the usual limitation for the computation of adjoints in most realistic applications. The formulation is shown to be amenable to partitioned solution methods for the adjoint equations. It also poses no restrictions to the nonlinear physics in either the fluid or structural field, other than the existence of a converged solution to the primal problem from which to compute the adjoints. We demonstrate the applicability of this procedure and the accuracy of the computed gradients on coupled problems involving viscous flows with geometrical and material nonlinearities in the structural domain.     

Computation of the time‐history response of dynamic problems using the boundary element method and modal techniques

The boundary element method in combination with modal techniques is used to calculate the response of transient excited structures in the time domain numerically. If the system matrices of a structure are evaluated with a fundamental solution in the frequency domain these matrices become functions of frequency which normally cannot be expressed analytically. The associated eigenvalue problem therefore is non‐linear and difficult to solve. For simplification a series expansion formula for the fundamental solution is used in different frequency ranges. Then the eigenvalue problem can be linearized and solved by direct or iterative methods. By using the orthogonal properties of the eigenfunctions, the normal modes of the dynamic problem can be uncoupled as is well known in vibration analysis. That way the transient response of a dynamic excited system in the time domain can be determined without difficulties. Displacements and stresses at different points of the structure are the result. Difficulties in the formulation of time‐dependent problems using the boundary element method can be avoided. There is no problem in considering modal damping factors, for general damping characteristics the associated fundamental solutions have to be found. Several examples are studied in the paper to illustrate how the new method can be applied. Copyright © 1999 John Wiley & Sons, Ltd.     

Interface‐reduction for the Craig–Bampton and Rubin method applied to FE–BE coupling with a large fluid–structure interface

Component mode‐based model‐order reduction (MOR) methods like the Craig–Bampton method or the Rubin method are known to be limited to structures with small coupling interfaces. This paper investigates two interface‐reduction methods for application of MOR to systems with large coupling interfaces: for the Craig–Bampton method a direct reduction method based on strain energy considerations is investigated. Additionally, for the Rubin method an iterative reduction scheme is proposed, which incrementally constructs the reduction basis. Hereby, attachment modes are tested if they sufficiently enlarge the spanned subspace of the current reduction basis. If so, the m‐orthogonal part is used to augment the basis. The methods are applied to FE–BE coupled systems in order to predict the vibro‐acoustic behavior of structures, which are partly immersed in water. Hereby, a strong coupling scheme is employed, since for dense fluids the feedback of the acoustic pressure onto the structure is not negligible. For two example structures, the efficiency of the reduction methods with respect to numerical effort, memory consumption and computation time is compared with the exact full‐order solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Monolithic modelling of electro‐mechanical coupling in micro‐structures

The purpose of the present work is to model and to simulate the coupling between the electric and mechanical fields. A new finite element approach is proposed to model strong electro‐mechanical coupling in micro‐structures with capacitive effect. The proposed approach is based on a monolithic formulation: the electric and the mechanical fields are solved simultaneously in the same formulation. This method provides a tangent stiffness matrix for the total coupled problem which allows to determine accurately the pull‐in voltage and the natural frequency of electro‐mechanical systems such as MEMs. To illustrate the methodology results are shown for the analysis of a micro‐bridge. Copyright © 2005 John Wiley & Sons, Ltd.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.