Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

The paper presents a gradient‐based topology optimization formulation that allows to solve acoustic–structure (vibro‐acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface‐coupling integrals, however, such a formulation does not allow for free material re‐distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u /p‐formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two‐dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of hybrid Trefftz finite element method to non‐linear problems of minimal surface

This investigation provides a hybrid Trefftz finite element approach for analysing minimal surface problems. The approach is based on combining Trefftz finite element formulation with radial basis functions (RBF) and the analogue equation method (AEM). In this method, use of the analogue equation approach avoids the difficulty of treating the non‐linear terms appearing in the soap bubble equation, making it possible to solve non‐linear problems with the Trefftz method. Global RBF is used to approximate the inhomogeneous term induced from non‐linear functions and other loading terms. Finally, some numerical experiments are implemented to verify the efficiency of this method. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large‐scale problems. This paper discusses an application of FMM to three‐dimensional boundary integral equation method for elastostatic crack problems. The boundary integral equation for many crack problems is discretized with FMM and Galerkin's method. The resulting algebraic equation is solved with generalized minimum residual method (GMRES). The numerical results show that FMM is more efficient than conventional methods when the number of unknowns is more than about 1200 and, therefore, can be useful in large‐scale analyses of fracture mechanics. Copyright © 2001 John Wiley & Sons, Ltd.     

An assumed‐gradient finite element method for the level set equation

The level set equation is a non‐linear advection equation, and standard finite‐element and finite‐difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed‐distance function. For some interface‐evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity‐capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed‐gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level‐set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton–Jacobi equation with convex/non‐convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite‐element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd.     

A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity

In this paper, a new and simple boundary‐domain integral equation is presented for heat conduction problems with heat generation and non‐homogeneous thermal conductivity. Since a normalized temperature is introduced to formulate the integral equation, temperature gradients are not involved in the domain integrals. The Green's function for the Laplace equation is used and, therefore, the derived integral equation has a unified form for different heat generations and thermal conductivities. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the normalized temperature using a series of basis functions and polynomials in global co‐ordinates. Numerical examples are given to demonstrate the robustness of the presented method. Copyright © 2005 John Wiley & Sons, Ltd.     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

Hybrid BEM for the early stage of unsteady transport process

This paper presents a new approach for solving the early stage of 3D time‐dependent transport problems in non‐homogeneous fractured porous media in which the initial distribution of concentration presents discontinuous jump at the interface of two regions of transport properties differing in several orders of magnitude. The goal of this formulation is to overcome the problems of different scales and apparent large flux during early time by combining the 3D dual reciprocity boundary element method and a semi‐analytical solution of the time‐dependent advection–diffusion equation, employing a two‐level finite difference time integration scheme. Theoretical background, validation results and practical applications for the advection–diffusion equation in fractured and continuous porous media are reported. The results show advantages for relatively large‐scale models and complex geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Large‐eddy simulation of separated leading‐edge flow in general co‐ordinates

An incompressible separated transitional boundary‐layer flow on a flat plate with a semi‐circular leading edge has been simulated and a very good agreement with the experimental data has been obtained, demonstrating how this technique may be applied even when finite difference formulae are used in the periodic direction. The entire transition process has been elucidated and vortical structures have been identified at different stages during the transition process. Efficient numerical methods for the large‐eddy simulation (LES) of turbulent flows in complex geometry are developed. The methods used are described in detail: body‐fitted co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered mesh with a dynamic subgrid‐scale model in general co‐ordinates. The main source of computational expense in simulations for incompressible flows is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. Fourier techniques can be employed to speed up the pressure solution significantly for a flow which is periodic in one dimension. With simple conditions fulfilled, it is possible to Fourier transform a discrete elliptic equation such as the Poisson equation for the pressure field, decomposing the problem into a set of two‐dimensional problems of similar type (Poisson‐like). Even when a complex geometry and body‐fitted curvilinear co‐ordinates are used in the other two dimensions, as in the present case, the resulting Fourier‐transformed 2D problems are much more efficiently solved than the 3D problem by iterative means. Copyright © 2000 John Wiley & Sons, Ltd.     

Convergence of the fixed point theorem with random parameters

Solving stochastic non‐linear dynamical problems represents a formidable task which, in many cases, can be achieved solely through numerical simulation techniques. This is true for high dimensional as well as low dimensional problems. One method to deal with the non‐linearity is to use the fixed point theorem which gives the convergence conditions of the iterative scheme towards the equilibrium point of the equation. In this paper we look at the particular case where the equilibrium equation depends on a random variable. This case arises for instance in the study of coupled non‐linear dynamical systems when structural uncertainties are introduced in the dynamical systems. We give almost sure and L ^p convergence conditions for the simulation iterative scheme. Copyright © 2003 John Wiley & Sons, Ltd.     

Direct simulations of the linear elastic displacements field based on a lattice Boltzmann model

In this paper, a lattice Boltzmann model for simulating linear elastic Lame equation is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on displacement distribution function in lattice Boltzmann equation. By using the technique of the higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales, we obtain the Lame equation with fourth‐order truncation errors. Based on this model, some problems with small deflection are simulated. The comparisons between the numerical results and the analytical solutions are given in detail. The numerical examples show that the lattice Boltzmann model can be used to solve problems of the linear elastic displacement field with small deflection. Copyright © 2015 John Wiley & Sons, Ltd.     

Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media

Weight‐adjusted inner products are easily invertible approximations to weighted L² inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.     

Scalar wave equation by the boundary element method: A D‐BEM approach with constant time‐weighting functions

A D‐BEM approach, based on time‐weighting residuals, is developed for the solution of two‐dimensional scalar wave propagation problems. The modified basic equation of the D‐BEM formulation is generated by weighting, with respect to time, the basic D‐BEM equation, under the assumption of linear and cubic time variation for the potential and for the flux. A constant time‐weighting function is adopted. The time integration reduces the order of the time‐derivative that appears in the domain integral; as a consequence, the initial conditions are directly taken into account. An assessment of the potentialities of the proposed formulation is provided by the examples included at the end of the work. Copyright © 2009 John Wiley & Sons, Ltd.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

On the singularities of Green's formula and its normal derivative,with an application to surface‐wave–body interaction problems

The paper presents the non‐singular forms of Green's formula and its normal derivative of exterior problems for three‐dimensional Laplace's equation. The main advantage of these modified formulations is that they are amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss flux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problems. The efficacy of the modified formulations is examined by a sphere, in an infinite domain, subject to Neumann and Dirichlet conditions, respectively. The modified integral formulations are further applied to a practical problem, i.e. surface‐wave–body interactions. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem. Removing the ‘irregular’ frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added‐mass and damping coefficients and wave exciting forces on a floating hemisphere. Comparing the numerical results with that by other approaches demonstrates the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

GPU‐accelerated boundary element method for Helmholtz' equation in three dimensions

Recently, the application of graphics processing units (GPUs) to scientific computations is attracting a great deal of attention, because GPUs are getting faster and more programmable. In particular, NVIDIA's GPUs called compute unified device architecture enable highly mutlithreaded parallel computing for non‐graphic applications. This paper proposes a novel way to accelerate the boundary element method (BEM) for three‐dimensional Helmholtz' equation using CUDA. Adopting the techniques for the data caching and the double–single precision floating‐point arithmetic, we implemented a GPU‐accelerated BEM program for GeForce 8‐series GPUs. The program performed 6–23 times faster than a normal BEM program, which was optimized for an Intel's quad‐core CPU, for a series of boundary value problems with 8000–128000 unknowns, and it sustained a performance of 167 Gflop/s for the largest problem (1 058 000 unknowns). The accuracy of our BEM program was almost the same as that of the regular BEM program using the double precision floating‐point arithmetic. In addition, our BEM was applicable to solve realistic problems. In conclusion, the present GPU‐accelerated BEM works rapidly and precisely for solving large‐scale boundary value problems for Helmholtz' equation. Copyright © 2009 John Wiley & Sons, Ltd.     

A method for structural dynamic contact problems with friction and wear

A method for structural dynamic contact problems with friction and wear is suggested. The method is obtained by including wear in the non‐smooth contact dynamics method of Moreau. A comparison of the method to the discrete energy‐momentum method of Simo and Tarnow is also outlined briefly. The fully discrete equations are treated using the augmented Lagrangian approach, where a non‐smooth Newton method is used as the equation solver. Two two‐dimensional examples are solved by the method. It is investigated how solutions of contact, friction and wear are influenced by inertia. It is shown that the quasi‐static assumption might be questionable for solving contact problems with friction and wear. Copyright © 2003 John Wiley & Sons, Ltd.     

Implementation of meshless LBIE method to the 2D non‐linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.     

Skeletal reduction of boundary value problems

Boundary value problems posed over thin solids are amenable to a dimensional reduction in that one or more spatial variables may be eliminated from the governing equation, resulting in significant computational gains with minimal loss in accuracy. Extant dimensional reduction techniques rely on representing the solid as a hypothetical mid‐surface plus a possibly varying thickness. Such techniques are however hard to automate since the mid‐surface is often ill‐defined and ambiguous. We propose here a skeletal representation based dimensional reduction of boundary value problems. The proposed technique has the computational advantages of mid‐surface reduction, but can be easily automated. A systematic methodology is presented for reducing boundary value problems to lower‐dimensional problems over the skeleton of a solid. The theoretical properties of the proposed method are derived, and supported by representative numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.