An efficient finite element time‐domain formulation for the elastic second‐order wave equation: A non‐split complex frequency shifted convolutional PML

The perfectly matched layer (PML) technique has demonstrated very high efficiency as absorbing boundary condition for the elastic wave equation recast as a first‐order system in velocity and stress in attenuating non‐grazing bulk and surface waves. This paper develops a novel convolutional PML formulation based on the second‐order wave equation with displacements as the only unknowns to annihilate spurious reflections from near‐grazing waves. The derived variational form allows for the use of e.g. finite element and the spectral element methods as spatial discretization schemes. A recursive convolution update scheme of second‐order accuracy is employed such that highly stable, effective time integration with the Newmark‐beta (implicit and explicit with mass lumping) method is achieved. The implementation requires minor modifications of existing displacement‐based finite element software, and the stability and efficiency of the proposed formulation is verified by relevant two‐dimensional benchmarks that accommodate bulk and surface waves. Copyright © 2011 John Wiley & Sons, Ltd.     

Fast direct solution of the Helmholtz equation with a perfectly matched layer or an absorbing boundary condition

We consider the efficient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) finite‐element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction‐type fast direct solver. We present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the reflection. Copyright © 2003 John Wiley & Sons, Ltd.     

The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation

We investigate the feasibility of using the perfectly matched layer (PML) as an absorbing boundary condition for the ultra weak variational formulation (UWVF) of the 3D Helmholtz equation. The PML is derived using complex stretching of the spatial variables. This leads to a modified Helmholtz equation for which the UWVF can be derived. In the standard discrete UWVF, the approximating subspace is constructed from local solutions of the Helmholtz equation. In previous studies plane wave basis functions have been advocated because they simplify the building of the UWVF matrices. For the PML domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections. The method is validated by numerical experiments. In comparison to a low‐order absorbing boundary condition, the PML shows superior performance. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient time‐domain perfectly matched layers formulation for elastodynamics on spherical domains

Many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain. One widely used approach for truncating the infinite domain is the so‐called method of perfectly matched layers (PMLs). Most existing PML formulations are developed for finite difference methods based on the first‐order velocity‐stress form of the elasticity equations, and they are not straight‐forward to implement using standard finite element methods (FEMs) on unstructured meshes. Some of the problems with these formulations include the application of boundary conditions in half‐space problems and in the treatment of edges and/or corners for time‐domain problems. Several PML formulations, which do work with FEMs have been proposed, although most of them still have some of these problems and/or they require a large number of auxiliary nodal history/memory variables. In this work, we develop a new PML formulation for time‐domain elastodynamics on a spherical domain, which reduces to a two‐dimensional formulation under the assumption of axisymmetry. Our formulation is well‐suited for implementation using FEMs, where it requires lower memory than existing formulations, and it allows for natural application of boundary conditions. We solve example problems on two‐dimensional and three‐dimensional domains using a high‐order discontinuous Galerkin (DG) discretization on unstructured meshes and explicit time‐stepping. We also study an approach for stabilization of the discrete equations, and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves

The use of a perfectly matched layer (PML) model is an efficient approach toward the bounded‐domain modelling of wave propagation on unbounded domains. This paper formulates a three‐dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement‐based finite element setting. The novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration, thus making it practicable for use in large‐scale three‐dimensional dynamic analyses. An efficient method of calculating the strain terms in the PML is developed in order to take advantage of the lack of the overhead of solving equations at each time step. The PML formulation is studied and validated first for a semi‐infinite bar and then for the classical soil–structure interaction problems of a square flexible footing on a (i) half‐space, (ii) layer on a half‐space and (iii) layer on a rigid base. Numerical results for these problems demonstrate that the PML models produce highly accurate results with small bounded domains and at low computational cost and that these models are long‐time stable, with critical time step sizes similar to those of corresponding fully elastic models. Copyright © 2008 John Wiley & Sons, Ltd.     

Unified formulation of radiation conditions for the wave equation

A family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second‐order wave equations on the radiating surface. Several well‐established radiation boundary conditions appear as special cases, corresponding to different choices of the coefficients in the rational approximation. The relation between these choices is established, and an explicit formulation in terms of selected directions with ideal transmission is presented. A mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface. Numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

An adaptive finite element method for second‐order plate theory

We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P₂‐approximations on simplices for the out‐of‐plane deformations, using C⁰‐continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient indirect boundary element technique for multi‐frequency acoustic analysis

An efficient indirect boundary element solution procedure for the analysis of multi‐frequency acoustic problems is developed by incorporating techniques that improve the efficiency of the integration and matrix solution phases of the computing process. The integration phase is made efficient by computing the system matrices at few predetermined key frequencies only and then evaluating the matrices at other intermediate frequencies by quadratic interpolation. The matrix solution process is made efficient by iterating the solutions using the factored form of the key frequency matrices. The effectiveness of the present development is confirmed by solving a number of example problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A two‐and‐a‐half‐dimensional displacement‐based PML for elastodynamic wave propagation

This paper presents a perfectly matched layer (PML) technique for the numerical simulation of three‐dimensional linear elastodynamic problems, where the geometry is invariant in the longitudinal direction. Examples include transportation infrastructure, dams, lifelines, and alluvial valleys. For longitudinally invariant geometries, a computationally efficient two‐and‐a‐half‐dimensional (2.5D) approach can be applied, where the Fourier transform from the longitudinal coordinate to the wavenumber domain allows for the representation of the three‐dimensional radiated wave field on a two‐dimensional mesh. In this 2.5D framework, the equilibrium equations of a PML continuum are formulated in a weak form for an isotropic elastodynamic medium and discretized using a Galerkin approach. The 2.5D PML methodology is validated by computing the Green's displacements of a homogeneous halfspace, demonstrating that the 2.5D PML absorbs all propagating waves for different angles of incidence. Furthermore, the dynamic stiffness of a rigid strip foundation and the efficiency of a vibration isolating screen are computed. The examples demonstrate that the PML methodology is computationally efficient, especially when only the response of the structure or the near field response is of interest.Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order doubly asymptotic open boundaries for scalar wave equation

High‐order doubly asymptotic open boundaries are developed for transient analyses of scalar waves propagating in a semi‐infinite layer with a constant depth and a circular cavity in a full‐plane. The open boundaries are derived in the frequency domain as doubly asymptotic continued fraction solutions of the dynamic stiffness of the unbounded domains. Each term of the continued fraction is a linear function of the excitation frequency. The constants of the continued fraction solutions are determined recursively. The continued fraction solution is expressed in the time domain as ordinary differential equations, which can be solved by standard time‐stepping schemes. No parameters other than the orders of the low‐ and high‐frequency expansions need to be selected by users. Numerical experiments demonstrate that evanescent waves and long‐time (low‐frequency) responses are simulated accurately. In comparison with singly asymptotic open boundaries, significant gain in accuracy is achieved at no additional computational cost. Copyright © 2009 John Wiley & Sons, Ltd.     

Partition experimental designs for sequential processes: Part II—second‐order models

Second‐order experimental designs are employed when an experimenter wishes to fit a second‐order model to account for response curvature over the region of interest. Partition designs are utilized when the output quality or performance characteristics of a product depend not only on the effect of the factors in the current process, but the effects of factors from preceding processes. Standard experimental design methods are often difficult to apply to several sequential processes. We present an approach to building second‐order response models for sequential processes with several design factors and multiple responses. The proposed design expands current experimental designs to incorporate two processes into one partitioned design. Potential advantages include a reduction in the time required to execute the experiment, a decrease in the number of experimental runs, and improved understanding of the process variables and their influence on the responses. Copyright © 2002 John Wiley & Sons, Ltd.     

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

High‐order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high‐order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial‐boundary value problems, eigenvalue problems, and high‐order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high‐order differential equations and time‐dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high‐order accuracy, while maintaining the same or similar stability conditions of the standard high‐order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non‐standard high‐order methods is also considered. Copyright © 2008 John Wiley & Sons, Ltd.     

Three‐dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries

We report on a generalization of the Bayliss–Gunzburger–Turkel non‐reflecting boundary conditions to arbitrarily shaped convex artificial boundaries. For elongated scatterers such as submarines, we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three‐dimensional acoustic scattering problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time‐dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first‐order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non‐reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high‐order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical solution of second‐order,two‐point boundary value problems using continuous genetic algorithms

Second‐order, two‐point boundary‐value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non‐linear problems, these methods require some major modifications that include the use of some root‐finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non‐linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h²), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient second‐order characteristic finite element method for non‐linear aerosol dynamic equations

In the paper we consider the non‐linear aerosol dynamic equation on time and particle size, which contains the advection process of condensation growth and the process of non‐linear coagulation. We develop an efficient second‐order characteristic finite element method for solving the problem. A high accurate characteristic method is proposed to treat the condensation advection while a second‐order extrapolation along the characteristics is proposed to approximate the non‐linear coagulation. The method has second‐order accuracy in time and the optimal‐order accuracy of finite element spaces in particle size, which improves the first‐order accuracy in time of the classical characteristic method. Numerical experiments show the efficient performance of our method for problems of log‐normal distribution aerosols in both the Euler coordinates and the logarithmic coordinates. Copyright © 2009 John Wiley & Sons, Ltd.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.