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.     

DRBEM solution of exterior nonlinear wave problem using FDM and LSM time integrations

The nonlinear wave equation is solved numerically in an exterior region. For the discretization of the space derivatives dual reciprocity boundary element method (DRBEM) is applied using the fundamental solution of Laplace equation. The time derivative and the nonlinearity are treated as the nonhomogenity. The boundary integrals coming from the far boundary are eliminated using rational and exponential interpolation functions which have decay properties far away from the region of interest. The resulting system of ordinary differential equations in time are solved using finite difference method (FDM) with a relaxation parameter and least squares method (LSM). The proposed methods are examined with numerical test problems in which the behaviours of solutions are known. Although it gives almost the same accuracy with the DRBEM+FDM procedure, DRBEM+LSM solution procedure is preferred, since it is a direct method without the need of a parameter.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.     

Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation

When numerical methods such as the finite element method (FEM) are used to solve the Helmholtz equation, the solutions suffer from the so‐called pollution effect which leads to inaccurate results, especially for high wave numbers. The main reason for this is that the wave number of the numerical solution disagrees with the wave number of the exact solution, which is known as dispersion. In order to obtain admissible results a very high element resolution is necessary and increased computational time and memory capacity are the consequences. In this paper a meshfree method, namely the radial point interpolation method (RPIM), is investigated with respect to the pollution effect in the 2D‐case. It is shown that this methodology is able to reduce the dispersion significantly. Two modifications of the RPIM, namely one with polynomial reproduction and another one with a problem‐dependent sine/cosine basis, are also described and tested. Numerical experiments are carried out to demonstrate the advantages of the method compared with the FEM. For identical discretizations, the RPIM yields considerably better results than the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

An error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40 (1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.     

Weighted Smolyak algorithm for solution of stochastic differential equations on non‐uniform probability measures

This paper deals with numerical solution of differential equations with random inputs, defined on bounded random domain with non‐uniform probability measures. Recently, there has been a growing interest in the stochastic collocation approach, which seeks to approximate the unknown stochastic solution using polynomial interpolation in the multi‐dimensional random domain. Existing approaches employ sparse grid interpolation based on the Smolyak algorithm, which leads to orders of magnitude reduction in the number of support nodes as compared with usual tensor product. However, such sparse grid interpolation approaches based on piecewise linear interpolation employ uniformly sampled nodes from the random domain and do not take into account the probability measures during the construction of the sparse grids. Such a construction based on uniform sparse grids may not be ideal, especially for highly skewed or localized probability measures. To this end, this work proposes a weighted Smolyak algorithm based on piecewise linear basis functions, which incorporates information regarding non‐uniform probability measures, during the construction of sparse grids. The basic idea is to construct piecewise linear univariate interpolation formulas, where the support nodes are specially chosen based on the marginal probability distribution. These weighted univariate interpolation formulas are then used to construct weighted sparse grid interpolants, using the standard Smolyak algorithm. This algorithm results in sparse grids with higher number of support nodes in regions of the random domain with higher probability density. Several numerical examples are presented to demonstrate that the proposed approach results in a more efficient algorithm, for the purpose of computation of moments of the stochastic solution, while maintaining the accuracy of the approximation of the solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Improvements for the ultra weak variational formulation

In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two‐dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

Assessment and applications of point interpolation methods for computational mechanics

Two meshfree point interpolation methods (PIMs), which are based on the polynomial and the radial basis functions, have been proposed recently in addition to the earlier work with the moving least‐squares (MLS) approximation for the field function approximation. However, it is found that PIMs cannot automatically ensure the compatibility of the solution when they are used together with the energy principles. In this paper, issues related to the compatibility of PIMs are studied. A technique of background cell‐based nodal selections and a penalty method are proposed to enforce the compatibility of the solution of PIMs. The patch test is studied in great detail. The convergences and performances are investigated for both conforming and non‐conforming PIMs. It is found that those methods of the PIM family are very easy to implement, and are very efficient in obtaining numerical solutions for problems of computational mechanics. Copyright © 2004 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.     

SPEED: SPectral Elements in Elastodynamics with Discontinuous Galerkin: a non‐conforming approach for 3D multi‐scale problems

This work presents a new high performance open‐source numerical code, namely SPectral Elements in Elastodynamics with Discontinuous Galerkin, to approach seismic wave propagation analysis in visco‐elastic heterogeneous three‐dimensional media on both local and regional scale. Based on non‐conforming high‐order techniques, such as the discontinuous Galerkin spectral approximation, along with efficient and scalable algorithms, the code allows one to deal with a non‐uniform polynomial degree distribution as well as a locally varying mesh size. Validation benchmarks are illustrated to check the accuracy, stability, and performance features of the parallel kernel, whereas illustrative examples are discussed to highlight the engineering applications of the method. The proposed method turns out to be particularly useful for a variety of earthquake engineering problems, such as modeling of dynamic soil structure and site‐city interaction effects, where accounting for multiscale wave propagation phenomena as well as sharp discontinuities in mechanical properties of the media is crucial. Copyright © 2013 John Wiley & Sons, Ltd.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.     

A spline strip kernel particle method and its application to two‐dimensional elasticity problems

In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two‐dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh‐free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B₃‐spline function in the longitudinal direction. The formulation is validated on several beam and semi‐infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd.     

Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

Time‐domain vector potential technique for the meshless radial point interpolation method

A time‐domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second‐order vector potential wave equation intrinsically enforces the divergence‐free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first‐order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence‐free formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

A high‐order accurate particle‐in‐cell method

We propose the use of high‐order weighted essentially non‐oscillatory interpolation and moving‐least‐squares approximation schemes alongside high‐order time integration to enable high‐order accurate particle‐in‐cell methods. The key insight is to view the unstructured set of particles as the underlying representation of the continuous fields; the grid used to evaluate integro–differential coupling terms is purely auxiliary. We also include a novel regularization term to avoid the accumulation of noise in the particle samples without harming the convergence rate. We include numerical examples for several model problems: advection–diffusion, shallow water, and incompressible Navier–Stokes in vorticity formulation. The implementation demonstrates fourth‐order convergence, shows very low numerical dissipation, and is competitive with high‐order Eulerian schemes. Copyright © 2012 John Wiley & Sons, Ltd.