One‐dimensional dispersion analysis for the element‐free Galerkin method for the Helmholtz equation

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd.     

Mechanically based models: Adaptive refinement for B‐spline finite element

This article presents two new methods for adaptive refinement of a B‐spline finite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively refining a B‐spline finite element solution are a local variant of np‐refinement and a local variant of h‐refinement. The key component in the np‐refinement is the linear co‐ordinate transformation introduced into the refined element. The transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements. Therefore, the assembly proceeds as with classic finite elements, while the solution approximation conforms exactly along the inter‐element boundaries. For the h‐refinement, this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical B‐spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C⁰ as well as the Hermite B‐spline C¹ continuous shapes. For sculptured solids, C⁰ continuity only is considered in this article. The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two‐dimensional Poisson equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 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.     

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

Abstract

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?² u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.     

Higher‐order responses of three‐dimensional elastic plate structures and their numerical illustration by p‐FEM

The displacements of three‐dimensional linearly elastic plate domains can be expanded as a compound power‐series asymptotics, when the thickness parameter ε tends to zero. The leading term u ⁰ in this expansion is the well‐known Kirchhoff–Love displacement field, which is the solution to the limit case when ε→0. Herein, we focus our discussion on plate domains with either clamped or free lateral boundary conditions, and characterize the loading conditions for which the leading term vanishes. In these situations the first non‐zero term u ^k in the expansion appears for k=2, 3 or 4 and is denoted as higher‐order response of order 2,3 or 4, respectively. We provide herein explicit loading conditions under which higher order responses in three‐dimensional plate structures are visible, and demonstrate the mathematical analysis by numerical simulation using the p‐version finite element method. Owing to the need for highly accurate results and ‘needle elements’ (having extremely large aspect ratio up to 10000), a p‐version finite element analysis is mandatory for obtaining reliable and highly accurate results. Copyright © 2001 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.     

An adaptive strategy for the bivariate solution of finite element problems using multivariate nested Padé approximants

Most engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so‐called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well‐sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto‐poro‐acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd.     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

The mixed finite element method for the quasi‐static and dynamic analysis of viscoelastic timoshenko beams

The quasi‐static and dynamic responses of a linear viscoelastic Timoshenko beam on Winkler foundation are studied numerically by using the hybrid Laplace–Carson and finite element method. In this analysis the field equation for viscoelastic material is used. In the transformed Laplace–Carson space two new functionals have been constructed for viscoelastic Timoshenko beams through a systematic procedure based on the Gâteaux differential. These functionals have six and two independent variables respectively. Two mixed finite element formulations are obtained; TB12 and TB4. For the inverse transform Schapery and Fourier methods are used. The numerical results for quasi‐static and dynamic responses of several visco‐elastic models are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

Linear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.     

A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.     

New hybrid Laplace transform/finite element method for three-dimensional transient heat conduction problem

The paper presents results obtained by the implementation of a new hybrid Laplace transform/finite element method developed by the authors. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite element method. Previously reported hybrid Laplace transform/finite element methods¹ have been confined to one nodal solution at a time. When applied to many nodes it takes an excessive amount of computer time. By using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems. Test examples are used to show that the basic accuracy is comparable to that obtainable by analytical, finite difference and finite element methods.     

A new stabilized finite element method for reaction–diffusion problems: The source‐stabilized Petrov–Galerkin method

This paper proposes a new stabilized finite element method to solve singular diffusion problems described by the modified Helmholtz operator. The Galerkin method is known to produce spurious oscillations for low diffusion and various alternatives were proposed to improve the accuracy of the solution. The mostly used methods are the well‐known Galerkin least squares and Galerkin gradient least squares (GGLS). The GGLS method yields the exact nodal solution in the one‐dimensional case and for a uniform mesh. However, the behavior of the method deteriorates slightly in the multi‐dimensional case and for non‐uniform meshes. In this work we propose a new stabilized finite element method that leads to improved accuracy for multi‐dimensional problems. For the one‐dimensional case, the new method leads to the same results as the GGLS method and hence provides exact nodal solutions to the problem on uniform meshes. The proposed method is a Galerkin discretization used to solve a modified equation that includes a term depending on the gradient of the original partial differential equation. Copyright © 2008 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

The scaled boundary finite element method in structural dynamics

The scaled boundary finite element method is extended to solve problems of structural dynamics. The dynamic stiffness matrix of a bounded (finite) domain is obtained as a continued fraction solution for the scaled boundary finite element equation. The inertial effect at high frequencies is modeled by high‐order terms of the continued fraction without introducing an internal mesh. By using this solution and introducing auxiliary variables, the equation of motion of the bounded domain is expressed in high‐order static stiffness and mass matrices. Standard procedures in structural dynamics can be applied to perform modal analyses and transient response analyses directly in the time domain. Numerical examples for modal and direct time‐domain analyses are presented. Rapid convergence is observed as the order of continued fraction increases. A guideline for selecting the order of continued fraction is proposed and validated. High computational efficiency is demonstrated for problems with stress singularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Non‐local dispersive model for wave propagation in heterogeneous media: multi‐dimensional case

Three non‐dispersive models in multi‐dimensions have been developed. The first model consists of a leading‐order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non‐local model, contains a fourth‐order spatial derivative and thus requires C¹ continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C⁰ continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd.     

A partition of unity FEM for time‐dependent diffusion problems using multiple enrichment functions

An enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right‐hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p‐version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time‐dependent diffusion applications efficiently and with an appropriate level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

In this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u , p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem ( P +λ Q +λ² R ) d = 0 is obtained, where the saddle‐point‐type matrices P , Q , R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval ??(λ) ∈ (?0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley & Sons, Ltd.