An improved meshless method for analyzing the time dependent problems in solid mechanics

In this paper, the local radial point interpolation method (LRPIM) is developed for the investigation of time dependent problems in solid mechanics. By a new integration scheme considered for the obtained meshless weak form, integrands are approximated up to the second order of the Taylor series and the integrals are evaluated on some points, which are located inside the local quadrature domains, called integration points. In order to show the efficiency of the suggested method, some time dependent mechanical problems are considered for the engineering structures such as beams and plates, which are subjected to dynamical loads, the deflections and stresses are evaluated. Finally, it has been shown that using the proposed method greatly reduces the number of integration points without affecting the accuracy of the results.     

A localized differential quadrature (LDQ) method and its application to the 2D wave equation

 Differential Quadrature (DQ) is a numerical technique of high accuracy, but it is sensitive to grid distribution and requires that the number of grid points cannot be too large. These two requirements greatly restrict wider applications of DQ method. Through a simplified stability analysis in this paper, it is concluded that these two limitations are due to stability requirements. This analysis leads us to propose to localize differential quadrature to a small neighbourhood so as to keep the balance of accuracy and stability. The derivatives at a grid point are approximated by a weighted sum of the points in its neighbourhood rather than of all grid points. The method is applied to the one- and two-dimensional wave equations. Numerical examples show the present method produces very accurate results while maintaining good stability. The proposed method enables us to solve more complicated problems and enhance DQ's flexibility significantly. Received: 23 October 2001 / Accepted: 3 July 2002     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli

 The radial basis functions (RBFs) have been proven to have excellent properties for interpolation problems, which can be considered as an efficient scheme for function approximation. In this paper, we will explore another type of approximation problem, that is, the derivative approximation, by the RBFs. A new approach, which is based on the differential quadrature (DQ) approximation for the derivative with RBFs as test functions, is proposed to approximate the first, second, and third order derivatives of a function. The performance of three commonly-used RBFs for some typical expressions of derivatives as well as the computation of one-dimensional Burgers equation are studied. Furthermore, the proposed method is applied to simulate natural convection in a concentric annulus by solving Navier–Stokes equations. The obtained results are compared well with exact data or benchmark solutions. Received: 27 June 2001 / Accepted: 29 July 2002     

Recursive solution of inverse natural convection problems by means of mode reduction

 A recursive method based on the Kalman filtering is developed to solve inverse natural convection problems of estimating the unsteady nonuniform wall heat flux from temperature measurements in the flow. By employing the Karhunen–Loève Galerkin procedure that reduces the Boussinesq equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the partial differential equations are overcome. The present method is assessed through several numerical experiments, and is found to yield satisfactory results. Received 20 January 2001 / Accepted 31 May 2001     

3D fracture analysis by the symmetric Galerkin BEM

 The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method. Received 6 November 2000     

A mixed least squares method for solving problems in linear elasticity: theoretical study

 In a previous paper we proposed a mixed least squares method for solving problems in linear elasticity. The solution to the equations of linear elasticity was obtained via minimization of a least squares functional depending on displacements and stresses. The performance of the method was tested numerically for low order elements for classical examples with well known analytical solutions. In this paper we derive a condition for the existence and uniqueness of the solution of the discrete problem for both compressible and incompressible cases, and verify the uniqueness of the solution analytically for two low order piece-wise polynomial FEM spaces. Received: 20 January 2001 / Accepted: 14 June 2002 The authors gratefully acknowledge the financial support provided by NASA George C. Marshall Space Flight Centre under contract number NAS8-38779.     

Solving thermal and phase change problems with the eXtended finite element method

 The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented. Of particular interest is the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface. This is effected through the use of a time-dependent basis formed from the union of traditional shape functions with a set of evolving enrichment functions. The enrichment is constructed through the partition of unity framework, so that the system of equations remains sparse and the resulting approximation is conforming. In this manner, local solutions and arbitrary discontinuities that cannot be represented by the standard shape functions are captured with the enrichment functions. A standard time-projection algorithm is employed to account for the time-dependence of the enrichment, and an iterative strategy is adopted to satisfy local interface conditions. The separation of the approximation into classical shape functions that remain fixed in time and the evolving enrichment leads to a very efficient solution strategy. The robustness and utility of the method is demonstrated with several benchmark problems involving moving heat sources and phase transformations. Received 20 May 2001 / Accepted 19 December 2001     

Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals

 Fractional differential equations of degree 1<α<2 are considered in this paper. A summary of numerical schemes for the time-domain solution of such problems is given. While all these methods require evaluation of the history of the state variables, an alternative concept recently published by Yuan and Agrawal (2002), which is computationally more efficient, is further developed. This scheme is based on a transformation of the original integro-differential problem into a system of linear differential equations. Here, parallels to the theory of internal variables are drawn. Received: 18 June 2002 / Accepted: 12 July 2002     

A fast implementation of explicit time-stepping algorithms with the finite element method for a class of nonlinear evolution problems

In this paper a class of nonlinear evolution problems is considered. It is shown that, under special conditions, the application of the product approximation method for nonlinear problems in the finite element method results in constant (i.e. time-independent) matrices. In those cases the amount of computing required to solve these equations with an explicit time-stepping algorithm is decreased considerably compared to the standard Galerkin formulation in which the matrices are time-dependent. The method is applied to two practical two-dimensional problems: the shallow water equations and a nonlinear heat conduction problem.