N. Vu-Bac¹, H. Nguyen-Xuan², L. Chen³, S. Bordas⁴, P. Kerfriden⁴, R.N. Simpson⁴, G.R. Liu⁵, T. Rabczuk¹

CMES-Computer Modeling in Engineering & Sciences, Vol.73, No.4, pp. 331-356, 2011, DOI:10.3970/cmes.2011.073.331

Abstract This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains. As a result, we avoid integration of the stress singularity at the crack tip. It is not necessary to divide elements cut by cracks when we replace interior integration… More >

Jin Li ¹, De-hao Yu ²

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.4, pp. 331-346, 2011, DOI:10.3970/cmes.2011.071.331

Abstract The composite rectangle (midpoint) rule for the computation of multi-dimensional singular integrals is discussed, and the superconvergence results is obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate one order higher than the global one. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well. More >

Phong B.H. Le¹, Timon Rabczuk², Nam Mai-Duy¹, Thanh Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.66, No.1, pp. 25-52, 2010, DOI:10.3970/cmes.2010.066.025

Abstract A novel meshless method based on Radial Basis Function networks (RBFN) and variational principle (global weak form) is presented in this paper. In this method, the global integrated RBFN is localized and coupled with the moving least square method via the partition of unity concept. As a result, the system matrix is symmetric, sparse and banded. The trial and test functions satisfy the Kronecker-delta property, i.e. Φ_i(x_j) = δ_ij. Therefore, the essential boundary conditions are imposed in strong form as in the FEMs. Moreover, the proposed method is applicable to scattered nodes and arbitrary domains. The method is examined with… More >

V. Sladek¹, J. Sladek¹, Ch. Zhang²

CMES-Computer Modeling in Engineering & Sciences, Vol.63, No.3, pp. 243-264, 2010, DOI:10.3970/cmes.2010.063.243

Abstract The paper deals with diminishing the prolongation of the computational time due to procedural evaluation of the shape functions and their derivatives in weak formulations implemented with meshless approximations. The proposed numerical techniques are applied to problems of stationary heat conduction in functionally graded media. Besides the investigation of the computational efficiency also the accuracy and convergence study are performed in numerical tests. More >

M. Ferronato¹, A. Mazzia¹, G. Pini¹

CMES-Computer Modeling in Engineering & Sciences, Vol.62, No.2, pp. 205-224, 2010, DOI:10.3970/cmes.2010.062.205

Abstract In the engineering practice meshing and re-meshing complex domains by Finite Elements (FE) is one of the most time-consuming efforts. Meshless methods avoid this task but are computationally more expensive than standard FE. A somewhat natural improvement can be attempted by combining the two techniques with the aim at emphasizing the respective merits. The present work describes a FE enrichment by the Meshless Local Petrov-Galerkin (MLPG) method. The basic idea is to add a limited number of moving MLPG points over a fixed coarse FE grid, in order to improve the solution accuracy in specific regions of the domain with… More >

G. Formica¹, F. Milicchio²

CMES-Computer Modeling in Engineering & Sciences, Vol.60, No.3, pp. 199-220, 2010, DOI:10.3970/cmes.2010.060.199

Abstract This manuscript introduces a novel sufficient condition for the unconditionally stable convergence of the general class of trapezoidal integrators. Contrary to standard energy-based approaches, this convergence criterion is derived from the power principles, in terms of both balance and dissipation. This result improves the well-known condition of stable convergence based on the energy method, extending its applicative spectrum to a variety of physical problems, whose constitutive prescriptions may be more appropriately characterized by means of evolving field equations. Our treatment, tailored for generalized trapezoidal integrators, addresses both linear and nonlinear problems, extending its applicability to contexts where standard energy-based schemes… More >

Lorenzo Codecasa¹, Francesco Trevisan²

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.2, pp. 155-180, 2010, DOI:10.3970/cmes.2010.059.155

Abstract Electromagnetic problems spatially discretized by the so called Discrete Geometric Approach are considered, where Discrete Counterparts of Constitutive Relations are discretized within an Energetic Approach. Pairs of oriented dual grids are considered in which the primal grid is composed of (oblique) parallelepipeds, (oblique) triangular prisms and tetrahedra and the dual grid is obtained according to the barycentric subdivision. The focus of the work is the evaluation of the constants bounding the approximation error of the electromagnetic field; the novelty is that such constants will be expressed in terms of the geometrical details of oriented dual grids. A numerical analysis will… More >

Lorenzo Codecasa¹, Francesco Trevisan²

CMES-Computer Modeling in Engineering & Sciences, Vol.58, No.1, pp. 15-44, 2010, DOI:10.3970/cmes.2010.058.015

Abstract This paper starts from the spatial discretization of an electromagnetic problem over pairs of oriented grids, one dual of the other, according to the so called Discrete Geometric Approach(DGA) to computational electromagnetism; the Cell Method or the Finite Integration Technique are examples of such an approach. The core of the work is providing for the first time a convergence analysis when the discrete counter-parts of constitutive relations are computed by means of an energetic framework. More >

Hua Li¹, Shantanu S. Mulay¹, Simon See²

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.1, pp. 43-82, 2009, DOI:10.3970/cmes.2009.048.043

Abstract Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed… More >

Briti Sundar Deb¹, Srinivasan Natesan²

CMES-Computer Modeling in Engineering & Sciences, Vol.38, No.2, pp. 179-200, 2008, DOI:10.3970/cmes.2008.038.179

Abstract This paper presents an almost second--order uniformly convergent Richardson extrapolation method for convection- dominated coupled system of boundary value problems. First, we solve the system by using the classical finite difference scheme on the layer resolving Shishkin mesh, and then we construct the Richardson approximation solution using the solutions obtained on N and 2N mesh intervals. Second-order parameter--uniform error estimate is derived. The proposed method is applied to a test example for verification of the theoretical results for the case ε ≤ N⁻¹. More >