Simple modifications for stabilization of the finite point method

A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill‐conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non‐monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Complex variable moving least‐squares method: a meshless approximation technique

Based on the moving least‐squares (MLS) approximation, we propose a new approximation method—the complex variable moving least‐squares (CVMLS) approximation. With the CVMLS approximation, the trial function of a two‐dimensional problem is formed with a one‐dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of the MLS approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two‐dimensional elasticity problems—the complex variable meshless method (CVMM)—and the formulae of the CVMM for two‐dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd.     

Concept of stress dead zone in cracked plates: Theoretical,experimental, and computational studies

This paper presents a study on stress dead zone implications and its characterization with uniaxial tensile testing on MT polycarbonate plate specimens. To obtain the experimental solution, digital image correlation (DIC) is used. Numerically, based on LEFM, the problem is solved using advanced discretization techniques, finite element, and meshless methods. Considering stress dead zone notion, it aims to determine alternative analytical solution of the SIF. Thus, the compliance method is adopted, which is associated with the specimen's dead zone region that does not involve the cracking resistance. Hence, LEFM formulations complying with strain energy release rate criterion are assumed. A proper comparison is made amongst results obtained from all theoretical, experimental, and numerical analyses. Furthermore, this work focuses on the variational fields' evaluation and, in particular, verification on the theoretical assumption of dead zone characteristics. The obtained results support adopted methodologies and are encouragingly robust and cost‐effective.     

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

This paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time-marching of first-order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one-, two-, and three-dimensional cases show that the proposed hybrid KDC-GFDM scheme allows big time step size for a long-time dynamic simulation and has a great potential for the problems with complex boundaries. In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method.     

An accurate and fast regularization approach to thermodynamic topology optimization

In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii.     

TVD finite point method for two-dimensional conservation equation

A new meshless method, called total variation diminishing (TVD) finite point method (TVDFP), is proposed. The TVDFP method is developed on the least-square procedure which uses a global stencil of grid points and the two-dimensional (2D) TVD procedure for the approximation of fictitious interface directional fluxes. We present the accuracy of the TVDFP method and several 2D test computations.     

SENSITIVITY ANALYSIS IN THERMO-ELASTIC-PLASTIC PROBLEMS

The design sensitivity of a thermo-elastic-plastic system is considered. The continuum approach and resulting finite element formulations are described. Sensitivity is obtained by an incremental load approach for nonlinear response analysis. Kinematic and isotropic hardening in thermo-elasto-plasticity are discussed. The control volume approach is to analyze shape and nonshape design. The direct differentiation method is adopted to obtain the design sensitivity expression for the response variables. Analytical examples demonstrate the developed sensitivity procedure. The design sensitivity formulas are discretized to implement it as a computer program with isoparametric finite elements. Numerical examples are presented to calculate the design sensitivity.     

Structural shape and topology optimization using a meshless Galerkin level set method

This paper aims to propose a meshless Galerkin level set method for shape and topology optimization of continuum structures. To take advantage of the implicit free boundary representation scheme, the design boundary is represented as the zero level set of a scalar level set function, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes. The meshless Galerkin method with global weak form is used to implement the discretization of the state equations. This provides a pathway to unify the two different numerical stages in most conventional level set methods: (1) the propagation of discrete level set function on a set of Eulerian grid and (2) the approximation of discrete equations on a set of Lagrangian mesh. The original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization, to which many efficient optimization algorithms can be applied. The proposed level set method can describe the moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization. Several benchmark examples are used to demonstrate the effectiveness of the proposed method. The numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods. It is straightforward to apply the proposed method to more advanced shape and topology optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Helmholtz问题的径向基无网格配置法的研究

通过引入一种新的径向基函数构造了求解Helmholtz方程配置型的无网格方法,证明了数值解的存在惟一性,并且将该方法用于二维Helmholtz问题的数值检验.与有限元法及其他径向基函数配置法相比较,该方法计算精度高,更加实用有效.     

A natural neighbour method for linear elastic problems based on Fraeijs de Veubeke variational principle

The natural neighbour method can be considered as belonging to the meshless methods. Classically, the development of this method is based on the virtual work principle. In the present paper, we use the natural neighbour method for 2D domains starting from the Fraeijs de Veubeke variational principle and we approximate separately the displacement field, the stress field and the strain field: the assumed strains and the assumed stresses are constant over each Voronoi cell, the assumed surface reactions are constant along the edge where the displacements are imposed, while the assumed displacements are interpolated by Laplace interpolants. In the absence of body forces, it is shown that the calculation of integrals on the area of the solid domain can be avoided: only integrals on the edges of the Voronoi cells are necessary. On the other hand, displacements can be imposed in the average sense on some boundaries of the domain. Patch tests and some applications in the elastic domain are given in the paper and show the effectiveness of the method. Copyright © 2007 John Wiley & Sons, Ltd.