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J R Xiao - One of the best experts on this subject based on the ideXlab platform.

  • two dimensional stress analysis of Functionally graded solids using the mlpg method with radial basis Functions
    Computational Materials Science, 2008
    Co-Authors: D F Gilhooley, J R Xiao, M A Mccarthy, R C Batra, John W Gillespie
    Abstract:

    Abstract The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of Functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis Function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test Functions, namely a fourth-order spline weight Function, and a Heaviside Step Function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.

  • local Heaviside weighted mlpg meshless method for two dimensional solids using compactly supported radial basis Functions
    Computer Methods in Applied Mechanics and Engineering, 2004
    Co-Authors: J R Xiao
    Abstract:

    Abstract Compactly supported radial basis Functions (CSRBF) are employed for constructing trial Functions in the local Heaviside weighted meshless local Petrov–Galerkin method for stress analysis of two-dimensional solids, where the Heaviside Step Function is used as the weighting Function over a local sub-domain. The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local sub-domain and interpolation domain on the performance of the present method are investigated. In this paper, the size of the support of the basis Function has been treated as a shape parameter, and then, the behaviour of this shape parameter has been systematically studied for six different CSRBFs. Example problems in elastostatics are presented and compared with closed-form solutions. Results show that the proposed method is highly accurate and possesses no numerical difficulties.

  • a local Heaviside weighted meshless method for two dimensional solids using radial basis Functions
    Computational Mechanics, 2003
    Co-Authors: J R Xiao, M A Mccarthy
    Abstract:

    A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside Step Function as the weighting Function over a local subdomain. Trial Functions are constructed using radial basis Functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties.

Ted Belytschko - One of the best experts on this subject based on the ideXlab platform.

  • a three dimensional large deformation meshfree method for arbitrary evolving cracks
    Computer Methods in Applied Mechanics and Engineering, 2007
    Co-Authors: Timon Rabczuk, Ted Belytschko
    Abstract:

    A new approach for modelling discrete cracks in meshfree particle methods in three dimensions is described. The cracks can be arbitrarily oriented, but their growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack’s topology is needed. The crack is modelled by a local enrichment of the test and trial Functions with a sign Function (a variant of the Heaviside Step Function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centered at the particles. The method is formulated for large deformations and arbitrary nonlinear and rate-dependent materials; cohesive laws govern the traction-crack opening relations. To reduce computational cost and since more accuracy around the crack tip is needed to obtain adequate results, h-adaptivity is incorporated in the method. The model is applied to several three-dimensional problems, some of which are compared to experimental data.

  • achieving minimum length scale in topology optimization using nodal design variables and projection Functions
    International Journal for Numerical Methods in Engineering, 2004
    Co-Authors: James K Guest, Jeanherve Prevost, Ted Belytschko
    Abstract:

    A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent Functions that are based upon the minimum length scale. A simple linear projection scheme and a non-linear scheme using a regularized Heaviside Step Function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user-defined length scale criterion without additional constraints, penalty Functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.

  • non planar 3d crack growth by the extended finite element and level sets part i mechanical model
    International Journal for Numerical Methods in Engineering, 2002
    Co-Authors: Nicolas Moes, Anthony Gravouil, Ted Belytschko
    Abstract:

    A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside Step Function via a partition of unity. In addition, branch Functions are introduced for all elements containing the crack front. The branch Functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance Functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.

  • non planar 3d crack growth by the extended finite element and level sets part i mechanical model
    International Journal for Numerical Methods in Engineering, 2002
    Co-Authors: Nicolas Moes, Anthony Gravouil, Ted Belytschko
    Abstract:

    A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside Step Function via a partition of unity. In addition, branch Functions are introduced for all elements containing the crack front. The branch Functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance Functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.

M A Mccarthy - One of the best experts on this subject based on the ideXlab platform.

  • two dimensional stress analysis of Functionally graded solids using the mlpg method with radial basis Functions
    Computational Materials Science, 2008
    Co-Authors: D F Gilhooley, J R Xiao, M A Mccarthy, R C Batra, John W Gillespie
    Abstract:

    Abstract The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of Functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis Function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test Functions, namely a fourth-order spline weight Function, and a Heaviside Step Function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.

  • a local Heaviside weighted meshless method for two dimensional solids using radial basis Functions
    Computational Mechanics, 2003
    Co-Authors: J R Xiao, M A Mccarthy
    Abstract:

    A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside Step Function as the weighting Function over a local subdomain. Trial Functions are constructed using radial basis Functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties.

Jin Hua Yang - One of the best experts on this subject based on the ideXlab platform.

  • Dynamic stability of piezoelectric laminated cylindrical shells with delamination
    Journal of Intelligent Material Systems and Structures, 2013
    Co-Authors: Jin Hua Yang, Jie Yang, Sritawat Kitipornchai
    Abstract:

    The dynamic stability of a composite laminated cylindrical shell containing a throughout delamination along circumferential direction integrated with piezoelectric layers at both inner and outer surfaces is investigated in this article. The Heaviside Step Function is used to describe the displacement components in the regions with and without delamination. Based on the classical shell theory, linear piezoelastic constitutive relationship, and variational principle, the governing equations of motion are derived and then solved by employing Rayleigh-Ritz method and Bolotin method to obtain the principal unstable region. Numerical results are presented in both tabular and graphical forms to show the effects of the piezoelectric layer; the length, depth, and location of the delamination; and the static axial force on the resonance frequency and the principal unstable region of the delaminated piezoelectric laminated shell.

  • Analysis of dynamic stability for composite laminated cylindrical shells with delaminations
    Composite Structures, 2007
    Co-Authors: Jin Hua Yang
    Abstract:

    Abstract By introducing the Heaviside Step Function into the assumed displacement components and using the Rayleigh–Ritz method for minimizing the total potential energy, a set of dynamic governing equations for the delaminated cylindrical shells is derived. Then, the dynamic governing equations are written as the Mathieu-type equations to describe the parametric vibrating behavior of the shells, and these equations are solved by employing the Bolotin method. The numerical results for the dynamic stability of laminated cylindrical shells with delaminations are presented. The effects of the amplitude of external excitation, the delamination size and location and the material properties on the natural frequency and the principal dynamic instability region of the delaminated cylindrical shells are discussed. Present results are compared with available data.

Anirut Luadsong - One of the best experts on this subject based on the ideXlab platform.

  • Two-field-variable meshless method based on moving kriging interpolation for solving simply supported thin plates under various loads
    Journal of King Saud University - Science, 2015
    Co-Authors: S. Kaewumpai, Anirut Luadsong
    Abstract:

    Abstract Meshless method choosing Heaviside Step Function as a test Function for solving simply supported thin plates under various loads is presented in this paper. The shape Functions using regular and irregular nodal distribution as well as order of polynomial basis choice are constructed by moving kriging interpolation. Alternatively, two-field-variable local weak forms are used in order to decompose the governing equation, biharmonic equation, into a couple of Poisson equations and then impose straightforward boundary conditions. Selected numerical examples are considered to examine the applicability, the easiness, and the accuracy of the proposed method. Comparing to an exact solution, this robust method gives significantly accurate numerical results, implementing by maximum relative error and root mean square relative error.

  • a local integral equation formulation based on moving kriging interpolation for solving coupled nonlinear reaction diffusion equations
    Advances in Mathematical Physics, 2014
    Co-Authors: Kanittha Yimnak, Anirut Luadsong
    Abstract:

    The meshless local Pretrov-Galerkin method (MLPG) with the test Function in view of the Heaviside Step Function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK) interpolation method for constructing nodal shape Function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time Step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.