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  • Open Access


    Solving Cauchy Issues of Highly Nonlinear Elliptic Equations Using a Meshless Method

    Chih-Wen Chang*

    CMC-Computers, Materials & Continua, Vol.72, No.2, pp. 3231-3245, 2022, DOI:10.32604/cmc.2022.024563

    Abstract In this paper, we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data. We also add the average classification as an approximate solution to the nonlinear operator part, without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution. The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described… More >

  • Open Access


    Two Phase Flow Simulation of Fractal Oil Reservoir Based on Meshless Method

    Xian Zhou1, Fei Wang2, Ziyu Wang3, Yunfeng Xu1,*

    Energy Engineering, Vol.119, No.2, pp. 653-664, 2022, DOI:10.32604/ee.2022.019072

    Abstract The reservoir is the networked rock skeleton of an oil and gas trap, as well as the generic term for the fluid contained within pore fractures and karst caves. Heterogeneity and a complex internal pore structure characterize the reservoir rock. By introducing the fractal permeability formula, this paper establishes a fractal mathematical model of oil-water two-phase flow in an oil reservoir with heterogeneity characteristics and numerically solves the mathematical model using the weighted least squares meshless method. Additionally, the method’s correctness is verified by comparison to the exact solution. The numerical results demonstrate that the… More >

  • Open Access


    An Efficient Meshless Method for Hyperbolic Telegraph Equations in (1 + 1) Dimensions

    Fuzhang Wang1,2, Enran Hou2,*, Imtiaz Ahmad3, Hijaz Ahmad4, Yan Gu5

    CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.2, pp. 687-698, 2021, DOI:10.32604/cmes.2021.014739

    Abstract Numerical solutions of the second-order one-dimensional hyperbolic telegraph equations are presented using the radial basis functions. The purpose of this paper is to propose a simple novel direct meshless scheme for solving hyperbolic telegraph equations. This is fulfilled by considering time variable as normal space variable. Under this scheme, there is no need to remove time-dependent variable during the whole solution process. Since the numerical solution accuracy depends on the condition of coefficient matrix derived from the radial basis function method. We propose a simple shifted domain method, which can avoid the full-coefficient interpolation matrix More >

  • Open Access


    Numerical Modeling of Material Deformation Responses Using Gradient Continuum Theory

    Jurica Sorić*, Boris Jalušić, Tomislav Lesičar, Filip Putar, Zdenko Tonković

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.23, No.1, pp. 1-1, 2021, DOI:10.32604/icces.2021.08043

    Abstract In modeling of material deformation responses, the physical phenomena such as stress singularity problems, strain localization and modeling of size effects cannot be properly captured by means of classical continuum mechanics. Therefore, various regularization techniques have been developed to overcome these problems. In the case of gradient approach the implicit gradient formulations are usually used when dealing with softening. Although the structural responses are mesh objective, they suffer from spurious damage growth. Therefore, a new formulation based on the strain gradient continuum theory, which includes both strain gradients and their stress conjugates, has been proposed.… More >

  • Open Access


    RBF-FD Method for Some Dispersive Wave Equations and Their Eventual Periodicity

    Marjan Uddin1,*, Hameed Ullah Jan1,*, Muhammad Usman2

    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 797-819, 2020, DOI:10.32604/cmes.2020.08717

    Abstract In this paper, we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of (IBVPs) for some dispersive wave equations on a bounded domain corresponding to periodic forcing. The constructed numerical scheme is based on radial kernels and local in nature like finite difference method. The temporal variable is executed through RK4 scheme. Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution. The results achieved are validated and examined with other methods accessible in the literature. More >

  • Open Access


    Crack Growth Modelling in Functionally Graded Materials by Mesh-Free Method

    P.H. Wen1, M.H. Aliabadi2

    Structural Durability & Health Monitoring, Vol.8, No.3, pp. 223-248, 2012, DOI:10.32604/sdhm.2012.008.223

    Abstract A mesh-free method for modelling crack growth in functionally graded materials is presented. Based on the variational principle of the potential energy, mesh-free method has been implemented with enriched radial bases interpolation functions to evaluate mixed-mode stress intensity factors, which are introduced to capture the singularity of stress at the crack tip. Paris law and the maximum principle stress criterion are adopted for defining the growth rate and direction of the fatigue crack growth respectively. The accuracy of the proposed method is assessed by comparison to other available solutions. More >

  • Open Access


    Meshless Method with Enriched Radial Basis Functions for Fracture Mechanics

    P.H. Wen1, M.H. Aliabadi2

    Structural Durability & Health Monitoring, Vol.3, No.2, pp. 107-120, 2007, DOI:10.3970/sdhm.2007.003.107

    Abstract In the last decade, meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. Based on the variation of potential energy, the element-free Galerkin method is developed on the basis of finite element method by the use of radial basis function interpolation. An enriched radial basis function is formulated to capture the stress singularity at the crack tip. The usual advantages of finite element method are retained in this method but now significant improvement of accuracy. Neither the connectivity of mesh in the domain by the More >

  • Open Access


    Structural Integrity Analysis Using the Numerical Green's Function and the Local Boundary Integral Equation Method

    L.S. Miers1, J.C.F. Telles2

    Structural Durability & Health Monitoring, Vol.1, No.3, pp. 225-232, 2005, DOI:10.3970/sdhm.2005.001.225

    Abstract The present paper aims at introducing the concept of Green's function type fundamental solutions (i.e., unit source fundamental solutions satisfying particular boundary conditions) into the context of meshless approaches, particularly dealing with the local boundary integral equation method (LBIE) derived from the classic boundary integral equation procedure. The Green's functions discussed here are mainly the so-called half-plane solution, corresponding to a unit source within a semi-plane bounded by a flux-free straight line and an infinite plane containing internal lines of potential discontinuity. The latter is here introduced in numerical fashion, as an extension of the More >

  • Open Access


    The Generalized Interpolation Material Point Method

    S. G. Bardenhagen1,2, E. M. Kober3

    CMES-Computer Modeling in Engineering & Sciences, Vol.5, No.6, pp. 477-496, 2004, DOI:10.3970/cmes.2004.005.477

    Abstract The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov–Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point(GIMP) methods. The generalizationpermits identification with aspects of other point or node based discrete solution techniques which do not use a body–fixed grid, i.e. the “meshless methods”. Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demonstrate and explain numerical artifact noise which can be expected inMPM calculations. Thisnoiseresultsin non-physical local More >

  • Open Access


    Meshless Local Petrov-Galerkin (MLPG) Mixed Finite Difference Method for Solid Mechanics

    S. N. Atluri1, H. T. Liu2, Z. D. Han2

    CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.1, pp. 1-16, 2006, DOI:10.3970/cmes.2006.015.001

    Abstract The Finite Difference Method (FDM), within the framework of the Meshless Local Petrov-Galerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A "mixed'' interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction More >

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