Yoshihiro Ochiai¹, Vladimir Sladek²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.6, pp. 525-536, 2004, DOI:10.3970/cmes.2004.006.525

Abstract The conventional boundary element method (BEM) uses internal cells for the domain integralsCwhen solving nonlinear problems or problems with domain effects. This paper is concerned with conversion of the domain integral into boundary ones and some non-integral terms in a three-dimensional BIEM, which does not require the use of internal cells. This method uses arbitrary internal points instead of internal cells. The method is based on a three-dimensional interpolation method in this paper by using a polyharmonic function with volume distribution. In view of this interpolation method, the three-dimensional numerical integration is replaced by boundary ones and preceding calculation of… More >

J. Sladek¹, V. Sladek¹, S.N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.5, pp. 477-490, 2004, DOI:10.3970/cmes.2004.006.477

Abstract A meshless method based on the local Petrov-Galerkin approach is proposed for solution of static and elastodynamic problems in a homogeneous anisotropic medium. The Heaviside step function is used as the test functions in the local weak form. It is leading to derive local boundary integral equations (LBIEs). For transient elastodynamic problems the Laplace transfor technique is applied and the LBIEs are given in the Laplace transform domain. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-d problems. The final form of local integral equations has a pure contour character only in… More >

S.F. Moreira^∗, J. Belinha^{∗,† ,‡}, L.M.J.S. Dinis^∗,†, R.M. Natal Jorge^∗,†

Molecular & Cellular Biomechanics, Vol.11, No.3, pp. 151-184, 2014, DOI:10.3970/mcb.2014.011.151

Abstract In this work the maxillary central incisor is numerically analysed with an advance discretization technique – Natural Neighbour Radial Point Interpolation Method (NNRPIM). The NNRPIM permits to organically determine the nodal connectivity, which is essential to construct the interpolation functions. The NNRPIM procedure, based uniquely in the computational nodal mesh discretizing the problem domain, allows to obtain autonomously the required integration mesh, permitting to numerically integrate the differential equations ruling the studied physical phenomenon. A numerical analysis of a tooth structure using a meshless method is presented for the first time. A two-dimensional model of the maxillary central incisor, based… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.25, No.3, pp. 239-264, 2011, DOI:10.3970/cmc.2011.025.239

Abstract For the computational applications in several areas, we propose a single-scale and a multi-scale diagonal preconditioners to reduce the condition number of Vandermonde matrix. Then a new algorithm is given to solve the inversion of the resulting coefficient matrix after multiplying by a preconditioner to the Vandermonde matrix. We apply the new techniques to the interpolation of data by using very high-order polynomials, where the Runge phenomenon disappears even the equidistant nodes are used. In addition, we derive a new technique by employing an m-order polynomial with a multi-scale technique to interpolate 2m+1 data. Numerical results confirm the validity of… More >

C. F. Loeffle¹, L. Zamprogno², W. J. Mansur³, A. Bulcão⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.113, No.3, pp. 367-387, 2017, DOI:10.3970/cmes.2017.113.387

Abstract This study evaluates the effectiveness of a new technique that transforms domain integrals into boundary integrals that is applicable to the boundary element method. Simulations were conducted in which two-dimensional surfaces were approximated by interpolation using radial basis functions with full and compact supports. Examples involving Poisson’s equation are presented using the boundary element method and the proposed technique with compact radial basis functions. The advantages and the disadvantages are examined through simulations. The effects of internal poles, the boundary mesh refinement and the value for the support of the radial basis functions on performance are assessed. More >

L. Dong¹, A. Alotaibi², S.A. Mohiuddine², S. N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.1, pp. 1-85, 2014, DOI:10.3970/cmes.2014.099.001

Abstract In this expository article, a variety of computational methods, such as Collocation, Finite Volume, Finite Element, Boundary Element, MLPG (Meshless Local Petrov Galerkin), Trefftz methods, and Method of Fundamental Solutions, etc., which are often used in isolated ways in contemporary literature are presented in a unified way, and are illustrated to solve a 4^th order ordinary differential equation (beam on an elastic foundation). Both the primal formulation, which considers the 4^th order ODE with displacement as the primitive variable, as well as two types of mixed formulations (one resulting in a set of 2 second-order ODEs, and the other resulting… More >

Jean-Philippe Jehl¹, Richard Kouitat Njiwa²

CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.5, pp. 345-358, 2014, DOI:10.3970/cmes.2014.102.345

Abstract Extended continuum mechanical approaches are now becoming increasingly popular for modeling various types of microstructured materials such as foams and porous solids. The potential advantages of the microcontinuum approach are currently being investigated in the field of biomechanical modeling. In this field, conducting a numerical investigation of the material response is evidently of paramount importance. This study sought to investigate the potential of the (constrained) microstretch modeling method. The problem’s field equations have been solved by applying a numerical approach combining the conventional isotropic boundary elements method with local radial point interpolation. Our resulting numerical examples demonstrated that the model… More >

M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 399-444, 2014, DOI:10.3970/cmes.2014.100.399

Abstract In this paper three numerical techniques are proposed for solving the system of N-coupled nonlinear Schrödinger (CNLS) equations. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via the forward finite difference formula, then for obtaining a full discretization scheme, we use the Kansa’s approach to approximate the spatial derivatives via radial basis functions (RBFs) collocation methodology. We introduce the moving least squares (MLS) approximation and radial point interpolation method (RPIM) with their shape functions, separately. It should be noted that the shape functions of RPIM unlike the shape functions of the MLS approximation have kronecker… More >

Martino Pani¹, Fulvia Taddei¹

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.4, pp. 279-300, 2013, DOI:10.3970/cmes.2013.094.279

Abstract The Cell Method (CM) is a numerical method to solve field equations starting from its direct algebraic formulation. For two-dimensional problems it has been demonstrated that using simplicial elements with an affine interpolation, the CM obtains the same fundamental equation of the Finite Element Method (FEM); using the quadratic interpolation functions, the fundamental equation differs depending on how the dual cell is defined. In spite of that, the CM can still provide the same convergence rate obtainable with the FEM. Particularly, adopting a uniform triangulation and basing the dual cells on the Gauss points of the primal edges, the CM… More >

W. Zhao^1,2, J.K. Liu³, X.Y. Li², Q.W. Yang⁴, Y.Y. Chen⁵

CMES-Computer Modeling in Engineering & Sciences, Vol.93, No.6, pp. 469-488, 2013, DOI:10.3970/cmes.2013.093.469

Abstract In order to obtain reliable structural design, it is of extreme importance to evaluate the failure probability, safety levels of structure (reliability analysis) and the effect of a change in a variable parameter on structural safety (sensitivity analysis) when uncertainties are considered. With a computationally cheaper approximation of the limit state function, various response surface methods (RSMs) have emerged as a convenient tool to solve this especially for complex problems. However, the traditional RSMs may produce large errors in some conditions especially for those highly non-linear limit state functions. Instead of the traditional least squares approximation, in the present paper,… More >