Abstracts:This work concerns numerical simulations of the frictional contact of two elastic deformable bodies and a novel treatment technique of incompatible contact nodes. These numerical simulations are based on a high order mesh-free method coupling the Moving Least Square (MLS) with the Asymptotic Numerical Method (ANM) and on a regularization technique. The obtained nonlinear problem is solved by using ANM applied on strong formulation in order to avoid numerical integrations. According to ANM, the Taylor series development of unknown variables on nodes of nonlinear contact problem leads to a sequence of linear systems to be solved. These linear systems are then discretized by a collocation mesh-free approach by using the MLS functions and a continuation method is adopted to evaluate the solution. The contact problem is identified to boundary conditions which are replaced by force-displacement relations through a regularization technique. The ability of the proposed approach is tested on some contact elastic deformable bi-dimensional examples. The obtained results are compared to an analytical solution and to a numerical solution obtained by the Newton–Raphson method coupled also with MLS approximation.

Abstracts:The generalized finite difference method (GFDM) is a relatively new meshless method for the numerical solution of certain boundary value problems. The method uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. In this paper, we document the first attempt to apply the GFDM for the numerical solution of two-dimensional (2D) multi-layered elastic problems. A multi-domain GFDM scheme is proposed to model the composite (layered) elastic materials. The composite material considered is decomposed into several sub-domains and, in each sub-domain, the solution is approximated by using the GFDM-type expansion. On the subdomain interface, compatibility of displacements and equilibrium of tractions are imposed. Preliminary numerical experiments show that the introduced multi-domain GFDM is very promising for accurate and efficient numerical simulations of multi-layered materials.

Abstracts:A typical meshfree point interpolation method (PIM) is presented to investigate the dispersion error in the numerical solutions of acoustic problems which is governed by the Helmholtz equation. It is well-known that those results from several numerical approaches, such as the finite element method (FEM) and several meshfree techniques, will suffer from the pollution effect, leading to the incorrect acoustic wave propagation for high wave numbers. The reason for this phenomenon is that the numerical solutions of wave number do not accord with the exact wave number, which is the so-called dispersion issue. In addition, to overcome the possible singularity issue in constructing the shape functions for the PIM with the polynomial basis functions (PBFs), the Gauss–Jordan elimination (GJE) technique is employed here. Several numerical examples concerning dispersion analysis and acoustic wave propagation are performed to verify the accuracy of results from the PIM. It is found that the PIM can reduce the dispersion error effectively and hence generate more accurate results than the FEM with the same set of nodes.

Abstracts:In this paper, the T-stress and stress intensity factor (SIF) of multiple cracks with arbitrary position in a finite plate is evaluated by the spline fictitious boundary element alternating method. The multi-crack problem is firstly divided into a simple model without crack which can be solved by the spline fictitious boundary element method and several infinite domains with one crack which can be solved by the fundamental solution of an infinite domain with a crack, namely Muskhelishvili's fundamental solutions. The technique is superior as no meshing is needed near crack face and the analytical solution for solving infinite domains with one crack is accurate and efficient. Then, instead of using the asymptotic expansion, the closed-form expression for calculating the T-stress in multi-crack problem is derived directly, which makes it convenient and accurate for calculating the T-stress. Besides, the SIF can be calculated using the analytical SIF expression in Muskhelishvili's fundamental solutions. Finally, T-stresses and SIFs in a numerical example with double cracks are computed to validate the accuracy of the presented method, And the other two examples with three cracks are further studied to investigate the influence of lengths and locations of multiple cracks on their T-stresses and SIFs.

Abstracts:This paper describes a new formulation of the dual reciprocity boundary element method (DRBEM) for two-dimensional transient convection–diffusion–reaction problems with variable velocity. The formulation decomposes the velocity field into an average and a perturbation part, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. The integral representation formula for the convection–diffusion–reaction problem with variable velocity is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. A finite difference method (FDM) is used to simulate the time evolution procedure for solving the resulting system of equations. Numerical applications are included for three different benchmark examples for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate its efficiency. Finally, results obtained show that the DRBEM results are in excellent agreement with the analytical solutions and do not present oscillations or damping of the wave front, as it appears in other numerical techniques.

Abstracts:This paper presents a practical approach for reliability analysis of steady-state seepage by modeling spatial variability of the soil permeability. The traditional semi-analytical method; named Scaled Boundary Finite-Element Method (SBFEM) is extended by a coded program to develop a stochastic SBFEM coupled with random field theory. The domain is discretized into several non-uniform SBFEM sub-domains. The flow quantities such as exit gradient, flow rate, and the reliability index of piping safety factor are estimated. The precision of the outputs and the accuracy of the method are verified with the Finite-Element Method (FEM). A set of stochastic analysis is performed in three illustrative examples to illuminate the applicability of the proposed method. In these examples, the effect of the variations in the position of the sub-domain discretization center, the cutoff location, and the cutoff length are investigated stochastically. Further, the influence of the permeability's Coefficient of Variation (COVk ) and the correlation length is evaluated. The results are shown acceptable agreement with those obtained by the conventional Stochastic Finite-Element Method (SFEM). The proposed approach has potential to model the complex geometries and cutoffs in different locations without additional efforts to deal with the spatial variability of the permeability.

Abstracts:In this paper, flow across a bluff cylinder with circular and equilateral triangular sections shapes at low Reynolds number (10 ≤ Re ≤ 250) is numerically investigated using the gradient smoothed method (GSM). The method was originally developed based on unstructured grids, which could be generated easily for complicated domains. For solving the incompressible flow, artificial compressibility terms are introduced in Navier–Stokes equations. The spatial derivatives of convective and viscous fluxes are obtained using the gradient smoothing operation. And the time marching is implemented based on the dual time stepping technique. The situations of steady flows with Re = 10–40 and unsteady flows with Re = 50∼250 are simulated. Analysis of the drag coefficients, root mean square (rms) value of lift coefficients and Strouhal number of circular and triangular cylinders has been carried out. Compared with both experimental and numerical reference solutions in the literatures, the accuracy of GSM results has been demonstrated.

Abstracts:In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source offset, source location, or a fictitious boundary, a generalized method of fundamental solutions (GMFS) is proposed. The crucial part of the GMFS is using a generalized fundamental solution approximation (GFSA), which adopts a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Then the numerical solution of the GMFS is decided by a group of offsets corresponding to an intervention-point diffusion (IPD), instead of the MFS’ offset of a single source. To demonstrate the effectiveness of the proposed approach, five numerical examples are given. The results have shown that the GMFS is more accurate, stable, and has a better convergence rate than the traditional MFS.

Abstracts:In this paper, an adaptive refinement strategy based on the scaled boundary finite element method on quadtree meshes for linear elasticity problems is discussed. Within this framework, the elements with hanging nodes are treated as polygonal elements and thus does not require special treatment. The adaptive refinement is supplemented with a novel error indicator. The local error is estimated directly from the solution of the scaled boundary governing equations. The salient feature is that it does not require any stress recovery techniques. The efficacy and the robustness of the proposed approach are demonstrated with a few numerical examples.

Abstracts:The Boundary Element Method, a numerical technique that is not based on domain discretization, faces difficulties to model piecewise homogeneous problems that do not appear in many other applications. Thus, in this paper it is presented an alternative methodology for solution of this kind of problem, previously tested successfully for the Laplace Equation, applied here to static cases of linear elasticity. It is substantially different from the classic sub-region technique, since it is based on the sum of elastic energy retained in each distinct sector. Several examples that include cases with irregular domains are simulated showing the robustness and adequacy of the proposed technique. In the absence of analytical solutions, Finite Element Method solutions are used as reference for error evaluation.