A meshless singular boundary method for three‐dimensional elasticity problems

Singular boundary method for 2D dynamic poroelastic problems

The application of the singular boundary method (SBM), a relatively new meshless boundary collocation method, to the inverse Cauchy problem in three-dimensional (3D) linear elasticity is investigated. The SBM involves a coupling between the non-singular boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS. Due to the boundary-only discretizations and its semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique, while the optimal regularization parameter is determined by the $L$-curve criterion. Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.

Infinite domain potential problems by a new formulation of singular boundary method

The method of fundamental solutions with excitation source (MFS‐ES) is a reliable method for determining the eigenvalues of a two‐dimensional hollow waveguide. However, the accuracy in MFS‐ES is extremely dependent on the choice of the auxiliary boundary. In this work, to avoid the problem, a regularized MFS‐ES (RMFS‐ES) is proposed. First, to overcome the shortcomings of MFS, a regularized method of fundamental Solutions (RMFS) is proposed by borrowing the ideas of singular boundary method (SBM) and single layer regularized meshless method (SRMM). Second, the expressions for the fundamental solutions at the singularities are given by utilizing the subtraction and adding‐back technique (SAB). Third, the excitation source method is introduced to overcome the problem that the RMFS may also have spurious frequencies. Finally, to improve the accuracy of RMFS‐ES in solving polygonal waveguide eigenvalue, a wedge scheme (WS) is given. Numerical results show that the meshless RMFS‐ES is a promising numerical method for waveguide eigenvalue problems.

One considers a linear elastic composite material (CM, [1]), which consists of a homogeneous matrix containing the random set of heterogeneities. An operator form of the general integral equation (GIE, [2–6]) connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and defined at the inclusion interface by the unknown fields of both the displacement and traction. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs, and some particular cases, asymptotic representations, and simplifications of proposed GIEs are presented for the particular constitutive equations of linear thermoelasticity. In particular, we use a meshfree method [7] based on fundamental solutions basis functions for a transmission problem in linear elasticity. Numerical results were obtained for 2D CMs reinforced by noncanonical inclusions.

Singular boundary method: A review and computer implementation aspects

Singular boundary method for band structure calculations of in-plane waves in 2D phononic crystals

This paper describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the double layer fundamental solution of the Laplace equation. The collocation is used for determination of the expansion coefficients. This novel method does not require a fictitious boundary unlike the conventional Method of Fundamental Solutions (MFS). The source and collocation points thus coincide on the physical boundary of the system. The desingularised value of the fundamental solution is in case of the coincidence of the collocation and source points determined from the respective integral equations logic. The values of the derivatives of the double layer fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. A numerical example of potential flow around the two dimensional circular region is shown. The results with the MMFS are compared with the results of the classical single layer MFS with an artificial boundary, and the analytical solution. It is shown that the MMFS gives better accuracy of the velocity components as compared with the classical MFS.

Numerical treatment of acoustic problems with boundary singularities by the singular boundary method

A multi-domain singular boundary method for dynamic analysis of multilayered saturated porous media

On the choice of source points in the method of fundamental solutions

The singular boundary method (SBM) is a recent strong‐form boundary collocation method free of integration, mesh, and fictitious boundary. Although an extensive study has been reported in the literature on improving its accuracy and stability as well as its applications to diverse problems, little, however, has been done to analyze its convergence mathematically. The main purpose of this paper is to derive the explicit error bounds of the SBM for potential problems as well as to explain the essential difference between the origin intensity factor (OIF) in the SBM and the singular integration in the boundary element method (BEM). In the process of derivation, we also illustrate the physical meaning of OIF and explain the reason why the OIF has the function to correct the discretization error on the boundary. Finally, several benchmark examples are given to verify the effectiveness of the conclusions obtained from this article, as well as to investigate the different convergence behaviors between the SBM and BEM. It can be found that the SBM has the explicit error bound and is mathematically a stable technique.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1987–2004, 2017

This study considers the numerical comparison on three treatments for eliminating the singularities of acoustic fundamental solutions in singular boundary method (SBM). The singular boundary method is a recent meshless boundary collocation method, which introduces the concept of source intensity factors to eliminate the singularity of the fundamental solutions. Recently, three treatments, the inverse interpolation technique (IIT), the semi-analytical technique with boundary IIT (SAT1) and the semi-analytical technique with integral mean value (SAT2), have been proposed to determine the source intensity factors for removing the singularities of acoustic fundamental solutions at origin. This study compares numerical efficiency and stability of these three approaches on two benchmark examples under two-dimensional exterior acoustic problems.

On the application of the method of fundamental solutions to boundary value problems with jump discontinuities

A nonlinear rigid-plastic analysis for metal forming problem using the rigid-plastic point collocation method