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A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the near singularity of integrands resulting from the very small distance between the source and the integration points. By taking advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae, for both the 2D and 3D elastic problems. IntroductionThe accurate numerical evaluation of nearly singular boundary integrals has challenged researchers since the advent of the boundary element method. The problem generally refers to the boundary layer effect, a phenomenon that the boundary integrals cannot be computed accurately with the ordinary Gaussian quadrature if the computing point is located closely adjacent to the boundary [1-3]. Similar to the case of singular boundary integrals, the boundary layer effect comes from the properties of fundamental solutions and their derivatives, having the denominator of the Euclidean distance, r, between the source and the field points of various orders [4]. In consequence, the values of the integrals over a boundary element have a reverse correlation with this distance relative to the size of the element, so the system matrices are well behaved so enabling the boundary integrals to be evaluated accurately.The accurate numerical evaluation of nearly singular boundary integrals is crucial for many engineering problems, including, for example, the analysis of the thin or shell-like bodies as occur in many structural applications [5][6][7], the crack problems when the crack tip is deformed with an opening displacement [8], the contact problem [9] when the distance between the two contacting bodies is very small, as well as the sensitivity problem [10]. The importance of this subject area is considered second only to the singular case of the boundary element method and this explains the great attention and effort that has been focused on it in recent years [11][12][13][14][15][16][17][18][19][20][21][22][23].The difficulty encountered in the numerical evaluation results mainly from the fact that the integrands of nearly singular boundary integrals vary drastical...

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This paper aims to propose a meshless Galerkin level set method for shape and topology optimization of continuum structures. To take advantage of the implicit free boundary representation scheme, the design boundary is represented as the zero level set of a scalar level set function, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes. The meshless Galerkin method with global weak form is used to implement the discretization of the state equations. This provides a pathway to unify the two different numerical stages in most conventional level set methods: (1) the propagation of discrete level set function on a set of Eulerian grid and (2) the approximation of discrete equations on a set of Lagrangian mesh. The original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization, to which many efficient optimization algorithms can be applied. The proposed level set method can describe the moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization. Several benchmark examples are used to demonstrate the effectiveness of the proposed method. The numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods. It is straightforward to apply the proposed method to more advanced shape and topology optimization problems.