| Meshless methods were widely developed and applied in engineering in the past decades. Compared with finite element method (FEM) and boundary element method (BEM), these kind of methods can reduce the human-labor costs required for meshing the domains of complex shape and can be applied in problems with large deformation or crack propagation. It is also very suitable for simulation of composites in engineering. The hybrid boundary node method (Hybrid BNM) is one of the boundary type meshless methods, which can reduce the spatial dimensions by one like BEM.However, the Hybrid BNM has a dense and unsymmetrical system matrix, which requires O(N2) operations for an iterative solver or O(N3) operations for a direct solver, where N is the degrees of freedom (DOFs). Thus, it is impossible to apply the Hybrid BNM for large scale computation on a personal computer. The fast multipole method (FMM) is one of the most widely investigated and applied methods for accelerating.The Hybrid BNM for3D elasticity problems accelerated by the FMM is proposed, called as fast multipole hybrid boundary node method (FM-HBNM). The preconditioned Generalized Minimum Residual method (GMERS) is employed to solve the resulting system of equations. At each iteration step of the GMERS, the matrix-vector multiplication is accelerated by the fast multipole method (FMM). The fundamental solution of three-dimensional elasticity problem is expanded in terms of series. An oct-tree data structure is adopted to subdivide the computational domain into well-separated cells hierarchically and to invoke the multipole expansion approximation. The FM-HBNM is an efficient algorithm that can reduce both the computer costs and the human-labor. The computational complexity is estimated and an O(N) algorithm is obtained. Numerical examples show that the FM-HBNM is very efficient and accurate.In FMM, the major obstacle in achieving reasonable efficiency with high accuracy is the large number of the multipole to local (M2L) translations and there is still room to further reduce the computational time of FMM. Based on the original FM-HBNM for3D elasticity problems, a new FM-HBNM is derived by using a new diagonal form and the computational time can be further reduced by the new FM-HBNM. Results from the numerical examples are compared by using both the original and new FM-HBNM.The FM-HBNM is applied to simulate3D composite materials in structural engineering. In order to deal with composite materials, a multi-domain solver is used. In the multi-domain solver, the equilibrium and continuity conditions on the interfaces are used and the final algebraic equation is obtained by assembling the algebraic equation for each single sub-domain. The proposed multi-domain technique is capable to deal with interface and multi-medium problems and results in a block sparsity of the coefficient matrix.A new formulation based on the Hybrid BNM is proposed for the analysis of composites in structural engineering. In the new formulation, continuity conditions are used as the conventional multi-domain solver and the unknowns of the interfaces are assembled only once in the final system equation, which can reduce both the computational time and memory required. The new formulation is quite suitable for the inclusion-based composites, especially for the case when the inclusions are solid and totally embedded in the matrix domain.The original and new FM-HBNM are coupled with the multi-domain solver and new formulation for the analysis of3D composites in heat conduction problems and elasticity problems. Numerical examples show that the methods are very suitable for the analysis of3D composite materials.The study shows that the both the original and new FM-HBNM are very efficient and accurate. The methods are especially suitable for large scale computation and make the Hybrid BNM can be applied in more engineering areas. |
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