In this paper, the properties of various boundary integral operators are investigated for error estimation in adaptive BEM. It is found that the residual of the hyper-singular boundary integral equation (BIE) can be used for a-posteriori error estimation for different kinds of problems. Based on this result, a new a-posteriori error indicator is proposed which is a measure of the difference of two solutions for boundary stresses in elastic BEM. The first solution is obtained by the conventional boundary stress calculation method, and the second one by use of the regularized hyper-singular BIE for displacement derivative. The latter solution has recently been found to be of high accuracy and can be easily obtained under the most commonly used C 0 continuous elements. This new error indicator is defined by a L 1 norm of the difference between the two solutions under Mises stress sense. Two typical numerical examples have been performed for two-dimensional (2D) elasticity problems and the results show that the proposed error indicator successfully tracks the real numerical errors and effectively leads a h-type mesh refinement procedure.Keywords a-posteriori error estimation, Adaptive boundary element method, Hyper-singular boundary integral equation, Adaptive mesh refinement IntroductionError estimation and the consequent adaptive mesh refinement are crucially important in numerical engineering analysis. In the finite element method (FEM), started in 1970s, this field has been extensively studied and the present state emphasizes both academic research and the application of the developed algorithms to engineering softwares to improve the quality and reliability in analysis [1,2]. In the boundary element method, research in this field began in the early 1980s [3][4][5]. Similar in FEM, the adaptive schemes in BEM comprise h-type [6-8], p-type [9-11], r-type [12-15], and their combinations [16-20], and there are diverse strategies for the local and global error estimations, see further [21-23].The adaptive BEM has also been applied to different kinds of engineering problems [8,15,24,25]. Even though, the efficiency and flexibility of the above developed BEM algorithms deserve further research. As most of them are based on collocation BEM and they lack mathematical analysis. On the other hand, there are numerous researches in this topic on Galerkin BEM, which are generally with rigorous mathematical analysis but with simpler boundary condition and geometry boundary, see e.g. [3,[26][27][28][29][30][31].Recently some researchers [32][33][34][35] have tried to introduce the mathematical analysis to the error estimation in collocation BEM. The key idea of their works is to use a certain norm of the residual of the hyper-singular BIE as a-posteriori error indicator, and the mathematical analysis is to construct a relationship between the hypersingular residual and the error of the given approximate solution. Obviously, these researches are important improvements in the BEM error estimation and mesh refinement. However, t...
This paper presents a fast multipole boundary element approach to the 3D acoustic shape sensitivity analysis based on the discrete adjoint variable method. The Burton–Miller formula is used to conquer the fictitious eigenfrequency problem of the original boundary integral equation method in solving exterior acoustic problems. Constant elements are used to discretize the boundary, so that the hypersingular integrals contained in the formulae can be evaluated explicitly and directly. Numerical examples are presented to demonstrate the accuracy and efficiency of the present approach.
For thin structures immersed in water, a full interaction between the structural domain and the fluid domain needs to be taken into account. In this work, the finite element method (FEM) is used to model the structure parts, while the boundary element method (BEM) is applied to the exterior acoustic domain. A coupling algorithm based on FEM and the wideband fast multipole BEM (FEM/Wideband FMBEM) is used for the simulation of acoustic-structure interaction. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. Structural-acoustic design sensitivity analysis is performed based on the coupling formulation. The design variable can be chosen as the material parameters, structure and fluid parameters, such as the fluid density, structural density, Poisson's ratio, Young's modulus, structural shape size and so on. Furthermore, the impact of sound-absorbing material on the scattering problem for structures underwater is researched. The acoustic admittance of the soundabsorbing material has also been chosen as the design variable for the sensitivity analysis. Numerical example is presented to demonstrate the validity and efficiency of the proposed algorithm.
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