Initial stress formulation for three-dimensional elastoplastic analysis by the triple-reciprocity boundary element method

The conventional boundary element method (BEM) needs a domain integral in heat conduction analysis with heat generation or initial temperature distribution. This paper shows that the three-dimensional heat conduction problem can be solved effectively by using the triple-reciprocity boundary element method without internal cells. In this method, the distribution of heat generation and initial temperature are interpolated by using integral equations. In this method, time-dependant fundamental solutions are used. A new computer program was developed and applied to solving several problems.

The conventional boundary element method (BEM) uses internal cells for the domain integrals when solving nonlinear problems or problems with domain effects. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a three-dimensional meshless BEM, which does not require the use of internal cells. This method uses arbitrary internal points instead of internal cells. In this paper, a three-dimensional interpolation method is proposed. Furthermore, three-dimensional numerical integration is proposed using the above-mentioned three-dimensional interpolation method. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.

In general, internal cells are required to solve deformation of thin plate with arbitrary distributed load using conventional boundary element methods (BEM). However, in this case, the merit of BEM, which is the easy preparation of data, is lost. In this paper, it is shown that deformation analysis of thin plate with arbitrary distributed load can be solved without the use of internal cells by using the triple-reciprocity boundary element method. Distribution of arbitrary load is interpolated using boundary integral equations. The problem of thin plate according to Kirchhoff's theory is formulated by means of two coupled Poisson equations to be expressed in integral form using the second theorem of Green in the classical way. A new computer program was developed and applied to several example problems.

In this paper, a new and simple boundary element method without internal cells is presented for the analysis of elastoplastic problems, based on an effective transformation technique from domain integrals to boundary integrals. The strong singularities appearing in internal stress integral equations are removed by transforming the domain integrals to the boundary. Other weakly singular domain integrals are transformed to the boundary by approximating the initial stresses with radial basis functions combined with polynomials in global coordinates. Three numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.

Elastic and inelastic analysis of reinforced concrete structures using the boundary element method

O presente trabalho considera o uso do Método dos Elementos de Contorno (MEC) na avaliação da carga de colapso de problemas em estado plano de deformação, usando o teorema estático da análise-limite. Essa análise envolve o modelo constitutivo rígido-plástico, onde uma formulação não-linear do MEC faz-se necessária. Análises não-lineares com o MEC normalmente requerem o uso de uma discretização auxiliar no domínio através de células internas. Para evitar essa discretização de domínio, o presente desenvolvimento baseia-se em uma nova formulação não-linear do MEC, recentemente desenvolvida por Noronha e Pereira, usando apenas discretização de contorno (interfaces entre subregiões da zona plastificada). Os resultados obtidos pela discretização com a nova formulação do MEC são comparados com os obtidos pelo MEC convencional e pelo Método dos Elementos Finitos (MEF). A comparação entre os diferentes resultados é feita com os valores da função de escoamento e com o valor obtido para a carga de ruptura.

Linear stress analysis without body force can be easily solved by means of the boundary element method. Some cases of linear stress analysis with body force can also be solved without a domain integral. However, domain integrals are generally necessary to solve the linear stress problem with arbitrary body forces. This paper shows that the linear stress problem with arbitrary body forces can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of arbitrary body forces can be interpolated by the integral equation. A new computer program is developed and applied to several problems.

The computation of the acoustic scattering from objects in the ocean is useful to analyze the characteristics of the targets, which can be numerically modelled by acoustic boundary element method (BEM). The BEM has a defect of fictitious frequency problem, Burton-Miller method is studied to overcome the singular problems. In this paper, the Hadamard finite-part integral method based on collocation method is studied to overcome the singularity. It is found that, the near singularity is also quite significant and deteriorates largely the accuracy of BEM. A method based on polar transformation is proposed in this paper to overcome such a defect. Once the problems of singularity and near-singularity are solved, the conventional BEM model is established to serve as a foundation for fast BEM.It is well known that conventional BEM yields a full matrix, which is of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) memory complexity. Furthermore, the time complexity could be up to O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), if Gaussian elimination method is adopted to solve the linear equation of BEM. Many fast methods are proposed to solve the problems, such as the GMRES iterative solver, fast multipole method and H-matrix (Hierarchical Matrices) method. H-matrix method is applied to overcome the faults of conventional BEM, by using the low-rank approximations to compress blocks of the matrix, which is a pure algebraic method. The ACA (Adaptive Cross Approximation) method is proposed to compute the outer-product form of low-rank matrices, instead of SVD (Singular Value Decomposition), which consumes high CPU time and is of no usability. ACA-BEM is used to predict the acoustic scattering from objects in the ocean, especially the large-scale problems. The directivities of a rigid sphere and a rigid ellipsoid are computed by ACA BEM, the time consumed is very short. For a problem of more than 20,000 DOFs, it can be solved within 2 minutes. Therefore, ACA BEM can be used to solve large-scale acoustic scattering problems very efficiently.

If the initial temperature is assumed to be constant, a domain integral is not required for the solution of unsteady thermal stress problems without heat generaton using the boundary element method (BEM). However, under heat generation or nonuniform initial temperature distribution, the domain integral is necesary. In this report it is shown that the problem of unsteady heat conduction with heat generation and nonuniform initial temperature can be approximately solved without using the domain integral by the triple-reciprocity BEM. In this method, the distributions of heat generation and nonuniform initial temperature are interpolated using integral equations.

Homogeneous heat conduction analysis can be easily solved by means of the boundary-element method. However, domain integrals are generally necessary to solve the heat conduction problem in the functionally gradient materials. This paper shows that the heat conduction problem in the functionally gradient materials can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of domain effects is interpolated by integral equations. A new computer program is developed and applied to several problems.

The Boundary Element Method (BEM) has been an effective method for modelling a number of particular problems which may be described by potential flows or, in a quite different vein, by creeping motions. However when it comes to moving boundary flows the BEM requires a significant amount of computational time. In the context of accelerating the BEM, the Fast Multipole Method (FMM) has been used for two and three-dimensional problems but they are not yet reported for axisymmetric ones. Our purpose is to present a multipole based boundary element method (MM-BEM) for moving boundary problems in the axisymmetric case. The proposed method takes advantage of grouping the ring sources in the axisymmetric domain, in order to reduce the amount of direct computations. Direct computations are only performed when the ring sources are located close to the evaluation point. Here, MM-BEM is implemented to simulate the impact of a drop onto a liquid surface, modelled assuming potential flow. It is shown that the proposed method provides the same results as the conventional BEM.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

In this paper a total incremental method for solving nonlinear system equation due to plasticity of shear deformable plates is presented. The material is assumed to undergo small strains. The von Mises criterion is used to evaluate the plastic zone and elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. The domain integral due to material nonlinearity is evaluated using a cell discretization technique. Several examples are presented and comparisons are made to demonstrate the validity and the accuracy of the total incremental method to solve the nonlinear system of equation due to plasticity.

Fast algorithm for multi-frequency numerical integration in the simulation of acoustic scattering from rigid object by the boundary element method is presented. Normal derivative of the free-space Greens function is partially approximated with the unknown variable by a set of shape functions. As a result, the numerical integral is independent of frequency and need be calculated only at the first frequency step. Singular integral can be computed using the same procedure as that applied in the conventional boundary element method. Computational efficiency and accuracy of the new technique are demonstrated by an example. Numerical results obtained using the new technique are compared with the corresponding analytical solutions and numerical results obtained using the conventional boundary element method. The new technique works well and saves a lot of computational time in the process of generation of coefficient matrices for multi-frequency analysis.

NURBS-enhanced boundary element method based on independent geometry and field approximation for 2D potential problems