| Computer Aided Engineering(CAE) technology plays an important role in the design and analysis of engineering structures. And boundary integral equation method(BIEM) is one of the most important ??numerical methods of CAE analysis. As a precise geometric BIEM, the boundary face method(BFM) has many advantages,such as high accuracy, error-free geometry, dimension reduction, easy to implement integrated CAD\CAE etc. Dimensionality reduction is one of the most important advantages of the BIEM. In BIEM, only boundary of the model should be discretized.This will reduce the difficulty of the domain mesh, the computation time of integration and the scale of the problem. Then the BFM would have more advantages over the types of non-boundary integral method. While for the nonhomogeneous problems, such as the transient heat conduction problem including heat sources and the elastic problems including body forces, the domain integrals will appear in the BIE inevitably. These domain integrals can be divided into two types according to whether the distribution can be expressed by analytical function or not, namely analytical function type and non analytical function type. If we discretize the domain to calculate the domain integral directly, the advantage of dimensionality reduction of BIE will be weakened. If the analytical fundamental solution for the governing equation is not available, only the incomplete fundamental solution can be used. In this case, domain integrals including unknown variables will appear in BIE, for example elastic buckling problem. Then the unknown variables on domain nodes will be included in the equation set. The dimensionality of it will increase significantly,which leads the BIE to lose its advantage of dimensionality reduction completely.Therefore, the domain integral transformation methods in BIEM is very important.Currently several methods are proposed to transform the domain integral into boundary integral and applied in the potential problems and elastic problems etc.Because the fundamental solutions are included in the domain integral and the transformation methods are related to the governing equation, the domain integral is hard to be transformed for some problems in which the governing equation and the fundamental solutions are complex, such as the three-dimensional(3D) transient heat conduction problem and the elastic buckling problem. In these problems, the computation accuracy and efficiency of the domain integral should be improved.In this paper, in order to analyze the problems without any domain element and internal node, we will find the most effective way to transform the domain integral of three types in engineering problem based on the BFM. The main research achievements are as follows:(1) For the domain integral of analytical function type, a new multiple reciprocity formulation is proposed with the modified Helmholtz fundamental solution. Using this new formulation, the domain integrals of analytical function type are converted into the equivalent boundary integral successfully. This new multiple reciprocity formulation breaks the dependence on the fundamental solution of Laplace equation, overcomes the difficulties of convergence and estimating the error in the traditional multiple reciprocity method(MRM). Moreover, due to the relationship of the modified Helmholtz fundamental solution and its high order forms are just the simple multiple, the series of the boundary integral after transformed are very concise.No additional coefficient matrix is added, which saves the cost of computation and storage. This paper applies the new MRM in frequency-domain to the BIE for the transient heat conduction problem. Firstly, we employ the Laplace transformation to reduce the complex of the governing equation and fundamental solution. The corresponding BIE can be obtained by using the fundamental solution of the modified Helmholtz equation. Then, we employ the MRM formulation in frequency domain to convert the domain integrals. A pure BIE is obtained. For the discretized model, only the boundary mesh is needed and domain nodes are not necessary. Therefore, the difficulty of model discretization is reduced. The 4D time dependent problem is transformed to the time-independent problem which only needs the boundary mesh.The advantage of dimensionality reduction is persisted.(2) For the domain integral of non analytical function type, the triple reciprocity method(TRM) in frequency domain is derived. After approximating the non analytically physical distribution by the triple reciprocity interpolation, the domain integral can be converted to the boundary integral successfully. Comparing with the traditional TRM of time domain, the fundamental solutions in frequency domain are time-independent. The high-order fundamental solutions are simpler and the time integral is avoided. The form of the boundary integral after transformation is more concise, which saves the cost of computation and storage. In this paper, the TRM in frequency domain is employed in 3D transient heat conduction problem. The domain integrals of non analytical function type are converted into the boundary integral successfully and the goal of dimensional reduction is achieved. Finally, this new TRMformulation, combined with variable substitution method, is used to solve the transient heat conduction problem of functionally graded materials. The more complex domain integral of non analytical function type after variable substitution is also transformed successfully. Finally, the method which combines the MRM and TRM in frequency domain is obtained. All types of domain integral of known field distribution can be transformed effectively in transient heat conduction problem.(3) For the domain integral including unknown variables, the dual reciprocity method(DRM) based on the radial basis function(RBF) interpolation is derived. The domain discretization is avoided and the dimension of the problem is reduced. In the proposed method of this paper, the RBF is used to interpolate the domain integral function which includes fundamental solutions, non analytical distribution and the unknown variables. The unknown variables in domain integral are separated successfully which are only related to the boundary nodes. The dimension of the problem is reduced. No supplementary equations of the domain source points are needed. Finally, the domain integral in BIE is converted into boundary integral by DRM. The domain discretization is avoided. In this paper, we apply the DRM to the elastic buckling problem to convert the domain integral of non analytical function type. Firstly a new control equation of elastic buckling problem is derived based on complete solid theory, in which the deformation assumptions for rods, beams, plates and shells are all discarded. The entire structure, including all its local small-sized features, is modeled as a three dimensional solid according to its real shape and dimensions. Secondly, the domain integral of non analytical function type in the corresponding BIE is converted by DRM based on the RBF interpolation. Considering that the shape of models in buckling analysis is always strip or flat, only the boundary nodes are chosen as the interpolation points in RBF interpolation. The separated variables of displacement are boundary-only variables. Our method not only avoids the domain discretization, but also reduces the dimension of the problem. The supplementary equations of the domain source points are not necessary any more.Finally, the dual reciprocity BFM of the elastic buckling problem can be implemented numerically without any internal element or internal interpolation points. The goal of dimensionality reduction is achieved.
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