A Vehicle Body Design Oriented Rapid Computational Method Of Large-scale Problems

Vehicle body design phase is a time-consuming process which requires the solution of a large-scale problem with many parameters. A new computational method is proposed in the paper to achieve a rapid way of solving the large-scale linear algebraic equations with many parameters in vehicle design process with respect to the computational cost and the computational accuracy.The basic idea of the method is that although the field variable generally belongs to the infinite-dimensional space associated with the underlying partial differential equation, indeed, it resides on a low-dimensional subspace induced by the parametric dependence. In computational practice, the large-scale system is reduced to a smaller one by Galerkin projection based on that subspace. The structural behaviors are predicted using the reduced system. The physical property of the problem is retained during the projection.The research is carried out in following sections.First, the finite element formulation of shell element is analyzed and the element stiffness matrix is decomposed to achieve a parametrized formulation on the element level. Then the element matrices with the same parametric properties are assembled based on the finite element method via coordinate transforming. The parametric property is retained in the form of the explicit parametrized finite element formulation.Second, a new storage method of large-scale matrix named six-dimensional method is proposed to save the computational cost and improve the computational accuracy. In the new storage formulation, the global stiffness matrix is stored according to the node information which indicates the physical properties of each node. The components of the element stiffness matrices are analyzed first and the locations of the element stiffness matrices in six-dimensional matrix are determined according to the order of the nodes. The global stiffness matrix in six-dimensional form is achieved by assembling the element stiffness in the corresponding locations. Then, the solving method of large-scale linear algebraic equations in six-dimensional form is developed by Cholesky decomposition. As the matrix manipulation is needed in the Galerkin projection, large-scale matrix manipulating method in six-dimensional formulation is studyed to improve the speed of numerical computation.Third, a direct orthogonal method is proposed to construct a subspace which is used in the Galerkin projection. The method is used in some vehicle design problems to evaluate the computational cost and accuracy. When constructing the subspace, the finite element solutions derived from the sampling points are orthogonalized and normalized by using the single value decomposion technique. The left single vectors are orthogonal and used as the basis of the subspace. The subspace constructed by the direct orthogonal method is computational efficient compared with the subapace derived by the traditional Greedy adaptive method.Fourth, to further improve the computational efficiency, a hierarchical adaptive method is developed to construct the subspace based on the greedy adaptive method and the direct orthogonal method. The process of constructing the subspace is devided into two phases, which are expressed as the initial subspace and the ultimate subspace. The initial subspace is constructed using the direct orthogonal method and the ultimate subspace is constructed by updating the initial subspace using the greedy adaptive method. The computational accuracy is improved and the computational cost is saved using the hierarchical adaptive method. The effect of the dimension of the initial subspace is analyzed with respect to the computational cost and accuracy.Finally, an automatic and rapid computational method is proposed to simplifying the computational procedure. The automatic method is developed based on the automatic parameter domain discretization, finite element method, subspace construction and subspace updating techniques. The computational procedure and computational error evaluation of traditional reduced-basis method are changed completely and the offline-online phase is cancelled. It is a new method with real-time subspace construction, real-time subspace updating and real-time error evaluation.The proposed methods are tested on different vehicle design problems. It is found that the proposed methods are of higher accuracy and lower computational cost compared with the traditional reduced-basis methods. It is proved that the proposed methods are efficient in vehicle body design problems and are applicable to many other design contexts, especially in the real-time situation. For shell structural optimization in engineering practice, the proposed techniques provide an efficient way to achieve an optimal design solution. It is also suitable for other structures with a little change in the decomposition of finite element formulation, such as solid element and beam element.