An Efficient Reduced Basis Method Based On Least Squares Mapping And Its Application

It is difficult even impossible to obtain analytical solutions of problems considered in engineering practice and scientific research due to its irregular geometry or inhomogeneous medium and other reasons. So it is generally solved by numerical method. As the development of science and technology, the problems become more and more complex, and the computational scale of the structure also expanded. Despite the continuous development of computer hardware technology and the application of parallel computing, but for large-scale structure problems, the traditional method is still difficult to meet the engineering needs. So it is necessary to develop efficient algorithms.In this paper,reduced basis method(RBM) is deeply studied.And a new methods is suggested:least squares mapping reduced basis method(LS-RBM). Furthermore,this methods is applied to the structural computation and inverse problems analysis. The research is carried out in following sections.First, the basic principle of reduced basis method was studied, and parameters were extracted from the structure, then the calculation was divided into two stages: off-line and inline. The formulation of beam element and shell element are analyzed, and element stiffness matrix is decomposed to achieve a parameterized formulation on the element level. Then the element matrices with the same parametric properties are assembled. The parametric property is retained in the form of the explicit parameterized finite element formulation.Secondly, a new reduced-basis method based on the least squares mapping is suggested to improve the efficiency of solving the complex problems in mechanical engineering. In this method, sample points are obtained from the parameter domain, and a reduced-basis space is constructed by computing responses of the problem at these points. Then the least squares mapping is employed to conduct projection from original space onto the reduced basis space. A reduced system is obtained and can be solved efficiently. By projecting the reduced solution back into the original space, the approximate solution of the original system is obtained efficiently and accurately. Even with new variables, the solution can be fast obtained by solving the reduced system.Thirdly,a new parameter identification method based on t LS-RBM is suggested for reducing the time consume of repeated calls of forward computation. Due to the application of LS-RMB, the scale of the forward problem is reduced and the calculation efficiency is greatly improved. Then it's easy to identify parameters with high efficiency.The proposed method is tested on car frame analysis and cab of body in white analysis. The numerical examples demonstrate that this method is validity and reliability.