A Real-time Algorithm In Structure Dynamic Analysis Based On Reduced Basis Method

As the development of science and technology, the problems considered in engineering practice become more and more complex, and the computational scale of the structure also expanded. It is difficult to satisfy the efficiency demand of the complex engineering problems by improving the computer techniques. Thus it is necessary to develop an efficient numerical algorithm, which not only can significantly reduce the problem size and computational cost but also retain the reliability of the solution. High efficiency can be achieved in large scale structure analysis. In this case, many researchers began to study on real-time calculation methods and obtain a lot of achievement. This thesis introduces one of the real-time algorithms named reduced-basis method, and applies it to the analysis of structure dynamics. The works are summarized as following:1. A reduced-basis method in harmonic response of linear elastic structures with damping is developed. Firstly discretize the frequency domain into limited points. And a logarithmic sampling method is used to select samples. The greedy algorithm is suggested to train the samples adaptively and a series vectors is obtained. A Gram-Schmidt algorithm is used to orthogonalize the vectors in order to guarantee their linear independence, and then an optimum reduced-basis space is constructed. After the Galerkin projection of matrices into the reduced basis space, a reduced system is obtained and can be solved efficiently. The reduced solution is obtained. By projecting the reduced solution back into the original space, the approximate solution of the original system subjected to a harmonic load is obtained efficiently and accurately.2. A posteriori error estimation for reduced-basis method and FEM is proposed, and a method for constructing the reduced space based on greedy algorithm is studied. The harmonic response of structure with damping is analyzed using different sampling method. The efficiency and accuracy of reduced-basis method are validated by computational time and posteriori error.3. A reduced-basis method in transient response of piezoelectric laminates subjected to coupled electro-mechanical loads is studied. The laminates were discretized in the thickness direction by plate elements, and the governing equations were obtained. The governing equations in wavenumber domain were obtained by Fourier transformation. The Newmark direct integral method was employed to calculate the displacement responses and the potential responses in wavenumber domain. The reduced-basis method was introduced during the using of Newmark direct integral method and a reduced basis space was constructed. The equivalent stiffness matrix,mass matrix and load vector were projected into the space. The reduced Newmark formulation was obtained and the responses in wavenumber domain were calculated efficiency. The transient responses were obtained by using the inverse Fourier transformation.The proposed method is tested on structural dynamic analysis of damping structure and piezoelectric laminates. It is found that the proposed method is of higher accuracy and lower computational cost compare with the traditional method.