Based on the eigenvector expansion idea, the Multiscale Eigenelement Method(MEM)was proposed by the author and co-workers. MEM satisfies two equivalent conditions, one condition is the equivalence of strain energy, an...Based on the eigenvector expansion idea, the Multiscale Eigenelement Method(MEM)was proposed by the author and co-workers. MEM satisfies two equivalent conditions, one condition is the equivalence of strain energy, and the other is the deformation similarity. These two equivalent conditions character the structure-preserving property of a multiscale analysis method. The equivalence of strain energy is necessary for achieving accurate macro behaviors such as lower order frequencies, while the deformation similarity is essential for predicting accurate micro behaviors such as stresses. The MEM has become a powerful multiscale method for the analysis of composite structures because of its high accuracy and efficiency. In this paper, the research advances of MEM are reviewed and all types of eigenelement methods are compared, focusing on superiorities and deficiencies from practical viewpoint. It is concluded that the eigenelement methods with smooth shape functions are more suitable for the analysis of macro behaviors such as lower order frequencies, and the eigenelement methods with piecewise shape functions are suitable for the analysis of both macro and micro behaviors.展开更多
In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we ...In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.展开更多
基金supported by the National Natural Science Foundation of China (Nos. 11672019, 11372021 and 37686003)the Academic Excellence Foundation of BUAA for PhD Students (No. 2017038)
文摘Based on the eigenvector expansion idea, the Multiscale Eigenelement Method(MEM)was proposed by the author and co-workers. MEM satisfies two equivalent conditions, one condition is the equivalence of strain energy, and the other is the deformation similarity. These two equivalent conditions character the structure-preserving property of a multiscale analysis method. The equivalence of strain energy is necessary for achieving accurate macro behaviors such as lower order frequencies, while the deformation similarity is essential for predicting accurate micro behaviors such as stresses. The MEM has become a powerful multiscale method for the analysis of composite structures because of its high accuracy and efficiency. In this paper, the research advances of MEM are reviewed and all types of eigenelement methods are compared, focusing on superiorities and deficiencies from practical viewpoint. It is concluded that the eigenelement methods with smooth shape functions are more suitable for the analysis of macro behaviors such as lower order frequencies, and the eigenelement methods with piecewise shape functions are suitable for the analysis of both macro and micro behaviors.
文摘In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.
