On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an exampl...On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.展开更多
In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.
This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)...This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.展开更多
文摘On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.
文摘In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.
基金supported by Natural Science Foundation of China under Great Nos.11072053 and 11372068,and the National Basic Research Program of China under Grant No.2014CB74410.
文摘This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.
