In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Co...In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite ele- ment methods （FEMs） or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches com- putationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method （RBCM） motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the pro- posed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the increment al-it erative process.展开更多
Strong form collocation with radial basis approximation is introduced for the numerical solution of transient dynamics.Von Neumann stability analysis of this radial basis collocation method is performed to obtain the ...Strong form collocation with radial basis approximation is introduced for the numerical solution of transient dynamics.Von Neumann stability analysis of this radial basis collocation method is performed to obtain the stability conditions for second order wave equation with central difference temporal discretization.The shape parameter of the radial basis functions not only has strong influence on the spatial stability and accuracy,but also has profound influence on the temporal stability.Numerical studies are conducted and show reasonable agreement with stability analysis.Conclusions of selecting shape parameters as well as spatial discretization for solution stability are also presented.展开更多
基金Project supported by the Ministry of Science and Technology of Taiwan(No.MOST 104-2221-E-009-193)
文摘In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite ele- ment methods （FEMs） or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches com- putationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method （RBCM） motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the pro- posed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the increment al-it erative process.
文摘Strong form collocation with radial basis approximation is introduced for the numerical solution of transient dynamics.Von Neumann stability analysis of this radial basis collocation method is performed to obtain the stability conditions for second order wave equation with central difference temporal discretization.The shape parameter of the radial basis functions not only has strong influence on the spatial stability and accuracy,but also has profound influence on the temporal stability.Numerical studies are conducted and show reasonable agreement with stability analysis.Conclusions of selecting shape parameters as well as spatial discretization for solution stability are also presented.
