以強形式架構求解幾何非線性邊界值問題

Abstract

強形式配置法係一無網格法，此方法引入函數近似並以節點做離散，故不需要建立網格，因此可避免求解大變形問題時，因結構變形後產生不連續情形的數值模擬誤差。 本研究首先提出強形式增量疊代流程求解幾何非線性問題，將徑向基函數結合強形式配置法求解非線性邊界值問題。在非線性彈性力學分析中，使用總體拉格朗日表述法描述物體之平衡狀態，引入徑向基函數近似，以強形式表述法推導增量式之邊界值問題，包含控制方程與邊界條件，接著使用牛頓疊代法建立增量疊代演算。本研究最後以受拉力軸力桿問題與受彎矩懸臂梁問題驗證所提出之強形式增量疊代流程求解幾何非線性問題。

The strong form collocation method is a truly meshfree method, which introduces function approximation at nodes and uses direct collocation without using background mesh. As a consequence, it avoids mesh related issues when structures are subjected to large deformation and encounter discontinuity. In this study, we propose a strong-form formulation for performing the incremental-iterative process to solve geometric nonlinear problems, in which the radial basis collocation method is adopted for solving nonlinear boundary value problems. In the analysis of nonlinear elasticity, we first describe the equilibrium of a body by using the total Lagrangian formulation. Then, we introduce the radial basis function approximation together with the strong form formulation to derive the incremental equation of the nonlinear boundary value problem, which includes the governing equation and boundary conditions. Finally, we establish an incremental-iterative algorithm by using the Newton-Raphson iteration scheme. To demonstrate the proposed framework for solving geometric nonlinear problems, three benchmark problems including tensile bar problems and a cantilever beam under pure bending are solved.