径向基函数配点法分析三维功能梯度材料板的静力和动力问题

Radial basis collocation method for the static and dynamic problems of three dimensional functionally graded plate

传统配点法在求解动力学问题时会存在误差随时间累积的问题,而无网格径向基函数配点法在全域内采用具有无限连续性的径向基函数作为近似函数,结合配点法构建方程,通过最小二乘法进行求解。无网格径向基函数配点法不仅在数值计算过程中不需要任何网格,是真正的无网格法,而且易于离散,精度高,不需要积分,计算效率高;径向基函数的近似函数仅与距中心点的距离有关,非常适宜于求解三维问题。对于这种方法,本文先离散空间域,然后再离散时间域,并在每一时间步内施加边界条件,来分析三维功能梯度材料板的静力和动力问题,据此可解决传统配点方法在求解动力问题时误差随时间累积的问题。数值分析表明,材料性能呈梯度分布会导致其力学性能在梯度方向呈现非线性变化,不同的梯度分布模式会导致力学性能非线性变化的幅度不同。

Meshfree radial basis collocation method （RBCM）is introduced to study the static and dynamic problems of the three dimensional （3D）functionally graded plate.Radial basis functions which possess infinite continuity are employed to be the ap-proximation,collocation method is utilized for discretization,and least squares approach is adopted to solve the discretized equa-tions .No mesh will be required in the discretization and resolution and therefore RBCM is a truly meshfree method.Discretization scheme of RBCM is quite simple and high accuracy can be obtained.Radial basis approximation only depends on the distance from the center which makes RBF a good candidate to solve 3D problems.Conventional collocation method introduces error accumula-tion on the boundaries.In this paper,the spatial domain is discretized first,and then temporal domain is discretized.Boundary conditions are imposed in each time step.Therefore,error accumulation on the boundaries can be overcome.Numerical simula-tions demonstrate that graded distribution of material properties will lead to nonlinear variation of the mechanical properties.The magnitude of the nonlinear variation will be different for different graded distributions.