热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

MESHLESS METHOD BASED ON INTERPOLATING MOVING LEAST SQUARE SHAPE FUNCTIONS FOR DYNAMIC COUPLED THERMOELASTICITY ANALYSIS

该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin(MLPG)法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kronecker delta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。

The two-dimensional structural dynamic coupled thermoelastic problem is solved by meshless local Petrov-Galerkin (MLPG) method based on the interpolating moving least-squares (IMLS) method. The local weak forms are developed using the weighted residual method from the modified Fourier heat conduction equations and elastodynamic equations, in which the Heaviside step function is used as the test function in each sub-domain. Then the second-order ordinary differential equations describing the coupled thermoelasticity problem are obtained. Using the differential algebraic method, these second-order ordinary differential equations can be transformed into ordinary differential algebraic systems, in which temperature and displacement are chosen as auxiliary variables. The Newmark step-integration method is used to solve the ordinary differential system. The temperature and displacement numerical results can be obtained directly without the Laplace transform. Since the shape functions constructed from the IMLS method possess the Kronecker delta property, the essential boundary conditions can be implemented directly. Finally, two numerical examples are studied to illustrate the effectiveness of this method.