非线性热传导问题的基于滑动Kriging插值的MLPG法

利用基于滑动Kriging插值的无网格局部Petrov-Galerkin(MLPG)法来求解二维非线性稳态和瞬态热传导问题,Heaviside分段函数作为局部弱形式的权函数,并通过加权余量法推导相应的离散方程.该问题考虑了材料热传导系数随温度的线性变化,并通过拟线性法来求解非线性问题的解,时间域的离散通过向后差分法来实现.基于滑动Kriging插值构造MLPG中的形函数由于满足克罗内克δ性质,因此可以直接准确地施加本质边界条件.在构造刚度矩阵过程中,只涉及边界积分,不涉及区域积分和奇异积分.将数值计算结果与有限元法得到的结果加以对比可以看出,基于滑动Kriging插值的MLPG法能够很好地解决此类热传导问题.

MLPG method based on moving Kriging interpolation for solving nonlinear heat conduction problems

A meshless local Petrov-Galerkin (MLPG) method based on the moving Kriging interpolation is employed for solving two-dimensional nonlinear steady and transient heat conduction problems in which the Heaviside step function is used as the test function in each sub-domain and the local weak forms are developed using the weighed residual method locally from the partial differential equations of heat conduction problems. The thermal conductivity of the material is assumed to vary linearly with temperature. A quasi-linearization scheme is adopted to avoid the iteration for nonlinear solution and the backward difference method is selected for the time discretization scheme. The essential boundary conditions can be implemented directly as the shape functions possessing the Kronecker delta property which is constructed based on moving Kriging interpolation. This method does not involve the sub-domain and singularity integral in constructing the stiffness matrix except for the boundary integral. Numerical examples have shown that the MLPG method based on the moving Kriging interpolation can handle these problems very well compared with the numerical solutions obtained from the finite element method.