基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟

为数值求解描述不同物质间相位分离现象的高阶非线性Cahn-Hilliard(C-H)方程,发展了一种基于局部加密纯无网格有限点集法(local refinement finite pointset method,LR-FPM).其构造过程为:1)将C-H方程中四阶导数降阶为两个二阶导数,连续应用基于Taylor展开和加权最小二乘法的FPM离散空间导数;2)对区域进行局部加密和采用五次样条核函数以提高数值精度;3)局部线性方程组求解中准确施加含高阶导数Neumann边值条件.随后,运用LR-FPM求解有解析解的一维/二维C-H方程,分析粒子均匀分布/非均匀分布以及局部粒子加密情况的误差和收敛阶,展示了LR-FPM较网格类算法在非均匀布点情况下的优点.最后,采用LR-FPM对无解析解的一维/二维C-H方程进行了数值预测,并与有限差分结果相比较.数值结果表明,LR-FPM方法具有较高的数值精度和收敛阶,比有限差分法更易数值实现,能够准确展现不同类型材料间相位分离非线性扩散现象随时间的演化过程.

Simulation of nonlinear Cahn-Hilliard equation based on local refinement pure meshless method

The phase separation phenomenon between different matters plays an important role in many science fields.And the high order nonlinear Cahn-Hilliard(C-H)equation is often used to describe the phase separation phenomenon between different matters.However,it is difficult to solve the high-order nonlinear C-H equations by the theorical methods and the grid-based methods.Therefore,in this work the meshless methods are addressed,and a local refinement finite pointset method(LR-FPM)is proposed to numerically investigate the high-order nonlinear C-H equations with different boundary conditions.Its constructive process is as follows.1)The fourth derivative is decomposed into two second derivatives,and then the spatial derivative is discretized by FPM based on the Taylor series expansion and weighted least square method.2)The local refinement and quintic spline kernel function are employed to improve the numerical accuracy.3)The Neumann boundary condition with high-order derivatives is accurately imposed when solving the local linear equation sets.The1 D/2 D C-H equations with different boundary conditions are first solved to show the ability of the LR-FPM,and the analytical solutions are available for comparison.Meanwhile,we also investigate the numerical error and convergence order of LR-FPM with uniform/non-uniform particle distribution and local refinement.Finally,1 D/2 D C-H equation without analytical solution is predicted by using LR-FPM,and compared with the FDM result.The numerical results show that the implement of the boundary value condition is accurate,the LRFPM indeed has a higher numerical accuracy and convergence order,is more flexible and applicable than the grid-based FDM,and can accurately predict the time evolution of nonlinear diffusive phase separation phenomenon between different materials time.