A novel method to impose boundary conditions for higher-order partial differential equations
A novel method to impose boundary conditions for higher-order partial differential equations
复制标题
一种为高阶偏微分方程施加边界条件的新方法
DOI：
10.1016/j.cma.2021.114526
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发表时间：
2022
影响因子：
7.2
通讯作者：
Gomez, Hector
中科院分区：
文献类型：
--
作者：
Hu, Tianyi;Leng, Yu;Gomez, Hector
偏微分方程（PDE）的空间导数的阶数高于2正受到越来越多的关注，部分是由于目前流行的相场方法。在有限元界，等几何分析和其他离散化方案的增长，采用高阶连续性的近似空间，促进了使用变分公式，避免使用辅助未知数表示的衍生物的解决方案。然而，这种方法的一个警告是边界条件的准确和有效的强加，特别是在复杂的几何形状。本文提出了一种新的方法，自然地施加边界条件的弱形式的高阶偏微分方程。我们的方法是基于一个专门设计的弱形式，其中要施加的边界条件加权的权重函数的导数。这需要通过部分进行多次积分，由于基函数的平滑性，这是允许的。以Cahn-Hilliard方程和等温Navier-Stokes-Norteweg方程为例，验证了该方法的有效性。我们证明了，如果偏微分方程的解是足够光滑的，建议的变分方程和原来的偏微分方程是等价的。我们用等几何分析法离散变分方程。该方法的收敛性与基函数的最佳逼近误差一致。数值例子说明了该方法的适用性映射几何与非共形网格。
Partial differential equations (PDEs) with spatial derivatives of order higher than two are receiving increasing attention, partially due to the current popularity of the phase-field method. In the finite element community, the growth of Isogeometric Analysis and other discretization schemes that employ approximation spaces with high-order continuity has fostered the use of variational formulations that avoid the use of auxiliary unknowns representing derivatives of the solution. However, one of the caveats of this approach is the accurate and efficient imposition of boundary conditions, especially on complex geometries. This paper proposes a new method to impose boundary conditions naturally in the weak form of higher-order PDEs. Our method is based on a specially-designed weak form in which the boundary conditions to be imposed are weighted by derivatives of the weight functions. This requires multiple integrations by parts which are allowed due to the smoothness of the basis functions. The Cahn–Hilliard equation and the isothermal Navier–Stokes–Norteweg equations are used as examples to demonstrate the proposed method. We show that if the solutions of the PDEs are smooth enough, the proposed variational equations and the original PDEs are equivalent. We discretize the variational equations using Isogeometric Analysis. Convergence results of the proposed method agree with the best approximation errors of the basis functions. Numerical examples illustrate the applicability of the approach to mapped geometries with non-conformal grids.