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首页> 外文期刊>Journal of Computational Physics >The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes
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The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes

机译：多面体网格上3D静磁场问题的模拟有限差分法

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摘要

We extend the mimetic finite difference (MFD) method to the numerical treatment of magnetostatic fields problems in mixed div-curl form for the divergence-free magnetic vector potential. To accomplish this task, we introduce three sets of degrees of freedom that are attached to the vertices, the edges, and the faces of the mesh, and two discrete operators mimicking the curl and the gradient operator of the differential setting. Then, we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div-curl variational formulation of the magnetostatic equations. This construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the MFD method. We also discuss the linear algebraic form of the resulting MFD scheme, its practical implementation, and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. The accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem.
机译：我们将模拟有限差分法（MFD）扩展到混合div-curl形式的无静磁场矢量势的静磁场问题的数值处理。为了完成此任务，我们引入了三个自由度集，它们分别附加到网格的顶点，边和面，以及两个离散算子，它们模仿微分设置的卷曲和梯度算子。然后，我们为静磁方程的div-curl变分公式的域积分进行数值离散化，提出了两个合适的正交规则。此构造基于代数一致性条件，该条件概括了MFD方法的内积的常规构造。我们还讨论了所得MFD方案的线性代数形式及其实际实现，并通过将Hyman和Shashkov的逻辑矩形或三次网格的概念推广到非结构化多面体网格的情况，讨论了数值解的存在性和唯一性。通过数值解决一组学术问题和一个实际工程问题来说明该方法的准确性。

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