Polynomial inclusions: Definitions, applications, and open problems
Polynomial inclusions: Definitions, applications, and open problems
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DOI：
10.1016/j.jmps.2023.105440
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发表时间：
2023-09
影响因子：
5.3
通讯作者：
Tianyu Yuan;Liping Liu
中科院分区：
文献类型：
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作者：
Tianyu Yuan;Liping Liu
物理科学和工程中的预测建模主要基于求解某些偏微分方程，其中解的复杂性由域的几何形状决定。由于球面和椭球面域显式解的广泛应用，特别是弹性中的Eshelby解，我们提出了椭球面形状的一种推广，称为多项式包含。k次多项式包含（或简称p-包含）定义为光滑、连通、有界的物体，其牛顿势是该物体内部的k次多项式。从这个观点出发，椭球体被确定为唯一的二阶p包体；在各种物理环境中，许多基本问题对于一般p包体和椭球体都有简单的闭型解。因此，我们预计p-内含物将在预测材料模型、优化设计和反问题等应用中发挥作用。然而，二阶以上p-包含的存在性并不明显，更不用说它们的显式代数参数化了。在这项工作中，我们在势理论的背景下探索了p-内含物的替代定义和性质。基于变分不等式理论，我们证明了p-包含确实存在于某些多项式中，尽管完整的表征仍然是开放的。为了便于数值模拟和研究p包体的几何性质，我们将p包体表面的确定重新表述为非局部几何流动。在二维空间中，利用保角映射的方法得到了p-包含的显式代数参数化。我们也提出了一些开放的问题，这些问题的解决将加深我们对领域几何、牛顿势和一般偏微分方程解之间关系的理解。最后，我们给出了p-内含物在Eshelby内含物问题和磁体设计中的应用实例。
Predictive modeling in physical science and engineering is mostly based on solving certain partial differential equations where the complexity of solutions is dictated by the geometry of the domain. Motivated by the broad applications of explicit solutions for spherical and ellipsoidal domains, in particular, the Eshelby’s solution in elasticity, we propose a generalization of ellipsoidal shapes called polynomial inclusions. A polynomial inclusion (or p-inclusion for brevity) of degree k is defined as a smooth, connected and bounded body whose Newtonian potential is a polynomial of degree k inside the body. From this viewpoint, ellipsoids are identified as the only p-inclusions of degree two; many fundamental problems in various physical settings admit simple closed-form solutions for general p-inclusions as for ellipsoids. Therefore, we anticipate that p-inclusions will be useful for applications including predictive materials models, optimal designs, and inverse problems. However, the existence of p-inclusions beyond degree two is not obvious, not to mention their explicit algebraic parameterizations. In this work, we explore alternative definitions and properties of p-inclusions in the context of potential theory. Based on the theory of variational inequalities, we show that p-inclusions do exist for certain polynomials, though a complete characterization remains open. We reformulate the determination of surfaces of p-inclusions as nonlocal geometric flows which are convenient for numerical simulations and studying geometric properties of p-inclusions. In two dimensions, by the method of conformal mapping we find an explicit algebraic parameterization of p-inclusions. We also propose a few open problems whose solution will deepen our understanding of relations between domain geometry, Newtonian potentials, and solutions to general partial differential equations. We conclude by presenting examples of applications of p-inclusions in the context of Eshelby inclusion problems and magnet designs.