论文标题
弹性方程中的点力及其在多维中的替代方案
Point Forces in Elasticity Equation and Their Alternatives in Multi Dimensions
论文作者
论文摘要
我们考虑了模拟细胞施加的力的模型的几个数学问题。由于细胞的大小远小于计算域的大小，因此通常会考虑通过在细胞边界段上进行迪拉克三角洲分布建模的点力。在当前的论文中，我们处理针对细胞边界并针对细胞中心的力的力。由于可以证明没有平滑的解决方案，至少在$ h^1 $中，对于控制动量平衡方程的解决方案，我们分析了近似值的收敛性和质量。此外，我们得到的预期有限元问题需要仔细检查替代模型公式，例如使用平滑的狄拉克分布，或所谓的平滑粒子方法以及所谓的“孔”方法，其中通过使用（自然）边界条件对细胞力进行建模。在本文中，我们调查并试图量化各种方法之间一致性的条件。这导致了基于带有拉格朗日基础函数的Galerkin原理的数值解决方案的$ H^1 $ norm中的错误分析。本文还从存在和独特性方面解决了良好的态度。在胡克定律的假设下，已经对线性稳态（因此忽略惯性和阻尼）动量方程进行了当前的分析。
We consider several mathematical issues regarding models that simulate forces exerted by cells. Since the size of cells is much smaller than the size of the domain of computation, one often considers point forces, modelled by Dirac Delta distributions on boundary segments of cells. In the current paper, we treat forces that are directed normal to the cell boundary and that are directed toward the cell centre. Since it can be shown that there exists no smooth solution, at least not in $H^1$ for solutions to the governing momentum balance equation, we analyse the convergence and quality of the approximation. Furthermore, the expected finite element problems that we get necessitate scrutinizing alternative model formulations, such as the use of smoothed Dirac distributions, or the so-called smoothed particle approach as well as the so-called 'hole' approach where cellular forces are modelled through the use of (natural) boundary conditions. In this paper, we investigate and attempt to quantify the conditions for consistency between the various approaches. This has resulted in error analyses in the $H^1$-norm of the numerical solution based on Galerkin principles that entail Lagrangian basis functions. The paper also addresses well-posedness in terms of existence and uniqueness. The current analysis has been performed for the linear steady-state (hence neglecting inertia and damping) momentum equations under the assumption of Hooke's law.
