Point Forces in Elasticity Equation and Their Alternatives in Multi Dimensions

TL;DR

This paper analyzes mathematical models of cellular forces in elasticity, comparing point force representations and alternative formulations, and assesses their convergence, accuracy, and well-posedness in multi-dimensional elasticity problems.

Contribution

It provides a detailed error analysis and conditions for consistency among various force modeling approaches in elasticity, including point, smoothed, and boundary condition methods.

Findings

Error bounds in the $H^1$-norm for different models

Conditions for model consistency and convergence

Well-posedness results for the linear elasticity equations

Abstract

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.