Point Forces in Elasticity Equation and Their Alternatives in Multi Dimensions

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 H1 for solutions to the
governing momentum balance equation, we analyse the convergence and quality of
approximation. Furthermore, the expected finite element problems that we get
necessitate to scrutinize 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 into error analyses in the H1-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.