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We consider several mathematical issues regarding models that simulate forcesexerted by cells. Since the size of cells is much smaller than the size of thedomain of computation, one often considers point forces, modelled by DiracDelta distributions on boundary segments of cells. In the current paper, wetreat forces that are directed normal to the cell boundary and that aredirected toward the cell centre. Since it can be shown that there exists nosmooth solution, at least not in $H^1$ for solutions to the governing momentumbalance equation, we analyse the convergence and quality of the approximation.Furthermore, the expected finite element problems that we get necessitatescrutinizing alternative model formulations, such as the use of smoothed Diracdistributions, or the so-called smoothed particle approach as well as theso-called 'hole' approach where cellular forces are modelled through the use of(natural) boundary conditions. In this paper, we investigate and attempt toquantify the conditions for consistency between the various approaches. Thishas resulted in error analyses in the $H^1$-norm of the numerical solutionbased on Galerkin principles that entail Lagrangian basis functions. The paperalso addresses well-posedness in terms of existence and uniqueness. The currentanalysis has been performed for the linear steady-state (hence neglectinginertia and damping) momentum equations under the assumption of Hooke's law.

Point Forces in Elasticity Equation and Their Alternatives in Multi Dimensions