Point forces in elasticity equation and their alternatives in multi dimensions

Abstract

Deep dermal wounds induce skin contraction as a result of the traction forcing exerted by (myo)fibroblasts on their immediate environment. These (myo)fibroblasts are skin cells that are responsible for the regeneration of collagen that is necessary for the integrity of skin We consider several mathematical issues regarding models that simulate traction forces exerted by (myo)fibroblasts. Since the size of cells (e.g. (myo)fibroblasts) 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 to simulate the traction forces exerted by the skin cells. In the current paper, we treat the forces that are directed normal to the cell boundary and 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 approximation. Furthermore, the expected finite element problems that we get necessitate to scrutinize alternative model formulations, such as the use of smoothed Dirac Delta 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 L 2-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.