Some phenotypes of biological cells exert mechanical forces on their direct environment during their development and progression. In this paper the impact of cellular forces on the surrounding tissue is considered. Assuming the size of the cell to be much smaller than that of the computational domain, and assuming small displacements, linear elasticity (Hooke's Law) with point forces described by Dirac delta distributions is used in momentum balance equation. Due to the singular nature of the Dirac delta distribution, the solution does not lie in the classical $H^1$ finite element space for multi-dimensional domains. We analyze the $L^2$-convergence of forces in a superposition of line segments across the cell boundary to an integral representation of the forces on the cell boundary. It is proved that the $L^2$-convergence of the displacement field away from the cell boundary matches the quadratic order of convergence of the midpoint rule on the forces that are exerted on the curve or surface that describes the cell boundary.

翻译：生物细胞在其发育和进展过程中，某些表型会对直接环境施加机械力。本文研究了细胞力对周围组织的影响。假设细胞尺寸远小于计算域尺寸，并假设位移较小，在动量平衡方程中采用线性弹性理论（胡克定律），其中点力由狄拉克δ分布描述。由于狄拉克δ分布的奇异性，其解不位于多维域经典$H^1$有限元空间中。我们分析了细胞边界上线段叠加力向细胞边界力积分表示的$L^2$收敛性。证明远离细胞边界位移场的$L^2$收敛性与施加于描述细胞边界的曲线或曲面上力的中点法则二次收敛阶相匹配。