TY - JOUR

T1 - Numerical methods to compute stresses and displacements from cellular forces

T2 - Application to the contraction of tissue

AU - Peng, Qiyao

AU - Vermolen, Fred

PY - 2022/4/30

Y1 - 2022/4/30

N2 - We consider a mathematical model for skin contraction, which is based on solving a momentum balance under the assumptions of isotropy, homogeneity, Hooke's Law, infinitesimal strain theory and point forces exerted by cells. However, point forces, described by Dirac Delta distributions lead to a singular solution, which in many cases may cause trouble to finite element methods due to a low degree of regularity. Hence, we consider several alternatives to address point forces, that is, whether to treat the region covered by the cells that exert forces as part of the computational domain or as ‘holes’ in the computational domain. The formalisms develop into the immersed boundary approach and the ‘hole’ approach, respectively. Consistency between these approaches is verified in a theoretical setting, but also confirmed computationally. However, the ‘hole’ approach is much more expensive and complicated for its need of mesh adaptation in the case of migrating cells while it increases the numerical accuracy, which makes it hard to adapt to the multi-cell model. Therefore, for multiple cells, we consider the polygon that is used to approximate the boundary of cells that exert contractile forces. It is found that a low degree of polygons, in particular triangular or square shaped cell boundaries, already give acceptable results in engineering precision, so that it is suitable for the situation with a large amount of cells in the computational domain.

AB - We consider a mathematical model for skin contraction, which is based on solving a momentum balance under the assumptions of isotropy, homogeneity, Hooke's Law, infinitesimal strain theory and point forces exerted by cells. However, point forces, described by Dirac Delta distributions lead to a singular solution, which in many cases may cause trouble to finite element methods due to a low degree of regularity. Hence, we consider several alternatives to address point forces, that is, whether to treat the region covered by the cells that exert forces as part of the computational domain or as ‘holes’ in the computational domain. The formalisms develop into the immersed boundary approach and the ‘hole’ approach, respectively. Consistency between these approaches is verified in a theoretical setting, but also confirmed computationally. However, the ‘hole’ approach is much more expensive and complicated for its need of mesh adaptation in the case of migrating cells while it increases the numerical accuracy, which makes it hard to adapt to the multi-cell model. Therefore, for multiple cells, we consider the polygon that is used to approximate the boundary of cells that exert contractile forces. It is found that a low degree of polygons, in particular triangular or square shaped cell boundaries, already give acceptable results in engineering precision, so that it is suitable for the situation with a large amount of cells in the computational domain.

UR - https://research.tudelft.nl/en/publications/332208e5-f375-43d9-9e10-9633e8a1eac3

U2 - 10.1016/j.cam.2021.113892

DO - 10.1016/j.cam.2021.113892

M3 - Journal article

VL - 404

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 113892

ER -