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A Multi-Dimensional Mathematical Model for Surface Exerted Point Forces in Elastic Media

Sabia Asghar, Qiyao Peng, Fred J. Vermolen

Abstract

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.

A Multi-Dimensional Mathematical Model for Surface Exerted Point Forces in Elastic Media

Abstract

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 finite element space for multi-dimensional domains. We analyze the -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 -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.

Paper Structure

This paper contains 11 sections, 10 theorems, 72 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Linear Elasticity Equation
  4. Fundamental Solutions to Linear Elasticity Equations
  5. Singularity Removing Technique and Convergence Analysis
  6. Singularity Removing Technique
  7. Convergence between two Forms of Cellular Forces in Immersed Boundary Method
  8. Numerical Results
  9. Two Dimensional Simulations
  10. Three Dimensional Simulations
  11. Discussion and Conclusions

Key Result

Lemma 1

Let $\Omega$ be an open, bounded and connected Lipschitz domain in $\mathbb{R}^n$ with piecewise smooth boundary $\partial \Omega$. Then there exists a positive constant $K$, such that for any vector--valued function ${\bf u} \in {\bf H}_0^1(\Omega)$, we have

Figures (3)

  • Figure 1: a) Geometry and distribution of mesh points in $\mathbb{R}^2$. The circular cell boundary (red-circular marker) with radius $\epsilon$ is divided into finite number of line-segments (mesh points), at the mid-point of each of which point forces are applied (shown with blue-arrows). The outer circular boundary (black circular marker) with radius R over which we evaluate the two-dimensional Green's function. b) For three dimensions, the cell boundary is approximated by mesh points and connecting triangles. Then the three-dimensional fundamental solution for the displacement is evaluated.
  • Figure 2: Displacement field around a circular cell, whose boundary is indicated by the red circle, in two dimensions over a grid of $10 \times 10$. The displacement vectors are indicated by the blue arrows. The length of the arrows indicate the magnitude of the displacement, but does not reflect the real magnitude of the displacement. The input parameters can be found in Table \ref{['Table:para_2D']}.
  • Figure 3: Geometry in $\mathbb{R}^3$ where a spherical cell is centered at the origin i.e. $(0,0,0)$. a) A spherical cell and a line segment which is lying outside the cell (non-intersecting). b) A spherical cell and a plane which is lying outside the cell (non- intersecting). c) A line segment with sub-segments (or -divisions) on it (in context of applying the midpoint rule to the line segment). d) A plane segment divided into sub-rectangles (in context of applying the midpoint rule to the plane segment).

Theorems & Definitions (15)

  • Lemma 1: Korn's Inequality braess2001finite
  • Lemma 2: Lax Milgram's Lemma braess2001finite
  • Lemma 3: Trace Theorem adams2003sobolev
  • Lemma 4: Trace Extension Theorem adams2003sobolev
  • Lemma 5
  • Lemma 6
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • ...and 5 more