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The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...

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Related Experiment Video

Updated: May 16, 2026

Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces

Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces

Published on: September 9, 2022

Convergence of the Immersed Interface Method in Linear Elasticity.

Sabia Asghar^1,2, Qiyao Peng^3,4, Etelvina Javierre⁵

¹Department of Mathematics and Statistics, Computational Mathematics Group, University of Hasselt, Diepenbeek, Hasselt, 3590 Belgium.

La Matematica

|May 15, 2026

Summary

This study analyzes numerical methods for linear elasticity problems with forces on an interface. It proves that quadrature approximation errors directly impact solution accuracy, guiding more precise computational models.

Keywords:

Convergence Dirac delta distribution Fundamental solutions Linear elasticity Point forces Singularity removal technique

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Area of Science:

Computational Mechanics
Numerical Analysis
Solid Mechanics

Background:

The immersed interface method approximates forces on an interface using numerical quadrature.
Exact integration of forces over interfaces is often intractable in computational mechanics.

Purpose of the Study:

To analyze the error introduced by quadrature approximations of interface forces in linear elasticity.
To establish a theoretical bound for the difference between exact and quadrature-approximated solutions.

Main Methods:

Considered two linear elasticity problems: one with exact interface integral forces, another with quadrature approximation.
Utilized fundamental solutions for linear elasticity and singularity removal principles.
Applied the Extended Trace Theorem for convergence analysis in the L^2-norm.

Main Results:

Proved that the L^2-norm difference between solutions is of the same order as the quadrature error.
Demonstrated convergence in L^2-norm on curves and manifolds for bounded and unbounded domains.
Numerical experiments confirmed theoretical findings, highlighting additional errors in finite element methods.

Conclusions:

Quadrature rule accuracy is critical for the fidelity of immersed interface method simulations.
The theoretical framework provides a basis for error estimation in computational solid mechanics.
Understanding these errors is essential for improving the accuracy of numerical simulations involving interfaces.