Minimization of peak stresses with the shape derivative

Area of Science:

• Solid Mechanics
• Computational Engineering
• Applied Mathematics

Background:

• Minimizing peak stresses is crucial for structural integrity in linear elasticity.
• Traditional methods often rely on L2-norm regularization, which may not effectively handle stress concentrations.

Purpose of the Study:

• To develop and evaluate a novel method for minimizing the maximal von Mises stress in elastic bodies.
• To address the challenges posed by non-smooth shape functionals in stress minimization problems.

Main Methods:

• Derivation of the shape derivative for the maximal von Mises stress functional.
• Application of the Clarke sub-differential for non-smooth optimization.
• Implementation of a steepest descent algorithm for numerical simulations.

Main Results:

• The proposed method successfully minimizes peak stresses by targeting the maximal von Mises stress.
• Numerical simulations demonstrate superior performance compared to standard L2-norm regularization techniques.
• The approach effectively handles the non-smooth nature of the objective functional.

Conclusions:

• Minimizing the maximal von Mises stress offers a more effective strategy for peak stress reduction in linear elasticity.
• The developed shape derivative and optimization framework provide a robust tool for non-smooth variational problems in mechanics.
• This work contributes to advancing computational methods for structural optimization and design.