NumericalRelaxation.jl

This package is an outcome of the research of the Chair of Mechanics - Continuum Mechanics Ruhr-University Bochum in collaboration with Chair of Computational Mathematics - University of Augsburg.

The goal of this project is the (semi-)convexification of some generalized energy density function $W$, which often becomes non-convex in dissipative continuum mechanics or non-linear elasticity. In both cases per incremental time step a potential should become stationary

\[\Pi = \int_{\mathcal{B}} W(\boldsymbol{F}) \ dV - \Pi^{\text{ext}} \rightarrow \text{stationary}.\]

with the corresponding Euler-Lagrange equation

\[\delta \Pi = \int_{\mathcal{B}} \delta \boldsymbol{F} \cdot \partial_{\boldsymbol{F}} W(\boldsymbol{F}) \ dV - \Pi^{\text{ext}} = 0.\]

The solution to it only possesses minimizers if $W$ is convex in the one-dimensional case and polyconvex in the multi-dimensional case. This package implements currently three different (semi)convexification procedures. Two methods implement the convexification in the one-dimensional case and one method for the rank-one convexification in the multi-dimensional case.

• GrahamScan
• AdaptiveGrahamScan

for one-dimensional convexification and

• R1Convexification
• HROC

for the rank-one convexification.

This package only provides the (semi)convex envelope of $W$, the minimization by means of finite elements must be realized by e.g. Ferrite.jl.