The metric-restricted inverse design problem

TL;DR

This paper investigates a new class of solid mechanics design problems related to embedding Riemannian manifolds, deriving conditions for existence, and analyzing dimension reduction in a variational framework with $ ext{Gamma}$-convergence.

Contribution

It introduces a necessary and sufficient condition for the existence of solutions, extending classical embedding conditions to a broader context, and studies dimension reduction via $ ext{Gamma}$-convergence.

Findings

Derived a system of total differential equations for existence conditions.

Provided an algebraic description of integrability through successive differentiation.

Established a $ ext{Gamma}$-convergence result for 3D to 2D dimension reduction.

Abstract

We study a class of design problems in solid mechanics, leading to a variation on the classical question of equi-dimensional embeddability of Riemannian manifolds. In this general new context, we derive a necessary and sufficient existence condition, given through a system of total differential equations, and discuss its integrability. In the classical context, the same approach yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor. In the present situation, the equations do not close in a straightforward manner, and successive differentiation of the compatibility conditions leads to a new algebraic description of integrability. We also recast the problem in a variational setting and analyze the infimum of the appropriate incompatibility energy, resembling the "non-Euclidean elasticity." We then derive a $\Gamma$-convergence result for the dimension reduction from $3$d to $2$d in the Kirchhoff energy scaling regime.