Optimal design of plane elastic membranes using the convexified F\"{o}ppl's model

TL;DR

This paper introduces a convexified Föppl's model-based optimal design framework for planar elastic membranes, enabling convex dual formulations and efficient numerical simulations for minimizing membrane compliance.

Contribution

It presents a novel convex variational formulation of membrane design using the convexified Föppl's model, allowing dual problem analysis and computational implementation.

Findings

Existence and regularity of solutions are established.

A finite element scheme is developed for the dual problems.

Numerical simulations demonstrate the method's effectiveness.

Abstract

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane's compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified F\"{o}ppl's model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain-displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. In the primal problem a linear functional is maximized with respect to displacement functions while enforcing that point-wisely the strain lies in an unbounded closed convex set. The dual problem consists in finding equilibrated stresses that are to minimize a convex integral functional of linear growth defined on the space of Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.