On material optimisation for nonlinearly elastic plates and shells

Abstract

This paper investigates the optimal distribution of hard and soft material on elastic plates. In the class of isometric deformations stationary points of a Kirchhoff plate functional with incorporated material hardness function are investigated and a compliance cost functional is taken into account. Under symmetry assumptions on the material distribution and the load it is shown that cylindrical solutions are stationary points. Furthermore, it is demonstrated that the optimal design of cylindrically deforming, clamped rectangular plates is non trivial, i.e. with a material distribution which is not just depending on one axial direction on the plate. Analytical results are complemented with numerical optimization results using a suitable finite element discretization and a phase field description of the material phases. Finally, using numerical methods an outlook on the optimal design of non isometrically deforming plates and shells is given.

Finite Element Approximation of Large-Scale Isometric Deformations of Parametrized Surfaces

“…Note that this flat case is already covered by Bartels [1]. However, as mentioned above, our numerical method differs by the enforcing of a nodal-wise isometry constraint as in [11] instead of the linearization of the contraint in a gradient descent. In Table 1, for decreasing grid size h, the minimal discrete energy, the isometry error in L 1 , the L 1 -norm of the discrete Gauss-curvature K h rψ h s " detpgrψ h s ´1∇θ h rψ h s ¨nrψ h sq with ∇θ h rψ h s ¨nrψ h s " ´ř3 l"1 n l rψ h sB k θ j h rψ l h s ¯k,j"1,2 and the L 2 approximate error in the Hessian of the energy are shown.…”

“…for h small enough. From this, the continuity of E h r¨s on W 3 h , and the norm property of ψ h Þ Ñ ∇θ h rψ h s L 2 pωq the existence of a minimizer ψ h of E h r¨s follows for h sufficiently small and ∇θ h rψ h s L 2 pωq ď C. Then, Poincaré's inequality yields θ h rψ h s L 2 pωq ď C. Applying once more the norm equivalence estimates (11) we obtain ∇ψ h L 2 pωq ď C . Now, we consider the lim inf inequality.…”

“…. Now, using the triangle inequality, Young's inequality, the nodal isometry constraint, and the norm equivalence estimates (11) we obtain…”

“…With respect to the definition of a recovery sequence, we consider a function ψ P H 3 pω; R 3 qXA. For h ą 0, let ψ h " I DKT ψ P W 3 h be the interpolation of ψ defined on every triangle T P T h by ψ h pzq " ψpzq and ∇ψ h pzq " ∇ψpzq for all vertices z P N h X T. Taking into account (10), (11), and (13) we have for every…”

“…Furthermore, Bartels considered a linearization of the isometry constraint and a discrete gradient flow approach to minimize the energy. In [11], Hornung et al applied the DKT element for a material optimization problem on thin elastic plates, where the isometry constraint was strictly enforced in a second order scheme.…”