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The metric-restricted inverse design problem

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journal contribution
posted on 15.12.2004, 00:00 by Amit Acharya, Marta Lewicka, Mohammad Reza Pakzad

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 more sophisticated algebraic description of integrability. We also recast the problem in a variational setting and analyze the infimum value of the appropriate incompatibility energy, resembling the “non-Euclidean elasticity”.