Adjoint optimization of pressurized membrane structures using automatic differentiation tools

This paper presents an adjoint-based method for solving optimization problems involving pressurized membrane structures subject to external pressure loads. Shape optimization of pressurized membranes is complicated by the fact that, lacking bending stiffness, their three-dimensional shape must be sustained by the internal pressure of the inflation medium. The proposed method treats the membrane structure as an immersed manifold and employs a total Lagrangian kinematic description with an analytical pressure–volume relationship for the inflating medium. To demonstrate the proposed method, this paper considers hydrostatically loaded inflatable barriers and develops an application-specific shape parametrization based on the analytical inhomogeneous solution for the inflated shape of cylindrical membranes. Coupling this shape parametrization approach with the adjoint method for computing the gradients of functionals enables a computationally efficient optimization of pressurized membrane structures. Numerical examples include minimization and minimax problems with inequality and state constraints, which are solved considering both plane strain and general plane stress conditions. The numerical implementation leverages the high-level mathematical syntax and automatic differentiation features of the finite-element library FEniCS and related library dolfin-adjoint. The overall techniques generalize to a broad range of structural optimization problems involving pressurized membrane and thin shell structures.