Abstract 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.

中文翻译：

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使用自动微分工具对加压膜结构进行伴随优化

摘要 本文提出了一种基于伴随法的方法，用于求解涉及受外压载荷作用的加压膜结构的优化问题。由于缺乏弯曲刚度，加压膜的形状优化很复杂，它们的三维形状必须由膨胀介质的内部压力来维持。所提出的方法将膜结构视为浸入式歧管，并采用总拉格朗日运动学描述和膨胀介质的分析压力-体积关系。为了演示所提出的方法，本文考虑了静压加载的充气屏障，并基于圆柱膜充气形状的解析非均匀解开发了一种特定于应用程序的形状参数化。将这种形状参数化方法与计算泛函梯度的伴随方法相结合，可以实现加压膜结构的计算效率优化。数值示例包括具有不等式和状态约束的最小化和最小化问题，这些问题是在考虑平面应变和一般平面应力条件的情况下解决的。数值实现利用了有限元库 FEniCS 和相关库 dolfin-adjoint 的高级数学语法和自动微分功能。整体技术推广到广泛的结构优化问题，包括加压膜和薄壳结构。