离散几何上非线性Naghdi壳的形状优化
Shape optimisation of nonlinear Naghdi shells on discrete geometries
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中文总结 AI 辅助
本文提出嵌入非线性Naghdi壳响应的自动化形状优化框架，在离散三角剖分上实现，经基准验证可有效降低壳的弹性应变能与平均挠度。
中文摘要 AI 辅助
薄壳通过其形状承受载荷，在有限挠度下其刚度会随载荷本身变化，因此用线性模型找到的形状最优解可能远非真正最优。我们提出了一个自动化框架，将完全几何非线性的壳响应嵌入形状优化循环中。正向模型是一个五参数非线性Naghdi壳，通过部分选择性缩减积分抑制锁死效应，直接在离散（ faceted ）三角剖分上运行，并带有数值恢复的 director 场——无需等几何方法中精确的中面参数化，而当几何本身作为设计变量时，该参数化就不复存在。该模型在 Firedrake 中实现，自动生成其残差、一致切线和伴随项；通过对完整载荷延续求解过程进行算法微分计算得到的形状导数，驱动 Fireshape/ROL 信赖域优化器。正向求解器重现了 Sze/Abaqus 关于受点载荷的夹持半圆柱壳的基准测试结果，捕捉到几何线性模型无法重现的渐进硬化现象。优化过程通过 COMSOL 基准测试（受弯曲的金属薄板支架）得到验证：该框架形成了与基准相同的离中面波纹机制，在规定的位移预算内实现了弹性应变能降低87%，与基准的量级和面积变化匹配。将其应用于弯曲半圆柱时，该框架形成了光滑的硬化折痕，使壳在载荷下的平均挠度降低了78%。
英文摘要
A thin shell carries load through its shape and, at finite deflections, its stiffness changes with the load itself, so a shape optimum found with a linear model can be far from optimal. We present an automated framework that embeds the fully geometrically nonlinear shell response in the shape-optimisation loop. The forward model, a five-parameter nonlinear Naghdi shell stabilised against locking by partial selective reduced integration, operates directly on a discrete (faceted) triangulation with a numerically recovered director field -- dispensing with the exact mid-surface parameterisation of isogeometric approaches, a chart that ceases to exist once the geometry itself is the design variable. Implemented in Firedrake, the model generates its residual, consistent tangent and adjoint automatically; shape derivatives, computed by algorithmic differentiation through the full load-continuation solve, drive the Fireshape/ROL trust-region optimiser. The forward solver reproduces the Sze/Abaqus benchmark for a clamped semi-cylindrical shell under a point load, capturing the progressive stiffening that a geometrically linear model cannot reproduce. The optimisation is validated against the COMSOL benchmark, a sheet-metal bracket under bending: the framework develops the same off-mid-plane corrugation mechanism and attains an 87% reduction of elastic strain energy within the prescribed displacement budget, matching the benchmark's magnitude and area change. Applied to the curved semi-cylinder, it forms a smooth stiffening crease that reduces the shell's average deflection under load by 78%.
