An Adaptive Computational Framework for Structural Shape Optimization: Deterministic and Robust Formulations

Date

2026-8-28

Author

Altıntaş, Oğuz Han

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This thesis develops, analyzes, and implements a computationally efficient, fully adaptive algorithmic framework for robust structural shape optimization problems governed by the linear elasticity system, with the objective of minimizing the expected compliance of structures subjected to uncertainties in the loading and material data. The evolving geometry is represented implicitly by a level-set function on a fixed working domain, together with the ersatz material approach. On the theoretical side, the existence of an optimal shape is established within the class of admissible domains satisfying the $\varepsilon$-cone condition, the shape sensitivities are derived in the distributed (volumetric) form using the formal Lagrange method, and an $H^1$-regular descent field is obtained by projecting the shape derivative through a coercive deformation equation. The computational burden is attacked simultaneously on three fronts. First, a multi-goal a posteriori error estimator based on the dual-weighted residual method controls both the finite element error of the elasticity state and the discretization error of the auxiliary deformation problem defining the descent direction, and drives goal-oriented adaptive mesh refinement. Second, the computational cost of standard Monte Carlo sampling is reduced through an adaptive strategy in which an augmented inner-product test dynamically selects the smallest reliable batch size within each stochastic gradient descent iteration. Third, the descent step length is adapted automatically through an algebraic estimate of the local Lipschitz constant based on the geometric discrepancy between consecutive shapes, eliminating manual step-size tuning. Numerical experiments on a cantilever benchmark and on a leg-like touchdown structure under uncertain contact loads demonstrate that the combined framework attains designs statistically and topologically equivalent to those obtained from brute-force fixed-mesh, full-sampling computations, while reducing the measured computational effort by up to two orders of magnitude. Finally, to stabilize the level-set transport, a model-based anisotropic diffusion tensor is derived from an LES-inspired subgrid kinematic closure and incorporated into the Lax-Friedrichs and streamline upwind Petrov-Galerkin (SUPG) formulations.

Subject Keywords

shape optimization, topology optimization, adaptive mesh refinement, adaptive sampling, monte carlo method, level-set method

URI

https://hdl.handle.net/11511/120714

Collections

Graduate School of Applied Mathematics, Thesis