gradient-based shape optimization

In engineering design, the non-differentiability of conventional CAE workflows—particularly mesh generation and physics simulation—hinders gradient-based high-dimensional optimization. To address this, we propose an end-to-end differentiable shape optimization framework: geometry is represented via signed distance fields (SDFs), and a 3D U-Net serves as a full-field surrogate model that directly learns the mapping from SDFs to physical fields (e.g., pressure, velocity), thereby bypassing non-differentiable components without requiring differentiable solvers or adjoint methods. The surrogate is embedded within a differentiable optimization pipeline, enabling backpropagation to compute gradients with respect to design parameters. Evaluated on aerodynamic shape optimization, our method achieves fully gradient-driven, efficient iterative design refinement. Results demonstrate substantial improvements in optimization efficiency and validate the framework’s feasibility and advantages in complex engineering applications.

This work proposes the first differentiable geometry processing system that seamlessly integrates with modern machine learning frameworks, addressing the longstanding challenge of combining geometric algorithms—typically non-differentiable and reliant on complex control flow—with gradient-based optimization. By unifying the adjoint method with a scatter-gather mesh processing paradigm, the system enables efficient gradient computation for existing geometric algorithms without requiring algorithmic reimplementation. It supports state-of-the-art solvers such as local-global and ADMM schemes and provides native differentiability for classical operations including curvature flows and conformal parameterizations. Evaluated on multiple inverse geometry problems, the approach significantly reduces both memory consumption and computational overhead, outperforming general-purpose differentiable optimization tools in runtime efficiency while dramatically lowering implementation effort.

This work addresses the challenge in aerodynamic inverse design, where high-dimensional geometry and computationally expensive simulations hinder the simultaneous optimization of performance and geometric plausibility. To overcome this, the authors propose a unified framework that integrates optimal design points with design distributions by combining optimization and guided generative modeling. Key innovations include a novel loss function for cost predictor training, a density gradient-based optimization strategy, and an efficient approximate conditional covariance estimation algorithm that enables a guidance generation framework without additional training. The approach is implemented with OpenFOAM simulations and offline reinforcement learning, and validated through 3D-printed wind tunnel experiments. It demonstrates significant performance improvements on both 2D control tasks and high-fidelity 3D benchmarks for automotive and aerospace applications, showcasing both effectiveness and practicality.

本文针对型材挤出模具设计中的复杂流道几何问题，提出了一种基于伴随形状优化的自动设计框架，并采用非边界一致几何表示方法解决了网格生成难题。

Traditional heuristic approaches struggle to precisely control the nonlinear mechanical behavior of pneumatic soft actuators to achieve desired deformations. To address this challenge, this work proposes the first gradient-based inverse design framework that integrates nonlinear finite element modeling, three-dimensional shape parameterization, and pneumatic actuation mechanics. By leveraging gradient-based optimization, the method directly tailors the actuator’s geometric configuration to realize complex, target deformation patterns. This approach overcomes the limitations of conventional design strategies, enabling high-fidelity customization of soft actuator behavior. Experimental validation demonstrates excellent agreement between simulated and measured deformations of the designed actuators, significantly enhancing the accuracy and capability of demand-driven soft actuator design.

This work addresses the inability of traditional linear shell models to accurately capture stiffness variations under large deformations, which often leads to distorted shape optimization results. For the first time, a fully geometrically nonlinear Naghdi shell formulation is implemented directly on discrete triangular meshes for shape optimization, eliminating the need for mid-surface parametrization and circumventing limitations inherent in isogeometric analysis. The approach integrates a five-parameter nonlinear shell model—stabilized via selectively reduced integration—with automated residual and tangent operator generation in Firedrake, adjoint-based sensitivity analysis enabled by automatic differentiation, and the Fireshape/ROL trust-region optimizer. The framework successfully reproduces benchmark cases from Sze and Abaqus, accurately capturing stiffness stiffening effects, and achieves an 87% reduction in elastic strain energy for a sheet-metal bracket and a 78% decrease in average deflection for a hemicylindrical shell.

This work addresses the lack of a unified, open-source, and modular platform for collaboratively exploring shape and topology optimization methods in both teaching and research. The authors present an object-oriented, MATLAB-based open-source framework that employs abstract base classes to define core interfaces, enabling seamless integration of parametric and level-set-based shape optimization alongside density-based, level-set, and topological sensitivity approaches to topology optimization. By directly mapping mathematical formulations to executable code, the framework allows users to extend objective functionals or constraints simply by deriving new classes without modifying the core implementation. Highly modular and reproducible, the framework bridges the gap between shape and topology optimization, offering a continuous research pathway. Its effectiveness and flexibility are demonstrated through diverse numerical examples in both educational and research contexts.

This work addresses the lack of a unified modular framework for analyzing adaptive optimizers, which hinders a precise characterization of their behavior under constraints on directional reachability, information budgets, and update rules. We propose a geometric–non-geometric decoupled calculus for optimizers: the geometric module, constituted by a family of positive-definite cometrics, captures realizable descent directions, while the non-geometric module governs mechanisms such as information processing, memory, and control. Within this framework, we establish a direction expressivity theorem and a residual theory for constrained cometric families, disentangling directional expressiveness from condition-number complexity and recasting optimizer design as a Pareto optimization problem under modular budgets. Theoretically, we prove that fully positive-definite geometry exactly spans all strictly descending directions; experiments demonstrate that high-information full-metric probes attain numerical precision on deterministic quadratic problems, and a Muon-style implementation preliminarily validates the auditability of matrix-operator updates.

This study addresses the inefficiency of gradient computation in voxel-attribute-based surface optimization by proposing a differentiable surface-to-voxel conversion method. By deriving analytical gradients of winding numbers with respect to surface geometry, we establish a differentiable mapping from volumetric attributes to mesh parameters, enabling direct surface optimization via gradient descent. The proposed approach effectively handles complex tasks, including mesh self-intersection resolution, bandsaw manufacturability constraints, and 3D tessellation shape generation. Consequently, this method significantly enhances both the efficiency and accuracy of geometric optimization driven by voxel attributes. Furthermore, it introduces a novel paradigm for differentiable interaction between implicit and explicit representations, bridging the gap between volumetric processing and surface-based modeling in computational geometry and graphics.

本文提出了一种两阶段管道，通过机器学习代理来解决汽车引擎盖内板的设计问题，使其满足性能要求。