A Lagrangian shape and topology optimization framework based on semi-discrete optimal transport (2409.07873v1)

Abstract: This article revolves around shape and topology optimization, in the applicative context where the objective and constraint functionals depend on the solution to a physical boundary value problem posed on the optimized domain. We introduce a novel framework based on modern concepts from computational geometry, optimal transport and numerical analysis. Its pivotal feature is a representation of the optimized shape by the cells of an adapted version of a Laguerre diagram. Although such objects are originally described by a collection of seed points and weights, recent results from optimal transport theory suggest a more intuitive parametrization in terms of the seed points and measures of the associated cells. The polygonal mesh of the shape induced by this diagram serves as support for the deployment of the Virtual Element Method for the numerical solution of the physical boundary value problem at play and the calculation of the objective and constraint functionals. The sensitivities of the latter are derived next; at first, we calculate their derivatives with respect to the positions of the vertices of the Laguerre diagram by shape calculus techniques; a suitable adjoint methodology is then developed to express them in terms of the seed points and cell measures of the diagram. The evolution of the shape is realized by first updating the design variables according to these sensitivities and then reconstructing the diagram with efficient algorithms from computational geometry. Our shape optimization strategy is versatile: it can be applied to a wide gammut of physical situations. It is Lagrangian by essence, and it thereby benefits from all the assets of a consistently meshed representation of the shape. Yet, it naturally handles dramatic motions, including topological changes, in a very robust fashion. These features, among others, are illustrated by a series of 2d numerical examples.

• The paper introduces a novel Lagrangian framework that uses Laguerre diagrams and semi-discrete optimal transport to enable stable topology changes.
• It derives sensitivities via shape calculus for seed points and cell measures, guiding precise shape evolution under various physical constraints.
• Efficient numerical algorithms demonstrate the method’s capability in 2D, with promising extensions to 3D and integration with machine learning for automated designs.

A Lagrangian Shape and Topology Optimization Framework Based on Semi-Discrete Optimal Transport

This paper introduces a novel numerical framework for shape and topology optimization, leveraging modern techniques from computational geometry, optimal transport theory, and numerical analysis. The framework is centered around a Lagrangian method, which minimizes design criteria subject to physical boundary value problems, by representing the shape of interest with a Laguerre diagram. The approach is versatile, allowing application across a range of physical phenomena, such as thermal and structural mechanics, under single or multi-phase design configurations.

Summary and Key Contributions

Shape Representation and Practical Implementation: The optimized shape is described using a weighted partition of space through Laguerre diagrams, an extension of Voronoi diagrams. This representation allows the discrete polygonal nature of the shape to serve as a support for mechanical computations like the Virtual Element Method. The major advancement here is the parameterization of cell volumes, enabling stable topology changes through adaptation based on semi-discrete optimal transport results.

Derivation of Sensitivities: The paper applies shape calculus to derive sensitivities of optimization criteria with respect to seed points and cell measures of the diagram, which are pivotal for determining the direction of shape evolution during optimization.

Numerical Algorithms and Efficiency: Efficient numerical algorithms from computational geometry are leveraged to reconstruct diagrams after each iteration. This includes handling significant shape deformations robustly, as verified through various 2D numerical examples illustrating the flexibility and effectiveness of the proposed method.

Implications and Future Developments

1. Robust Topology Changes: The ability to naturally handle topology changes without needing an implicit Eulerian representation or complex remeshing strategies is a significant achievement, providing a stable framework for complex design spaces.
2. Extension to 3D: While the paper focuses on 2D implementations, extending the framework to 3D is a critical next step. The core numerical techniques utilized extend conceptually to 3D, promising a powerful tool for practical engineering and physics applications.
3. Integration with Machine Learning: Given the framework's capability to represent a wide range of shapes with relatively few parameters, integration with machine learning techniques for parameter optimization could be explored, opening new avenues in automated design processes.

Analytical Rigor and Technical Merit

The authors establish a solid theoretical foundation by combining concepts from diverse mathematical domains, ensuring that the proposed framework not only aligns with current methodologies but also expands them significantly. The novel integration of Laguerre diagrams with optimal transport theory and their application to shape and topology optimization are particularly noteworthy. Additionally, the application of this framework to classic optimization problems, such as minimizing eigenvalues of the Laplace operator and optimizing shapes subject to compliance criteria, exhibits the framework's broad applicability and effectiveness.

In conclusion, this work presents a rigorous, innovative method for shape and topology optimization, positioned at the intersection of theoretical elegance and practical applicability. The research contributes significantly to the computational methods available for solving complex optimization problems across engineering disciplines and sets the stage for future work in higher dimensions and interdisciplinary applications.