AEI-DFG: Design of FLEXible materials and structures via shape and topology optimizatION

• Applicant: Professor Antoine Laurain, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541557013

The general objective of this project is to advance the mathematical analysis of shape and topology optimization for materials described by nearly incompressible and incompressible hyperelasticity, applying these techniques for the design of flexible materials and structures, with a specific focus on utilizing shape and topological derivatives. This project represents the first opportunity to foster collaboration and synergy between research teams from Germany and Spain in the field of shape and topology optimization. The four specific objectives of the project are: (O1) Analyze the existence of minimizers for shape optimization problems in hyperelasticity: our objective is to determine appropriate assumptions on the hyperelasticity models and on the set of admissible shapes, and to prove, under these assumptions, existence of minimizers for shape optimization problems in the framework of nearly incompressible and incompressible hyperelasticity. (O2) Obtain shape and topological derivative for nearly incompressible and incompressible hyperelasticity models: this includes the formal computation of shape and topological derivatives and the identification of a mathematical framework, consisting of adequate assumptions and sufficiently simple hyperelasticity models, for which shape and topological differentiability can be rigorously proven. (O3) Develop efficient numerical tools for simulating incompressible hyperelasticity: the efficiency of new discretization schemes including advanced finite element, non-conforming, mixed formulations and stabilization techniques will be analyzed for optimization. In addition, deflation techniques will be incorporated in the optimization scheme to deal with bifurcations, to discover new disconnected branches and to provide stability in the design process. (O4) Design new compliant mechanisms and metamaterials with the use of topological derivatives (for the first time in hyperelasticity), shape derivatives and level-set methods. The aim is to significantly increase the performance of the current unrealistic designs by incorporating large deformations to the structure, especially in the hinges parts.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Professor Alex Ferrer, Ph.D.