Strong C1-coupling multi-patch isogeometric topology optimization of complex structures for strain gradient problems

This paper presents a novel framework for strong $C^1$-coupling multi-patch isogeometric topology optimization, tailored for addressing strain gradient problems governed by fourth-order partial differential equations (PDEs), including strain gradient elasticity and flexoelectric effects. The proposed method achieves $C^1$-continuity across multi-patch domains by reconstructing basis functions at patch interfaces, ensuring seamlessly matching first derivatives. We validated this method through structural analyses and optimizations on single and multi-patch domains with and without strain gradient elasticity. The $C^1$-locking phenomenon, attributed to over-constrained solution spaces at patch interfaces, are systematically investigated. Order elevation of basis functions effectively mitigates $C^1$-locking by providing sufficient degrees of freedom. Additionally, the optimization framework incorporates advanced filtering techniques to handle unstructured grids in multi-patch configurations. Neighbor domain filtering demonstrates stability but incurs high computational costs, whereas Helmholtz PDE filtering offers greater efficiency but requires careful tuning to mitigate oscillatory behavior. The practical utility of the framework is demonstrated through case studies involving the optimization of a connecting rod under strain gradient elasticity and a flexoelectric beam with circular voids. These examples highlight the framework’s potential in addressing complex engineering challenges, offering smoother solutions, reducing stress concentrations in strain gradient elasticity, and enhancing electromechanical coupling in flexoelectric materials. This work significantly broadens the scope of structural optimization for complex systems governed by high-order PDEs, paving the way for more efficient and accurate design methods in advanced engineering applications.