Shuo Yang, Assistant Professor, Beijing Institute of Mathematical Sciences and Applications

Wenbo Li, Associate Professor

Accelerated gradient flows for large bending deformations of nonlinear plates

10:30-11:30 January 25(Thursday), Z311

In recent years, there has been some numerical explorations of large bending deformations in nonlinear plates, including single layer, prestrained, and bilayer plates. The mathematical problems consist of minimizations of bending energy functionals while subject to nonlinear and non-convex metric constraints. In previous works, the computation of such non-convex constrained energy minimization problems has relied on gradient flows. While these approaches are known for their stability, they often exhibit slow convergence, especially when dealing with highly non-convex problems. In this presentation, we introduce new iterative schemes that surpass existing gradient flow algorithms in terms of speed and efficiency. In our accelerated flows, we linearize the metric constraint through incremental updates to a tangent plane, building upon the previous iteration. Rigorous analysis confirms the stability of these accelerated flows, and also proves the proper control over constraint violations. Furthermore, we discuss other related properties, including the convergence of flow.

We note that people have spatially discretized these problems using approaches such as Kirchhoff Finite Element Method (FEM), interior penalty Discontinuous Galerkin (DG), and local DG. In contrast, our work adopts Morley FEM for spatial discretization, offering a simpler formulation compared to previous methods. We establish a Gamma convergence theory specific to the Morley discretization of these problems, aligning with established analysis in previous works. We further present a series of numerical simulations that illustrate the acceleration effect and various properties of our new approach.

This is a joint work with Guozhi Dong (Central South University) and Hailong Guo (University of Melbourne).