Finite Element solver for kinematically incompatible non-simply connected Föppl-von Kármán plates

Reference No.	2025a033
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	Finite Element solver for kinematically incompatible non-simply connected Föppl-von Kármán plates
Principal Investigator	Edoardo Fabbrini（Kyushu University, Graduate School of Mathematics・PhD student）
Research Period	April 21,2025. - May 2,2025.
Keyword(s) of Research Fields	Finite Element, Discontinuous Galerkin, 2D elasticity, Volterra defects.
Abstract for Research Report	The proposed visit of Dr. León Baldelli is part of an ongoing scientific collaboration that began at IMI, Kyushu, in May 2024. This collaboration continued with a second visit by Mr. Fabbrini to CNRS-Sorbonne in November 2024. The project, led by Prof. Cesana, focuses on modeling thin structures where deformation is induced by the presence of topological defects in the lattice of a crystalline material. Applications include modeling graphene structures and thin shape memory alloys. My collaboration with Dr. Baldelli has already resulted in a research paper, which has been submitted to a peer-reviewed international journal [FLC], as well as the development of computer codes in the FEniCS environment based on the Finite Element Method (FEM) (available in open access at https://github.com/kumiori/disclinations). The goal of the proposed two-week visit is to further develop the computational code and produce a new research paper in collaboration at IMI. Specifically, we aim to develop a solver for fourth-order elliptic equations using the FEM in non-simply connected domains, incorporating non-classical boundary conditions. These boundary conditions model the presence of a topological defect at a singular length scale (on the order of nanometers). Our approach follows a two-step process. We first solve an independent "fundamental cell problem". The solution to this problem provides optimized boundary conditions that satisfy the zero-Neumann condition (zero normal stress) at the core of the defect. As a second step, using these optimized boundary conditions, we solve a fourth-order problem to determine the membrane stress potential. Fourth-order problems are inherently challenging, as they require the continuity of the gradient of the shape functions. To address this, we plan to use the C0-Discontinuous Galerkin method, extending the implementation in [FLC] to non-simply connected domains. Our approach aims to reduce the computational cost of FEM simulations, simplify the formulation, and result in a solver that is easier to implement. By the end of the project, we expect to deliver an innovative tool for studying the effects of defects in elastic materials. [FLC] Kinematically Incompatible Föppl-von Kármán Plates: Analysis and Numerics – Fabbrini, León Baldelli, Cesana. ArXiv: https://arxiv.org/pdf/2501.15959v1, submitted for publication in Applied Mathematical Modeling (peer-reviewed journal).
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Edoardo FABBRINI（Graduate School of Mathematics, Kyushu University・PhD student） Pierluigi CESANA（IMI department, Kyushu University・Associate Professor） Andrés Alessandro León Baldelli（d'Alembert Institute, CNRS, Sorbonne University・researcher）