Session: 13-03-03: General: Mechanics of Solids, Structures and Fluids III

Paper Number: 166852

A Meshless Analysis for Biharmonic Shell Bending Using the Mixed Fragile Points Method (Mixed-FPM) 

Shell bending analysis is crucial in engineering design, such as bridge decks, aircraft wings, and thin-walled vessels. Beyond the most basic cases, analytical bending solutions are unavailable, making numerical simulation such as the Finite Element Method (FEM) a primary tool for engineers and researchers. However, meshing, a critical step in all traditional element-based methods, remains computationally expensive despite advancements in mesh generation. Mesh quality significantly impacts accuracy, requiring meticulous inspection before interpretation. Additionally, large deformations lead to mesh distortion, compromising computational accuracy, while crack propagation or other analyses involving changes in domain continuity require mesh refinement or trial function enhancement at each step increment, further increasing computational costs. These computational bottlenecks, along with FEM's limitations in handling higher-order partial differential equations (PDEs), have driven the development of meshfree methods such as the Element-Free Galerkin (EFG) method, Meshless Local Petrov-Galerkin (MLPG) method, and the proposed Fragile Points Method (FPM).

Traditional meshless methods typically do not require fine meshing and overcome issues such as mesh distortion and locking, but suffer from tedious numerical integration. The MLPG method relies on Moving Least Squares (MLS) approximation, which can generate C¹ or higher-order continuous functions with appropriate weight functions. However, the trial functions are complicated rational polynomials with even more complex derivatives, making numerical integration challenging, and requiring a large number of integration points. Similarly, the EFG method employs implicitly defined MLS-based shape functions, yet their weak form integration remains complex and computationally demanding.

In contrast, the new Fragile Points Method (FPM) employs simple, polynomial, piecewise-continuous trial and test functions. This simplicity facilitates the derivation of weak form (Galerkin or Petrov-Galerkin) and makes integration trivially easy. Unlike the EFG or MLPG methods in which sometimes even dozens of integration points fail to offer a reliable integration, FPM only requires a single integration point in each discretized region of the problem domain.

This study formulates and applies the FPM to solve the fourth-order biharmonic equation governing plate bending behavior. Unlike traditional FEM, the FPM eliminates the need for fine meshing by utilizing a point cloud and performing the calculation on these easily created ‘fragile points.’ This approach makes it well-suited for large deformation analysis and crack propagation without the need for mesh refinement like FEM.  Local, discontinuous trial and test functions are employed in the Galerkin weak form, with continuity across subdomains ensured through numerical flux corrections using a penalty parameter. This approach enables C¹ continuity analysis without the need for complex trial functions with numerous degrees of freedom (DoFs), while also maintaining computational efficiency and accuracy.

A mixed FPM framework is formulated for thin shell bending analysis, treating the displacement and its curvature as independent unknown variables. Both isotropic homogeneous and functionally graded shells are studied. Various combinations of Dirichlet and Neumann boundary conditions are examined. As a case study, FPM results are validated against the analytical solution for an axisymmetric circular plate under self-weight, and an appropriate order of magnitude for the penalty parameters is reported.  Both randomly distributed fragile points (with Voronoi subdomains) and uniformly distributed fragile points (with structured hexagonal subdomains) are employed for discretization. While achieving consistent final results, their differences in accuracy and computational efficiency are compared and reported.

The application of FPM in shell structure analysis requires further exploration, including large deformation and large rotation analysis, dynamic analysis, and self-contact analysis, to fully leverage the benefits of FPM. Nevertheless, this study has established a solid foundation for applying FPM to the analysis of shell structures and other fourth-order or higher-order continuous structural problems.

Presenting Author: Arash Rahmati The University of Memphis

Presenting Author Biography: Arash is a PhD candidate in mechanical engineering at the University of Memphis, specializing in meshfree methods in finite element analysis (FEM). With a strong background in computational mechanics, he is developing a novel method to enhance FEM applications. His research interests include FEM, machine learning, and advanced manufacturing, and his research involves working with Abaqus and extracting simulation data from ODB files to analyze structural behavior.

Authors:

Arash Rahmati The University of Memphis
Yue Guan The University of Memphis