How do meshfree methods improve accuracy and efficiency in FEA?

One of the fundamental limitations of the Finite Element Method (FEM) is that it typically starts with a Partial Differential Equation (PDE) defined by continuum mechanics models. These models assume that materials can be represented as continuous media, which is inaccurate at small scales where molecular or atomistic effects or other discrete effects become significant. To model at small scales, methods like Molecular Dynamics, Multiscale Modeling, and the Discrete Element Method are more appropriate. These methods can capture the discrete nature of materials and provide insights that continuum-based FEM cannot achieve at small scales.

FEA/FEM in simplest terms is a numerical method used to solve differential equations. Finite Element methods (FEM) are versatile and can be applied to solve various types of differential equations encountered in engineering, physics, and other fields. The differential equations are discretized into algebraic equations, which transforms the differential equations into a system of linear equations that can be solved numerically. The versatility of FEA lies in its ability to handle complex geometries, material properties, boundary conditions, and coupled physics, making it a powerful tool for solving a wide range of differential equation problems encountered in engineering and science.

Most real-life engineering problems have no closed form solutions. Hence we are forced to resort to numerical methods like FEA. This enables us to solve problems involving complex shapes and complex Physics. In most organizations, commercial FEA software tools are used.

Mesh generation is one of the most important problems faced in FEA practice in the industry. "Why not use auto mesh"? Because for large assemblies, we intend to keep the model size as small as possible while satisfying the mesh quality and mesh size required for solution convergence in order to keep the solving time manageable. The bigger the assembly in terms of number of parts, usually, bigger the model size. Complexity of shape has the same effect. A coarse global mesh with a fine local mesh at the regions of interest is adopted to keep the model size manageable. But as the difference between the global size and the local size becomes large, it takes a good amount of skill to generate a mesh with acceptable quality.

A limitation of FEM is the need to model a finite domain. The Boundary Element Method (BEM) can overcome the difficulties of FEM meshing large volumes. BEM is based on transforming the original PDE into an equivalent boundary integral equation using fundamental solutions (Green’s functions). Unlike the FEM, which requires discretization of the entire problem domain, BEM only requires discretization of the surface of the volume. This can lead to a significant reduction in the size of the problem, especially for problems with infinite domains. FEM has now developed infinite elements and layering methods, among others, for efficiently handling infinite domains, so this limitation has been minimized.

Below are some advantages of meshfree methods: 1. Mesh Dependency: Structured mesh can be complex and time-consuming, especially for irregular geometries or problems. 2. Numerical Diffusion: Numerical diffusion refers to the smoothening of solution gradients across elements due to the interpolation functions used. Meshfree methods are better in problems with sharp gradients or discontinuities. 3. Adaptive Refinement: Meshfree methods can more easily adapt to local solution requirements without the need for global remeshing. 4. Large Deformations/Mesh Distortion: In problems involving large deformations, where mesh distortion and element quality degradation can become significant issues, meshfree methods be a better choice.

Adopting FEA the convergence of the displacements is faster than the stress. Radial basis functions can be used to differentiate the displacement field getting a better accuracy.