How do meshfree methods compare with traditional mesh-based methods in FEA?

2 Why use meshfree methods?

One of the main advantages of meshfree methods is that they can handle complex geometries and large deformations without the need for mesh generation and refinement. Mesh generation is often a time-consuming and error-prone process, especially for three-dimensional problems with irregular shapes, cracks, or holes. Mesh refinement is also necessary to capture the solution features and gradients in regions of high stress or strain. Meshfree methods can avoid these difficulties by adapting the node distribution and the basis functions to the problem characteristics.

3 How do meshfree methods compare with FEA?

FEA is one of the most widely used mesh-based methods for solving PDEs in engineering. FEA uses a mesh or grid to divide the domain into small elements, such as triangles or tetrahedra, and approximates the solution within each element by a polynomial function. The solution at each node is obtained by solving a system of linear or nonlinear equations that result from applying the variational or weak form of the PDE. FEA has many advantages, such as its versatility, robustness, and accuracy, but it also has some limitations, such as its dependence on the quality of the mesh, its difficulty in handling discontinuities, and its high computational cost for large problems.

Meshfree methods can overcome some of these limitations by using a more flexible and adaptive node distribution and basis function selection. However, meshfree methods also have some challenges, such as their lack of convergence and stability guarantees, their difficulty in enforcing boundary conditions, and their high computational cost for large numbers of nodes. Therefore, meshfree methods are not a replacement for FEA, but rather a complement or an alternative for some specific problems or applications.

4 What are some applications of meshfree methods?

Meshfree methods have been applied to a variety of engineering problems involving complex geometries, large deformations, or discontinuities. For instance, they can be used in fracture mechanics to model the propagation of cracks without requiring mesh updating or remeshing, which can be difficult for FEA. Additionally, meshfree methods can simulate the coupling between fluid and solid domains without needing mesh matching or interface tracking, making them ideal for fluid-structure interaction. Lastly, they can capture the deformation and stress distribution of soft tissues and organs without requiring mesh fitting or smoothing, which can be inaccurate for FEA; making them ideal for use in biomechanics.