What are the most suitable applications and domains for meshfree methods in FEA?

1 Benefits of meshfree methods

Meshfree methods have several benefits over FEA, especially for problems involving large deformations, complex geometries, moving boundaries, cracks, or discontinuities. Meshfree methods can handle these situations without suffering from mesh distortion, element quality degradation, or remeshing issues. Meshfree methods can also adapt to local features and refine the solution where needed, without affecting the global consistency. Moreover, meshfree methods can reduce the computational cost and time by using fewer nodes and simpler data structures than FEA.

2 Challenges of meshfree methods

However, meshfree methods also face some challenges, such as ensuring the stability, accuracy, and convergence of the solution, dealing with boundary conditions, and integrating the weak form of the PDEs. Meshfree methods often require more complex and nonlinear interpolation functions than FEA, which can lead to ill-conditioning, numerical instability, or spurious oscillations. Meshfree methods also need to define suitable support domains and weights for each node, which can affect the smoothness and completeness of the approximation. Furthermore, meshfree methods have to use special techniques, such as penalty methods, Lagrange multipliers, or Nitsche's method, to impose boundary conditions and enforce the weak form.

3 Types of meshfree methods

Meshfree methods are a type of numerical technique used for a variety of applications. Smoothed particle hydrodynamics (SPH) is a Lagrangian method that uses particles as nodes and smoothing kernels as interpolation functions, and is often used for fluid dynamics, free surface flows, and solid mechanics. Element-free Galerkin (EFG) is a Galerkin method that uses nodes as collocation points and moving least squares (MLS) as interpolation functions, making it suitable for elasticity, fracture mechanics, and heat transfer problems. Reproducing kernel particle method (RKPM) is a collocation method that uses nodes as interpolation points and reproducing kernels as interpolation functions, similar to EFG but with completeness and positivity of the approximation ensured. Meshless local Petrov-Galerkin (MLPG) is a Petrov-Galerkin method that uses local subdomains and trial and test functions as interpolation functions to achieve high accuracy and stability. All four meshfree methods can solve problems in solid mechanics, fluid mechanics, acoustics, diffusion, and electromagnetics.

4 Examples of meshfree applications

Meshfree methods have been applied to a variety of engineering and scientific domains, such as biomechanics, geomechanics, manufacturing, and aerospace. In biomechanics, they can model the deformation and damage of soft tissues and organs, as well as the fluid-structure interaction between blood and cardiovascular devices. In geomechanics, they can simulate the behavior of soils, rocks, landslides, earthquakes, and other geophysical phenomena. For manufacturing, meshfree methods can analyze the processes of metal forming, welding, cutting, machining, and additive manufacturing. Lastly, in aerospace they can study the aerodynamics, aeroelasticity and impact of aircrafts, rockets, satellites and re-entry vehicles.

5 How to choose a meshfree method

Choosing a meshfree method for a given problem is a complex decision, as it depends on many factors, such as the nature of the PDEs, the geometry of the domain, and the desired accuracy and efficiency of the solution. However, some general guidelines can be followed. Comparing different meshfree methods and FEA for a simple problem is a good way to assess their performance and suitability. Additionally, choosing an interpolation function that ensures smoothness and stability of the approximation is important, as well as selecting a support domain and weight function that balance local and global properties. Furthermore, a numerical integration scheme should be chosen that preserves accuracy and stability of the solution while also preserving any conservation laws or symmetries of the problem.