Meshless

Meshless methods, or meshfree methods, are numerical techniques for solving partial differential equations that do not require a mesh to discretize the domain. The solution is approximated from values at a set of scattered nodes, with field variables and derivatives obtained through moving least squares, radial basis functions, or other meshless interpolants. This contrasts with mesh-based methods like the finite element method, which rely on elements and a mesh connectivity.

Prominent approaches include moving least squares based local or global approximations; Element-Free Galerkin (EFG); Reproducing Kernel Particle Method (RKPM); and meshless finite difference methods such as RBF-FD. Smoothed particle hydrodynamics (SPH) is a well-known meshless method originally developed for fluids but extended to solids. Many methods combine meshless approximations with Galerkin or collocation frameworks.

Advantages include robust handling of large deformations and complex or evolving geometries, straightforward refinement by adding or removing nodes, and reduced meshing effort. Challenges include higher computational cost for large problems, sensitivity to node distribution and conditioning of system matrices, and complexity in imposing essential boundary conditions and enforcing compatibility. The lack of universally accepted standards can hinder comparison across methods.

Applications span solid and structural mechanics, fracture and crack propagation, fluid and aeroelastic problems, heat conduction, poroelasticity, and geotechnical and biomedical simulations. The field has matured since the late 20th century, with SPH and MLS-based methods playing central roles, and continues to evolve with hybrid approaches and improved stabilization techniques.