Meshless Methods in Computational Mechanics

Meshless methods constitute a class of numerical techniques for solving partial differential equations without relying on a predefined mesh. Instead of subdividing a domain into elements, these approaches employ a set of scattered nodes or points to represent geometry and approximate field variables. Early developments arose from moving least squares and radial basis function interpolation, addressing limitations of mesh generation in complex geometries and large deformation problems. Today, meshless schemes such as the Element-Free Galerkin method, Radial Point Interpolation Method and Natural Neighbour approaches offer high flexibility in handling evolving boundaries, crack propagation and multiscale phenomena. Their capacity to adapt the nodal distribution dynamically enhances accuracy in regions of steep gradients, discontinuities or singularities. Applications span solid mechanics, fluid–structure interaction, biomechanics and geomechanics, where traditional meshes impose prohibitive pre-processing or remeshing overheads. Despite higher computational cost per node, advances in point cloud generation, integration algorithms and parallel implementations have narrowed the performance gap with finite elements. Ongoing research focuses on robust stabilisation strategies, efficient numerical integration over arbitrary point sets and hybrid coupling with mesh-based methods to combine the strengths of each paradigm. These developments are driving meshless techniques towards routine use in industrial and biomedical simulations.

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Meshless Methods in Computational Mechanics publication trend

The graph below shows the total number of articles in meshless methods in computational mechanics across all publications each year (not limited to Nature Index journals).

Technical terms

Meshless method: A numerical technique that approximates field variables using scattered nodes rather than a connected element mesh.

Radial basis function (RBF): A mathematical function centred on nodes, used to construct smooth shape functions for interpolation.

Point cloud: An unstructured collection of nodes that discretises a computational domain for meshless analysis.

Natural Neighbour Radial Point Interpolation Method (NNRPIM): A meshless scheme combining natural neighbour interpolation with radial basis functions for multiscale problems.

Homogenisation: A procedure to compute effective macroscopic properties of heterogeneous materials through averaging of microstructural responses.

References

1. Point Cloud Generation for Meshfree Methods: An Overview. Archives of Computational Methods in Engineering (2022).
2. Elasto-Static Analysis of Composite Restorations in a Molar Tooth: A Meshless Approach. Polymers (2024).
3. Extending the Natural Neighbour Radial Point Interpolation Meshless Method to the Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core. Applied Sciences (2024).