Meshless and Finite Element Methods in Structural Analysis

In structural analysis, the Finite Element Method (FEM) has long served as the standard computational tool. By discretising complex geometries into interconnected elements, FEM solves equilibrium equations in their weak form to predict stresses, deformations and dynamic responses across engineering structures. Traditional FEM relies on a mesh to define element connectivity, which can pose challenges for highly intricate domains and evolving interfaces. Meshless methods have emerged to address these limitations by dispensing with fixed element meshes. Instead, they construct approximation functions directly from nodal distributions, often employing strong-form formulations or collocation techniques to enforce governing equations without domain integrals.

The interplay between meshless and mesh-based schemes has given rise to hybrid strategies that leverage the rigorous variational foundation of FEM alongside the flexibility of meshless interpolation. Such approaches facilitate rapid re-modelling in applications ranging from fracture mechanics to piezoelectric device design. Recent progress has focused on reducing computational cost and improving convergence through novel shape functions, enhanced enforcement of boundary conditions and adaptive refinement. Combined with developments in high-performance computing, these advances are enabling ever more detailed analysis of thin-walled shells, laminated composites and multifunctional materials. The global significance of these methods spans infrastructure, aerospace and renewable energy sectors, where accurate prediction of structural performance under complex loading is critical to safety and sustainability.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Meshless and Finite Element Methods in Structural Analysis publication trend

The graph below shows the total number of articles in meshless and finite element methods in structural analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Finite Element Method (FEM): A numerical approach that subdivides a structure into discrete elements linked by nodes, solving equilibrium equations in a variational (weak) form.

Meshless Method: A computational technique that dispenses with traditional element meshes, constructing approximations using nodes and shape functions without explicit connectivity.

Strong-form Formulation: A numerical strategy that enforces governing differential equations directly at discrete points rather than in an integral sense.

Collocation Method: A meshless approach that satisfies governing equations at selected nodes, producing algebraic equations without domain integration.

Mixed Finite Element Method: A formulation that employs multiple primary variables, such as stress and displacement, simultaneously to improve solution accuracy.

References

1. Method of matched sections in application to thin-walled and Mindlin rectangular plates. Mechanics and Advanced Technologies (2023).
2. Application of novel refined mixed finite elementmethod in the analysis of composite laminated beams. ASPS Conference Proceedings (2022).
3. A New Strong Form Technique for Thermo-Electro-Mechanical Behaviors of Piezoelectric Solids. Coatings (2021).