Meshfree and Finite Element Analysis of Functionally Graded Structures

Functionally graded materials (FGMs) exhibit a continuous variation in composition and microstructure, allowing a smooth transition of mechanical or thermal properties through their volume. Such gradation is tailored to withstand severe thermal gradients, resist wear and corrosion, and optimise stress distributions in aerospace, automotive, civil and biomedical applications. Computational analysis plays a central role in predicting the static, dynamic and stability behaviour of FGM plates, shells and nanostructures. Traditional finite element methods (FEM) discretise the domain into elements, applying shape functions and numerical integration to approximate displacements and stresses. Recent extensions incorporate nonlocal and strain gradient theories to capture size effects at the micro- and nano-scales, as well as cell-based smoothing and discrete shear gap techniques to suppress shear locking in thin or thick structures. In parallel, meshfree approaches—such as the natural element method (NEM) and generalised finite difference method (GFDM)—eschew a fixed mesh. They approximate the field variables via node-based interpolation, offering greater flexibility in handling large deformations, evolving discontinuities and irregular geometries without remeshing. Hybrid schemes increasingly couple meshfree interpolation with conventional FEM domains or enriched functions to model cracks, porosity distributions and carbon-nanotube or graphene reinforcements. These advances underpin reliable predictions of bending, vibration, buckling and post-buckling responses of FGMs over scales ranging from ultra-thin membranes to thick shells and nanoplates, thereby guiding the design of next-generation multifunctional structures.

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Meshfree and Finite Element Analysis of Functionally Graded Structures publication trend

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

Technical terms

Functionally graded material (FGM): A composite whose material properties vary continuously through one or more dimensions.

Meshfree method: A numerical scheme that approximates field variables using node-based interpolation without a predefined element mesh.

Finite element method (FEM): A discretisation technique that divides a domain into elements and applies shape functions to approximate differential equations.

Natural element method (NEM): A meshfree interpolation approach using Voronoi diagram and Laplace shape functions for a smooth approximation of field variables.

Generalised finite difference method (GFDM): A truly meshless domain-type method that represents derivatives via weighted combinations of neighbouring node values.

First-order shear deformation theory (FSDT): A plate or shell theory that accounts for transverse shear strains through a linear thickness-wise displacement assumption.

Nonlocal strain gradient theory: A continuum model incorporating material length scales to capture size-dependent mechanical behaviour at micro- and nano-scales.

References

1. Static stability of functionally graded porous nanoplates under uniform and non-uniform in-plane loads and various boundary conditions based on the nonlocal strain gradient theory. Results in Engineering (2025).
2. Analysis of Functionally Graded Material Plates Using Triangular Elements with Cell‐Based Smoothed Discrete Shear Gap Method. Mathematical Problems in Engineering (2014).
3. A Numerical Evaluation of SIFs of 2-D Functionally Graded Materials by Enriched Natural Element Method. Applied Sciences (2019).
4. Generalized Finite Difference Method for Plate Bending Analysis of Functionally Graded Materials. Mathematics (2020).