finiteelementanalyses

Finite element analyses (FEA) are numerical methods used to approximate the behavior of physical systems governed by partial differential equations. By subdividing a complex domain into smaller, simple elements connected at nodes, FEA converts a continuous problem into a discrete system that can be solved on a computer. The method is used to predict displacements, stresses, temperatures, and other field variables under prescribed loads and constraints.

Typical FEA workflow includes creating a mesh of the domain, selecting appropriate element types (for example, 1D, 2D, or 3D elements) and material models, applying boundary conditions and loads, assembling element equations into a global system, solving the resulting equations (linear or nonlinear, static or dynamic), and post-processing the results to extract quantities of interest.

Analyses can be linear or nonlinear, static or dynamic, and may couple multiple physics in a single model (multiphysics). Implicit methods are common for steady problems, while explicit methods handle highly nonlinear or transient events. Time integration, mass and stiffness matrices, and contact, plasticity, or fracture models extend capabilities to complex behaviors.

Mesh quality and refinement control accuracy, with convergence studies used to assess error. Material models may be elastic, plastic, viscoelastic, or temperature-dependent, among others. FEA finds wide use in civil and mechanical engineering, aerospace, automotive, biomechanics, electronics, and other fields for design optimization, safety assessment, and failure analysis.

Limitations include reliance on geometric and material assumptions, potential modeling errors, computational cost for large nonlinear problems, and sensitivity to mesh generation. The method was developed in the mid-20th century and has since become a standard tool in engineering analysis.