Finite Element Analysis in Contact Mechanics: Formulations, Algorithms, and Engineering Applications

Contact between deformable bodies is one of the most common and most difficult sources of nonlinearity in structural and manufacturing analysis. The contact area is not known in advance, the interface forces depend on the deformation they cause, and frictional response introduces path dependence. The finite element method has become the principal tool for resolving these difficulties, and the past four decades have produced a mature but still evolving body of formulations and algorithms. This paper reviews the usage of finite element analysis in contact mechanics, covering the variational statement of the contact problem, the discretization of the contact interface, the enforcement of the non-penetration and frictional constraints, and the nonlinear solution strategies that make these models converge in practice. Representative applications in additively manufactured lattice structures, metal forming, and biomechanics are discussed to show how modelling choices affect accuracy and cost. The paper closes with a discussion of open problems, including rough-surface contact, multiphysics coupling, and data-driven acceleration, and offers practical recommendations for analysts.