The finite element method (FEM) is a technique for obtaining approximate numerical solutions to boundary value problems to predict the response of physical systems subjected to external loads.

In general, finite element analysis (FEA) begins by generating a finite element model of a system. In this model, a subject structure is reduced into a number of node points that are connected together to form finite elements. The governing equations of motion are written in a discrete form, where the displacements of each node point are the unknown part of the solution. A simulated load or other external influence is applied to the system and the resulting effect is analyzed.

Two such mathematical methods are the implicit finite element method and the explicit finite element method. These methods can each be used to solve transient dynamic equations of motion. However, when a static equilibrium solutions is needed, omitting the effects of inertia, the implicit method can be used in the absence of rigid body nodes to directly solve the governing equations.

The implicit method is characterized by the formation of a stiffness matrix to represent the interaction of nodal motions within the structure. In the implicit solution process, the stiffness matrix is assembled, its inverse is computed and this inverse is multiplied by the nodal forces to produce a solution of nodal displacements. For nonlinear problems, these displacements are tested to verify that they satisfy the governing equations of motion. If equilibrium is not satisfied, an iterative procedure is applied until equilibrium is achieved. Successful convergence of these iterations is not guaranteed and can be very difficult in practice