Finite element method

Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.

FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems).

[1] To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements.

The subdivision of a whole domain into simpler parts has several advantages:[3] A typical approach using the method involves the following: The global system of equations uses known solution techniques and can be calculated from the initial values of the original problem to obtain a numerical answer.

It includes the use of mesh generation techniques for dividing a complex problem into smaller elements, as well as the use of software coded with a FEM algorithm.

FEA simulations provide a valuable resource, as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.

[4] While it is difficult to quote the date of the invention of FEM, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering.

Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Lord Rayleigh, Walther Ritz, and Boris Galerkin.

University of California Berkeley made the finite element programs SAP IV[10] and, later, OpenSees widely available.

[12] The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, such as electromagnetism, heat transfer, and fluid dynamics.

[13][14] A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures.

These algorithms are designed to exploit the sparsity of matrices that depend on the variational formulation and discretization strategy choices.

To meet the requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest.

When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the action of the analyst.

After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP.

Examples of methods that use higher degree piecewise polynomial basis functions are the hp-FEM and spectral FEM.

For second-order elliptic boundary value problems, piecewise polynomial basis function that is merely continuous suffice (i.e., the derivatives are discontinuous.)

If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh, which are continuous at each edge midpoint.

Yang and Lui introduced the Augmented-Finite Element Method, whose goal was to model the weak and strong discontinuities without needing extra DoFs, as PuM stated.

[15] The approach is "to make the discretization as independent as possible of the geometric description and minimize the complexity of mesh generation, while retaining the accuracy and robustness of a standard finite element method.

The hp-FEM combines adaptively elements with variable size h and polynomial degree p to achieve exceptionally fast, exponential convergence rates.

Extended finite element methods enrich the approximation space to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc.

It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).

Spectral methods are the approximate solution of weak-form partial equations based on high-order Lagrangian interpolants and used only with certain quadrature rules.

The crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters.

Metals can be regarded as crystal aggregates, which behave anisotropy under deformation, such as abnormal stress and strain localization.

[28] Various specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products.

Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments.

In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and minimizing weight, materials, and costs.

FEM software provides a wide range of simulation options for controlling the complexity of modeling and system analysis.