Finite element analysis (FEA) was developed decades ago to solve engineering structural problems. In its infancy, FEA was simply a matrix method to deal with simple structures that could be analyzed with rudimentary computer systems.
However, it has evolved into a widely used method for evaluating a variety of products in many engineering disciplines. FEA has been used for decades to solve complex structural, thermal, and fluid mechanics problems. Applications are typically found in civil and mechanical engineering, but FEA has also been used extensively in electrical engineering as well. Finite element analysis enables engineers to tackle real world complex assemblies and systems that are impossible to solve with simpler methods.
The finite element method (FEM) originated in the 1940’s. At this time, there was a need to solve more complex problems in elasticity and structural analysis. As the method evolved, various approaches were recognized that involved dividing a continuous domain into sub domains, or elements. Mathematically, the method involved solving an array of partial differential equations. During the 1950’s, further development of FEA continued in airframe and structural analysis. Research at several universities culminated in more efficient methods for solving the stiffness matrices that were the basis of the method. This research was concentrated in civil engineering applications during the 1960’s, but applications to other engineering disciplines were also beginning to appear at this time. During the late 60’s, three of the more recognized finite element codes were introduced: ANSYS, NASTRAN, and STARDYNE. Advances in computing power and software development over the years have culminated in the efficient, multi-purpose finite element (FE) programs that are available today.
The Finite Element Method
The finite element method consists of three major tasks. The first is pre-processing, where the analyst develops a mesh which consists of nodes and elements. This mesh is the basis for the analysis, and is supplemented with appropriate material properties, element properties, and boundary conditions. The second task is the solution phase, where the elements are assembled into matrices. These matrices are then solved for basic parameters, such as displacements or temperatures. The final step is post-processing, where the analyst checks the results. Part of this process involves reviewing the magnitudes and distributions of the primary solution parameters (such as displacements and stresses).
The Finite Element Model
An FE model consists of a finite number of points (nodes) and elements. The nodes are actually points in space that are used to define the elements. The elements consist of various numbers of nodes (typically from 1 to 20). These elements are defined in space by the location and connectivity of the nodes. Finite elements are actually mini structures themselves, with displacement functions that are defined by the element types. Each element is further defined by element and material properties. The density of the finite element mesh may vary throughout the model, depending on the stress gradients within the structure.
Answers from FE Models
Each node in an FE model is characterized by a specific number of degrees of freedom (DOF). For three dimensional structural problems, the maximum number of DOF is 6 (three translations and three rotations in a global coordinate system). The minimum number of DOF is one (such as temperature for a thermal model). In simple terms, static structural FE models solve the equation F = k x, where F is the applied force, k is the stiffness, and x is the displacement. All of this is done in matrix format, where thousands of DOF are calculated in a single FE model. The most common results from FEA are displacements and stresses. Nonlinear design analyses include large deflection, elastic-plastic deformation, and contact between adjacent structures.