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Introduction to Finite Element Analysis

Finite Element Analysis is a computerized method for predicting how a component/assembly will react to environmental factors such as forces, heat and vibration. It is called "analysis", but in the product design cycle it is used as a "virtual prototyping" tool to predict what is going to happen when the product is used.

Finite element analysis, as related to the mechanics of solids, is the solution of a finite set of algebraic matrix equations that approximate the relationships between load and deflection for static analysis as well as velocity, acceleration and time for dynamic analysis.

In 1678, Robert Hooke set down the basis for modern finite element stress analysis as Hooke's Law. Simply, an elastic body stretches (strain) in proportion to the force (stress) on it.

Mathematically:

F=kx.

• F = force
• k = proportional constant
• x = distance of stretching

This is the only equation you need to know to understand linear finite element stress analysis.

The finite element method works by breaking a real object down into a large number (1000's or 100,000's) of elements (e.g. cubes). The behavior of each little element, which is regular in shape, is readily predicted by set mathematical equations. The summation of the individual element behaviors produces the expected behavior of the actual object.

The "finite element" is a small, but not infinitesimal, part of the mechanical structure being modeled that incorporates complex strength of materials formulations into a relatively simple geometric shape. The simplest examples are rods, beams and triangular plates. More complicated elements include quadrilateral plates, curved shells and 3-dimensional solids such as hexahedrons (bricks).

The "Finite" in Finite Element Analysis comes from the idea that there are a finite number of elements in a finite element model. Previously, engineers employed integral and differential calculus techniques to solve engineering analysis problems. These techniques break objects down into an infinite number of elements, for problem solving.

Finite Element Analysis Process

Finite Element Modeling Finite Element Analysis begins with the finite-element modeler (sometimes called a mesher or preprocessor). The cost-effectiveness of FEA is heavily dependent on the Pre-Processor since the vast majority of human time involved in Finite Element Analysis is spent in creating the model for analysis. In order to effectively incorporate analysis into the design cycle, you must be able to quickly create the required models. The modeler creates the physical data necessary for analyis by creating a mesh of elements utilizing either an imported 3D CAD model or one generated internally.

There are two basic mesh types characterized by the connectivity of their points. Structured meshes have a regular connectivity, which means that each point has the same number of neighbors (for some grids a small number of points will have a different number of neighbors). Unstructured meshes have irregular connectivity (e.g. each point can have a different number of neighbors.)

Unstructured meshes have been developed mainly for the finite element method. There is a large range of possible shapes for finite elements: tetrahedra, prisms, blocks, and there can be arbitrary connectivity, leading to unstructured meshes. Meshes can be generated fully automatically using triangles in 2D and tetrahedra and now blocks in 3D.

Finite Element Solvers Solvers are the engines of finite-element analysis. They take the elements, boundary conditions, and loads and output a solution containing all of the information needed to review and understand results. Solvers may be divided into two categories: linear and nonlinear.

Linear FEA is differentiated from nonlinear in that all deflections are assumed small, no boundary conditions change during the analysis and material properties are linear (i.e., elastic).

Post Processing

Postprocessors--or visualizers--utilize the data generated by the solver to create easily understandable graphics and reports.

The Finite Element Method is employed to predict the behavior of things with respect to virtually all physical phenomena:

• Mechanical stress (stress analysis)
• Mechanical vibration
• Heat transfer - conduction, convection, radiation
• Fluid Flow - both liquid and gaseous fluids
• Various electrical and magnetic phenomena
• Acoustics