FEA and Convergence Basics...

In a typical design-engineering project one may be asked to predict stress, deflections, natural frequencies or temperature distribution in a part or assembly. The design may have to be modified or optimized based on these predictions.

The traditional methods involve idealizing your physically complex designs using simple equations, empirical formulas and tables of numbers from various handbooks and engineering texts. These equations have been derived from theory over the years by many great scientists, engineers and professors and have been tested and become the standard used by many engineers today. Empirical formulas and tables of constants have been derived from testing performed by many universities, professional societies and corporations over the years to handle problems that are not easily explained by theory. In most all of these cases the authors or researchers have published this information to cover the most general and simple cases. It is then left up to the engineers to perform the idealization of the real parts to best match the simple equations provided. These idealizations are required because in most cases the engineer is not going to find an equation that exactly fits his problem.

This idealization process has always been the 'art' of analysis and requires many years of mentoring and experience to become good at it. Most of the time the tendency is to over simplify the problem on the conservative side, which typically results in designs that are not well optimized. The computer aided Finite Element Method (FEM) was developed to help engineering perform analysis in much greater detail and with much more complex idealizations than would not be practical using hand calculation methods.

With FEM the real structure is idealized with a standard set of Finite Elements. This process is called Modeling and the result is the Finite Element Model. The set of finite elements is also called the Finite Element Mesh. Some computer programs use automatic element creation with a tool known as the Mesher. In Pro/MECHANICA the mesher is called AutoGEM (Automatic Geometric Element Meshing).

Each finite element has an equation that defines its stiffness [K] based on the physical properties, material and geometry of the element. The stiffness [K] describes the relationship between applied forces {F} and displacements {d} using the basic equation:

{d} = {F}/[K]

As a result of the set of finite elements used to idealize the structure you will have a set of equations as above. These equations are connected or tied together with another set of equations call the constraint equations or equilibrium equations. These equations enforce the displacements along common element boundaries to be equal. They also define the displacements at ground constraint locations.

Finite Element Analysis (FEA) is the process used by the computer program to solve for this set of equations. The end result is the deformed shape or displacements throughout the model.

The set of equations used to define stiffness in each element is based on the shape and properties of the element. These equations are most accurate for the ideal element shape. For shells or solids this is a perfect square or cube. If the elements are not nicely shaped the accuracy will diminish.

The derivative of displacement defines the strain in the element. This is used with special functions to calculate the stress field through the element. It is important to note that the stress field is calculated individually for each element in the mesh and will not be continuous across the entire mesh. In other words, two elements that share a common boundary will calculate a different stress along that boundary. This is not the case with displacements. Constraint or equilibrium equations force the displacements to be continuous across the mesh.

Most conventional FEA codes use linear equations to define the stiffness of the elements. Some codes may use second or third order equations. One of the things that make Pro/MECHANICA unique is that it uses variable polynomial equations (P-elements) to define stiffness. The equations can be from linear to ninth order.

If linear equations are used the displacements will be linear and the strain will be constant. If higher order equations are used the displacement tend to be more accurate within each element.

To achieve higher accuracy with a linear element model requires mesh refinement. Adding more elements to the mesh will result in smaller and smaller linear pieces to define a true picture of the displacement and stress field. To show evidence of solution convergence the user must run several models of various mesh densities for comparison.

To achieve higher accuracy with a P-element model requires the use of higher order equations. The element mesh may not need to change. As the equation order increases the displacement and stress field calculated will improve in accuracy. To show solution convergence the user needs to run the model at various polynomial levels (P-Levels) for comparison. This convergence process is easily automated and is built into the Pro/MECHANICA software.

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