The American Society of Mechanical Engineers (ASME) put out a newsletter with the following four articles. It gives an introduction to the new method of analysis called the "finite element method." The author is an expert in finite element analysis and works as a consultant and expert witness in engineering.
FINITE ELEMENT ANALYSIS: Pre-processing
by Roensch & Associates President Steve Roensch
This is the second of four parts.
As we talked about last month, there are three parts to finite element analysis: pre-processing, solution, and post-processing. The goals of pre-processing are to make a good finite element mesh, give the right properties to the materials, and set boundary conditions in the form of loads and restraints.
The finite element mesh breaks up the geometry into "elements," which are connected by "nodes." The nodes, which are just points in space, are usually at the corners of the element and sometimes near the middle. For a two-dimensional (2D) analysis or a three-dimensional (3D) thin shell analysis, the elements are mostly 2D, but they may be slightly "warped" to fit a 3D surface. The thin shell linear quadrilateral is an example. "Thin shell" means "basically classical shell theory," "linear" means "mathematical quantities can be interpolated across the element," and "quadrilateral" describes the shape. For a 3D solid analysis, the elements must have thickness in all three dimensions. Solid linear bricks and solid parabolic tetrahedral elements are two common examples. There are also a lot of special elements, like axisymmetric elements, which are used when the geometry, material, and boundary conditions are all the same on both sides of an axis.
At the nodes, the degrees of freedom (dof) of the model are set. Most solid elements have three translational degrees of freedom (dof) per node. Rotations are done by moving groups of nodes around in relation to other nodes. Thin shell elements, on the other hand, have three translations and three rotations per node, for a total of six dof per node. By adding rotational dof, it is possible to figure out things like bending stresses caused by the rotation of one node relative to another. So, for structures where the classical thin shell theory is a good approximation, adding extra degrees of freedom (dof) at each node makes it unnecessary to model the physical thickness. The class of analysis also affects how the nodal dof is given out. For a thermal analysis, for example, each node has only one temperature dof.
Most of the time, the most time-consuming part of FEA is making the mesh. In the past, the locations of nodes were manually typed in to get a rough idea of the geometry. The more modern way is to build the mesh directly on the CAD geometry, which will be either (1) "wireframe," with points and curves representing edges, (2) "surfaced," with surfaces defining boundaries, or (3) "solid," with the material showing where it is. Solid geometry is better, but a surfacing package can often make a blend that is too complicated for a solids package to handle. In terms of geometric detail, one of the most important rules of FEA is to "model what is there." However, simplifying assumptions must be used to keep models from getting too big. Analysts need to have a lot of experience.
A mapping algorithm or an automatic free-meshing algorithm is used to mesh the geometry. The first one puts a rectangular grid on a geometric area, which must have the right number of sides because of this. Mapped meshes can use the accurate and cheap solid linear brick 3D element, but applying it to complex geometries can take a long time or even be impossible. Free-meshing divides meshing regions into elements automatically. It has the benefits of being fast, easy to change mesh size (for a denser mesh in areas with a big gradient), and able to adapt. Disadvantages include making models that are too big, making elements that aren't straight, and, in 3D, using the expensive solid parabolic tetrahedral element. Before you solve a problem, you should always check for elemental distortion. A matrix singularity will be caused by an element that is badly out of shape, which will kill the solution. A less distorted element might work, but it might also give very bad answers. How much distortion is okay depends on the solver that is being used.
Different types of solutions need different kinds of materials. For a linear statics analysis, each material will need to have an elastic modulus, Poisson's ratio, and maybe a density. For a thermal analysis, you need to know what the thermal properties are. Declaring a nodal translation or temperature is an example of a restraint. Forces, pressures, and heat flux are all types of loads. To make it easier to use adaptive and optimization algorithms, it is best to put boundary conditions on the CAD geometry and have the FEA package move them to the underlying model. It's important to remember that the boundary conditions are often where the biggest mistake is made. As part of a sensitivity analysis, it may be necessary to run multiple cases.
In the article for next month, the solution phase of the finite element method will be talked about.
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