# Parametric FEA

Nov. 15, 2002
Parametric features are becoming more common in FEA packages.

Parametric features are becoming more common in FEA packages. The key benefit of parametric features is that they let users see the effects of design changes quickly. With adequate planning, users can define an FE model entirely in terms of variables or parameters. Even mesh characteristics can be defined as parameters.

To illustrate, consider a simple design-analysis problem. A simply supported steel beam deflects excessively under load. To stiffen the design, reinforcing strips will be spot welded to 500 already-cut beams.

Theory says that the addition would provide adequate stiffness if the two components were perfectly joined. But spot welds provide a discontinuous interface. Therefore, a new requirement is to find the minimum number of spots welds to limit maximum deflection of the welded beam to no more than 110% of the deflection of a solid beam with the same cross section.

The parametric finite-element model must be flexible enough to simulate welded beams with a varying number of spot welds. However, with little additional effort we can create a more general model that also simulates welded beams with a range of dimensions. The payoff is that we are better prepared to evaluate reinforcement bars with different cross sections, should that become necessary. In addition, we want to define mesh characteristics as parameters. This helps with convergence or mesh-refinement studies. All the dimensions are treated as parameters. To make the finite-element model as general as possible, top and bottom sections are allowed a different number of elements across their widths. However, the two sections must have the same number of elements along their lengths.

The model consists of two layers of quadrilateral shell elements. The bottom layer of nodes sits in the plane y = -(t1)/2 where t1 is the thickness of the reinforcement bar.

The sketch of corner nodes shows them at the bottom section of the model along with parameters that define two corner nodes. Coordinates (X, Y, Z) for node 1 are (0, -(t1)/2, (b1)/2). These coordinates become arguments to the first command which defines the first node of the model. The last node, nend, on the same edge is at coordinates (len, -(t1)/2, (b1)/2).

The value of nend is defined in terms of the number of welds and number of elements needed between welds. Coordinates for node nend become arguments to the second command. A Fill command simply fills in the nodes between nodes 1 and nend.

If we increase the number of welds or elements between welds, we increase the value of nend. In other words, we have automated part of the mesh generation. By continuing this process, we automate every aspect of the finite-element mesh. This gives us precise control over dimensions, nodes, and elements in the model.

Loads are also parameterized. For the example, the model is subjected to a unit distributed load. One parameter denotes the load. Another (beam length width) defines the load area on the upper section. Two other parameters calculate the pressure required for the upper surface. In this way, changing the model size does not affect the total load on the beam.

Even boundary conditions can be applied with parameters. For instance, we use parameters to select nodes at the ends of the upper layer. Thus, we are assured that the simply supported boundary conditions are correctly applied for each run of the study, regardless of the changes that might be made to either geometry or mesh.

In the example, parametric modeling also specifies the spot-weld locations. We want to spot welds to be placed at the beam's symmetry plane and equally spaced along its span. Linear constraint equations, automatically generated by the program, model the stiffening effect of spot welds.

Assigning parameters that describe linked nodes lets the model simulate any number of spot welds. Model dimensions and mesh density are changed with equal ease. Parametric models also help with convergence or mesh-refinement studies which estimate the discretization error, an accuracy indicator.

A convergence study lets us select the FE mesh that provides the quickest acceptable solution. This is important when the model is used many times in parametric studies. In such case, it's advantageous to generate several solutions with a finite, but acceptable error in each.

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