Lawrence S. Gould

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Will the structure break? Will the tie-rods buckle? Is the engine too hot? Finite element analysis (FEA) is the software tool that can answer these questions and more. Early in the design process, FEA can help determine the effectiveness of a design, show the best of several design alternatives, and identify where a design can be improved. As the design is completed, FEA can show whether the design will, in fact, work.

FEA helps manufacturers optimize parts for structural integrity, efficient material consumption, manufacturability, lowered product and after-sales costs, and ultimately for competitive advantage. Better, today's FEA systems keep the science behind FEA mostly transparent, making FEA one more powerful tool in the automotive engineering toolkit.

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**A World of Springs**

"Computer-aided design (CAD) only takes care of the size and shape—the geometry—of a design. It cannot predict whether a design will work in its intended environment. For that you need a `simulation tool,'" says Brian Shepherd, manager of Functional Simulation Products for Parametric Technology Corp. (Waltham, MA).

Enter FEA.

FEA is one of several numerical techniques for solving boundary value problems: how objects are held and stressed. It begins with a model of the part, the part's material properties, and its boundary conditions. The boundary conditions detail the environment in which the part will be used: how the part will be constrained and the characteristics of the load applied to the part. A mesher then uses this information to break the model into smaller chunks, called "finite elements." The type of finite element depends on the problem to be solved.

"The behavior of each little element, which is regular in shape, is readily predicted by a set of mathematical equations," says Shepherd. This set, made up of literally thousands of equations, is really a matrix version of Hooke's Law. "In 1678, Robert Hooke set down the basis for modern finite element stress: an elastic body stretches (strain) in proportion to the force (stress) on it," continues Shepherd. In mathematical terms, F = kx, where F represents force, k is a proportional constant, and x is the linear displacement of the spring. An FEA analyst applies a known force (F) to a part of known material and geometry (yielding k) and solves for x, the linear displacement of the part. Other physical properties, such as thermal conductivity, plasticity, fluid flow, acoustics and vibration, and magnetic conductivity, can be simulated in the same fashion.

"The `finite' in FEA comes from the idea that there are a finite, or countable, number of elements in a finite element model," explains Shepherd. "A computer then adds up all the individual behaviors to predict the behavior of the actual object." The computer is a necessity because using a calculator, while possible, would not get you very far; FEA is that tedious a technique. The FEA solver outputs a voluminous amount of data, which visualization tools then post process into graphical displays to show the stresses, strains, deformations, temperatures, and other details throughout the part.

The FEA model need not begin in CAD. In fact, the early days of FEA did not start with CAD at all. Instead, analysts created points out in space, called "nodes," and hooked them together creating a mesh around the geometry. Because CAD drawings already contain much of the tedious work of identifying points in space, automated meshers included in some CAD packages link these nodes together with 2D or 3D finite elements, depending on the source file and the problem to be solved. Once the geometry is meshed, the analyst is back to nodes and elements, and the CAD geometry is no longer important.

**Types of Chunks**

There are two basic types of finite elements: H (hierarchical), the older technology, and P (polynomial). The edges of H elements are described by a line, sometimes by a curve. The analyst divides the H elements into smaller elements when a more detailed analysis of the loading problem is needed; where there was one element, now there can be two. The more detailed the analysis, the more the H elements.

An experienced person will know where to divide the model into more elements. "You don't want to brazenly triple or quadruple all regions of the model because then your compute time goes through the roof," says John Whetstone, CAE Product Manager for EDS/Unigraphics (Cypress, CA). Plus, adding elements means having to remesh the model, which is time consuming.

In contrast, the edges of P elements are described mathematically; a polynomial describes the shape of the edge. Edge deformations are described by different polynomials, some with variables up to the ninth power. The more detailed the analysis, the higher the order of the polynomial. "P elements can capture pretty radical stress and displacement behavior in one element, without requiring more elements in that area," explains Shepherd.

Polynomials of higher orders, while accurate, are very complex descriptions that take a long time to solve. Consequently, some FEA vendors apply "locally adaptive P orders"—P-adaptive FEA tools that will automatically adapt themselves to high stress areas in the model by "raising the order" of the polynomial. "The beauty of the P elements is that if you set the problem up correctly, you will know within a certain confidence level that the answer is good," continues Whetstone. "The theory is that you will get a good quality answer without having to remesh."

**What's In A Shape?**

A finite element's "whole job in life is to connect together nodes with some kind of stiffness," says Shepherd. There are several classes of finite elements. A one-dimensional element represents line shapes, such as beams or springs. Two-dimensional elements represent quadrilateral elements, typically triangles or squares. Three-dimensional elements represent solid shapes, and come in two basic types: brick (also called "hexahedrons") and pyramids (called "tetras," for tetrahedrons). Other elements, like wedges, are variations on these themes.

Different finite elements apply to different applications. For example, a scaffolding consists of several one-dimensional (line) elements. Car bodies and other stamped or formed sheet metal parts are thin structures best modeled with 2D (shell or "plate") elements. These objects could have been meshed with 3D (solid) elements, but that would increase the number of nodes to analyze, which in turn increases the time needed for analysis.

On the other hand, chunky objects, such as an engine block, car suspension components, and other castings, do not follow thin element behavior. So thick (that is, solid) finite elements are used to model the object.

Sometimes the analyst will use a combination of finite elements.

The science of finite element shapes has evolved over the years. For example, tetras were unpopular for years because the triangular shape is inherently stiff—too stiff—and the mathematical formulation using those elements never seemed to accurately replicate the real world of deformations and stresses. Now, quadratic or "ten-noded" tetrahedrons can be used to follow the shape of a part more exactly.

Generally, solid elements provide better simulations and better FEA results than 2D shells. However, and as with any simulation, FEA solutions can be flawed, riddled with spurious results. "We don't see this typically in the type of element or solver technology," explains Joe Solecki, corporate fellow at ANSYS, Inc. (Houston, PA). Instead, the flaws typically occur in the definition of the problem and in using improper boundary conditions. For example, the analyst may use loads that don't make sense or hold a part in a way that just does not represent nature. "These are problems independent of FEA," comments Solecki.

**Selecting An FEA Package**

With today's FEA systems, you can mostly forget the math and physics because that is hidden for all but very specialized stress problems. Nevertheless, selecting an FEA package involves a number of criteria. Whetstone urges users to evaluate the class of problems to be solved. "Many times, people go off into the weeds implementing non-linear, transient crash analysis because of that outside chance their parts may have this permanent type of deformation. However, all they really need is a linear static analysis solution."

Once the need is determined, Whetstone suggests analysts invest in FEA code based on the design code they're using. While he admits this may cause them problems with their superiors, Whetstone says, "The design side of the house should drive the choice of the FEA. What you really want is associativity between the design and the simulation, the FEA."

The FEA package should be simple enough for the casual user, or generalist, to run. The FEA should have "enough abstraction into the language," according to Solecki, so that the design engineer can answer real-world questions, such as whether the automobile engine will get too hot or will the door lock break, without knowing exactly in a finite element sense what is going on."

By making FEA more accessible, part analysis can be done earlier in the product design lifecycle, translating into faster lifecycles, optimized part designs, and less costly analysis. Generally, building and breaking part and product prototypes will cost 10 to 100 times the cost of FEA, including hardware, software, and analyst, according to Solecki. Moreover, the cost of FEA will usually be less than 5% of the cost of manufacturing the final part or product.

A small price to pay for big savings in design verification, part production, and customer satisfaction.