Everything you ever wanted to know about FEA

John Dunec at COMSOL recently explained the various kinds of analysis. He says:

There are different classes of analysis. A large structure such as a bridge can be done with all of them. The most basic is the so-called "back of the envelope" analysis. It can be extremely sophisticated. An example comes from hiring a structural engineer to certify to the building dept. that your house design is safe. He performs this type of analysis. Basically, algebraic equations represent loading over spans and the engineer is calculating that.

The next level is call "lumped mass analysis." What it results in is ordinary differential equations (ODE). You might use this type of analysis, say, on the Tacoma Narrows bridge. It was the bridge that failed due to resonances. One thing the engineer wants to know is the overall weight of the thing. He makes an equivalent spring network of the bridge, and then applys a sinusoidal loading on it, and solves that. A lot of control theory is based on lumped mass analysis. Basically algebraic equations are moved to ODEs, normally in time. But, lumped mass does not understand spatial variations well. When you do an equivalent spring network for the bridge, it is basically cut in a bunch of sections. They all act independently, and there is no detail about what is going on in particular members and what the stress distribution in the holes is.

The next level of analysis includes time and space. They involve with partial differential equations (PDE). Unlike ODEs, there are very few known solutions for PFEs. There are known solutions to some fairly straightforward ODEs that the lumped mass analyses generate. When the problems are put into computers, the analysis software breaks the spatial and the time parts nto finite elements, finite volumes, or finite differences (which is actually close to finite volumes). Typically, structural codes are FE and historically fluids codes are have been finite volume. There were nuts and bolts reasons for this, which in the modern world are no longer valid.

In all these cases, any time you are trying to solve a problem with FE, FV, or FD you are busting the problem into blocks of different shapes. What is going on in the block is represented with a set of equations that get solved. An FE analysis knows what is going on in the corners of the block and throughout it. A finite difference code only knows what is going on in the nodes, and that is it. FE uses what are called shape functions to interpolate in between the nodes. So, it gives a much more detailed analysis of what is going on throughout the field. The drawback is that information costs in computer resources.

Lets say you are worried about a tanker truck hitting a bridge and you are worried about corrosion from the chemical spill. Now you have a loading, and impact loading, and a flow, and also a chemical diffusion and reaction process attacking the bridge. That is a classic multiphysics problem. With Comsol, when I have a chemical problem interacting with a flow interacting with a structure, under the hood, down in the math, it's just one more row in the matrix.

If you did, say, the Tacoma Narrows problem with Comsol, you would set up the airflow and the bridge structure all in one big matrix. The software solves the problem as if it is linear, finds the answer, and then repeats that procedure. In contrast, say you use another program that has great structural code, and yet another that has a great fluids code, but the two of them only talk through an iterative process. You would probably solve the airflow, and that would give you loading. Then you would hand the loading to the structure's code. The results then get meshed. This approach is often used for older programs with enormous lines of code. The developers don't want to recode, so they instead make sophisticated translation tools.

Comsol is an integrated code for multiphysics. Our experience, particularily in nonliner problems is that cycling processes can take forever to converge. You have to do the convergence for every single time step.

Let's say you are trying to do a vibration analysis of the bridge. It is a transient problem. Once you do the FE thing, what comes out is ODEs in time. So you solve the bridge problem for time = 1 second. You get that answer as a starting point and solve again for time = 2 seconds. Each time you march forward, you have to solve the entire spatial problem, which on a bridge could be four million equations. You end up with a set of algebraic equations. If they are non-linear, you linearize them and you end up with a huge matrix problem.

The degrees of freedom (DOF) are the number of unknowns that are being solved in your problem. In a FD problem, that is usually the value of what you are solving for at each of the nodes. In a FE problem, the DOFs are the parameters that define the shape functions. DOFs is how big the matrix is in one direction -- i.,e., the number of unknowns.

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