What is FEA (finite element analysis)?

Finite element analysis is the numerical solution of how a solid body responds to what is done to it. You supply the geometry, the material, where it is held and what is applied; it returns displacement everywhere, and from that the strain and stress everywhere.

The underlying statement is equilibrium: internal forces must balance applied ones at every point. The body is divided into elements joined at nodes. Within each element, displacement is described by a simple function of the nodal values, which turns the continuous problem into a matrix equation — conventionally written Ku = F, where K is the stiffness matrix, F the applied loads and u the displacements you are solving for.

Everything else in FEA is a variation on that. Modal analysis adds mass and solves for the frequencies at which the structure prefers to move. Nonlinear analysis rebuilds K as the answer changes. Transient analysis marches it through time.

• Linear static. One load case, small deflections, material within its elastic range. Answers “what is the peak stress and how much does it move”. Most structural questions are settled here, and it is cheap enough to run while you are still deciding.
• Nonlinear static. Needed when deflections are large enough to change the geometry’s stiffness, when the material yields, or when rubber-like materials are involved. Solved as a series of increments.
• Contact. Parts that touch, slide and separate. The stiffness depends on an answer you do not have yet, which makes it the first genuinely awkward convergence problem most people meet outside CFD.
• Modal. Natural frequencies and mode shapes, with no load at all. Fast, and the first thing to run if something vibrates, rattles or sings.
• Harmonic and transient dynamics. Response to steady vibration at frequency, or to a load that varies in time.
• Buckling. The load at which a slender structure stops being stiff. Critically, this can happen at a stress far below yield, so a static run that reports a comfortable safety factor tells you nothing about it.
• Thermal stress. Stress from constrained thermal expansion, using a temperature field from a thermal analysis.
• Fatigue, creep, fracture and damage. Life and failure over time rather than strength on the day.

Three failure modes account for most wrong structural results, and none announce themselves.

Stress singularities. At a sharp re-entrant corner, the mathematical solution has infinite stress. Refine the mesh there and the reported peak stress does not converge — it keeps climbing, because you are converging towards infinity. This is the one place where the usual advice inverts: a finer mesh gives a “worse” answer. Real parts have a fillet; model it, or read your stress a small distance away from the corner and say that you did.

Element choice. First-order tetrahedra are notoriously stiff in bending — they under-predict deflection and stress unless the mesh is very fine. Second-order elements cost more per element and are usually the cheaper route to a correct answer.

Boundary conditions. Fully fixing a face is easy and almost always wrong: nothing in reality is infinitely rigid, and over-constraint manufactures stress that does not exist. Most implausible FEA results are restraint problems rather than mesh or material problems.

The habit that catches all three is a mesh convergence check: run at two or three densities and watch the quantity you intend to quote. Deflection converges quickly; stress converges slowly; stress at a singularity never converges at all, and noticing which case you are in is the whole point.

If you have done CFD, the workflow transfers directly: geometry, mesh, material, boundary conditions, solve, check convergence, post-process. Two practical differences are worth knowing.

First, FEA meshes the solid itself, whereas CFD meshes the space around or inside it — the fluid volume. Second, a linear static FEA run is a single matrix solve rather than an iteration to convergence, so “did it converge” only becomes a live question once contact, plasticity or large deflection enter the picture.

The deepest of the open-source structural codes is code_aster, EDF’s research code, used for nuclear plant assessment. Its nonlinear, fracture and creep capability goes a long way past what a general-purpose FE code carries, and it has a correspondingly steeper learning curve — which is the gap a front end exists to close.

It sits in the guide to open-source solvers alongside everything else in the stack.