How the Finite Element Method Works
A seed for a note about the method behind almost every structural program: how a structure becomes a mesh of small elements, what the computer actually solves, and why the mesh and the supports matter more than the colours.
Open in the interactive Digital Garden
In Structural Statics I calculated beams and frames by hand. Real buildings are rarely that simple – slabs, walls, openings, odd shapes. For those, almost every program today uses the finite element method (FEM). I use it at work all the time, and I want to write down how it works in plain words.
The Idea in Two Sentences
A complicated shape is cut into many small, simple pieces – the elements – whose behaviour we know exactly. The computer writes one equation for every connection point between them and solves them all together: a huge system of equations, often with millions of unknowns, which is what computers are good at.
What I Want to Cover
- A bit of history: from mathematicians in the 1940s to aircraft engineers at Boeing in the 1950s, who needed it for wings.
- From one spring to a stiffness matrix – the step from Structural Statics to FEM.
- Types of elements: beams, shells, solids – and when to use which.
- Why a finer mesh changes the result, and when it doesn't.
- Example: a cantilever in FreeCAD FEM with CalculiX, compared to the hand calculation.
- The classic mistakes: wrong supports, point loads on a mesh, stress peaks at corners that are not real, and units.
The open-source programs are in my Structural Analysis resources.