How FEA works, briefly put.

No time to waddle through dozens of pages full of mathematics? Here’s how FEA works – the sticky note version!

  1. The governing equation is defined. In magnetostatics, the Ampere’s law \nabla \times \mathbf{H} = \mathbf{J} is combined with the vector potential \mathbf{H} = \nu \nabla \times \mathbf{A} to get
    \nabla \times \left( \nu \nabla \times \mathbf{A} \right) = \mathbf{J}.
  2. We define an approximative solution
    \mathbf{A} \approx \mathbf{\hat{A}} = \sum\limits_{k=1}^N a_k \mathbf{w}_k.
    In other words, known (shape) functions \mathbf{w} multiplied by some unknown coefficients a.
  3. The problem geometry is meshed, and the shape functions are defined based on the mesh.
  4. To solve the coefficients, we try to get “as good as possible” solution according to the following criterion:
    \int \mathbf{w}_i \nabla \times \left( \nu \nabla \times \mathbf{\hat{A}} \right) \mathrm{d}x\mathrm{d}y = \int \mathbf{w}_i \mathbf{J} \mathrm{d}x\mathrm{d}y for all i = 1 \ldots N.
  5. This gives us a system of N equations for our N unknowns.
  6. Profit!
  7. Lots of awesome figures are drawn (see below).
The results look pretty.


P.S. To get a unique solution, boundary conditions of course have to be fixed. Typically, the left-hand-side integral is integrated by parts (this also means that \mathbf{\hat{A}} doesn’t have to be twice differentiable anymore), after which Neumann boundary conditions (forces, fluxes) can be included in the boundary integral that appears. Dirichlet confitions (fixed displacement, temperature, potential) have to be fixed more manually, by solving only those unknowns that are not directly fixed by this condition.

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How FEA works – Post-it version

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