How engineers crash a car thousands of times without building one
The equations describing how metal bends, heat spreads or air flows around a wing are usually impossible to solve exactly for a real, awkward shape. The finite element method sidesteps the problem by chopping the object into thousands of tiny, simple pieces, solving each one roughly, and stitching the answers back together.
The first move is to cover the object with a mesh, often of triangles or similar shapes. Within each small element, the true behaviour is replaced by a simple approximation, typically a polynomial, chosen so the mismatch with the real equation is as small as possible. That trick turns calculus into algebra: for a steady situation you end up with ordinary algebraic equations, and for one that changes over time, a set of simpler differential equations to step forward numerically. The pieces are then assembled, node by node, into one enormous system that a computer can solve.
Chopping things up has practical perks. Odd geometry is easy to follow, different materials can sit side by side, and the mesh can be made fine only where detail matters. In a simulated head-on collision, engineers can concentrate precision at the front of the car and go coarse at the back to save computing time. Weather models can likewise focus effort on a developing cyclone rather than calm air. Each virtual test replaces a costly physical prototype.
Nobody can pin down an exact birthday. The method grew out of hard problems in civil and aeronautical structures, with Alexander Hrennikoff and Richard Courant laying groundwork in the early 1940s; Courant carved regions into triangles to study a twisted cylinder, building on ideas from Rayleigh, Ritz and Galerkin. Feng Kang rediscovered it independently in China around 1960 while calculating dams.
Momentum built in the 1960s and 1970s through groups at Stuttgart, Berkeley, Swansea, Paris and Cornell, while freely shared programs spread it widely. NASA backed the original NASTRAN, and the Norwegian ship classifier Det Norske Veritas wrote Sesam in 1969 for hull analysis. Gilbert Strang and George Fix put the mathematics on firm footing in 1973. Today the same approach handles electromagnetism, heat transfer and fluid dynamics.
Source: Finite element method
