A non-linear finite element primer
The Galerkin approach is used to convert the governing physics equation to a weak formulation. For the thermal problem, an energy balance is the governing equation, while for the mechanical problem it is a stress equilibrium. The weak formulation results in a nodal solution vector U of either temperatures or displacements, a residual vector R, and a stiffness matrix dR = dT. From an initial estimate of the solution U0, the Newton-Raphson method can be iteratively applied:
Equation 1
where the superscripts i and i + 1 refer to the previous and current iterations. Equation 1 is applied until an appropriate norm of the residual R is less than a specified tolerance. Each time step takes the solution from the preceding time step as an initial estimate. Additional details pertaining to the formulation can be found in "Model verification, validation and error analysis" topic.