FEA Solver background of Linear static analysis

The basic finite element equation to be solved for structures experiencing static loads can be expressed as:

Ku = P

where K is the stiffness matrix of the structure (an assemblage of individual element stiffness matrices).  The vector u is the displacement vector, and P is the vector of loads applied to the structure.  The above equation is the equilibrium of external and internal forces.

The stiffness matrix is singular, unless displacement boundary conditions are applied to fix the rigid body degrees of freedom of the model.

The equilibrium equation is solved either by a direct or an iterative solver.  By default, the direct solver is invoked, whereby the unknown displacements are simultaneously solved using a Gauss elimination method that exploits the sparseness and symmetry of the stiffness matrix, K, for computational efficiency. Alternatively, an iterative solver using the preconditioning conjugate gradient method may be used.  While the direct solver is very robust, accurate and efficient, the iterative solver is sometimes superior, in terms of speed, for thick-walled solid structures. 

Once the unknown displacements at the nodal points of the elements are calculated, the stresses can be calculated by using the constitutive relations for the material.  For linear static analysis where the deformations are in the elastic range, i.e.: the stresses, σ, are assumed to be linear functions of the strains, ε, Hooke’s law can be used to calculate the stresses.  Hooke’s law can be stated as:

σ = Cε

with the elasticity matrix C of the material.  The strains ε are a function of the displacements.