How the subject is organized
Every topic in the library sits on one of three layers: the continuum description, its finite-element discretization, or the numerical methods that solve the discrete system.
Continuum
Continuum mechanics
A solid or a fluid is treated as a continuous medium. Motion is measured by strain; internal forces by stress. Conservation of mass and momentum must hold — Cauchy’s first law ∇·σ + ρ b = ρ a in the current placement, or Div P + ρ₀ B = ρ₀ A on the reference mesh — and a constitutive model says how stress depends on deformation or on the rate of deformation.
That is enough to distinguish elasticity from plasticity, and a viscous fluid from an elastic solid, without yet choosing a numerical method.
Discretization
Finite elements
The field problem is posed on a mesh. Unknowns are interpolated with shape functions, and the strong form of the balance laws is replaced by a weak statement—virtual work, or a Galerkin projection. Element matrices assemble into a global system for the nodal degrees of freedom.
Whether that system is a faithful model depends on the element, the mesh, and the boundary conditions, not only on the continuum theory.
Solution
Numerical methods
Integrals on elements are evaluated by quadrature. Linear systems are factored or iterated. Nonlinear residuals are reduced by Newton’s method, which needs a consistent tangent. Time-dependent problems add a choice between implicit and explicit stepping.
Sparse storage, preconditioning, and the use of a GPU belong here: they decide whether a formulation that is correct on paper can be solved at all.