A differential equation that must hold inside a domain, together with conditions on the domain boundary. Static stress analysis, steady heat conduction and potential flow are all boundary value problems.
The differential equation as written, required to hold at every point. For a bar: −(EAu′)′=q. It needs u to be twice differentiable.
Weak form
The equation multiplied by a test function and integrated, with one derivative moved onto the test function by integration by parts. It holds "on average" and needs only first derivatives. In mechanics it is the principle of virtual work.
Prescribes the unknown itself, e.g. u=0 at a support. Must be built into the set of candidate solutions; it does not appear in the weak form on its own.
Natural (Neumann) boundary condition
Prescribes a derivative of the unknown, which in mechanics is a force or traction, e.g. EAu′(L)=F. It enters the weak form automatically through the boundary term of integration by parts.
Test function
The arbitrary weighting function v in the weak form. Mechanically, a virtual displacement compatible with the supports. The weak form must hold for every admissible v.
Principle of virtual work
A body is in equilibrium if, for every small displacement compatible with its supports, the work done by the internal stresses equals the work done by the external loads.
Approximate the solution and the test functions with the same finite set of basis functions, then require the weak form to hold for each of them. Gives a square, and for elasticity symmetric, linear system.
A basis function Ni attached to node i: equal to 1 at that node and 0 at all others, non-zero only on the elements touching it. The field is uh=∑iNiui.
Partition of unity
The shape functions add up to one everywhere: ∑iNi(x)=1. Guarantees that a rigid translation is represented exactly and produces no strain.
Degree of freedom
One scalar unknown of the discrete model, typically one displacement component at one node. A 3D solid node has three; a shell node has six (three translations, three rotations).
Stiffness matrix
The matrix K in Ku=f. Entry Kij is the force at degree of freedom i caused by a unit displacement at degree of freedom j. Symmetric and sparse.
Building the global matrix by adding each element matrix into the rows and columns of that element's degrees of freedom. Also called the direct stiffness method.
Rigid-body mode
A motion that produces no strain: translation or rotation of the whole part. 3D solids have six. If supports don't prevent all of them, K is singular and the solver fails or returns nonsense.
Positive definite
A symmetric matrix with xTKx>0 for every non-zero x: every deformation stores positive strain energy. Guarantees a unique solution.
Nodal forces obtained by integrating the distributed load against the shape functions, fi=∫qNidx. Not always an equal split: on an 8-node quadratic face a uniform pressure puts negative force on the corner nodes, and on a 6-node triangle face exactly zero.
Superconvergence
Points inside an element where derivatives (strain, stress) are more accurate than elsewhere, typically the Gauss points. FE codes compute stresses there and extrapolate to the nodes.
von Mises stress
A scalar combination of the stress components that predicts yielding of ductile metals: σvm=23s:s, with s the deviatoric stress. Always positive, so it says nothing about tension versus compression.
The normal stresses on the planes where shear vanishes: the eigenvalues σ1≥σ2≥σ3 of the stress tensor. The largest tensile one drives brittle fracture and fatigue cracks.
Low-order elements that cannot bend without also shearing pick up spurious shear strain energy in bending, which makes the model too stiff. Cured by quadratic elements, reduced integration or enhanced formulations.
Stress singularity
A point where linear elasticity predicts infinite stress: sharp re-entrant corners, point loads, point supports. The FE peak there grows without limit as the mesh is refined.
Saint-Venant's principle
Statically equivalent load distributions produce the same stresses far from where they are applied. "Far" means roughly one characteristic dimension of the loaded region.
An element whose shape functions are second-order polynomials, with nodes at corners and mid-edges: 6-node triangles, 10-node tetrahedra, 20-node bricks. Strain varies linearly inside each element.
h-refinement
Improving accuracy by making elements smaller (h is the element size). The alternative, p-refinement, raises the polynomial order instead.
Convergence study
Solving the same model on successively finer meshes and checking that the quantity of interest stops changing. The only practical way to estimate discretisation error.
Averaged nodal stress
Stress computed in each element, extrapolated to its nodes, then averaged between the elements sharing each node. Gives smooth contour plots and hides the jumps that reveal a coarse mesh.
Unaveraged (element) stress
Stress reported per element without smoothing across element boundaries. Jumps between neighbours are a direct, local measure of discretisation error.
Symmetry boundary condition
On a plane of geometric and load symmetry, the displacement normal to the plane is zero and in-plane displacements are free. Lets you model a half, quarter or eighth of the part.
Shell element
A 2D element living on a part's mid-surface, with the thickness as a property. Resolves bending of thin walls with one element through the thickness, at a fraction of the cost of solids.
Reaction force
The force a support exerts on the part. In equilibrium the reactions balance the applied loads exactly, which makes their sum a free sanity check.
Stress concentration factor
Kt=σpeak/σnominal at a notch, hole or fillet. Tabulated for standard geometries in Peterson's handbook; finite for any real radius.