Shape functions are interpolation functions that describe how field values (like displacement) vary within an element based on nodal values. At each node, the associated shape function equals 1 and equals 0 at all other nodes.

For a linear bar element, N₁ = 1 - x/L (equals 1 at node 1 where x=0, equals 0 at node 2 where x=L). Similarly, N₂ = x/L.

Stiffness matrix symmetry (Kᵢⱼ = Kⱼᵢ) follows from energy principles and Maxwell-Betti reciprocal theorem. This symmetry is a fundamental property of elastic systems, not dependent on mesh or element choices.

Total DOFs = 100 nodes × 2 DOFs = 200. After removing 10 constrained DOFs using the partitioning method, the reduced system has 200 - 10 = 190 active DOFs, giving a 190 × 190 matrix.

Quadratic elements can model curved boundaries exactly and capture higher-order displacement/stress variations. They converge faster than linear elements, often requiring fewer elements for the same accuracy.

For isoparametric elements with curved geometry, the integrals for stiffness matrices involve complex expressions of the Jacobian that cannot be evaluated analytically. Gauss quadrature provides accurate numerical integration.

Hourglassing occurs when reduced integration fails to detect certain deformation modes because strain is zero at the integration point. These 'hourglass modes' have zero strain energy and can propagate through the mesh.

The Jacobian matrix J = ∂(x,y)/∂(ξ,η) relates derivatives in physical coordinates to derivatives in natural coordinates. It enables integration over arbitrary element shapes by mapping to the simple parent element.

Iterative solvers use O(N) memory vs O(N^1.5 to N²) for direct solvers. For large 3D problems exceeding 1 million DOFs, iterative methods with good preconditioners (AMG) are often the only practical choice.

Verification asks 'Are we solving the equations right?' (convergence studies, benchmarks). Validation asks 'Are we solving the right equations?' (comparison with experiments, physical reality).

A mesh is considered converged when further refinement produces negligible changes in key results (typically <5% change). This demonstrates that discretization error is acceptably small.

Concentrated point loads create mathematical singularities where stress approaches infinity as mesh is refined. In reality, all loads are distributed over finite areas. Use pressure or distributed loads for realistic results.

Condition number κ = λmax/λmin measures how sensitive the solution is to numerical errors. High condition numbers (common in FEA) slow iterative solver convergence, which is why preconditioning is essential.

Von Mises stress is used for ductile materials because it's based on distortion energy theory, which predicts yielding in ductile materials. Principal stresses are used for brittle materials (maximum principal stress theory).

Symmetry allows modeling only half (or quarter) of a symmetric problem, reducing DOFs by 50-75%. This dramatically reduces memory and solve time while giving identical results to the full model.