One Integral Read as 0.1406 and 56.91 — Torsion Stress Function and Duct Laminar Flow
A square bar's torsion constant and a square duct's f·Re are one Poisson solution read twice. One linear-triangle FEM run gives both handbook numbers.
A 30° rigid rotation read -13% strain — conjugate stress-strain pairs in large-strain shells
Rotation alone fabricates strain in the engineering measure, and a wrong stress-strain pairing shifts the internal work rate by 5% or 54%.
Pushing the cell Péclet number to 2 drove the solution to -0.33 — where the energy to minimize disappeared
Advection skews the stiffness matrix, so the energy minimum is gone. At cell Péclet 2, Galerkin falls to -0.33; SUPG buys a bound, not symmetry.
One Fewer Quadrature Point and the Solution Blew Up — DG's Integration Floor and the Taylor Basis
For DG-P2 the volume integrand has degree 3, so two Gauss points match three points at 5.989e-3. One point leaves L2 error near 11.7 and order 0.13.
Skew the Element 45° and the Stiffness Grows 2.5x — Metric Tensors and Constitutive Tensor Transformation
In curvilinear coordinates, writing strain as covariant components means the constitutive tensor has to move too. Skip that step and every skewed element reports the wrong stiffness.
Thinner Means Stiffer — Shear Locking in Shell Elements and the MITC Tying Fix
A linear shell element gets stiffer, not weaker, as it thins out. The cause is one line: shear strain obtained by differentiating displacements.
Why Does the Bridge Sag Like That? The Direct Stiffness Method for Trusses
A bar with EA/L = 8e6 N/m moves 1.5 mm under 12 kN. The 12-member truss stays singular until four wall degrees of freedom are removed.
Discontinuous Galerkin Method (DGM): High-Precision Discretization for Unstructured Grid Compressible Flow Analysis
In 3D the degrees of freedom at K = 1, 2, 3 are 4, 10, and 20. Halving the mesh spacing should cut the error by 4, 8, and 16, not by 2.
FEM for Rigid Body Dynamics: Quaternion-based Rotation Handling and Inertia Tensors
Store inertia once in the body frame and rotate it by R(q). If |q| leaves 1, R stops being orthogonal and angular acceleration picks up the wrong axis.
FDM vs FEM vs FVM: The Essential Differences Between Three Discretization Techniques
FDM discretizes the operator, FEM projects onto a basis, and FVM balances face fluxes. That choice fixes conservation, shocks, and unstructured meshes.