ME 428/528 — Applied Finite Element Analysis

Applied finite elements for engineers, taught with a lab section and cross-listed as BME 528, with additional expectations at the graduate level.

The lectures build one continuous line. An axial bar problem in strong form and the direct stiffness method for bars and trusses; residuals and weighted residuals, and the function spaces that decide when a weak form is even well defined; Galerkin approximation and 1D elements of arbitrary polynomial order; then the 2D isoparametric element, where the master element, the Jacobian and quadrature produce the element stiffness matrix and load vector rather than being asserted as recipes.

What students build and check

A homework on the direct stiffness method has students implement a relative L² error norm themselves, run it over a sequence of meshes, and verify that the computed reactions balance the applied load.

The Python project extends the same code to a tapered bar. Students verify a closed-form solution by hand, run a convergence study at polynomial orders one, two and three, choose the coarsest mesh that keeps the error at a design point under one percent, and then use that mesh inside an optimization over the taper parameter: first by brute-force sampling, and at the graduate level with a bounded scalar minimizer. The closing question is whether a mesh accurate enough for a displacement is accurate enough for the design decision made from it.

The Abaqus labs

The labs are not driver’s-license training for a solver. The first project is a truss and a bar, where an exact solution is available and the FE prediction has to be defended against it, including what refinement and enrichment would and would not buy.

The final project contrasts a distributed traction with its resultant point force on an aluminum plate with boron carbide inclusions. It requires a convergence study in elastic strain energy, asks how well the converged solution satisfies each boundary condition it was given, and asks how much the loads could be scaled before yield at a safety factor of three. Graduate students estimate the stress concentration factor at the slot edge under both loading models and account for the difference.

That is the intended skill: diagnosing a wrong boundary condition, an inconsistent unit or an unconverged mesh — judgment about a model, rather than screenshots of a solution that happens to be right.

The midterm is written the same way. State the residual, say what H¹(0, L) requires of a function, and identify the residual functional and the energy functional that belong to the problem. The graduate question places a material interface mid-bar and asks why the pointwise residual stops making sense there while the residual functional does not.

Assessment is homework 10%, lab project reports 40%, midterm 25%, and a comprehensive take-home final 25%.