PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Research software practice, verification and validation · Research software practice, verification and validation
Scientific Computing for Physicists, Ch Are they the right equations? Solution verification, benchmarks and validation
Problem
The midpoint deflection w(0.5), in centimetres, of a clamped-clamped elastic beam under a uniform load is found by discretising the bending-energy functional on a finite-difference mesh and minimising it with Newton's method, stopped when the gradient norm satisfies ∣∇E∣=1.0×10−8 (energy units per cm). At the minimum found, the smallest eigenvalue of the energy Hessian is λmin=15.3 (energy units per cm2). The minimisation is repeated, to the same stopping criterion, on meshes with N=25, 50 and 100 interior nodes (r=2 between successive meshes), giving converged deflections wN=25=0.0234512cm,wN=50=0.0235102cm,wN=100=0.0235247cm.
((a)) Find the observed order p^, the extrapolated deflection wext and the grid convergence index at N=100 (Fs=1.25).
((b)) Near the minimum, where the energy is quadratic, the distance between the nodal displacements found and the exact minimiser is bounded by ∣∇E∣/λmin, in centimetres. Evaluate this bound at the stated stopping criterion and again at a hundredfold looser one (∣∇E∣=1.0×10−6), compare both with the absolute discretisation uncertainty from (a), and decide whether a benchmark comparison against a published analytic beam solution at an absolute tolerance of 1×10−4cm could detect that the stopping criterion had been loosened a hundredfold.