Damping without tuning: the measurable stability threshold of the optimality-criteria iteration, and a controller that tracks it

Abstract

Every density-based topology optimization run using the optimality-criteria (OC) update makes one algorithmic choice: the damping exponent eta, folklore-set to 1/2. We measure what it controls. Exactly projected finite-difference Jacobians of the full update, across meshes, boundary conditions, volume fractions, filters and penalizations, give three empirical laws. One: oscillation is a period-doubling instability at eta* = 2/s_max, the top of the spectrum of the log-log stiffness of the filtered sensitivity field; the local multipliers mu = 1 - eta*s predict measured onset and decay rates to three decimals. Two: that spectrum saturates at p+1 on top while its lower edge sits at s = 0, and an edge at zero pins the min-max (Richardson) optimum onto the flip boundary itself: at p = 3 the folklore value is that optimum. The content of the coincidence is the spectral shape, not the optimum's worst-case rate, which the edge modes leave nearly vacuous; the damping's real work is done on the bulk at s = 1. Three: the observed s_max belongs to the fixed point the dynamics selects, and standard damping self-limits to s_max <= 2/eta, separating from the naturally saturated family exactly at p = 3. The threshold is thus a moving target: under penalization continuation, fixed 1/2 ends oscillating in 12 of 32 runs. It need not be. The trajectory estimators read the flip branch precisely when it is active and the s = 0 creep edge otherwise - a regime dependence that estimator-driven relaxation schemes do not account for - so a four-line additive-increase/multiplicative-decrease controller can track the threshold unaided: with one frozen parameter set it converges in all 64 continuation runs, matches the best hand tuning across an 11-configuration suite, and composes with Anderson acceleration so that each covers the other's failures.