Mechanostat-type effective density correction for Carter-Hayes growth applied to topology optimization and its efficient interpolation for a target strain energy and volume fraction

TL;DR

The paper tackles manufacturability in topology optimization by integrating a mechanostat-inspired effective density correction with an erosion filter and a growth-space, logarithmic densification framework, enabling multiple feasible topologies for the same boundary conditions and target properties. It derives a density-update mechanism that couples volume and compliance, defines design spaces that capture growth-induced topology families, and introduces an erosion filter to enforce minimum thickness, improving robustness under uncertain loading. A D$^2$NN surrogate is developed to interpolate fast between curves of compliance and volume, offering large speedups (≈25×) while preserving mechanical performance. The approach also supports design for target natural frequencies and demonstrates versatility across varied loading scenarios, with potential extensions to 3D filtering, probabilistic loads, and embedded damage criteria for practical engineering use.

Abstract

The need for optimized structures with good mechanical performance for the minimum weight is common in industry. Solid Isotropic Material with Penalization (SIMP) is a Topology Optimization (TO) method offering a trade-off between minimum compliance (i.e., maximum stiffness) and a fixed material amount for a given set of boundary conditions. Since TO is a non-convex problem, its gradient can be tuned by filtering the topology's contour, creating sharper material profiles without necessarily compromising optimality. However, despite simplifying the layout, some filters fail to address manufacturability concerns such as capillarity (thin tweaks as struts) generated by uncertain loading, vibration or fatigue. A tailored density-based filtering strategy is offered to tackle this issue. Additionally, volume fraction is left unconstrained so material can be strategically replenished through a logarithmic rule acting on the updated compliance. In doing so, an interpolation space with three degrees of freedom (volume, compliance, minimum thickness) is created, yielding diverse topologies for the same boundary conditions and design values along different stages of evolving topological families with distinct features. The optimization process is further accelerated by introducing the volume-compliance iterative scheme as a physical loss function in a Double Distance Neural Network (D$^2$NN), obtaining similar results to 2,000 steps worth of vanilla iteration within 500 training epochs. This proposal offers a novel topology optimization design space based on minimum strut thickness - via filtering - and topological families defined by minimum volume fraction and compliance. The methodology is tested on several examples with diverse loading and boundary conditions, obtaining similarly satisfactory results, and then boosted via Machine Learning, acting as a fast and cheap surrogate.

Figures (20)

• Figure 1: Filter with lengths $l_i, 1 < i < 8$ (black arrows) outside a radius $r$ (red circle). Each element's color represents its physical density, $\rho_{Phys}$, from white (virtually void) to dark gray (full material).
• Figure 2: Filter of radius $r=5$ applied to the a 1000x200 cantilever at iteration 1000 (Figure \ref{['fig:comparison_it1000']}a). Dark blue represent void, yellow the erased (filtered) twigs and light blue the remaining structure.
• Figure 3: Topology optimization processes (rows) for a cantilever beam with different starting points: a randomly convoluted topology $\alpha$ (top) and the usual intermediate density block ($\rho$ = 0.5, bottom). The same iterations (column-wise) share their boundary conditions $\eta$, volume fraction $\rho$ and strain energy $\Psi$.
• Figure 4: Different optimized cantilever beams (right) with column-wise equal densities $\rho$ represented in the $\Lambda$ space (left). Comparison between SIMP (green dots at the end of vertical arrows) and the proposed logarithmic densification starting from quasi-void (blue dots on the left-side black thick curve) and a very low density ($\rho_0 = 0.1$, pink dots on the red curve).
• Figure 5: Topological dilation on the $\rho_0 = 0.3$ curve from Figure \ref{['fig:traditional_vs_log']}. Other examples can be found in Figures \ref{['fig:3PB_loginterp']} and \ref{['fig:f2-r1_dataset_PINN']}.