Normalized field product approach: A parameter-free density evaluation method for close-to-binary solutions in topology optimization with embedded length scale

TL;DR

This work introduces the normalized field product (nFP) approach as a parameter-free density evaluation method for topology optimization with embedded length scale, achieving close-to-binary designs without user-defined thresholds or continuation schemes. By mapping an auxiliary scalar field through a normalized field product, the method defines element densities via $ ho_i = 1 - \prod_{j\in\mathbb{N}_i} f(\beta_j)^{A(\Omega_j)/A(\Gamma_i)}$, and selects monotone, bounded $f(\beta_j)$ to ensure stable gradients $\dfrac{\partial \rho_i}{\partial \beta_j} = - (1-\rho_i) \dfrac{A(\Omega_j)}{A(\Gamma_i)} \left( \dfrac{1}{f(\beta_j)} \dfrac{d f(\beta_j)}{d \beta_j} \right)$. The framework is integrated with SIMP interpolation for material stiffness and demonstrated on 2D and 3D stiff structures and compliant mechanisms, including a 3D extension with truncated octahedron elements, showing mesh-independence, automatic transition-free topologies, and embedded minimum length scale in the solid phase. The method’s gradient-based optimization, via the adjoint method and MMA, yields near-binary solutions across varying volume fractions and length scales, without relying on density thresholds or heuristic continuation. Overall, nFP provides a versatile, parameter-free pathway to robust, feature-size-controlled topology optimization with broad applicability to complex 3D designs.

Abstract

This paper provides a normalized field product approach for topology optimization to achieve close-to-binary optimal designs. The method employs a parameter-free density measure that implicitly enforces a minimum length scale on the solid phase, allowing for smooth and transition-free topologies. The density evaluation does not rely on weight functions; however, the related density functions must have values between 0 and 1. The method combines the SIMP scheme and the introduced density function for material stiffness interpolation. The success and efficacy of the approach are demonstrated for designing both two- and three-dimensional designs, encompassing stiff structures and compliant mechanisms. The structure's compliance is minimized for the former, while the latter involves optimizing a multi-criteria objective. Numerical examples consider different volume fractions, length scales, and density functions. A volume-preserving smoothing and resolution scheme is implemented to achieve serrated-free boundaries. The proposed method is also seamlessly extended with advanced elements for solving 3D problems. The optimized designs obtained are close to binary without any user intervention while satisfying the desired feature size on the solid phase.

Figures (18)

• Figure 1: A schematic diagram for the proposed density formulation. (\ref{['fig:density1']}) elements with $\alpha =1$ are shown in black, whereas remnant elements are with $\alpha =0$. (\ref{['fig:density2']}) Suppose a set of neighborhood elements contains only the immediate elements. For example, elements $a_1,\,a_2,\,\cdots,\,a_8$ are the neighbor elements for element $a$. As element $a_2$ has, $\alpha =1$, $\rho_i|_{i= a,\,a_1,\,a_2,\,a_3,\,a_4,\,a_5,\,a_7,\, a_8} =1$, whereas $\rho_i|_{i= a_6} =0$ as per Eq. \ref{['Eq:BPP_mod']}. Likewise, one determines the density variable of other elements and plots the final density as shown in the right-side figure.
• Figure 2: $f(\beta_j)$ plots. $n=12$ is picked for the third function.
• Figure 3: \ref{['fig:ls=1']}, \ref{['fig:ls=2']} and \ref{['fig:ls=4']} depict discretization of the neighborhood $\Gamma(\boldsymbol{\mathrm{X}})$ for an interior point at different levels of refinement. (\ref{['fig:Boundary_neighborhood']}) indicates interaction of neighborhood with domain for points close to the domain boundary.
• Figure 4: Problem description for stiff structures
• Figure 5: Problem description for a displacement inverter.