Multiscale topology optimization of functionally graded lattice structures based on physics-augmented neural network material models

TL;DR

This work develops a framework for concurrent optimization of topology and relative density grading in functionally graded lattices by decoupling density into a topological field $\gamma$ and a microstructural field $\kappa$, and by employing a physics-augmented neural network to predict the isotropic lattice stiffness from microstructure inputs. The approach relaxes the NP-hard MINLP into a gradient-based problem using a SIMP-like penalty and adjoint sensitivities, while enforcing physical consistency through a Cholesky-based isotropic parameterization and a RVE-based data-driven material model trained on computational homogenization data. The method is demonstrated on 2D and 3D benchmarks (MBB, cantilever) and a jet-engine bracket, achieving substantial stiffness gains under manufacturability constraints and illustrating the practical potential for aerospace and lightweight design. The key contributions include: (i) a multiplicative density split for efficient optimization, (ii) a PANN-based isotropic material model grounded in homogenization, and (iii) a robust, gradient-based framework capable of handling functionally graded lattices within additive manufacturing constraints.

Abstract

We present a new framework for the simultaneous optimiziation of both the topology as well as the relative density grading of cellular structures and materials, also known as lattices. Due to manufacturing constraints, the optimization problem falls into the class of NP-complete mixed-integer nonlinear programming problems. To tackle this difficulty, we obtain a relaxed problem from a multiplicative split of the relative density and a penalization approach. The sensitivities of the objective function are derived such that any gradient-based solver might be applied for the iterative update of the design variables. In a next step, we introduce a material model that is parametric in the design variables of interest and suitable to describe the isotropic deformation behavior of quasi-stochastic lattices. For that, we derive and implement further physical constraints and enhance a physics-augmented neural network from the literature that was formulated initially for rhombic materials. Finally, to illustrate the applicability of the method, we incorporate the material model into our computational framework and exemplary optimize two-and three-dimensional benchmark structures as well as a complex aircraft component.

Figures (15)

• Figure 1: Strut-based body-centered cubic (BCC) unit cells for two different aspect ratios $a$. The volume filled with base material in the unit cell increases with the aspect ratio, implying the existence of a mapping function between the relative density and the aspect ratio, and vice versa.
• Figure 2: Admissible sets of relative density values and their effect on the optimized design. a) Simple 0-1 design. b) Design with graded density. c) Design with optimized topology and density grading.
• Figure 3: Multiplicative split of the relative density for the optimization of the topology and material grading of a lattice structure.
• Figure 4: Tetrahedral unit cell and cubic RVE composed of 2,845 unit cells. The aspect ratio is $a = 0.16$.
• Figure 5: Functional relation between the microstructural relative density $\kappa$ and the aspect ratio $a$ computed through least-squares fitting. The outcome of the naive approach deviates significantly from the data obtained through Monte Carlo integration.