LatticeGraphNet: A two-scale graph neural operator for simulating lattice structures

TL;DR

This study introduces a two-scale graph neural operator (GNO), namely, LatticeGraphNet, designed as a surrogate model for costly nonlinear finite-element simulations of three-dimensional latticed parts and structures, establishing the use of GNOs as efficient surrogate models for evaluating mechanical responses of lattices and structures.

Abstract

This study introduces a two-scale Graph Neural Operator (GNO), namely, LatticeGraphNet (LGN), designed as a surrogate model for costly nonlinear finite-element simulations of three-dimensional latticed parts and structures. LGN has two networks: LGN-i, learning the reduced dynamics of lattices, and LGN-ii, learning the mapping from the reduced representation onto the tetrahedral mesh. LGN can predict deformation for arbitrary lattices, therefore the name operator. Our approach significantly reduces inference time while maintaining high accuracy for unseen simulations, establishing the use of GNOs as efficient surrogate models for evaluating mechanical responses of lattices and structures.

Figures (7)

• Figure 1: A sample 3D printed elastomeric part with a combination of five different unit cells, demonstrating the variety of configurations achieved by a small number of initial repeat blocks.Carbon_lattice
• Figure 2: Overview of the LatticeGrapNet (LGN) Pipeline. The pipeline starts with A) an initial three-dimensional lattice represented by a tetrahedral mesh and is transformed to B) a reduced (skeletal) representation of the initial mesh. C) LGN-i (see \ref{['sec:lgn-i']}) runs inference on the reduced mesh to get the coarse displacement. D) LGN-ii (see \ref{['sec:lgn-ii']}) maps the reduced displacements of LGN-i to predict three-dimensional displacements on the tetrahedral mesh.
• Figure 3: Architecture of LGN-i. A) The input to the network is a reduced (graph) representation of the tetrahedral mesh. B) The node encoder ($\epsilon_N$) and edge encoder ($\epsilon_E$) encode the node and edge features to high-dimensional vectors, $n_i$ and $e_{ij}$, respectively. C) The message-passing block uses two processor MLPs: $P_e$ which updates $e_{ij}$ to $e'_{ij}$, and $P_n$ which updates $n_i$ to $n'_i$. LGN-i uses 15 message-passing steps with identical blocks. D) The two-part decoder contains MLPs $D_{\boldsymbol{u}}$ and $D_{\boldsymbol{\sigma}}$. $D_{\boldsymbol{u}}$ uses the updated node features to compute the displacement increment for node $i$, i.e., $\delta\tilde{\boldsymbol{u}}_i$. $D_{\sigma}$ uses the final $n'_i$ concatenated with $\boldsymbol{u}_i$ to calculate the change in the $I_1$ and $J_2$ stress invariants at node $i$, $\delta \tilde{\boldsymbol{\sigma}}_i$. E) Upon rollout, denoted by the orange lines, the calculated $\delta \tilde{\boldsymbol{u}}_i$ and $\delta \tilde{\boldsymbol{\sigma}}_i$ are integrated to obtain the node and edge features for the next timestep.
• Figure 4: Volumetric deformation prediction set up by the LGN-ii. At time $t$, an initial configuration of the $i^{th}$ strut node ($S_i$), its nearby tetrahedral nodes ($T_{ij}$), and its connected strut nodes ($S_1$, $S_2$) are known. LGN-i predicts the displacement of every strut node ($\delta \tilde{\boldsymbol{u}}_{S_i}$, $\delta \tilde{\boldsymbol{u}}_{S_1}$, $\delta \tilde{\boldsymbol{u}}_{S_2}$). Based on this information, LGN-ii predicts the fine local deformation of the $j^{th}$ tetrahedral node ($\delta \boldsymbol{U}_{ij}$). The final tetrahedral node displacement is $\delta \boldsymbol{u}_{T_{ij}} = \delta \boldsymbol{U}_{ij} + \delta \tilde{\boldsymbol{u}}_{S_i}$.
• Figure 5: Visualization of the (Left) original latticed puck, (Middle) predicted deformation at 25% compression from the LatticeGraphNet, and (Right) the distribution of displacement errors (mm) of volumetric nodes. Results are given for a) T1, b) T3, c) T5, and d) T7.