Deep neural networks for parameterized homogenization in concurrent multiscale structural optimization

Abstract

Concurrent multiscale structural optimization is concerned with the improvement of macroscale structural performance through the design of microscale architectures. The multiscale design space must consider variables at both scales, so design restrictions are often necessary for feasible optimization. This work targets such design restrictions, aiming to increase microstructure complexity through deep learning models. The deep neural network (DNN) is implemented as a model for both microscale structural properties and material shape derivatives (shape sensitivity). The DNN’s profound advantage is its capacity to distill complex, multidimensional functions into explicit, efficient, and differentiable models. When compared to traditional methods for parameterized optimization, the DNN achieves sufficient accuracy and stability in a structural optimization framework. Through comparison with interface-aware finite element methods, it is shown that sufficiently accurate DNNs converge to produce a stable approximation of shape sensitivity through back propagation. A variety of optimization problems are considered to directly compare the DNN-based microscale design with that of the Interface-enriched Generalized Finite Element Method (IGFEM). Using these developments, DNNs are trained to learn numerical homogenization of microstructures in two and three dimensions with up to 30 geometric parameters. The accelerated performance of the DNN affords an increased design complexity that is used to design bio-inspired microarchitectures in 3D structural optimization. With numerous benchmark design examples, the presented framework is shown to be an effective surrogate for numerical homogenization in structural optimization, addressing the gap between pure material design and structural optimization.

Appendices

Appendix 1: IGFEM shape sensitivity of the homogenized elasticity tensor

This section introduces the relevant IGFEM sensitivity analysis for the homogenized elasticity tensor in relation to material shape parameters. We begin with the energy-based expression of the homogenized elasticity tensor:

$$\begin{aligned}&\varvec{C}^{\text {H}} = {C}_{ij}^{\text {H}} =\frac{1}{ \left| Y \right| } \sum _{{e_{\mu} }=1}^{N_{\mu} } \left( \varvec{u}_{{e_{\mu} }}^{0(i)} - \varvec{u}_{{e_{\mu} }}^{(i)} \right) ^{\text {T}} \varvec{k}_{e_{\mu} } \left( \varvec{u}_{{e_{\mu} }}^{0(j)} - \varvec{u}_{{e_{\mu} }}^{(j)} \right) \end{aligned}$$

(35)

$$\begin{aligned}&\quad =\frac{1}{ \left| Y \right| } \sum _{{e_{\mu} }=1}^{N_{\mu} } \left( \varvec{\varepsilon }_{{e_{\mu} }}^{0(i)} - \varvec{\varepsilon }_{{e_{\mu} }}^{(i)} \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }_{{e_{\mu} }}^{0(j)} - \varvec{\varepsilon }_{{e_{\mu} }}^{(j)} \right) . \end{aligned}$$

(36)

If we simplify the expression to a single component of the homogenized tensor and omit the subscripts used to indicate microscale element quantities, we can write an element’s contribution to the homogenized tensor as

$$c = \int _{\varOmega _{e}} \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) {\text {d}}\varOmega .$$

(37)

We remark that in this expression, only the strain $\varvec{\varepsilon }$ is a function of the shape parameters. The strain $\varvec{\varepsilon }^{0}$ is prescribed, the element-wise constitute relation $\varvec{C}_{e_{\mu} }$ is not a function of the design variables in the IGFEM shape optimization formulation.

The strain $\varvec{\varepsilon }$ can be represented as the function $\varvec{\varepsilon }\left( \varvec{X}(\varvec{x}), \varvec{x} \right) = \mathbb {B}\left( \varvec{X}(\varvec{x}), \varvec{x} \right) \mathbb {U}\left( \varvec{X}(\varvec{x}), \varvec{x} \right)$ for the shape parameters $\varvec{x}$. Hereafter, we consider a single shape parameter $x_{i}$. We introduce the simple notation $\frac{\partial \mathbb {B}}{\partial x_{i}}$ as the expression for the shape derivative of $\mathbb {B}$. The defining feature of IGFEM is $\mathbb {B}\left( \varvec{X}(\varvec{x}), \varvec{x} \right)$, where the strain–displacement is a function of the shape parameters; for more information on the IGFEM implementation of $\frac{\partial \mathbb {B}}{\partial x_{i}}$, see Najafi et al. (2015, 2017, 2021) and Brandyberry et al. (2020). The shape material derivative of $\mathbb {U}$ is introduced as $\mathbb {U}_{i}^{*}$. Following these definitions, the shape derivative of $\varvec{\varepsilon }$ is expressed element-wise as

$$\frac{{\text {d}} \varvec{\varepsilon }}{{\text {d}} x_{i}} = \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*},$$

(38)

while the shape derivative $\mathbb {U}_{i}^{*}$ is evaluated through the following pseudo-problem:

$$\mathbb {K}\mathbb {U}_{i}^{*} = \mathbb {P}^{i}_{ps} = -\frac{\partial \mathbb {K}}{\partial x_{i}}\mathbb {U } + \frac{\partial \mathbb {F}}{\partial x_{i}}.$$

(39)

The pseudo-problem is assembled from the element quantities $\frac{\partial \varvec{K}_{e}}{\partial x_{i}}$ and $\frac{\partial \varvec{F}_{e}}{\partial x_{i}}$. We assert that the material derivative $\frac{\partial \varvec{C}_{e_{\mu} }}{\partial x_{i}}$ is zero, so the element stiffness derivative follows:

$$\begin{aligned}&\frac{\partial \varvec{K}_{e}}{\partial x_{i}} =\int _{\varOmega _{e}} \left( \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) {\text {d}}\varOmega , \end{aligned}$$

(40)

where we note that $\frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B}$ is symmetric and ${\text {div}}(\mathbb {V}_{i})$ follows from the shape velocity term (Najafi et al. 2015). For the homogenization case where $\frac{{\text {d}} \varvec{\varepsilon }_0}{{\text {d}} x_{i}} = 0$, the element force derivative is

$$\frac{\partial \varvec{F}_{e}}{\partial x_{i}} = \int _{\varOmega _{e}} \left( \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \varvec{\varepsilon }^{0} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \varvec{\varepsilon }^{0} {\text {div}}(\mathbb {V}_{i}) \right) {\text {d}}\varOmega ,$$

(41)

where

$$\varvec{\varepsilon }^{0} = \mathbb {B}\mathbb {U}_{e}^{0}.$$

(42)

Recalling that only $\varvec{\varepsilon }$ is a function of the design parameter with its shape sensitivity in (38), then the sensitivity expression of the homogenized elasticity tensor can be defined similar to (40):

$$\begin{aligned} \frac{{\text {d}} c}{{\text {d}} x_{i}}&= - \int _{\varOmega _{e}} \left[ \left( \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*} \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) \right. \\&\quad +\left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*} \right) \\&\quad \left. + \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) {\text {div}}(\mathbb {V}_{i})\right] {\text {d}}\varOmega . \end{aligned}$$

(43)

Next we target the term $\frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*}$. If we combine the expression for the pseudo-element with the relation $\mathbb {K}_{e} = \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} }\mathbb {B}$, the pseudo-element can be used to eliminate $\mathbb {B}\mathbb {U}_{ei}^{*}$ in (43) using the element-wise pseudo-force of (39):

$$\begin{aligned}&\mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \left( \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*} \right) \\&\quad = \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {P}^{i}_{pse} \\&\quad = \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} - \frac{\partial \varvec{K}_{e}}{\partial x_{i}}\mathbb {U } + \frac{\partial \varvec{F}_{e}}{\partial x_{i}} \\&\quad =\mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} \\&\qquad -\left( \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \mathbb {U}_{e} \\&\qquad + \left( \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \varvec{\varepsilon }^{0} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \varvec{\varepsilon }^{0} {\text {div}}(\mathbb {V}_{i}) \right) \\&\quad = \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} }\mathbb {B}\mathbb {U}^{0}_{e} - \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B}\mathbb {U}_{e} \\&\qquad + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i})\mathbb {U}^{0}_{e}-\mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i})\mathbb {U}_{e} \\&\quad = \left( \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} }\mathbb {B} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) . \end{aligned}$$

(44)

Applying the symmetry of $\frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}} \varvec{C}_{e_{\mu} }\mathbb {B}$ in (44), we conclude

$$\begin{aligned}&\mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \left( \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*} \right) \\&\quad =\left( \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} } \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) \\&\quad =\mathbb {B}^{\text {T}} \varvec{C}_{e_{\mu} }\left( \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) \\&\quad \Rightarrow \frac{\partial \mathbb {B}}{\partial x_{i}}\mathbb {U}_{e} + \mathbb {B}\mathbb {U}_{ei}^{*} = \left( \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) . \end{aligned}$$

(45)

Using (45) in the expression for constitutive sensitivity (43), we produce

$$\begin{aligned}&\frac{{\text {d}} c}{{\text {d}} x_{i}}=\int _{\varOmega _{e}} \left[ -\left( \left( \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) \right. \\&\qquad - \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \left( \frac{\partial \mathbb {B}}{\partial x_{i}} + \mathbb {B} {\text {div}}(\mathbb {V}_{i}) \right) \left( \mathbb {U}^{0}_{e}-\mathbb {U}_{e} \right) \right) \\&\qquad \left. + \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) {\text {div}}(\mathbb {V}_{i})\right] d\varOmega \\&\quad = -\int _{\varOmega _{e}} \left[ -\left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}}\varvec{C}_{e_{\mu} } \mathbb {B} \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \right. \\&\qquad - \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} } \mathbb {B}{\text {div}}(\mathbb {V}_{i}) \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \\&\qquad - \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}} \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \\&\qquad - \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} } \mathbb {B}{\text {div}}(\mathbb {V}_{i}) \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \\&\qquad \left. + \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) ^{\text {T}} \varvec{C}_{e_{\mu} } \left( \varvec{\varepsilon }^{0} - \varvec{\varepsilon } \right) {\text {div}}(\mathbb {V}_{i})\right] {\text {d}}\varOmega \\&\quad = \int _{\varOmega _{e}} \left[ \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \frac{\partial \mathbb {B}}{\partial x_{i}}^{\text {T}}\varvec{C}_{e_{\mu} } \mathbb {B} \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \right. \\&\qquad + \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} } \mathbb {B}{\text {div}}(\mathbb {V}_{i}) \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \\&\qquad \left. + \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}} \mathbb {B}^{\text {T}}\varvec{C}_{e_{\mu} }\frac{\partial \mathbb {B}}{\partial x_{i}} \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) \right] {\text {d}}\varOmega \\&\quad = \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) ^{\text {T}}\frac{\partial \varvec{K}_{e}}{\partial x_{i}} \left( \mathbb {U}^{0}_{e} - \mathbb {U}_{e} \right) . \end{aligned}$$

(46)

Using this element contribution to the shape sensitivity of the constitutive parameters, we recover the form in (29).

Appendix 2: Practical considerations

The appropriate DNN architecture and training procedure heavily depend on the application (that is, it depends on the function space to be emulated). For multiscale optimization problems employing homogenization, including applications in structural, thermal, and acoustic simulations that employ parameterized microstructures, this section may be used to generally guide the DNN training process. This section reviews some of the key issues associated with DNN training including vanishing/exploding gradients, batch size, and training dataset generation.

1.1 Gradient propagation

As the DNN trains, its weights are iteratively updated to improve some objective function. The back propagation procedure [cf. (9)] is used to update the weights and biases of the DNN. The convergence of these model parameters is not guaranteed; some combinations of model initialization and training procedures will produce unstable gradients, often referred to as vanishing or exploding gradients (Glorot and Bengio 2010; Goodfellow et al. 2016).

In this work, vanishing gradients were observed and are reported in Table 1 as the number of DNN hidden layers was increased past $L=3$. Figure 20 illustrates the propagation of the DNN’s Jacobian for a collection of architectures all trained with 667 IGFEM elliptical training examples with an initial learning rate of $10^{-4}$ over $10^{5}$ iterations of full-batch gradient descent. As the number of hidden layers increases, as in Fig. 20e, f, the Jacobian tends toward zero and caused the training failures reported in Table 1. In all examples, the vanishing gradient phenomena manifested during training and resulted in blatantly poor models. For the relatively small models in this work, if the training process was stable, then the DNN’s Jacobian was adequate for applications in multiscale optimization.

Fig. 20

The gradient produced through back propagation is represented as a histogram for a DNN of size $n=32$ and $L = 1, 2, 3, 4, 5$ or 6 for (a–f), respectively. Each DNN was trained to homogenize the IGFEM ellipse microstructure using full-batch training

1.2 Batch size

The batch size in a DNN training procedure refers to the number of training examples used to calculate the model’s sensitivity for a given training iteration (Goodfellow et al. 2016). Full-batch training was implemented in this work because the training datasets are relatively small (100s to 1000s of examples) and an efficient training procedure was desirable. Training in mini batches, generally samples of 4–32 training examples may improve generalization and robustness (Nikolakakis et al. 2022; Novak et al. 2018). Table 2 compares the objective values for two identical DNN architectures trained via full-batch gradient descent and small-batch gradient descent (batch size $=32$). Small-batch training did improve the DNN’s performance as parameterization increased. The sensitivity, shown in Fig. 21, was inconsistently improved. Based on this evidence, the gains achieved through small-batch training do not significantly outweigh the additional training cost. For more complicated systems that require highly parameterized models, however, small-batch training may be necessary to build accurate surrogate models Fig. 22.

Table 2 The Loss values of (8) are reported as $\mathcal {L}\times 10^{3}$ for different DNN frameworks to compare a full-batch and small-batch training strategies

Fig. 21

The shape sensitivities are compared for full-batch and batch size 32 DNNs trained to homogenized the IGFEM ellipse microstructure. RMSE error is reported using Eq. (30)

Fig. 22

The gradient produced through back propagation is represented as a histogram for a DNN of size $n=32$ and $L = 1, 2, 3,$ or 4 for (a–d), respectively. Each DNN was trained to homogenize the IGFEM ellipse microstructure using a batch size of 32

1.3 Training dataset size

A training dataset is necessary to construct a viable DNN surrogate model for engineering applications. The ideal training dataset captures the depth and complexity of the target function so that the DNN may learn a general and robust map within the function space. Whether due to excessively costly data generation or incalculable complexity, the ideal training dataset is not always feasible.

Parameterized homogenization is apt for building effective training datasets. Input parameters are bounded by geometric limits, and output parameters are bounded by the constitutive limits of the material. Given these conditions, it is possible to create a representative dataset with 100s to 1000s of examples that may be used to create a relatively small yet general surrogate model for homogenization. Figure 23 illustrates correlation between accurate execution and training dataset size for a DNN of $L=3$ and $n=32$. For more complicated geometric parameterizations and/or nonlinear physics, it is likely that more data are needed to capture the complexity of the feature space.

Fig. 23

The Ellipse microstructure (3 parameters per cell), the 2D BioTruss (8 parameters per cell), and the 3D BioTruss (30 parameters per cell) illustrate the correlation between number of training data and the accuracy of a trained DNN ($n=32$, $L=3$)

1.4 Homogenization in multiscale optimization

Homogenization assumes a significant separation of scales, approximating the local effects of a periodically varying microstructure (cf. Sect. 2). The examples presented in this work have largely focused on the numerical behavior of DNN surrogate models for homogenization in a selection of optimization exercises. Continued work through full-scale simulation and physical experimentation is necessary to judge the effects of homogenization on multiscale structures. This “Appendix” is presented as a short illustrative study to show the limits of homogenization-based multiscale design.

The test case for experimental validation is derived from a prescribed deformation problem characterized by

$$\begin{aligned} \mathbb {U}_{{\text {T}}1}&= \mathbb {U}_{{\text {T}}2} = \mathbb {U}_{{\text {T}}3} = \varvec{0}\end{aligned}$$

(47)

which targets the displacement of a zero Poisson’s ratio structure given the boundary conditions shown in Fig. 24a. Design optimization was performed using the FEM-informed DNN model for the 3D BioTruss, producing the $10\times 10\times 1$ structure shown in Fig. 24a after 100 iterations. Designs are compared using measured Poisson’s ratio of the macroscale structure

$$\nu = \frac{-\varepsilon _{\text {lat}}}{\varepsilon _{\text {long}}},$$

(48)

where the strains $\varepsilon _{\text {lat}}$ and $\varepsilon _{\text {long}}$ are the lateral and longitudinal strains measured along the specimen’s centroidal axes. The initial uniform specimen [$\varvec{\beta }^{(i)} =\{0.5, 0.5\}_{i = 1:12}$; ${\zeta }^{(i)} =0.5_{i = 1:6}$] has a Poisson’s ratio of 0.33 as evaluated by FEM-based homogenization. After 200 iterations of design optimization ($V_{x}=0.2$), the BioTruss design converged to a Poisson’s ratio of 0.00 (as evaluated by FEM-based homogenization). Because the design space reached the parameter limits imposed by the BioTruss geometry (Fig. 25), this specific microarchitecture formulation is likely unable to produce a negative Poisson’s ratio.

The design produced via DNN-driven multiscale optimization was manufactured using 3D printing of TPU 95a filament ($E_{1}=49$ MPa, $\nu = 0.32$ Lee et al. 2022) via fused deposition modeling (Fig. 24b, c). The properties of TPU 95a differ from the simulated fictitious material ($E_{1}=1$ Pa, $\nu = 0.30$), but because the structural deformation is displacement controlled, the deformation of both materials are sufficiently similar for comparison. Indeed both the fictitious material and TPU 95a produce an initial Poisson’s ratio of 0.33 for the uniform specimen and 0.00 for the optimized structure, as evaluated by FEM-based homogenization.

Fig. 24

a The boundary conditions and optimization result are shown for the test specimen with spatially varying microstructures (b). c The manufactured specimen is shown in its undeformed condition

Fig. 25

A selection of the BioTruss features is illustrated to explore the optimization result of Fig. 24a

The optimized design of TPU 95a microarchitectures was analyzed in the displacement controlled compression fixture shown in in Fig. 26. The Poisson’s ratio was measured experimentally using $\varepsilon _{\text {lat}}$ and $\varepsilon _{\text {long}}$ measured along the specimen’s respective centroidal axes. At $\varepsilon _{\text {long}} = -0.10$, the calculated Poisson’s ratio was $-0.06$, and at $\varepsilon _{\text {long}} = -0.20$, the measured Poisson’s ratio was $-0.02$. The variation between modeled ($\nu = 0.00$) and experimental Poisson’s ratios is attributed to localized buckling near the compression plates. A full exploration of the observed nonlinear behavior is well outside the scope of this work; we simply conclude that the optimized design did indeed approach the targeted displacement within the limits of its parameterized geometry provided the DNN’s evaluations and shape sensitivities. Beyond navigating the design space, a thorough validation of the final analysis would require full-scale simulation and experimentation as in Cheng et al. (2019).

Fig. 26

The compression fixture (buckling guides and compression plates) used to reproduce the boundary conditions for the auxetic design is shown at 20% longitudinal compression