# Optimization of Generatively Encoded Multi-Material Lattice Structures for Desired Deformation Behavior

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Evolutionary Algorithms

#### 2.2. Generative Encoding

_{i}denote the midpoint of the ith segment and let f

_{i}denote f (x

_{i}). To avoid aliasing, the values of k are limited to be k = 0, 1, 2, …, N—1. The integral (4) is approximated by the midpoint quadrature rule, yielding:

_{1}of the structure, should be evaluated. This also holds for the case of symmetrical loads and restraints, but without promoting the symmetry through appropriate encoding. In the approach presented in this paper, all the deflection error calculations were performed only for one edge, which proved enough, as will be explained in the following sections.

_{m}) of each strut. The position of c

_{m}for the strut i in the 3D density matrix defines the property of the material of this strut, proportional to the material density field. This procedure must be repeated for each strut in the lattice structure. Despite that, the computational overhead in doing this is not excessively intensive, because the topology of the lattice stays fixed over time, since the nodes remain fixed and there is no variation of the length of the struts. For that reason, no iteration over the structure is necessary, a lookup table is designed in advance holding the information of c

_{m}and compared to mass distribution matrix.

#### 2.3. Fitness Evaluation

## 3. Algorithm Details

## 4. Results

^{−3}within 2000 iterations. Case (b) showed that the deflection line was constructed to have a zero slope both on the clamped and the free edge.

^{−3}within 5000 iterations. Case (c) was for the step function, which proved to be hard to satisfy, especially in the region where discontinuities were located.

^{−1}at the end, compared to the previous two cases. The last case, (d), presents the slope discontinuity, which is composed of two piecewise linear functions. The algorithm yields quality results, with the RMSE values at 10

^{−2}found around the discontinuity of the profile.

## 5. Discussion

## 6. Conclusions and Future Work

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Response of the homogenous structure, and (

**b**) evolved heterogenous structure for given equal loads and restraints. Red struts—stiff material, blue struts—soft material. The structure (

**b**) is optimized for linear deflection.

**Figure 2.**Generative representation of material distribution for truss structure using three weighted sinusoids.

**Figure 4.**Difference between the optimized multi-material and unoptimized single material structures for RMSE minimization.

**Figure 5.**Evolved structures for different deflection profiles, with goal, actual, and normal profiles and deviation between target and actual profiles.

**Figure 6.**Evolution of the structure optimized for negative quadratic curvature: (

**a**) initial stage, (

**b**) 2500 generations, and (

**c**) at 5000 generations. Top right is the goal profile along with actual, nominal, and error plots for the final structure.

**Figure 7.**RMSE relation to different elastic moduli ratio $\xi =\raisebox{1ex}{${E}_{stiff}$}\!\left/ \!\raisebox{-1ex}{${E}_{soft}$}\right.$, for negative quadratic (S1) and step discontinuity specimens (S2).

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Ćurković, P.
Optimization of Generatively Encoded Multi-Material Lattice Structures for Desired Deformation Behavior. *Symmetry* **2021**, *13*, 293.
https://doi.org/10.3390/sym13020293

**AMA Style**

Ćurković P.
Optimization of Generatively Encoded Multi-Material Lattice Structures for Desired Deformation Behavior. *Symmetry*. 2021; 13(2):293.
https://doi.org/10.3390/sym13020293

**Chicago/Turabian Style**

Ćurković, Petar.
2021. "Optimization of Generatively Encoded Multi-Material Lattice Structures for Desired Deformation Behavior" *Symmetry* 13, no. 2: 293.
https://doi.org/10.3390/sym13020293