Quantum Computing and Tensor Networks for Laminate Design: A Novel Approach to Stacking Sequence Retrieval

TL;DR

This work derives a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem and demonstrates the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian.

Abstract

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters. To adapt this problem for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

Figures (5)

• Figure 1: a) Diagram of a laminated composite material, which consists of multiple layers that are oriented in different directions. b) A representation of the stress resultants $\vec{N}$, which for isotropic materials result in in-plane deformations $\vec{\varepsilon}^{\: 0}$. c) A pictographical representation of the moment resultants $\vec{M}$, which for isotropic materials result in out-of-plane bending $\vec{\kappa}$. Non-diagonal elements in the $\mathbf{ABD}$-matrix allow for coupling between all possible stress and moment resultants on the one hand, and all in- and out-of-plane deformations on the other hand.
• Figure 2: Parameterized quantum circuit for hardware-efficient approach
• Figure 3: Results from the simulations of QAOA. The top row shows the average Hamiltonian expectation value while the bottom row shows the average probability of obtaining the exact solution when measuring the final state. Shown are the average of the 6 trails for each individual set of target parameters (blue and green solid lines) and the average over all 15 sets of target parameters (dashed red lines). On the right, the stacking sequences corresponding to the exact solutions are shown.
• Figure 4: Results for the hardware-efficient approach (a) and DMRG (b). In both cases, the top row shows the average RMSE for the final results, and the bottom row shows the ratio of resulting stacks that coincided with the exact solution. The averages are taking over the results from all 15 sets of target lamination parameters with 5 trials performed for each.
• Figure 5: Results from DMRG with $N=200$ plies. a) Traces of expectation values of the loss function Hamiltonian, averaged over all samples and measurements after each sweep. Each line corresponds to a different bond dimension, denoted at the end of each trace. Red lines represent trials excluding the disorientation constraint and blue lines correspond to trials including the constraint. Solid lines represent the optimization phase with a constant maximum bond dimension. The successive reduction of the bond dimension at the final phase of the optimization is indicated by dotted lines that can be found at the right-hand end of each trace. b) Histogram of the RMSE of the solutions for maximum bond-dimension 32 and inward direction. Counted were the points from all target lamination parameters and initial MPS. The bin-size is $0.002$. The ranges of RMSE for the bottom half of the points, and up to the average RMSE are shown in both plots. Left: Without the disorientation constraint. Right: With the disorientation constraint. c) Average sweep duration as a function of bond dimension. Left and right plots respectively correspond to scenarios without and with the disorientation constraint. Each sweep direction is represented by distinct points in different colors. A third-order polynomial fit is included as dashed lines, to easier distinguish trends for different configurations.