[en] We propose a framework to solve non-linear and history-dependent mechanical problems based on a hybrid classical computer – quantum annealer approach. Quantum Computers are anticipated to solve particular operations exponentially faster. The available possible operations are however not as versatile as with a classical computer. However, quantum annealers (QAs) are well suited to evaluate the minimum state of a Hamiltonian quadratic potential. Therefore, we reformulate the elasto-plastic finite element problem as a double-minimisation process framed at the structural scale using the variational updates formulation. In order to comply with the expected quadratic nature of the Hamiltonian, the resulting non-linear minimisation problems are iteratively solved with the suggested Quantum Annealing-assisted Sequential Quadratic Programming (QA-SQP): a sequence of minimising quadratic problems is performed by approximating the objective function by a quadratic Taylor’s series. Each quadratic minimisation problem of continuous variables is then transformed into a binary quadratic problem. This binary quadratic minimisation problem can be solved on quantum annealing hardware such as the D-Wave system. The applicability of the proposed framework is demonstrated with one- and two-dimensional elasto-plastic numerical benchmarks. The current work provides a pathway of performing general non-linear finite element simulations assisted by quantum computing.
Research Center/Unit :
A&M - Aérospatiale et Mécanique - ULiège MolSys - Molecular Systems - ULiège
Disciplines :
Mechanical engineering Engineering, computing & technology: Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences: Multidisciplinary, general & others
Author, co-author :
Nguyen, Van Dung ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Wu, Ling ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Remacle, Françoise ; Université de Liège - ULiège > Département de chimie (sciences) > Laboratoire de chimie physique théorique
Noels, Ludovic ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Language :
English
Title :
A quantum annealing-sequential quadratic programming assisted finite element simulation for non-linear and history-dependent mechanical problems
F.R.S.-FNRS - Fonds de la Recherche Scientifique ULiège - University of Liège
Funding text :
V.-D.N acknowledges the support of the Fonds National de la Recherche (F.R.S.-FNRS,
Belgium).
F.R. acknowledges the support of the Fonds National de la Recherche (F.R.S.-FNRS,
Belgium), #T0205.20.
This work is partially supported by a “Strategic Opportunity” grant from the University
of Liege.
NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in European Journal of Mechanics - A/Solids 105 (2024) , DOI: 10.1016/j.euromechsol.2024.105254
Anon, T., D-wave system documentation. 2022 https://docs.dwavesys.com/docs/latest/index.html. (Accessed: 5 January 2024).
Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., Biswas, R., Boixo, S., Brandao, F.G., Buell, D.A., et al. Quantum supremacy using a programmable superconducting processor. Nature 574:7779 (2019), 505–510.
Bleyer, J., Automating the formulation and resolution of convex variational problems: Applications from image processing to computational mechanics. ACM Trans. Math. Software, 46(3), 2020, 10.1145/3393881.
Bleyer, J., Applications of conic programming in non-smooth mechanics. J. Optim. Theory Appl., 2022, 10.1007/s10957-022-02105-z.
Bonnans, J.-F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A., Numerical Optimization: Theoretical and Practical Aspects. 2006, Springer Science & Business Media.
Borle, A., Lomonaco, S.J., Analyzing the quantum annealing approach for solving linear least squares problems. WALCOM: Algorithms and Computation: 13th International Conference, WALCOM 2019, Guwahati, India, February 27–March 2, 2019, Proceedings 13, 2019, Springer, 289–301.
Born, M., Fock, V., Beweis des adiabatensatzes. Z. Phys. 51:3–4 (1928), 165–180.
Brassart, L., Stainier, L., Doghri, I., Delannay, L., A variational formulation for the incremental homogenization of elasto-plastic composites. J. Mech. Phys. Solids 59:12 (2011), 2455–2475.
Brooke, J., Bitko, D., Rosenbaum, I., Aeppli, G., Quantum annealing of a disordered magnet. Science 284:5415 (1999), 779–781.
Castelvecchi, D., IBM quantum computer passes calculation milestone. rvtNat 618 (2023), 656–657, 10.1038/d41586-023-01965-3.
Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S.C., Endo, S., Fujii, K., McClean, J.R., Mitarai, K., Yuan, X., Cincio, L., et al. Variational quantum algorithms. Nat. Rev. Phys. 3:9 (2021), 625–644.
Childs, A.M., Kothari, R., Somma, R.D., Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM J. Comput. 46:6 (2017), 1920–1950.
Conley, R., Choi, D., Medwig, G., Mroczko, E., Wan, D., Castillo, P., Yu, K., Quantum optimization algorithm for solving elliptic boundary value problems on D-wave quantum annealing device. Quantum Computing, Communication, and Simulation III, vol. 12446, 2023, SPIE, 53–63.
Criado, J.C., Spannowsky, M., Qade: solving differential equations on quantum annealers. Quantum Sci. Technol., 8(1), 2022, 015021, 10.1088/2058-9565/acaa51.
Date, P., Arthur, D., Pusey-Nazzaro, L., QUBO formulations for training machine learning models. Sci. Rep., 11(1), 2021, 10029.
de Souza Neto, E.A., Peric, D., Owen, D.R., Computational Methods for Plasticity: Theory and Applications. 2011, John Wiley & Sons.
Endo, K., Matsuda, Y., Tanaka, S., Muramatsu, M., A phase-field model by an ising machine and its application to the phase-separation structure of a diblock polymer. Sci. Rep., 12(1), 2022, 10794, 10.1038/s41598-022-14735-4.
Fancello, E., Ponthot, J.-P., Stainier, L., A variational framework for nonlinear viscoelastic models in finite deformation regime. J. Comput. Appl. Math. 215:2 (2008), 400–408.
Hager, W.W., Zhang, H., A survey of nonlinear conjugate gradient methods. Pacific J. Optim. 2:1 (2006), 35–58.
Halphen, B., Nguyen, Q.S., Sur les matériaux standard généralisés. J. Méc. 14:1 (1975), 39–63.
Harrow, A.W., Hassidim, A., Lloyd, S., Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103(15), 2009, 150502.
Jiang, S., Britt, K.A., McCaskey, A.J., Humble, T.S., Kais, S., Quantum annealing for prime factorization. Sci. Rep., 8(1), 2018, 17667.
Kadowaki, T., Nishimori, H., Quantum annealing in the transverse ising model. Phys. Rev. E, 58(5), 1998, 5355.
Kintzel, O., Mosler, J., A coupled isotropic elasto-plastic damage model based on incremental minimization principles. Tech. Mech.-Eur. J. Eng. Mech. 30:1–3 (2010), 177–184.
Kintzel, O., Mosler, J., An incremental minimization principle suitable for the analysis of low cycle fatigue in metals: A coupled ductile–brittle damage model. Comput. Methods Appl. Mech. Engrg. 200:45–46 (2011), 3127–3138.
Mandal, A., Roy, A., Upadhyay, S., Ushijima-Mwesigwa, H., 2020. Compressed quadratization of higher order binary optimization problems. In: Proceedings of the 17th ACM International Conference on Computing Frontiers. pp. 126–131.
McGeoch, C.C., Theory versus practice in annealing-based quantum computing. Theoret. Comput. Sci. 816 (2020), 169–183, 10.1016/j.tcs.2020.01.024 URL https://www.sciencedirect.com/science/article/pii/S0304397520300529.
Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numerical Methods Eng. 55:11 (2002), 1285–1322.
Montanaro, A., Pallister, S., Quantum algorithms and the finite element method. Phys. Rev. A, 93(3), 2016, 032324.
Mosler, J., Bruhns, O., On the implementation of rate-independent standard dissipative solids at finite strain–Variational constitutive updates. Comput. Methods Appl. Mech. Engrg. 199:9–12 (2010), 417–429.
National Academies of Sciences, Engineering, and Medicine, J., Grumbling, E., Horowitz, M., (eds.) Quantum Computing: Progress and Prospects, 2019, The National Academies Press, Washington, DC, 10.17226/25196.
Nguyen, V.D., Wu, L., Remacle, F., Noels, L., Data of “a quantum annealing-sequential quadratic programming assisted finite element simulation for non-linear and history-dependent mechanical problems”. 2024, 10.5281/zenodo.10451584.
Nielsen, M.A., Chuang, I.L., Quantum Computation and Quantum Information: 10th Anniversary Edition. 2010, Cambridge University Press, 10.1017/CBO9780511976667.
O'Malley, D., Vesselinov, V.V., Toq. jl: A high-level programming language for d-wave machines based on julia. 2016 IEEE High Performance Extreme Computing Conference, HPEC, 2016, IEEE, 1–7.
Ortiz, M., Stainier, L., The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Engrg. 171:3–4 (1999), 419–444.
Pednault, E., Gunnels, J., Maslov, D., Gambetta, J., On “quantum supremacy”. IBM Res. Blog, 2019 URL https://go.nature.com/34Qh4OP.
Radovitzky, R., Ortiz, M., Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comput. Methods Appl. Mech. Engrg. 172:1–4 (1999), 203–240.
Raisuddin, O.M., De, S., FEqa: Finite element computations on quantum annealers. Comput. Methods Appl. Mech. Engrg., 395, 2022, 115014, 10.1016/j.cma.2022.115014 URL https://www.sciencedirect.com/science/article/pii/S0045782522002444.
Rajak, A., Suzuki, S., Dutta, A., Chakrabarti, B.K., Quantum annealing: An overview. Phil. Trans. R. Soc. A, 381(2241), 2023, 20210417.
Rosenberg, I.G., Reduction of bivalent maximization to the quadratic case. Cahiers Centre Etudes Rech. Oper. 17 (1975), 71–74.
Ruszczynski, A., Nonlinear Optimization. 2011, Princeton University Press.
Srivastava, S., Sundararaghavan, V., Box algorithm for the solution of differential equations on a quantum annealer. Phys. Rev. A, 99, 2019, 052355, 10.1103/PhysRevA.99.052355 URL https://link.aps.org/doi/10.1103/PhysRevA.99.052355.
Stainier, L., Ortiz, M., Study and validation of a variational theory of thermo-mechanical coupling in finite visco-plasticity. Int. J. Solids Struct. 47:5 (2010), 705–715.
Tanaka, S., Tamura, R., Chakrabarti, B.K., Quantum Spin Glasses, Annealing and Computation. 2017, Cambridge University Press.
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G.H., et al. The variational quantum eigensolver: a review of methods and best practices. Phys. Rep. 986 (2022), 1–128.
Tosti Balducci, G., Chen, B., Möller, M., Gerritsma, M., De Breuker, R., Review and perspectives in quantum computing for partial differential equations in structural mechanics. Front. Mech. Eng., 8, 2022, 10.3389/fmech.2022.914241 URL https://www.frontiersin.org/articles/10.3389/fmech.2022.914241.
Vassoler, J., Reips, L., Fancello, E., A variational framework for fiber-reinforced viscoelastic soft tissues. Internat. J. Numer. Methods Engrg. 89:13 (2012), 1691–1706.
Weinberg, K., Mota, A., Ortiz, M., A variational constitutive model for porous metal plasticity. Comput. Mech. 37 (2006), 142–152.
Willsch, D., Willsch, M., Gonzalez Calaza, C.D., Jin, F., De Raedt, H., Svensson, M., Michielsen, K., Benchmarking advantage and D-wave 2000Q quantum annealers with exact cover problems. Quantum Inf. Process., 21(4), 2022, 141.
Wriggers, P., Nonlinear Finite Element Methods. 2008, Springer Science & Business Media.
Yang, Q., Stainier, L., Ortiz, M., A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. J. Mech. Phys. Solids 54:2 (2006), 401–424.
Yarkoni, S., Raponi, E., Bäck, T., Schmitt, S., Quantum annealing for industry applications: Introduction and review. Rep. Progr. Phys., 2022.
