Abstract
The utilization of surrogate models to approximate complex systems has recently gained increased popularity. Because of their capability to deal with black-box problems and lower computational requirements, surrogates were successfully utilized by researchers in various engineering and scientific fields. An efficient use of surrogates can bring considerable savings in computational resources and time. Since literature on surrogate modelling encompasses a large variety of approaches, the appropriate choice of a surrogate remains a challenging task. This review discusses significant publications where surrogate modelling for finite element method-based computations was utilized. We familiarize the reader with the subject, explain the function of surrogate modelling, sampling and model validation procedures, and give a description of the different surrogate types. We then discuss main categories where surrogate models are used: prediction, sensitivity analysis, uncertainty quantification, and surrogate-assisted optimization, and give detailed account of recent advances and applications. We review the most widely used and recently developed software tools that are used to apply the discussed techniques with ease. Based on a literature review of 180 papers related to surrogate modelling, we discuss major research trends, gaps, and practical recommendations. As the utilization of surrogate models grows in popularity, this review can function as a guide that makes surrogate modelling more accessible.
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References
Abadi M, Barham P, Chen J et al (2016) Tensorflow: a system for large-scale machine learning. In: 12th {USENIX} symposium on operating systems design and implementation ({OSDI} 16), pp 265–283
Abueidda DW, Koric S, Sobh NA (2020) Topology optimization of 2d structures with nonlinearities using deep learning. Comput Struct 237(106):283. https://doi.org/10.1016/j.compstruc.2020.106283
Acar E (2015) Effect of error metrics on optimum weight factor selection for ensemble of metamodels. Expert Syst Appl 42(5):2703–2709. https://doi.org/10.1016/j.eswa.2014.11.020
Al Kajbaf A, Bensi M (2020) Application of surrogate models in estimation of storm surge: a comparative assessment. Appl Soft Comput 91(106):184
Alizadeh R, Allen JK, Mistree F (2020) Managing computational complexity using surrogate models: a critical review. Res Eng Des 31:275–298. https://doi.org/10.1007/s00163-020-00336-7
Asteris PG, Cavaleri L, Ly HB et al (2021) Surrogate models for the compressive strength mapping of cement mortar materials. Soft Comput 25(8):6347–6372
Babaei M, Pan I (2016) Performance comparison of several response surface surrogate models and ensemble methods for water injection optimization under uncertainty. Comput Geosci 91:19–32. https://doi.org/10.1016/j.cageo.2016.02.022
Baeck T, Fogel D, Michalewicz Z (1997) Handbook of evolutionary computation. Taylor & Francis, London
Benaouali A, Kachel S (2019) Multidisciplinary design optimization of aircraft wing using commercial software integration. Aerosp Sci Technol 92:766–776. https://doi.org/10.1016/j.ast.2019.06.040
Berthelson P, Ghassemi P, Wood JW et al (2021) A finite element-guided mathematical surrogate modeling approach for assessing occupant injury trends across variations in simplified vehicular impact conditions. Med Biol Eng Comput 59:1065–1079. https://doi.org/10.1007/s11517-021-02349-3
Bhosekar A, Ierapetritou M (2018) Advances in surrogate based modeling, feasibility analysis, and optimization: a review. Comput Chem Eng 108:250–267. https://doi.org/10.1016/j.compchemeng.2017.09.017
Bischl B, Mersmann O, Trautmann H et al (2012) Resampling methods for meta-model validation with recommendations for evolutionary computation. Evol Comput 20:249–275. https://doi.org/10.1162/EVCO_a_00069
Bonfiglio L, Perdikaris P, del Águila J et al (2018) A probabilistic framework for multidisciplinary design: application to the hydrostructural optimization of supercavitating hydrofoils. Int J Numer Meth Eng 116(4):246–269. https://doi.org/10.1002/nme.5923
Booker A, Dennis J, Frank P et al (1998) A rigorous framework for optimization of expensive functions by surrogates. Struct Optim 17:1–13. https://doi.org/10.1007/BF01197708
Bouhlel M, He S, Martins J (2020) Scalable gradient-enhanced artificial neural networks for airfoil shape design in the subsonic and transonic regimes. Struct Multidiscip Optim 61:1363–1376. https://doi.org/10.1007/s00158-020-02488-5
Bouhlel MA, Hwang JT, Bartoli N et al (2019) A python surrogate modeling framework with derivatives. Adv Eng Soft. https://doi.org/10.1016/j.advengsoft.2019.03.005
Boukouvala F, Floudas CA (2017) Argonaut: Algorithms for global optimization of constrained grey-box computational problems. Optim Lett 11(5):895–913. https://doi.org/10.1007/s11590-016-1028-2
Box E, Draper N (1987) Empirical model building and response surfaces. Wiley, New York
Bramerdorfer G, Zăvoianu AC (2017) Surrogate-based multi-objective optimization of electrical machine designs facilitating tolerance analysis. IEEE Trans Magn 53(8):1–11. https://doi.org/10.1109/TMAG.2017.2694802
Broomhead D, Lowe D (1988) Multivariable functional interpolation and adaptive networks. Complex Syst 2:321–355
Brown A, Montgomery J, Garg S (2021) Automatic construction of accurate bioacoustics workflows under time constraints using a surrogate model. Appl Soft Comput 113(107):944
Bunnell S, Thelin C, Gorrell S et al (2018) Rapid visualization of compressor blade finite element models using surrogate modeling. p v07AT30A011. https://doi.org/10.1115/GT2018-77188
Cernuda C, Llavori I, Zăvoianu AC et al (2020) Critical analysis of the suitability of surrogate models for finite element method application in catalog-based suspension bushing design. In: 2020 25th IEEE international conference on emerging technologies and factory automation (ETFA), pp 829–836. https://doi.org/10.1109/ETFA46521.2020.9212166
Chatterjee T, Chakraborty S, Chowdhury R (2019) A critical review of surrogate assisted robust design optimization. Arch Comput Methods Eng 26:245–274. https://doi.org/10.1007/s11831-017-9240-5
Chen H, Loeppky JL, Sacks J et al (2016) Analysis methods for computer experiments: how to assess and what counts? Stat Sci 31(1):40–60. https://doi.org/10.1214/15-STS531
Christelis V, Regis RG, Mantoglou A (2017) Surrogate-based pumping optimization of coastal aquifers under limited computational budgets. J Hydroinf 20(1):164–176. https://doi.org/10.2166/hydro.2017.063
Chu L, Shi J, de Cursi ES (2019) Kriging surrogate model for resonance frequency analysis of dental implants by a Latin hypercube-based finite element method. Appl Bionics Biomech. https://doi.org/10.1155/2019/3768695
Costa A, Nannicini G (2018) RBFOpt: an open-source library for black-box optimization with costly function evaluations. Math Program Comput 10:597–629. https://doi.org/10.1007/s12532-018-0144-7
Cozad A, Sahinidis N, Miller D (2014) Learning surrogate models for simulation-based optimization. AIChE J 60:2211–2227. https://doi.org/10.1002/aic.14418
Crombecq K, Laermans E, Dhaene T (2011) Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling. Eur J Oper Res 214:683–696. https://doi.org/10.1016/j.ejor.2011.05.032
Deb K, Pratap A, Agarwal S et al (2002) A fast and elitist multiobjective genetic algorithm: Nsga-II. IEEE Trans Evol Comput 6(2):182–197. https://doi.org/10.1109/4235.996017
Deng Y, Di Bucchianico A, Pechenizkiy M (2020) Controlling the accuracy and uncertainty trade-off in RUL prediction with a surrogate wiener propagation model. Reliab Eng Syst Saf 196(106):727
De’ath G (2007) Boosted trees for ecological modeling and prediction. Ecology 88(1):243–51. https://doi.org/10.1890/0012-9658(2007)88[243:BTFEMA]2.0.CO;2
Dong J, Qin Q, Xiao Y (2020) Nelder–Mead optimization of elastic metamaterials via machine-learning-aided surrogate modeling. Int J Appl Mech 12(2050):011. https://doi.org/10.1142/S1758825120500118
Eason J, Cremaschi S (2014) Adaptive sequential sampling for surrogate model generation with artificial neural networks. Comput Chem Eng 68:220–232. https://doi.org/10.1016/j.compchemeng.2014.05.021
Easum JA, Nagar J, Werner DH (2017) Multi-objective surrogate-assisted optimization applied to patch antenna design. In: 2017 IEEE international symposium on antennas and propagation USNC/URSI national radio science meeting, pp 339–340. https://doi.org/10.1109/APUSNCURSINRSM.2017.8072212
Eigel M, Gruhlke R (2021) A local hybrid surrogate-based finite element tearing interconnecting dual-primal method for nonsmooth random partial differential equations. Int J Numer Methods Eng 122(4):1001–1030. https://doi.org/10.1002/nme.6571
Fan X, Wang P, Hao F (2019) Reliability-based design optimization of crane bridges using kriging-based surrogate models. Struct Multidisc Optim 59:993–1005. https://doi.org/10.1007/s00158-018-2183-0
Fatahi L (2021) Surrogate-based sensitivity analysis and finite element model updating of welded plates. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2021.1907006
Forrester A, Keane A (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45:50–79. https://doi.org/10.1016/j.paerosci.2008.11.001
Friedman J (2001) Greedy function approximation: a gradient boosting machine. Ann Stat 29:1189–1232
Garud S, Karimi I, Kraft M (2017) Smart sampling algorithm for surrogate model development. Comput Chem Eng 96:103–114. https://doi.org/10.1016/j.compchemeng.2016.10.006
Gaspar B, Teixeira A, Soares CG (2014) Assessment of the efficiency of kriging surrogate models for structural reliability analysis. Probab Eng Mech 37:24–34. https://doi.org/10.1016/j.probengmech.2014.03.011
Gaspar B, Teixeira A, Guedes Soares C (2017) Adaptive surrogate model with active refinement combining kriging and a trust region method. Reliab Eng Syst Saf 165:277–291. https://doi.org/10.1016/j.ress.2017.03.035
Ghorbel H, Zannini N, Cherif S et al (2019) Smart adaptive run parameterization (SARP): enhancement of user manual selection of running parameters in fluid dynamic simulations using bio-inspired and machine-learning techniques. Soft Comput 23(22):12031–12047
Goel T, Haftka R, Shyy W et al (2007) Ensemble of surrogates. Struct Multidiscip Optim 33:199–216. https://doi.org/10.1007/s00158-006-0051-9
Gogu C, Passieux JC (2013) Efficient surrogate construction by combining response surface methodology and reduced order modeling. Struct Multidiscip Optim 47:821–837. https://doi.org/10.1007/s00158-012-0859-4
Goldberg D, Holland J (1988) Genetic algorithms and machine learning. Mach Learn 3:95–99. https://doi.org/10.1023/A:1022602019183
Gorissen D, Couckuyt I, Demeester P et al (2010) A surrogate modeling and adaptive sampling toolbox for computer based design. J Mach Learn Res 11:2051–2055
Grama A, Kumar V, Gupta A et al (2003) Introduction to parallel computing. Addison-Wesley, Pearson Education, Reading
Gutmann HM (2001) A radial basis function method for global optimization. J Glob Optim 19:201–227. https://doi.org/10.1023/A:1011255519438
Hadigol M, Doostan A (2018) Least squares polynomial chaos expansion: a review of sampling strategies. Comput Methods Appl Mech Eng 332:382–407. https://doi.org/10.1016/j.cma.2017.12.019
Haeri A, Fadaee MJ (2016) Efficient reliability analysis of laminated composites using advanced kriging surrogate model. Compos Struct 149:26–32. https://doi.org/10.1016/j.compstruct.2016.04.013
Han Z (2016) SurroOpt: a generic surrogate-based optimization code for aerodynamic and multidisciplinary design. In: Proceedings of the 30th congress of the international council of the aeronautical sciences, DCC, Daejeon, Korea, 25–30
Hassan AKS, Etman AS, Soliman EA (2018) Optimization of a novel nano antenna with two radiation modes using kriging surrogate models. IEEE Photonics J 10(4):1–17. https://doi.org/10.1109/JPHOT.2018.2848593
Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer, New York
Ho T (1998) The random subspace method for constructing decision forests. IEEE Trans Pattern Anal Mach Intell 20:832–844. https://doi.org/10.1109/34.709601
Hooke R, Jeeves TA (1961) “Direct search’’ solution of numerical and statistical problems. J ACM 8(2):212–229. https://doi.org/10.1145/321062.321069
Huang C, Radi B, Hami A (2016) Uncertainty analysis of deep drawing using surrogate model based probabilistic method. Int J Adv Manuf Technol 86:3229–3240. https://doi.org/10.1007/s00170-016-8436-4
Hung Y (2011) Penalized blind kriging in computer experiments. Stat Sin 21(3):1171–1190
Jin SS (2021) Accelerating gaussian process surrogate modeling using compositional kernel learning and multi-stage sampling framework. Appl Soft Comput 104(106):909
Jin SS, Jung HJ (2016) Sequential surrogate modeling for efficient finite element model updating. Comput Struct 168:30–45. https://doi.org/10.1016/j.compstruc.2016.02.005
Jin Y, Wang H, Chugh T et al (2019) Data-driven evolutionary optimization: an overview and case studies. IEEE Trans Evol Comput 23(3):442–458. https://doi.org/10.1109/TEVC.2018.2869001
Johnson M, Moore L, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plann Inference 26:131–148. https://doi.org/10.1016/0378-3758(90)90122-B
Jones D, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13:455–492. https://doi.org/10.1023/A:1008306431147
Jones DR, Perttunen CD, Stuckman B (1993) Lipschitzian optimization without the Lipschitz constant. J Optim Theory Appl 79:157–181. https://doi.org/10.1007/BF00941892
Joseph V, Hung Y, Sudjianto A (2008) Blind kriging: a new method for developing metamodels. J Mech Des 130(031):102. https://doi.org/10.1115/1.2829873
Kamiński B (2015) A method for the updating of stochastic kriging metamodels. Eur J Oper Res 247(3):859–866. https://doi.org/10.1016/j.ejor.2015.06.070
Karban P, Pánek D, Orosz T et al (2021) Fem based robust design optimization with agros and \(\bar{{\rm A}}\)rtap. Comput Math Appl 81:618–633. https://doi.org/10.1016/j.camwa.2020.02.010, development and Application of Open-source Software for Problems with Numerical PDEs
Kaya M, Hajimirza S (2018) Surrogate based modeling and optimization of plasmonic thin film organic solar cells. Int J Heat Mass Transf 118:1128–1142. https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.044
Kaymaz I (2005) Application of kriging method to structural reliability problems. Struct Saf 27(2):133–151. https://doi.org/10.1016/j.strusafe.2004.09.001
Kazikova A, Pluhacek M, Senkerik R (2020) Why tuning the control parameters of metaheuristic algorithms is so important for fair comparison? MENDEL 26(2):9–16. https://doi.org/10.13164/mendel.2020.2.009
Kersaudy P, Sudret B, Varsier N et al (2015) A new surrogate modeling technique combining kriging and polynomial chaos expansions-application to uncertainty analysis in computational dosimetry. J Comput Phys 286:103–117
Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. https://doi.org/10.1126/science.220.4598.671
Kleijnen JP (2017) Regression and kriging metamodels with their experimental designs in simulation: a review. Eur J Oper Res 256(1):1–16
Knowles J (2006) ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans Evol Comput 10(1):50–66. https://doi.org/10.1109/TEVC.2005.851274
Koza J (1992) Genetic programming—on the programming of computers by means of natural selection. MIT Press, Cambridge
Krige D (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. J S Afr Inst Min Metall 52(6):119–139
Ktari Z, Leitão C, Prates PA et al (2021) Mechanical design of ring tensile specimen via surrogate modelling for inverse material parameter identification. Mech Mater 153(103):673. https://doi.org/10.1016/j.mechmat.2020.103673
Kudela J (2019) Minimum-volume covering ellipsoids: Improving the efficiency of the Wolfe-Atwood algorithm for large-scale instances by pooling and batching. MENDEL 25(2):19–26. https://doi.org/10.13164/mendel.2019.2.019
Kudela J, Matousek R (2022) Lipschitz-based surrogate model for high-dimensional computationally expensive problems. arXiv preprint arXiv:2204.14236
Kudela J, Popela P (2020) Pool & discard algorithm for chance constrained optimization problems. IEEE Access 8:79397–79407. https://doi.org/10.1109/ACCESS.2020.2990726
Lai X, Wang X, Nie Y et al (2020) An efficient parameter estimation method for nonlinear high-order systems via surrogate modeling and cuckoo search. Soft Comput 24(22):17065–17079
Lal A, Datta B (2017) Modelling saltwater intrusion processes and development of a multi-objective strategy for management of coastal aquifers utilizing planned artificial freshwater recharge. Model Earth Syst Environ 4:111–126. https://doi.org/10.1007/s40808-017-0405-x
Lee S, Chen W (2008) A comparative study of uncertainty propagation methods for black-box-type problems. Struct Multidiscip Optim 37:239–253. https://doi.org/10.1007/s00158-008-0234-7
Leifsson L, Hermannsson E, Koziel S (2015) Optimal shape design of multi-element trawl-doors using local surrogate models. J Comput Sci 10:55–62. https://doi.org/10.1016/j.jocs.2015.01.006
Leser PE, Hochhalter JD, Warner JE et al (2017) Probabilistic fatigue damage prognosis using surrogate models trained via three-dimensional finite element analysis. Struct Health Monit 16(3):291–308. https://doi.org/10.1177/1475921716643298
Li H, Liu T, Wang M et al (2017) Design optimization of stent and its dilatation balloon using kriging surrogate model. BioMedical Eng OnLine 16. https://doi.org/10.1186/s12938-016-0307-6
Li S, Trevelyan J, Wu Z et al (2019) An adaptive SVD–Krylov reduced order model for surrogate based structural shape optimization through isogeometric boundary element method. Comput Methods Appl Mech Eng 349:312–338. https://doi.org/10.1016/j.cma.2019.02.023
Liang L, Liu M, Martin C, et al (2018a) A deep learning approach to estimate stress distribution: a fast and accurate surrogate of finite-element analysis. J R Soc Interface 15(138):20170844. https://doi.org/10.1098/rsif.2017.0844
Liang L, Liu M, Martin C et al (2018b) A machine learning approach as a surrogate of finite element analysis-based inverse method to estimate the zero-pressure geometry of human thoracic aorta. Int J Numer Methods Biomed Eng 34(8):e3103. https://doi.org/10.1002/cnm.3103, e3103 CNM-Dec-17-0318, https://arxiv.org/abs/https://onlinelibrary.wiley.com/doi/pdf/10.1002/cnm.3103
Lim DK, Woo DK, Yeo HK et al (2015) A novel surrogate-assisted multi-objective optimization algorithm for an electromagnetic machine design. IEEE Trans Magn 51(3):1–4. https://doi.org/10.1109/TMAG.2014.2359452
Lin Y, Yang Q, Guan G (2019) Scantling optimization of FPSO internal turret area structure using RBF model and evolutionary strategy. Ocean Eng 191(106):562. https://doi.org/10.1016/j.oceaneng.2019.106562
Lirio RB, Camejo D, Loubes JM et al (2014) Estimation of covariance functions by a fully data-driven model selection procedure and its application to Kriging spatial interpolation of real rainfall data. Stat Methods Appl 23(2):149–174. https://doi.org/10.1007/s10260-013-0250-7
Liu Z, Zhu C, Zhu P et al (2018) Reliability-based design optimization of composite battery box based on modified particle swarm optimization algorithm. Compos Struct 204:239–255. https://doi.org/10.1016/j.compstruct.2018.07.053
Loshchilov I, Schoenauer M, Sebag M (2010) Dominance-based pareto-surrogate for multi-objective optimization. In: Simulated evolution and learning. SEAL 2010. Lecture notes in computer science, vol 6457. Springer, Berlin. https://doi.org/10.1007/978-3-642-17298-4_24
Matousek R, Dobrovsky L, Kudela J (2022) How to start a heuristic? Utilizing lower bounds for solving the quadratic assignment problem. Int J Ind Eng Comput 13(2):151–164
McKay M, Beckman R, Conover W (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–245. https://doi.org/10.2307/1268522
Mendes MHS, Soares GL, Coulomb JL et al (2013) A surrogate genetic programming based model to facilitate robust multi-objective optimization: a case study in magnetostatics. IEEE Trans Magn 49(5):2065–2068. https://doi.org/10.1109/TMAG.2013.2238615
Meng D, Yang S, Zhang Y et al (2019) Structural reliability analysis and uncertainties-based collaborative design and optimization of turbine blades using surrogate model. Fatigue Fract Eng Mater Struct 42(6):1219–1227. https://doi.org/10.1111/ffe.12906
Morris M, Mitchell T (1995) Exploratory designs for computational experiments. J Stat Plan Inference 43:381–402. https://doi.org/10.1016/0378-3758(94)00035-T
Mueller J (2014) MATSuMoTo: the MATLAB surrogate model toolbox for computationally expensive black-box global optimization problems. arXiv:1404.4261
Mukhopadhyay T, Naskar S, Dey S et al (2016) On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells. Compos Struct 140:798–805. https://doi.org/10.1016/j.compstruct.2015.12.037
Müller J, Piché R (2011) Mixture surrogate models based on Dempster-Shafer theory for global optimization problems. J Glob Optim 51:79–104. https://doi.org/10.1007/s10898-010-9620-y
Nabian MA, Meidani H (2019) A deep learning solution approach for high-dimensional random differential equations. Probab Eng Mech 57:14–25. https://doi.org/10.1016/j.probengmech.2019.05.001
Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7(4):308–313. https://doi.org/10.1093/comjnl/7.4.308
Ninic J, Freitag S, Meschke G (2017) A hybrid finite element and surrogate modelling approach for simulation and monitoring supported TBM steering. Tunn Undergr Space Technol 63:12–28. https://doi.org/10.1016/j.tust.2016.12.004
Nobari A, Ouyang H, Bannister P (2015) Uncertainty quantification of squeal instability via surrogate modelling. Mech Syst Signal Process 60:887–908
Nocedal J, Wright S (2006) Numerical optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York
Novak L, Novak D (2018) Polynomial chaos expansion for surrogate modelling: theory and software. Beton- und Stahlbetonbau 113(S2):27–32. https://doi.org/10.1002/best.201800048
Nyshadham C, Rupp M, Bekker B et al (2019) Machine-learned multi-system surrogate models for materials prediction. npj Comput Mater 5:51. https://doi.org/10.1038/s41524-019-0189-9
Omairey SL, Dunning PD, Sriramula S (2019) Multiscale surrogate-based framework for reliability analysis of unidirectional FRP composites. Compos B Eng 173(106):925. https://doi.org/10.1016/j.compositesb.2019.106925
Owen NE, Challenor P, Menon PP et al (2017) Comparison of surrogate-based uncertainty quantification methods for computationally expensive simulators. SIAM/ASA J Uncertain Quantif 5(1):403–435
Papadopoulos V, Soimiris G, Giovanis D et al (2018) A neural network-based surrogate model for carbon nanotubes with geometric nonlinearities. Comput Methods Appl Mech Eng 328:411–430. https://doi.org/10.1016/j.cma.2017.09.010
Park HJ, Yeo HK, Jung SY et al (2018) A robust multimodal optimization algorithm based on a sub-division surrogate model and an improved sampling method. IEEE Trans Magn 54(3):1–4. https://doi.org/10.1109/TMAG.2017.2755073
Paszke A, Gross S, Chintala S et al (2017) Automatic differentiation in pytorch. In: NIPS 2017 workshop autodiff decision program chairs
Pavlíček K, Kotlan V, Doležel I (2019) Applicability and comparison of surrogate techniques for modeling of selected heating problems. Comput Math Appl 78(9):2897–2910. https://doi.org/10.1016/j.camwa.2019.02.013, applications of Partial Differential Equations in Science and Engineering
Pedregosa F, Varoquaux G, Gramfort A et al (2011) Scikit-learn: machine learning in python. J Mach Learn Res 12(85):2825–2830
Pfrommer J, Zimmerling C, Liu J et al (2018) Optimisation of manufacturing process parameters using deep neural networks as surrogate models. Procedia CIRP 72:426–431. https://doi.org/10.1016/j.procir.2018.03.046, 51st CIRP Conference on Manufacturing Systems
Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimization. Swarm Intell 1:33–57. https://doi.org/10.1007/s11721-007-0002-0
Provost F, Jensen D, Oates T (1999) Efficient progressive sampling. In: Proceedings of the fifth ACM SIGKDD international conference on knowledge discovery and data mining, pp 23–32. https://doi.org/10.1145/312129.312188
Putra N, Palar PS, Anzai H et al (2018) Multiobjective design optimization of stent geometry with wall deformation for triangular and rectangular struts. Med Biol Eng Comput 57:15–26. https://doi.org/10.1007/s11517-018-1864-6
Qian J, Yi J, Cheng Y et al (2020) A sequential constraints updating approach for kriging surrogate model-assisted engineering optimization design problem. Eng Comput 36:993–1009. https://doi.org/10.1007/s00366-019-00745-w
Qin S, Zhang Y, Zhou YL et al (2018) Dynamic model updating for bridge structures using the kriging model and PSO algorithm ensemble with higher vibration modes. Sensors. https://doi.org/10.3390/s18061879
Qiu N, Gao Y, Fang J et al (2018) Crashworthiness optimization with uncertainty from surrogate model and numerical error. Thin-Walled Struct 129:457–472. https://doi.org/10.1016/j.tws.2018.05.002
Rafiee V, Faiz J (2019) Robust design of an outer rotor permanent magnet motor through six-sigma methodology using response surface surrogate model. IEEE Trans Magn 55(10):1–10. https://doi.org/10.1109/TMAG.2019.2923160
Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408. https://doi.org/10.1016/j.probengmech.2004.04.003
Rasmussen C, Williams C (2006) Gaussian processes for machine learning. MIT Press, US
Rios LM, Sahinidis N (2013) Derivative-free optimization: a review of algorithms and comparison of software implementations. J Glob Optim 56:1247–1293. https://doi.org/10.1007/s10898-012-9951-y
Rocas M, García-González A, Zlotnik S et al (2021) Nonintrusive uncertainty quantification for automotive crash problems with VPS/Pamcrash. Finite Elem Anal Des 193(103):556
Rocas M, García-González A, Larrayoz X et al (2022) Adaptive surrogates of crashworthiness models for multi-purpose engineering analyses accounting for uncertainty. Finite Elem Anal Des 203(103):694
Sacks J, Welch WJ, Mitchell TJ et al (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423
Sanchez F, Budinger M, Hazyuk I (2017) Dimensional analysis and surrogate models for the thermal modeling of multiphysics systems. Appl Therm Eng 110:758–771. https://doi.org/10.1016/j.applthermaleng.2016.08.117
Schmidt M, Lipson H (2009) Distilling free-form natural laws from experimental data. Science 324(5923):81–85. https://doi.org/10.1126/science.1165893
Schulz M, Dittmann J, Böl M (2019) Modeling the mechanical behavior of semi-flexible polymer chains using a surrogate model based on a finite-element approach to Brownian polymer dynamics. J Mech Phys Solids 130:101–117. https://doi.org/10.1016/j.jmps.2019.05.016
Shi H, Ma T, Chu W et al (2017) Optimization of inlet part of a microchannel ceramic heat exchanger using surrogate model coupled with genetic algorithm. Energy Convers Manag 149:988–996. https://doi.org/10.1016/j.enconman.2017.04.035
Shi J, Chu L, Braun R (2019) A kriging surrogate model for uncertainty analysis of graphene based on a finite element method. Int J Mol Sci. https://doi.org/10.3390/ijms20092355
Shi M, Li H, Liu X (2017) Multidisciplinary design optimization of dental implant based on finite element method and surrogate models. J Mech Sci Technol 31:5067–5073. https://doi.org/10.1007/s12206-017-0955-x
Shi R, Liu L, Long T et al (2017) Surrogate assisted multidisciplinary design optimization for an all-electric geo satellite. Acta Astronaut 138:301–317. https://doi.org/10.1016/j.actaastro.2017.05.032, the Fifth International Conference on Tethers in Space
Silber S, Koppelstätter W, Weidenholzer G et al (2018) Reducing development time of electric machines with symspace. In: 2018 8th international electric drives production conference (EDPC), pp 1–5. https://doi.org/10.1109/EDPC.2018.8658312
Slot RM, Sørensen JD, Sudret B et al (2020) Surrogate model uncertainty in wind turbine reliability assessment. Renew Energy 151:1150–1162
Sobester A (2003) Enhancements to global design optimization techniques. PhD thesis, University of Southampton
Song X, Lv L, Li J et al (2018) An advanced and robust ensemble surrogate model: extended adaptive hybrid functions. J Mech Des 140(041):402. https://doi.org/10.1115/1.4039128
Steuben J, Turner C (2014) Adaptive surrogate-model fitting using error monotonicity
Stork J, Friese M, Zaefferer M et al (2020) Open issues in surrogate-assisted optimization. Springer, Cham, pp 225–244. https://doi.org/10.1007/978-3-030-18764-4_10
Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11:341–359. https://doi.org/10.1023/A:1008202821328
Su G, Peng L, Hu L (2017) A gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis. Struct Saf 68:97–109. https://doi.org/10.1016/j.strusafe.2017.06.003
Sun G, Wang S (2019) A review of the artificial neural network surrogate modeling in aerodynamic design. Proc Inst Mech Eng Part G J Aerosp Eng 233(16):5863–5872. https://doi.org/10.1177/0954410019864485
Tan F, Wang L, Yin M et al (2019) Obtaining more accurate convective heat transfer coefficients in thermal analysis of spindle using surrogate assisted differential evolution method. Appl Therm Eng 149:1335–1344. https://doi.org/10.1016/j.applthermaleng.2018.12.124
Tan Z, Song X, Cao W et al (2015) DFIG machine design for maximizing power output based on surrogate optimization algorithm. IEEE Trans Energy Convers 30(3):1154–1162. https://doi.org/10.1109/TEC.2015.2411153
Taran N, Ionel DM, Dorrell DG (2018) Two-level surrogate-assisted differential evolution multi-objective optimization of electric machines using 3-d FEA. IEEE Trans Magn 54(11):1–5. https://doi.org/10.1109/TMAG.2018.2856858
Tie Y, Hou Y, Li C et al (2020) Optimization for maximizing the impact-resistance of patch repaired CFRP laminates using a surrogate-based model. Int J Mech Sci 172(105):407. https://doi.org/10.1016/j.ijmecsci.2019.105407
Torkzadeh P, Fathnejat H, Ghiasi R (2016) Damage detection of plate-like structures using intelligent surrogate model. Smart Struct Syst 18:1233–1250. https://doi.org/10.12989/sss.2016.18.6.1233
Tripathy RK, Bilionis I (2018) Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification. J Comput Phys 375:565–588
Vapnik V (1995) The nature of statistical learning theory. Springer, New York
Vega MA, Todd MD (2020) A variational Bayesian neural network for structural health monitoring and cost-informed decision-making in miter gates. Struct Health Monit. https://doi.org/10.1177/1475921720904543
Viana FA, Goel T (2010) Surrogates toolbox user’s guide. In: Gainesville, FL, USA
Viana FA, Picheny V, Haftka R (2010) Using cross validation to design conservative surrogates. AIAA J 48:2286–2298. https://doi.org/10.2514/1.J050327
Wang B, Yan L, Duan X et al (2022) An integrated surrogate model constructing method: Annealing combinable gaussian process. Inf Sci
Wang N, Chang H, Zhang D (2021) Efficient uncertainty quantification for dynamic subsurface flow with surrogate by theory-guided neural network. Comput Methods Appl Mech Eng 373(113):492
Wang T, Shao M, Guo R et al (2021b) Surrogate model via artificial intelligence method for accelerating screening materials and performance prediction. Adv Funct Mater 31(8). https://doi.org/10.1002/adfm.202006245
Wang Y, Zhang Y, Zhao H et al (2018) Identifying interphase properties in polymer nanocomposites using adaptive optimization. Compos Sci Technol 162:146–155. https://doi.org/10.1016/j.compscitech.2018.04.017
Watts S, Arrighi W, Kudo J et al (2019) Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-019-02297-5
Wee H, Reid J, Chinchilli V et al (2016) Finite element-derived surrogate models of locked plate fracture fixation biomechanics. Ann Biomed Eng 45:668–680. https://doi.org/10.1007/s10439-016-1714-3
Westermann P, Evins R (2019) Surrogate modelling for sustainable building design—a review. Energy Build 198:170–186. https://doi.org/10.1016/j.enbuild.2019.05.057
White DA, Arrighi WJ, Kudo J et al (2019) Multiscale topology optimization using neural network surrogate models. Comput Methods Appl Mech Eng 346:1118–1135. https://doi.org/10.1016/j.cma.2018.09.007
Wiener N (1938) The homogeneous. Am J Math 60:897–936
Wu MC, Kamensky D, Wang C et al (2017) Optimizing fluid-structure interaction systems with immersogeometric analysis and surrogate modeling: application to a hydraulic arresting gear. Comput Methods Appl Mech Eng 316:668–693. https://doi.org/10.1016/j.cma.2016.09.032, special Issue on Isogeometric Analysis: Progress and Challenges
Xiu D, Karniadakis G (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24:619–644. https://doi.org/10.1137/S1064827501387826
Xu J, Han Z, Yan X et al (2019) Design optimization of a multi-megawatt wind turbine blade with the NPU-MWA airfoil family. Energies. https://doi.org/10.3390/en12173330
Yan C, Yin Z, Shen X et al (2020) Surrogate-based optimization with improved support vector regression for non-circular vent hole on aero-engine turbine disk. Aerosp Sci Technol 96(105):332. https://doi.org/10.1016/j.ast.2019.105332
Yang S, Tian W, Cubi E et al (2016) Comparison of sensitivity analysis methods in building energy assessment. Proc Eng 146:174–181. https://doi.org/10.1016/j.proeng.2016.06.369, the 8th international cold climate HVAC Conference
Ye D, Zun P, Krzhizhanovskaya V et al (2022) Uncertainty quantification of a three-dimensional in-stent restenosis model with surrogate modelling. J R Soc Interface 19(187):20210864
Yong H, Wang L, Toal D et al (2019) Multi-fidelity kriging-assisted structural optimization of whole engine models employing medial meshes. Struct Multidisc Optim 60:1209–1226. https://doi.org/10.1007/s00158-019-02242-6
Yoo K, Bacarreza O, Aliabadi MHF (2020) A novel multi-fidelity modelling-based framework for reliability-based design optimisation of composite structures. Eng Comput. https://doi.org/10.1007/s00366-020-01084-x
Zerpa L, Queipo N, Pintos SA et al (2005) An optimization methodology of alkaline-surfactant-polymer flooding processes using field scale numerical simulation and multiple surrogates. J Petrol Sci Eng 47:197–208. https://doi.org/10.1016/j.petrol.2005.03.002
Zhang J, Au F (2014) Calibration of initial cable forces in cable-stayed bridge based on kriging approach. Finite Elem Anal Des 92:80–92
Zhao Z, Dai K, Lalonde ER et al (2019) Studies on application of scissor-jack braced viscous damper system in wind turbines under seismic and wind loads. Eng Struct 196(109):294. https://doi.org/10.1016/j.engstruct.2019.109294
Zhou X, Zhang G, Hao X et al (2016) Enhanced differential evolution using local Lipschitz underestimate strategy for computationally expensive optimization problems. Appl Soft Comput 48(C):169–181. https://doi.org/10.1016/j.asoc.2016.06.044
Zhou Y, Lu Z (2020) An enhanced kriging surrogate modeling technique for high-dimensional problems. Mech Syst Signal Process 140(106):687. https://doi.org/10.1016/j.ymssp.2020.106687
Acknowledgements
This work was supported by The Ministry of Education, Youth and Sports of the Czech Republic project No. CZ.02.1.01/0.0/0.0/16_026/0008392 “Computer Simulations for Effective Low-Emission Energy” and by IGA BUT: FSI-S-20-6538.
Funding
The Ministry of Education, Youth and Sports of the Czech Republic project No. CZ.02.1.01/0.0/0.0/16_026/0008392 “Computer Simulations for Effective Low-Emission Energy” and by IGA BUT: FSI-S-20-6538.
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Kudela, J., Matousek, R. Recent advances and applications of surrogate models for finite element method computations: a review. Soft Comput 26, 13709–13733 (2022). https://doi.org/10.1007/s00500-022-07362-8
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DOI: https://doi.org/10.1007/s00500-022-07362-8