{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T21:01:56Z","timestamp":1780606916340,"version":"3.54.1"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12272354"],"award-info":[{"award-number":["12272354"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["52375270"],"award-info":[{"award-number":["52375270"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the Natural Science Foundation of Henan","award":["222300420547"],"award-info":[{"award-number":["222300420547"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s00366-024-01991-3","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T13:02:16Z","timestamp":1715691736000},"page":"3717-3734","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Corner error reduction by Chebyshev transformed orthogonal grid"],"prefix":"10.1007","volume":"40","author":[{"given":"Zebin","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shizhao","family":"Jing","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yaohui","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xianzong","family":"Meng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,5,14]]},"reference":[{"key":"1991_CR1","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.1007\/s00158-017-1864-4","volume":"57","author":"C Jiang","year":"2018","unstructured":"Jiang C, Zheng J, Han X (2018) Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: A review. Struct Multidisc Optim 57:2485\u20132502","journal-title":"Struct Multidisc Optim"},{"key":"1991_CR2","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1007\/s00158-020-02825-8","volume":"63","author":"X Zhang","year":"2021","unstructured":"Zhang X, Wu Z, Ma H (2021) An effective Kriging-based approximation for structural reliability analysis with random and interval variables. Struct Multidisc Optim 63:2473\u20132491","journal-title":"Struct Multidisc Optim"},{"issue":"2","key":"1991_CR3","doi-asserted-by":"crossref","first-page":"1617","DOI":"10.1007\/s00366-021-01557-7","volume":"39","author":"P Ye","year":"2023","unstructured":"Ye P, Pan G (2023) Multi-surrogates and multi-points infill strategy-based global optimization method. Eng Comput 39(2):1617\u20131636","journal-title":"Eng Comput"},{"key":"1991_CR4","first-page":"4919","volume":"6","author":"HY Qian","year":"2022","unstructured":"Qian HY (2022) Numerical simulation of stress-based topological optimization of continuum structures under casting constraints. Eng Comput 6:4919\u20134945","journal-title":"Eng Comput"},{"key":"1991_CR5","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1007\/s11831-019-09336-w","volume":"3","author":"Z Tang","year":"2020","unstructured":"Tang Z, Hu X, P\u00e9riaux J (2020) Multi-level hybridized optimization methods coupling local search deterministic and global search evolutionary algorithms. Arch Comput Methods Eng 3:939\u2013975","journal-title":"Arch Comput Methods Eng"},{"key":"1991_CR6","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s10898-020-00952-6","volume":"3","author":"DR Jones","year":"2021","unstructured":"Jones DR, Martins JRRA (2021) The direct algorithm: 25 years later. J Global Optim 3:521\u2013566","journal-title":"J Global Optim"},{"key":"1991_CR7","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1016\/j.apm.2023.05.018","volume":"2","author":"Y Pang","year":"2023","unstructured":"Pang Y, Lai X, Zhang S, Wang Y, Yang L, Song X (2023) A Kriging-assisted global reliability-based design optimization algorithm with a reliability-constrained expected improvement. Appl Math Model 2:611\u2013630","journal-title":"Appl Math Model"},{"key":"1991_CR8","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.matcom.2022.04.016","volume":"200","author":"GY Dong","year":"2022","unstructured":"Dong GY, Lemieux C (2022) Dependence properties of scrambled Halton sequences. Math Comput Simul 200:240\u2013262","journal-title":"Math Comput Simul"},{"key":"1991_CR9","volume":"204","author":"T Zhang","year":"2020","unstructured":"Zhang T, Yang C, Chen H, Sun L, Deng K (2020) Multi-objective optimization operation of the green energy island based on Hammersley sequence sampling. Energy Convers Manag 204:112316","journal-title":"Energy Convers Manag"},{"key":"1991_CR10","doi-asserted-by":"crossref","first-page":"856","DOI":"10.3103\/S1062873822070085","volume":"7","author":"AA Belov","year":"2022","unstructured":"Belov AA, Tintul MA (2022) Calculating multidimensional cubatures on Sobol sequences. Bull Russ Acad Sci Phys 7:856\u2013860","journal-title":"Bull Russ Acad Sci Phys"},{"key":"1991_CR11","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1198\/004017006000000453","volume":"49","author":"T Cioppa","year":"2007","unstructured":"Cioppa T, Lucas TW (2007) Efficient nearly orthogonal and space-filling latin hypercubes. Technometrics 49:45\u201355","journal-title":"Technometrics"},{"key":"1991_CR12","doi-asserted-by":"crossref","unstructured":"Zhang ZB, Demory B, et al (2014) Space infill study of Kriging meta-model for multi-objective optimization of an engine cooling fan. In: ASME Turbo Expo 2014: Turbine Technical Conference and Exposition","DOI":"10.1115\/GT2014-25281"},{"key":"1991_CR13","first-page":"020029\u201314","volume":"1","author":"Z Zhang","year":"2019","unstructured":"Zhang Z, Han Z, Ferrand P (2019) High anisotropy space exploration with co-Kriging method. AIP Conf Proc 1:020029\u201314","journal-title":"AIP Conf Proc"},{"key":"1991_CR14","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s00366-020-01145-1","volume":"5","author":"Z Zhai","year":"2022","unstructured":"Zhai Z, Li H, Wang X (2022) An adaptive sampling method for Kriging surrogate model with multiple outputs. Eng Comput 5:277\u2013295","journal-title":"Eng Comput"},{"key":"1991_CR15","doi-asserted-by":"crossref","unstructured":"Sanchez SM, Hong W (2011) Better than a petaflop: The power of efficient experimental design. In: Proceedings of the 2011 Winter Simulation Conference (WSC)","DOI":"10.1109\/WSC.2011.6147863"},{"key":"1991_CR16","volume-title":"Uniform design and uniform design table","author":"K Fang","year":"1994","unstructured":"Fang K (1994) Uniform design and uniform design table. Science Press, Beijing"},{"key":"1991_CR17","first-page":"275","volume":"237","author":"K Fang","year":"2002","unstructured":"Fang K, Ma C, Winker P (2002) Centered l$$_2$$-discrepancy of random sampling and latin hypercube design, and construction of uniform designs. Math Comput 237:275\u2013296","journal-title":"Math Comput"},{"key":"1991_CR18","first-page":"1430","volume":"444","author":"KQ Ye","year":"1998","unstructured":"Ye KQ (1998) Orthogonal column latin hypercubes and their application in computer experiments. J Am Stat Assoc 444:1430\u20131439","journal-title":"J Am Stat Assoc"},{"key":"1991_CR19","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/s00158-017-1766-5","volume":"2","author":"X Wu","year":"2018","unstructured":"Wu X, Zhang W, Song S (2018) Robust aerodynamic shape design based on an adaptive stochastic optimization framework. Struct Multidiscip Optim 2:639\u2013651","journal-title":"Struct Multidiscip Optim"},{"key":"1991_CR20","first-page":"6226","volume":"11","author":"S L\u00e4mmle","year":"2023","unstructured":"L\u00e4mmle S, Bogoclu C, Cremanns K, Roos D (2023) Gradient and uncertainty enhanced sequential sampling for global fit. Comput Methods Appl Mech Eng 11:6226","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1991_CR21","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.1007\/s00158-020-02759-1","volume":"3","author":"H Fang","year":"2021","unstructured":"Fang H, Gong C, Li C, Zhang Y, Ronch AD (2021) A sequential optimization framework for simultaneous design variables optimization and probability uncertainty allocation. Struct Multidiscip Optim 3:1307\u20131325","journal-title":"Struct Multidiscip Optim"},{"key":"1991_CR22","first-page":"323","volume":"4","author":"YH Li","year":"2014","unstructured":"Li YH, Wu YZ, Huang ZD (2014) An incremental Kriging method for sequential optimal experimental design. CMES-Comput Model Eng Sci 4:323\u2013357","journal-title":"CMES-Comput Model Eng Sci"},{"key":"1991_CR23","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1007\/s00366-019-00745-w","volume":"3","author":"J Qian","year":"2020","unstructured":"Qian J, Yi J, Cheng Y, Liu J, Zhou Q (2020) A sequential constraints updating approach for Kriging surrogate model-assisted engineering optimization design problem. Eng Comput 3:993\u20131009","journal-title":"Eng Comput"},{"key":"1991_CR24","first-page":"6199","volume":"11","author":"K Li","year":"2023","unstructured":"Li K, Huang H, Fu S, Ma C, Fan Q, Zhu Y (2023) A multi-strategy enhanced northern goshawk optimization algorithm for global optimization and engineering design problems. Comput Methods Appl Mech Eng 11:6199","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1991_CR25","first-page":"1","volume":"1","author":"T Ullrich","year":"2008","unstructured":"Ullrich T (2008) Smolyak\u2019s algorithm, sampling on sparse grids and sobolev spaces of dominating mixed smoothness. East J Approx 1:1\u201338","journal-title":"East J Approx"},{"key":"1991_CR26","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1023\/B:NUMA.0000040060.56819.a7","volume":"3","author":"L Plaskota","year":"2004","unstructured":"Plaskota L, Wasilkowski GW (2004) Smolyak\u2019s algorithm for integration and l$$_1$$ -approximation of multivariate functions with bounded mixed derivatives of second order. Numer Algor 3:229\u2013246","journal-title":"Numer Algor"},{"key":"1991_CR27","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1007\/s10898-018-0643-0","volume":"4","author":"CA Kieslich","year":"2018","unstructured":"Kieslich CA, Boukouvala F, Floudas CA (2018) Optimization of black-box problems using smolyak grids and polynomial approximations. J Global Optim 4:845\u2013869","journal-title":"J Global Optim"},{"key":"1991_CR28","doi-asserted-by":"crossref","first-page":"3190","DOI":"10.1016\/j.cma.2007.03.003","volume":"33\u201334","author":"HG Beyer","year":"2007","unstructured":"Beyer HG, Sendhoff B (2007) Robust optimization\u2014a comprehensive survey. Comput Methods Appl Mech Eng 33\u201334:3190\u20133218","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1991_CR29","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0378-3758(90)90122-B","volume":"2","author":"ME Johnson","year":"1990","unstructured":"Johnson ME, Moore LM, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Infer 2:131\u2013148","journal-title":"J Stat Plan Infer"},{"key":"1991_CR30","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1287\/ijoc.1050.0136","volume":"3","author":"JPC Kleijnen","year":"2005","unstructured":"Kleijnen JPC, Sanchez SM, Lucas TW (2005) State-of-the-art review: A user\u2019s guide to the brave new world of designing simulation experiments. INFORMS J Comput 3:263\u2013289","journal-title":"INFORMS J Comput"},{"key":"1991_CR31","first-page":"1042","volume":"5","author":"SA Smolyak","year":"1963","unstructured":"Smolyak SA (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk SSSR 5:1042\u20131045","journal-title":"Dokl Akad Nauk SSSR"},{"key":"1991_CR32","volume-title":"Approximation theory and approximation practice","author":"LN Trefethen","year":"2013","unstructured":"Trefethen LN (2013) Approximation theory and approximation practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia"},{"key":"1991_CR33","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.compstruc.2014.09.006","volume":"2","author":"J Wu","year":"2015","unstructured":"Wu J, Luo Z, Zhang N, Zhang Y (2015) A new interval uncertain optimization method for structures using Chebyshev surrogate models. Comput Struct (C) 2:185\u2013196","journal-title":"Comput Struct (C)"},{"key":"1991_CR34","first-page":"6755","volume":"11","author":"C Fu","year":"2022","unstructured":"Fu C, Zhu W, Yang Y, Zhao S, Lu K (2022) Surrogate modeling for dynamic analysis of an uncertain notched rotor system and roles of Chebyshev parameters. J Sound Vib 11:6755","journal-title":"J Sound Vib"},{"key":"1991_CR35","volume":"206","author":"C Fu","year":"2020","unstructured":"Fu C, Wang P, Zhao L, Wang X (2020) A distance correlation-based Kriging modeling method for high-dimensional problems. Knowl-Based Syst 206:106356","journal-title":"Knowl-Based Syst"},{"key":"1991_CR36","doi-asserted-by":"crossref","first-page":"2","DOI":"10.2514\/1.32308","volume":"46","author":"J Laurenceau","year":"2008","unstructured":"Laurenceau J, Sagaut P (2008) Building efficient response surfaces of aerodynamic functions with Kriging and Cokriging. Am Inst Aeronaut Astronaut 46:2","journal-title":"Am Inst Aeronaut Astronaut"},{"key":"1991_CR37","first-page":"7","volume":"51","author":"Z Zebin","year":"2019","unstructured":"Zebin Z, Pengfei Z, Hong G, Yong L (2019) Implementation of Kriging model based sequential design on the optimization of sliding bearing. J Harbin Inst Technol 51:7","journal-title":"J Harbin Inst Technol"},{"key":"1991_CR38","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/4235.771163","volume":"2","author":"X Yao","year":"1999","unstructured":"Yao X (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 2:82\u2013102","journal-title":"IEEE Trans Evol Comput"},{"key":"1991_CR39","unstructured":"Adorio EP, Diliman UP (2005) MVF-multivariate test functions library in C for unconstrained global optimization"},{"key":"1991_CR40","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1147\/rd.165.0504","volume":"5","author":"FH Branin","year":"1972","unstructured":"Branin FH (1972) Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. IBM J Res Dev 5:504\u2013522","journal-title":"IBM J Res Dev"},{"key":"1991_CR41","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1007\/s00158-013-0919-4","volume":"3","author":"V Picheny","year":"2013","unstructured":"Picheny V, Wagner T, Ginsbourger D (2013) A benchmark of Kriging-based infill criteria for noisy optimization. Struct Multidiscip Optim 3:607\u2013626","journal-title":"Struct Multidiscip Optim"},{"key":"1991_CR42","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1145\/962437.962444","volume":"4","author":"M Gaviano","year":"2003","unstructured":"Gaviano M, Kvasov DE, Lera D, Sergeyev YD (2003) Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans Math Softw 4:469\u2013480","journal-title":"ACM Trans Math Softw"},{"key":"1991_CR43","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.apm.2022.05.033","volume":"2","author":"Z Zhang","year":"2022","unstructured":"Zhang Z, Li Y (2022) Exploration of anisotropic design space by using unified taylor-cokriging method. Appl Math Model 2:45\u201360","journal-title":"Appl Math Model"},{"key":"1991_CR44","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1093\/comjnl\/3.3.175","volume":"3","author":"HH Rosenbrock","year":"1960","unstructured":"Rosenbrock HH (1960) A automatic method for finding the greatest or least value of a function. Comput J 3:174\u2013184","journal-title":"Comput J"},{"key":"1991_CR45","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1002\/nav.3800200316","volume":"3","author":"JK Hartman","year":"1973","unstructured":"Hartman JK (1973) Some experiments in global optimization. Naval Res Logist Quart 3:569\u2013576","journal-title":"Naval Res Logist Quart"},{"key":"1991_CR46","doi-asserted-by":"crossref","first-page":"1464","DOI":"10.1109\/5.58325","volume":"9","author":"T Kohonen","year":"1990","unstructured":"Kohonen T (1990) The self-organizing map. Proc IEEE 9:1464\u20131480","journal-title":"Proc IEEE"},{"issue":"3","key":"1991_CR47","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1051\/jnwpu\/20203830677","volume":"38","author":"ZB Zhang","year":"2020","unstructured":"Zhang ZB, Zhang PF, Li RZ (2020) Som-based high-dimensional design spaces mapping for multi-objective optimization. J Northwestern Polytech Univ 38(3):677\u2013684","journal-title":"J Northwestern Polytech Univ"},{"key":"1991_CR48","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.compfluid.2013.04.009","volume":"2","author":"L Soulat","year":"2013","unstructured":"Soulat L, Ferrand P, Moreau S, Aubert S, Buisson M (2013) Efficient optimisation procedure for design problems in fluid mechanics. Comput Fluids 2:73\u201386","journal-title":"Comput Fluids"},{"key":"1991_CR49","doi-asserted-by":"crossref","first-page":"2959","DOI":"10.1007\/s00158-021-03067-y","volume":"5","author":"JRRA Martins","year":"2021","unstructured":"Martins JRRA, Kennedy GJ (2021) Enabling large-scale multidisciplinary design optimization through adjoint sensitivity analysis. Struct Multidiscip Optim 5:2959\u20132974","journal-title":"Struct Multidiscip Optim"},{"key":"1991_CR50","doi-asserted-by":"crossref","first-page":"4437","DOI":"10.2514\/1.J060415","volume":"11","author":"M Ohashi","year":"2021","unstructured":"Ohashi M, Tokugawa N (2021) Adjoint-based sensitivity analysis for airfoil flow control aiming at lift-to-drag ratio improvement. Am Inst Aeronaut Astronaut 11:4437\u20134448","journal-title":"Am Inst Aeronaut Astronaut"},{"key":"1991_CR51","first-page":"2","volume":"2","author":"Z Zhang","year":"2015","unstructured":"Zhang Z, Buisson M, Ferrand P, Henner M, Gillot F (2015) Meta-model based optimization of a large diameter semi-radial conical hub engine cooling fan. Mech Ind 2:2","journal-title":"Mech Ind"},{"key":"1991_CR52","volume":"3","author":"M Ribaud","year":"2020","unstructured":"Ribaud M, Blanchet-Scalliet C, Helbert C, Gillot F (2020) Robust optimization: a Kriging-based multi-objective optimization approach. Reliab Eng Syst Saf 3:106913","journal-title":"Reliab Eng Syst Saf"},{"key":"1991_CR53","volume-title":"Genetic algorithms in search, optimization and machine learning","author":"DE Goldberg","year":"1989","unstructured":"Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning, 1st edn. Longman Publishing Co., Inc, Boston","edition":"1"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-024-01991-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-024-01991-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-024-01991-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,3]],"date-time":"2024-12-03T16:24:52Z","timestamp":1733243092000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-024-01991-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,14]]},"references-count":53,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["1991"],"URL":"https:\/\/doi.org\/10.1007\/s00366-024-01991-3","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,14]]},"assertion":[{"value":"21 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no Conflict of interest regarding the publication of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}