{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T05:46:01Z","timestamp":1761975961507,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,28]],"date-time":"2022-01-28T00:00:00Z","timestamp":1643328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Nowadays, product development times are constantly decreasing, while the requirements for the products themselves increased significantly in the last decade. Hence, manufacturers use Computer-Aided Design (CAD) and Finite-Element (FE) Methods to develop better products in shorter times. Shape optimization offers great potential to improve many high-fidelity, numerical problems such as the crash performance of cars. Still, the proper selection of optimization algorithms provides a great potential to increase the speed of the optimization time. This article reviews the optimization performance of two different algorithms and frameworks for the structural behavior of a b-pillar. A b-pillar is the structural component between a car\u2019s front and rear door, loaded under static and crash requirements. Furthermore, the validation of the algorithm includes a feasibility constraint. Recently, an optimization routine was implemented and validated for a Non-dominated Sorting Genetic Algorithm (NSGA-II) implementation. Different multi-objective optimization algorithms are reviewed and methodically ranked in a comparative study by given criteria. In this case, the Gap Optimized Multi-Objective Optimization using Response Surfaces (GOMORS) framework is chosen and implemented into the existing Institut f\u00fcr Konstruktionstechnik Optimizes Shapes (IKOS) framework. Specifically, the article compares the NSGA-II and GOMORS directly for a linear, non-linear, and feasibility optimization scenario. The results show that the GOMORS outperforms the NSGA-II vastly regarding the number of function calls and Pareto-efficient results without the feasibility constraint. The problem is reformulated to an unconstrained, three-objective optimization problem to analyze the influence of the constraint. The constrained and unconstrained approaches show equal performance for the given scenarios. Accordingly, the authors provide a clear recommendation towards the surrogate-based GOMORS for costly and multi-objective evaluations. Furthermore, the algorithm can handle the feasibility constraint properly when formulated as an objective function and as a constraint.<\/jats:p>","DOI":"10.3390\/a15020046","type":"journal-article","created":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T01:41:59Z","timestamp":1643420519000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Test and Validation of the Surrogate-Based, Multi-Objective GOMORS Algorithm against the NSGA-II Algorithm in Structural Shape Optimization"],"prefix":"10.3390","volume":"15","author":[{"given":"Yannis","family":"Werner","sequence":"first","affiliation":[{"name":"Institute for Engineering Design, Technische Universit\u00e4t Braunschweig, Hermann-Blenk-Str. 42, 38108 Brunswick, Germany"}]},{"given":"Tim","family":"van Hout","sequence":"additional","affiliation":[{"name":"Technische Universit\u00e4t Braunschweig, 38106 Brunswick, Germany"}]},{"given":"Vijey Subramani","family":"Raja Gopalan","sequence":"additional","affiliation":[{"name":"Technische Universit\u00e4t Braunschweig, 38106 Brunswick, Germany"}]},{"given":"Thomas","family":"Vietor","sequence":"additional","affiliation":[{"name":"Institute for Engineering Design, Technische Universit\u00e4t Braunschweig, Hermann-Blenk-Str. 42, 38108 Brunswick, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Feldhusen, J., and Grote, K.-H. (2013). Pahl\/Beitz Konstruktionslehre, Springer. Methoden und Anwendung Erfolgreicher Produktentwicklung.","DOI":"10.1007\/978-3-642-29569-0"},{"key":"ref_2","unstructured":"Schumacher, A., Seibel, M., Zimmer, H., and Sch\u00e4fer, M. (2005, January 20\u201321). New optimization strategies for crash design. Proceedings of the 4th LS-DYNA Anwenderforum, Bamberg, Germany."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Stein, E., de Borst, R., and Hughes, T.J.R. (2018). Shape Optimization. Encyclopedia of Computational Mechanics, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781119176817"},{"key":"ref_4","first-page":"1","article-title":"Erweiterte Knotenfunktionalit\u00e4t im parametrischen Entwurfswerkzeug SFE Concept","volume":"Nr.172","author":"Zimmer","year":"2002","journal-title":"FAT"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Paas, M.H.J.W., and van Dijk, H.C. (2017). Multidisciplinary Design Optimization of Body Exterior Structures, Springer.","DOI":"10.1007\/978-3-319-67988-4_2"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Schumacher, A., Vietor, T., Fiebig, S., Bletzinger, K.-U., and Maute, K. (2018). Advances in Structural and Multidisciplinary Optimization, Springer International Publishing.","DOI":"10.1007\/978-3-319-67988-4"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s00158-007-0130-6","article-title":"Multidisciplinary optimization of car bodies","volume":"35","author":"Duddeck","year":"2008","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1080\/13588265.2013.832720","article-title":"Geometrical compatibility in structural shape optimisation for crashworthiness","volume":"19","author":"Rayamajhi","year":"2013","journal-title":"Int. J. Crashworthiness"},{"key":"ref_9","unstructured":"Ryberg, A.-B., Domeij B\u00e4ckryd, R., and Nilsson, L. (2012). Metamodel-Based Multidisciplinary Design Optimization for Automotive Applications, Link\u00f6ping University Electronic Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1080\/03052150903386674","article-title":"Trends, features, and tests of common and recently introduced global optimization methods","volume":"42","author":"Younis","year":"2010","journal-title":"Eng. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Palar, P.S., Liem, R.P., Zuhal, L.R., and Shimoyama, K. (2019, January 13\u201317). On the use of surrogate models in engineering design optimization and exploration. Proceedings of the Genetic and Evolutionary Computation Conference Companion, New York, NY, USA.","DOI":"10.1145\/3319619.3326813"},{"key":"ref_12","unstructured":"Vahid, G. (2020). Adaptive Search Approach in Multidisciplinary Optimization of Lightweight Structures Using Hybrid-Metaheuristics, Technische Universit\u00e4t Braunschweig."},{"key":"ref_13","unstructured":"Rayamajhi, M., Hunkeler, S., Duddeck, F., Zarroug, M., and Rota, L. (2012, January 5\u20136). Robust Shape Optimization for Crashworthiness via a Sub-structuring Approach. Proceedings of the 9th ASMO UK\/ISSMO Conference on Engineering Design Optimization, Product and Process Improvement, Cork, Ireland."},{"key":"ref_14","unstructured":"Rayamajhi, M. (2014). Efficient Methods for Robust Shape Optimisation for Crashworthiness, Technische Universit\u00e4t M\u00fcnchen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4050","DOI":"10.15282\/ijame.14.1.2017.17.0327","article-title":"Multidisciplinary design optimisation methods for automotive structures","volume":"14","author":"Ryberg","year":"2017","journal-title":"Int. J. Automot. Mech. Eng."},{"key":"ref_16","first-page":"45","article-title":"Penalty Function Methods for Constrained Optimization with Genetic Algorithms","volume":"10","author":"Yeniay","year":"2005","journal-title":"Math. Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Malen, D.E. (2020). Fundamentals of Automobile Body Structure Design, SAE International.","DOI":"10.4271\/9781468601756"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1080\/0305215X.2020.1837791","article-title":"Multidisciplinary design optimization of a generic b-pillar under package and design constraints","volume":"53","author":"Werner","year":"2021","journal-title":"Eng. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Computat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/j.procir.2021.05.118","article-title":"From package and design surfaces to optimization\u2014How to apply shape optimization under geometrical constraints","volume":"100","author":"Werner","year":"2021","journal-title":"Procedia CIRP"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10898-015-0270-y","article-title":"Multi objective optimization of computationally expensive multi-modal functions with RBF surrogates and multi-rule selection","volume":"64","author":"Akhtar","year":"2016","journal-title":"J. Glob. Optim."},{"key":"ref_22","unstructured":"Duddeck, F., and Zimmer, H. (2012, January 18\u201320). New Achievements on Implicit Parameterization Techniques for Combined Shape and Topology Optimization for Crashworthiness based on SFE CONCEPT. Proceedings of the Shape and Technology Optimization for Crashworthiness, Int. Crashworthiness Conf. ICRASH2012, Milano, Italy."},{"key":"ref_23","unstructured":"Ghaffarimejlej, V., T\u00fcrck, E., and Vietor, T. (2016, January 26\u201330). Finding the best material combinations through multi-material joining, using genetic algorithm. Proceedings of the European Conference on Composite Materials (ECCM 2016), Munich, Germany."},{"key":"ref_24","unstructured":"Rayamajhi, M., Hunkeler, S., and Duddeck, F. (2013, January 19\u201324). Efficient Robust Shape Optimization for Crashworthiness. Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, USA."},{"key":"ref_25","unstructured":"Hillmann, J. (2009). On the Development of a Process Chain for Structural Optimization in Vehicle Passive Safety. [Ph.D. Thesis, Technische Universit\u00e4t Berlin]."},{"key":"ref_26","unstructured":"Schmitt, B.I. (2015). Konvergenzanalyse f\u00fcr die Partikelschwarmoptimierung. Ausgezeichnete Informatikdissertationen 2015, Gesellschaft f\u00fcr Informatik."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"977","DOI":"10.12785\/amis\/080306","article-title":"Metaheuristic Algorithms: Optimal Balance of Intensification and Diversification","volume":"8","author":"Yang","year":"2014","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s00158-013-1033-3","article-title":"Aeroelastic tailoring and scaling using Bacterial Foraging Optimisation","volume":"50","author":"Georgiou","year":"2014","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_29","unstructured":"Yang, X.-S., Deb, S., and He, X. (2013, January 22\u201325). Eagle Strategy with Flower Algorithm. Proceedings of the 2013 International Conference on Advances in Computing, Communications and Informatics (ICACCI), Mysore, India."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1109\/TEVC.2013.2281535","article-title":"An Evolutionary Many-Objective Optimization Algorithm Using Reference-point Based Non-dominated Sorting Approach, Part I: Solving problems with Box Constraints","volume":"18","author":"Deb","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1177\/0954407017724636","article-title":"Multi-objective crashworthiness optimization of vehicle body using particle swarm algorithm coupled with bacterial foraging algorithm","volume":"232","author":"Wang","year":"2018","journal-title":"Proc. Inst. Mech. Eng. Part D J. Automob. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"22272","DOI":"10.1109\/ACCESS.2017.2764047","article-title":"A Simple Multi-Objective Optimization Based on the Cross-Entropy Method","volume":"5","author":"Haber","year":"2017","journal-title":"IEEE Access"},{"key":"ref_33","unstructured":"Belakaria, S., Deshwal, A., and Doppa, J.R. (2019, January 8\u201314). Max-value Entropy Search for Multi-Objective Bayesian Optimization. Proceedings of the International Conference on Neural Information Processing Systems (NeurIPS), Vancouver, BC, Canada."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1109\/TEVC.2005.851274","article-title":"ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems","volume":"10","author":"Knowles","year":"2006","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3137","DOI":"10.1007\/s00500-017-2965-0","article-title":"A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms","volume":"23","author":"Chugh","year":"2019","journal-title":"Soft Comput."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Yang, X.-S. (2014). Nature-Inspired Optimization Algorithms, Elsevier.","DOI":"10.1016\/B978-0-12-416743-8.00005-1"},{"key":"ref_37","unstructured":"Raja Gopalan, V.S., Werner, Y., and van Hout, T. (2021, January 27). GOMORS Implementation in Python 3.7 Using Pysot Package. Available online: https:\/\/github.com\/Vijey-Subramani-Raja-Gopalan\/GOMORS_Python3.7_PYSOT0.2.0\/tree\/v1.0.1."},{"key":"ref_38","unstructured":"Georgios, K. (2007, January 28\u201330). Shape and parameter optimization with ANSA and LS-OPT using a new flexible interface. Proceedings of the 6th European LS-DYNA Conference, Gothenburg, Sweden."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Cavagna, L., Ricci, S., and Riccobene, L. (2009, January 4\u20137). A Fast Tool for Structural Sizing, Aeroelastic Analysis and Optimization in Aircraft Conceptual Design. Proceedings of the 50th AIAA\/ASME\/ASCE\/AHS\/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, USA.","DOI":"10.2514\/6.2009-2571"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.asoc.2017.09.030","article-title":"State-of-the-art in aerodynamic shape optimisation methods","volume":"62","author":"Skinner","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_41","unstructured":"Katz, J.J. (2015). Race-Car Aerodynamics, McGraw-Hill Professional."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/2\/46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:10:04Z","timestamp":1760134204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/2\/46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,28]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["a15020046"],"URL":"https:\/\/doi.org\/10.3390\/a15020046","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,1,28]]}}}