{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T01:59:17Z","timestamp":1776650357109,"version":"3.51.2"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,22]],"date-time":"2019-03-22T00:00:00Z","timestamp":1553212800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/EME-APL\/29713\/2017 (CENTRO-01-0145-FEDER-029713)"],"award-info":[{"award-number":["PTDC\/EME-APL\/29713\/2017 (CENTRO-01-0145-FEDER-029713)"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Metals"],"abstract":"<jats:p>The development of full-field measurement methods has enabled a new trend of heterogeneous mechanical tests. The inhomogeneous strain fields retrieved from these tests are being widely used in the calibration of constitutive models for sheet metals. However, today, there is no mechanical test able to characterize the material in a large range of strain states. The aim of this work is to present a heterogeneous mechanical test with an innovative tool\/specimen shape, capable of producing rich heterogeneous strain paths and thus providing extensive information on material behavior. The proposed specimen is found using a shape optimization process where an index that evaluates the richness of strain information is used. In this work, the methodology and results are extended to non-specimen geometry dependence and to the non-dependence of the geometry parametrization through the use of the Ritz method for boundary value problems. Different curve models, such as splines, B-splines, and NURBS, are used, and C1 continuity throughout the specimen is guaranteed. Moreover, several deterministic and stochastic optimization methods are used in order to find the method or the combination of methods able to minimize the cost function effectively. Results demonstrated that the solution is dependent on the geometry definition, as well as on the optimization methodology. Nevertheless, the obtained solutions provided a wider spectrum of strain states than standard tests.<\/jats:p>","DOI":"10.3390\/met9030371","type":"journal-article","created":{"date-parts":[[2019,3,27]],"date-time":"2019-03-27T05:03:12Z","timestamp":1553662992000},"page":"371","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["On the Design of Innovative Heterogeneous Sheet Metal Tests Using a Shape Optimization Approach"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3988-5606","authenticated-orcid":false,"given":"Ant\u00f3nio","family":"Andrade-Campos","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Centre for Mechanical Technology and Automation, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"given":"Jos\u00e9","family":"Aquino","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Centre for Mechanical Technology and Automation, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3420-9219","authenticated-orcid":false,"given":"Jo\u00e3o M. P.","family":"Martins","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Centre for Mechanical Technology and Automation, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4542-1215","authenticated-orcid":false,"given":"Bernardete","family":"Coelho","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Centre for Mechanical Technology and Automation, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ablat, M.A., and Qattawi, A. (2016). Numerical simulation of sheet metal forming: A review. Int. J. Adv. Manuf. Technol., 1235\u20131250.","DOI":"10.1007\/s00170-016-9103-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/j.ijmecsci.2009.12.003","article-title":"Identification of sheet metal plastic anisotropy using heterogeneous biaxial tensile tests","volume":"52","author":"Teaca","year":"2010","journal-title":"Int. J. Mech. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1007\/s11340-014-9983-y","article-title":"Potential of the Cross Biaxial Test for Anisotropy Characterization Based on Heterogeneous Strain Field","volume":"55","author":"Zhang","year":"2015","journal-title":"Exp. Mech."},{"key":"ref_4","unstructured":"Souto, N. (2015). Computational Design of a Mechanical Test for Material Characterization by Inverse Analysis. [Ph.D. Thesis, University of Aveiro]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/s11340-008-9148-y","article-title":"Overview of identification methods of mechanical parameters based on full-field measurements","volume":"48","author":"Avril","year":"2008","journal-title":"Exp. Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s00158-008-0267-y","article-title":"Optimization of a mechanical test on composite plates with the virtual fields method","volume":"38","author":"Toussaint","year":"2009","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/s12289-016-1284-9","article-title":"Mechanical design of a heterogeneous test for material parameters identification","volume":"10","author":"Souto","year":"2017","journal-title":"Int. J. Mater. Form."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1016\/j.ymssp.2010.08.013","article-title":"3D digital image correlation methods for full-field vibration measurement","volume":"25","author":"Helfrick","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/S0924-0136(97)00366-X","article-title":"Determination of the elasto-plastic properties of aluminium using a mixed numerical\u2013experimental method","volume":"75","author":"Meuwissen","year":"1998","journal-title":"J. Mater. Process. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s12289-010-0992-9","article-title":"Advances in anisotropy and formability","volume":"3","author":"Banabic","year":"2010","journal-title":"Int. J. Mater. Form."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1007\/s11340-011-9555-3","article-title":"Out-of-plane Testing Procedure for Inverse Identification Purpose: Application in Sheet Metal Plasticity","volume":"52","author":"Pottier","year":"2012","journal-title":"Exp. Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.ijmecsci.2015.07.026","article-title":"Design of an indicator to characterize and classify mechanical tests for sheet metals","volume":"101\u2013102","author":"Souto","year":"2015","journal-title":"Int. J. Mech. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4152963","DOI":"10.1155\/2016\/4152963","article-title":"Inverse Strategies for Identifying the Parameters of Constitutive Laws of Metal Sheets","volume":"2016","author":"Prates","year":"2016","journal-title":"Adv. Mater. Sci. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"090002","DOI":"10.1063\/1.5034928","article-title":"Experimental validation of a new heterogeneous mechanical test design","volume":"1960","author":"Aquino","year":"2018","journal-title":"AIP Conf. Proc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Souto, N., Thuillier, S., and Andrade-Campos, A. (2016). Design of a mechanical test to characterize sheet metals\u2014Optimization using B-splines or cubic splines. AIP Conf. Proc., 1769.","DOI":"10.1063\/1.4963427"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"110001","DOI":"10.1063\/1.5034958","article-title":"On the design of innovative heterogeneous tests using a shape optimization approach","volume":"1960","author":"Aquino","year":"2018","journal-title":"AIP Conf. Proc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.jmatprotec.2015.01.017","article-title":"Material parameter identification within an integrated methodology considering anisotropy, hardening and rupture","volume":"220","author":"Souto","year":"2015","journal-title":"J. Mater. Process. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.ijsolstr.2017.06.029","article-title":"Evaluation and calibration of anisotropic yield criteria in shear Loading: Constraints to eliminate numerical artefacts","volume":"151","author":"Abedini","year":"2018","journal-title":"Int. J. Solids Struct."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Piegl, L., and Tiller, W. (1997). The NURBS Book, Springer.","DOI":"10.1007\/978-3-642-59223-2"},{"key":"ref_20","unstructured":"Smith, M. (2009). ABAQUS\/Standard User\u2019s Manual, Version 6.9, Simulia."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF00933141","article-title":"Convergence of the Control Parameterization Ritz Method for Nonlinear Optimal Control Problems","volume":"29","author":"Sirisena","year":"1979","journal-title":"J. Optim. Theory Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1137\/S1052623497330392","article-title":"On the Global Convergence of Derivative-Free Methods for Unconstrained Optimization","volume":"13","author":"Lucidi","year":"2002","journal-title":"SIAM J. Optim."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/BF00547132","article-title":"A review of techniques for parameter sensitivity analysis of environmental models","volume":"32","author":"Hamby","year":"1994","journal-title":"Environ. Monit. Assess."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1007\/s10898-012-9951-y","article-title":"Derivative-free optimization: A review of algorithms and comparison of software implementations","volume":"56","author":"Rios","year":"2013","journal-title":"J. Glob. Optim."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1145\/321062.321069","article-title":"\u201cDirect Search\u201d Solution of Numerical and Statistical Problems","volume":"8","author":"Hooke","year":"1961","journal-title":"J. ACM"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1111\/j.1475-3995.2005.00503.x","article-title":"Comparison of deterministic and stochastic approaches to global optimization","volume":"12","author":"Liberti","year":"2005","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Cavazzuti, M. (2013). Optimization Methods: From Theory to Design: Scientific and Technological Aspects in Mechanics, Springer.","DOI":"10.1007\/978-3-642-31187-1"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1162\/106365601750190398","article-title":"Completely Derandomized Self-Adaptation in Evolution Strategies","volume":"9","author":"Hansen","year":"2001","journal-title":"Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1137\/060671267","article-title":"Erratum: Mesh Adaptive Direct Search Algorithms for Constrained Optimization","volume":"18","author":"Audet","year":"2008","journal-title":"SIAM J. Optim."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s11340-007-9039-7","article-title":"Calibration of Stress-triaxiality Dependent Crack Formation Criteria: A New Hybrid Experimental\u2013Numerical Method","volume":"47","author":"Mohr","year":"2007","journal-title":"Exp. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s11340-016-0211-9","article-title":"Fracture Characterization of Rolled Sheet Alloys in Shear Loading: Studies of Specimen Geometry, Anisotropy, and Rate Sensitivity","volume":"57","author":"Abedini","year":"2017","journal-title":"Exp. Mech."}],"container-title":["Metals"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-4701\/9\/3\/371\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:40:00Z","timestamp":1760186400000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-4701\/9\/3\/371"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,22]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["met9030371"],"URL":"https:\/\/doi.org\/10.3390\/met9030371","relation":{},"ISSN":["2075-4701"],"issn-type":[{"value":"2075-4701","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,22]]}}}