{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:46:25Z","timestamp":1740123985817,"version":"3.37.3"},"reference-count":76,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,8,13]],"date-time":"2019-08-13T00:00:00Z","timestamp":1565654400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,8,13]],"date-time":"2019-08-13T00:00:00Z","timestamp":1565654400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Italian Ministry of Education, University and Research","award":["\u2018Departments of Excellence\u2019 grant L.232\/2016"],"award-info":[{"award-number":["\u2018Departments of Excellence\u2019 grant L.232\/2016"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s10957-019-01572-1","type":"journal-article","created":{"date-parts":[[2019,8,13]],"date-time":"2019-08-13T07:04:39Z","timestamp":1565679879000},"page":"613-629","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Optimal Prediction of the Stress Field Associated with Discrete Element Models"],"prefix":"10.1007","volume":"187","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2562-881X","authenticated-orcid":false,"given":"Ada","family":"Amendola","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,8,13]]},"reference":[{"key":"1572_CR1","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1016\/S0022-5096(98)00051-9","volume":"47","author":"VB Shenoy","year":"1999","unstructured":"Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics\u2014the quasicontinuum method. J. Mech. Phys. Solids 47, 611\u2013642 (1999)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR2","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1016\/S0022-5096(01)00034-5","volume":"49","author":"J Knap","year":"2001","unstructured":"Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899\u20131923 (2001)","journal-title":"J. Mech. Phys. Solids"},{"issue":"1","key":"1572_CR3","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1615\/IntJMultCompEng.v1.i1.100","volume":"1","author":"T Belytschko","year":"2003","unstructured":"Belytschko, T., Xiao, S.P.: Coupling methods for continuum model with molecular model. Int. J. Multiscale Comput. Eng. 1(1), 115\u2013126 (2003)","journal-title":"Int. J. Multiscale Comput. Eng."},{"key":"1572_CR4","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1016\/j.cma.2003.12.037","volume":"193","author":"B Liu","year":"2004","unstructured":"Liu, B., Huang, Y., Jiang, H., Qu, S., Hwang, K.C.: The atomic-scale finite element method. Comput. Methods Appl. Mech. Eng. 193, 1849\u20131864 (2004)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"1572_CR5","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/07069969X","volume":"7","author":"S Badia","year":"2008","unstructured":"Badia, S., Parks, M., Bochev, P., Gunzburger, M., Lehoucq, R.: On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7(1), 381\u2013406 (2008)","journal-title":"Multiscale Model. Simul."},{"key":"1572_CR6","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.cma.2012.05.010","volume":"237\u2013240","author":"Q Yang","year":"2012","unstructured":"Yang, Q., Biyikli, E., Zhang, P., Tian, R., To, A.C.: Atom collocation method. Comput. Methods Appl. Mech. Eng. 237\u2013240, 67\u201377 (2012)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1572_CR7","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.cma.2013.01.014","volume":"258","author":"Q Yang","year":"2013","unstructured":"Yang, Q., Biyikli, E., To, A.C.: Multiresolution molecular mechanics: statics. Comput. Methods Appl. Mech. Eng. 258, 26\u201338 (2013)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1572_CR8","doi-asserted-by":"crossref","first-page":"053001","DOI":"10.1088\/0965-0393\/17\/5\/053001","volume":"17","author":"RE Miller","year":"2009","unstructured":"Miller, R.E., Tadmor, E.B.: A unified framework and performance benchmark of fourteen multiscale atomistic\/continuum coupling methods. J. Model. Simul. Mater. Sci. Eng. 17, 053001 (2009)","journal-title":"J. Model. Simul. Mater. Sci. Eng."},{"key":"1572_CR9","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1166\/jctn.2008.2485","volume":"5","author":"ZC Tu","year":"2008","unstructured":"Tu, Z.C., Ou-Yang, Z.C.: Elastic theory of low-dimensional continua and its application in bio- and nano-structures. J. Comput. Theor. Nanosci. 5, 422\u2013448 (2008)","journal-title":"J. Comput. Theor. Nanosci."},{"issue":"1","key":"1572_CR10","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.jmps.2010.09.004","volume":"59","author":"F Fraternali","year":"2010","unstructured":"Fraternali, F., Blegsen, M., Amendola, A., Daraio, C.: Multiscale mass-spring models of carbon nanotube foams. J. Mech. Phys. Solids 59(1), 89\u2013102 (2010)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR11","doi-asserted-by":"crossref","first-page":"3013","DOI":"10.1016\/j.compstruct.2011.04.034","volume":"93","author":"JR Raney","year":"2011","unstructured":"Raney, J.R., Fraternali, F., Amendola, A., Daraio, C.: Modeling and in situ identification of material parameters for layered structures based on carbon nanotube arrays. Compos. Struct. 93, 3013\u20133018 (2011)","journal-title":"Compos. Struct."},{"key":"1572_CR12","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.mechrescom.2012.07.006","volume":"45","author":"T Blesgen","year":"2012","unstructured":"Blesgen, T., Fraternali, F., Raney, J.R., Amendola, A., Daraio, C.: Continuum limits of bistable spring models of carbon nanotube arrays accounting for material damage. Mech. Res. Commun. 45, 58\u201363 (2012)","journal-title":"Mech. Res. Commun."},{"issue":"8","key":"1572_CR13","first-page":"383","volume":"8","author":"P Trovalusci","year":"1998","unstructured":"Trovalusci, P., Augusti, G.: A continuum model with microstructure for materials with flaws and inclusions. J. Phys. IV 8(8), 383\u2013390 (1998)","journal-title":"J. Phys. IV"},{"key":"1572_CR14","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.compositesb.2017.06.015","volume":"128","author":"P Trovalusci","year":"2017","unstructured":"Trovalusci, P., De Bellis, M.L., Masiani, R.: A multiscale description of particle composites: from lattice microstructures to micropolar continua. Compos. Part B Eng. 128, 164\u2013173 (2017)","journal-title":"Compos. Part B Eng."},{"key":"1572_CR15","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.compositesb.2017.10.017","volume":"136","author":"E Reccia","year":"2018","unstructured":"Reccia, E., De Bellis, M.L., Trovalusci, P., Masiani, R.: Sensitivity to material contrast in homogenization of random particle composites as micropolar continua. Compos. Part B Eng. 136, 39\u201345 (2018)","journal-title":"Compos. Part B Eng."},{"key":"1572_CR16","doi-asserted-by":"crossref","first-page":"2997","DOI":"10.1103\/PhysRevB.56.2997","volume":"56","author":"Z Zhou","year":"1997","unstructured":"Zhou, Z., Jo\u00f3s, B.: Mechanisms of membrane rupture: from cracks to pores. Phys. Rev. B 56, 2997\u20133009 (1997)","journal-title":"Phys. Rev. B"},{"key":"1572_CR17","volume-title":"Statistical Mechanics of Membranes and Surfaces","year":"2004","unstructured":"Nelson, D., Piran, T., Weinberg, S. (eds.): Statistical Mechanics of Membranes and Surfaces, 2nd edn. World Scientific, Singapore (2004)","edition":"2"},{"key":"1572_CR18","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.physrep.2006.08.003","volume":"434","author":"M Muller","year":"2006","unstructured":"Muller, M., Katsov, K., Schick, M.: Biological and synthetic membranes: what can be learned from a coarse-grained description? Phys. Rep. 434, 113\u2013176 (2006)","journal-title":"Phys. Rep."},{"key":"1572_CR19","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1016\/j.msec.2005.08.020","volume":"26","author":"M Dao","year":"2006","unstructured":"Dao, M., Li, J., Suresh, S.: Molecular based analysis of deformation of spectrin network and human erythrocyte. Mater. Sci. Eng. 26, 1232\u20131244 (2006)","journal-title":"Mater. Sci. Eng."},{"key":"1572_CR20","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.jcp.2011.09.017","volume":"231","author":"F Fraternali","year":"2012","unstructured":"Fraternali, F., Lorenz, C., Marcelli, G.: On the estimation of the curvatures and bending rigidity of membrane networks via a local maximum-entropy approach. J. Comput. Phys. 231, 528\u2013540 (2012)","journal-title":"J. Comput. Phys."},{"key":"1572_CR21","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.jmps.2011.09.003","volume":"60","author":"B Schmidt","year":"2012","unstructured":"Schmidt, B., Fraternali, F.: Universal formulae for the limiting elastic energy of membrane networks. J. Mech. Phys. Solids 60, 172\u2013180 (2012)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR22","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1007\/s10237-012-0376-9","volume":"11","author":"F Fraternali","year":"2012","unstructured":"Fraternali, F., Marcelli, G.: A multiscale approach to the elastic moduli of biomembrane networks. Biomech. Model. Mechanobiol. 11, 1097\u20131108 (2012)","journal-title":"Biomech. Model. Mechanobiol."},{"issue":"4","key":"1572_CR23","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1080\/01418619108213953","volume":"64","author":"S Kohlhoff","year":"1991","unstructured":"Kohlhoff, S., Gumbsch, P., Fischmeister, F.: Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model. Philos. Mag. A 64(4), 851\u2013878 (1991)","journal-title":"Philos. Mag. A"},{"key":"1572_CR24","doi-asserted-by":"crossref","first-page":"1318","DOI":"10.1016\/j.jmps.2010.06.001","volume":"58","author":"RE Jones","year":"2010","unstructured":"Jones, R.E., Zimmerman, J.A.: The construction and application of an atomistic J-integral via Hardy estimates of continuum fields. J. Mech. Phys. Solids 58, 1318\u20131337 (2010)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR25","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1137\/S0036142995296102","volume":"35","author":"C Davini","year":"2000","unstructured":"Davini, C., Pitacco, I.: Relaxed notions of curvature and a lumped strain method for elastic plates. SIAM J. Numer. Anal. 35, 677\u2013691 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"1572_CR26","doi-asserted-by":"crossref","first-page":"6211","DOI":"10.1016\/S0020-7683(02)00472-9","volume":"39","author":"F Fraternali","year":"2002","unstructured":"Fraternali, F., Angelillo, M., Fortunato, A.: A lumped stress method for plane elastic problems and the discrete-continuum approximation. Int. J. Solids Struct. 39, 6211\u20136240 (2002)","journal-title":"Int. J. Solids Struct."},{"key":"1572_CR27","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107050310","volume-title":"The Stone Skeleton","author":"J Heyman","year":"1995","unstructured":"Heyman, J.: The Stone Skeleton. Cambridge University Press, Cambridge (1995)"},{"key":"1572_CR28","first-page":"187","volume":"73","author":"D O\u2018Dwyer","year":"1999","unstructured":"O\u2018Dwyer, D.: Funicular analysis of masonry vaults. Int. J. Solids Struct. 73, 187\u2013197 (1999)","journal-title":"Int. J. Solids Struct."},{"issue":"2","key":"1572_CR29","first-page":"77","volume":"46","author":"A Kilian","year":"2005","unstructured":"Kilian, A., Ochsendorf, J.: Particle-spring systems for structural form finding. IASS J. 46(2), 77\u201385 (2005)","journal-title":"IASS J."},{"issue":"3","key":"1572_CR30","first-page":"167","volume":"48","author":"P Block","year":"2007","unstructured":"Block, P., Ochsendorf, J.: Thrust network analysis: a new methodology for three-dimensional equilibrium. J. IASS 48(3), 167\u2013173 (2007)","journal-title":"J. IASS"},{"key":"1572_CR31","unstructured":"Block, P.: Thrust network analysis: exploring Three-dimensional equilibrium. Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, USA (2009)"},{"key":"1572_CR32","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.mechrescom.2009.12.010","volume":"37","author":"F Fraternali","year":"2010","unstructured":"Fraternali, F.: A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech. Res. Commun. 37, 198\u2013204 (2010)","journal-title":"Mech. Res. Commun."},{"key":"1572_CR33","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.mechrescom.2011.03.008","volume":"38","author":"F Fraternali","year":"2011","unstructured":"Fraternali, F.: A mixed lumped stress\u2013displacement approach to the elastic problem of masonry walls. Mech. Res. Commun. 38, 176\u2013180 (2011)","journal-title":"Mech. Res. Commun."},{"key":"1572_CR34","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.compositesb.2017.03.004","volume":"118","author":"F Greco","year":"2017","unstructured":"Greco, F., Leonetti, L., Luciano, R., Trovalusci, P.: Multiscale failure analysis of periodic masonry structures with traditional and fiber-reinforced mortar joints. Compos. Part B Eng. 118, 75\u201395 (2017)","journal-title":"Compos. Part B Eng."},{"key":"1572_CR35","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1002\/stc.8","volume":"9","author":"RE Skelton","year":"2002","unstructured":"Skelton, R.E.: Structural systems: a marriage of structural engineering ans system science. J. Struct. Control 9, 113\u2013133 (2002)","journal-title":"J. Struct. Control"},{"issue":"10","key":"1572_CR36","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1007\/s10439-005-4698-y","volume":"33","author":"C Vera","year":"2005","unstructured":"Vera, C., Skelton, R.E., Bosscns, F., Sung, L.A.: 3-D nanomechanics of an erythrocyte junctional complex in equibiaxial and anisotropic deformations. Ann. Biomed. Eng. 33(10), 1387\u20131404 (2005)","journal-title":"Ann. Biomed. Eng."},{"volume-title":"Cytoskeletal Mechanics: Models and Measurements","year":"2006","key":"1572_CR37","unstructured":"Mofrad, M.R.K., Kamm, R.D. (eds.): Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press, Cambridge (2006)"},{"key":"1572_CR38","volume-title":"Tensegrity Systems","author":"RE Skelton","year":"2010","unstructured":"Skelton, R.E., de Oliveira, M.C.: Tensegrity Systems. Springer, Berlin (2010)"},{"key":"1572_CR39","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1016\/j.jmps.2012.02.007","volume":"60","author":"F Fraternali","year":"2012","unstructured":"Fraternali, F., Senatore, L., Daraio, C.: Solitary waves on tensegrity lattices. J. Mech. Phys. Solids 60, 1137\u20131144 (2012)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR40","doi-asserted-by":"crossref","first-page":"5728","DOI":"10.1016\/j.ces.2008.08.006","volume":"63","author":"HP Zhu","year":"2008","unstructured":"Zhu, H.P., Zhou, Z.Y., Yang, R.Y., Yu, A.B.: Discrete particle simulation of particulate systems: a review of mayor applications and findings. Chem. Eng. Sci. 63, 5728\u20135770 (2008)","journal-title":"Chem. Eng. Sci."},{"key":"1572_CR41","doi-asserted-by":"crossref","first-page":"036603","DOI":"10.1103\/PhysRevE.82.036603","volume":"82","author":"C Daraio","year":"2010","unstructured":"Daraio, C., Ngo, D., Nesterenko, V.F., Fraternali, F.: Highly nonlinear pulse splitting and recombination in a two-dimensional granular network. Phys. Rev. E 82, 036603 (2010)","journal-title":"Phys. Rev. E"},{"key":"1572_CR42","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s11340-011-9544-6","volume":"53","author":"A Leonard","year":"2013","unstructured":"Leonard, A., Fraternali, F., Daraio, C.: Directional wave propagation in a highly nonlinear square packing of spheres. Exp. Mech. 53, 327\u2013337 (2013)","journal-title":"Exp. Mech."},{"issue":"11","key":"1572_CR43","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1007\/s00419-008-0269-7","volume":"79","author":"P Trovalusci","year":"2009","unstructured":"Trovalusci, P., Capecchi, D., Ruta, G.: Genesis of the multiscale approach for materials with microstructure. Arch. Appl. Mech. 79(11), 981\u2013997 (2009)","journal-title":"Arch. Appl. Mech."},{"issue":"11","key":"1572_CR44","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1007\/s00419-010-0502-z","volume":"81","author":"D Capecchi","year":"2011","unstructured":"Capecchi, D., Ruta, G., Trovalusci, P.: Voigt and Poincar\u00e9\u2019s mechanistic-energetic approaches to linear elasticity and suggestions for multiscale modelling. Arch. Appl. Mech. 81(11), 1573\u20131584 (2011)","journal-title":"Arch. Appl. Mech."},{"key":"1572_CR45","series-title":"CISM (International Centre for Mechanical Sciences) Series","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/978-3-7091-1812-2_7","volume-title":"Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects","author":"P Trovalusci","year":"2014","unstructured":"Trovalusci, P.: Molecular approaches for multifield continua: origins and current developments. In: Sadowski, T., Trovalusci, P. (eds.) Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects. CISM (International Centre for Mechanical Sciences) Series, vol. 556, pp. 211\u2013278. Springer, Berlin (2014)"},{"key":"1572_CR46","first-page":"74","volume":"32","author":"J Schlaich","year":"1987","unstructured":"Schlaich, J., Sch\u00e4fer, K., Jennewein, M.: Toward a consistent design of structural concrete. J. Prestress. Concr. Inst. 32, 74\u2013150 (1987)","journal-title":"J. Prestress. Concr. Inst."},{"key":"1572_CR47","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10659-010-9249-6","volume":"100","author":"NC Admal","year":"2010","unstructured":"Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecular systems. J. Elast. 100, 63\u2013143 (2010)","journal-title":"J. Elast."},{"issue":"3","key":"1572_CR48","first-page":"552","volume":"10","author":"S Morante","year":"2012","unstructured":"Morante, S., Rossi, G., Testa, M.: The stress tensor of an atomistic system. Cent. Eur. J. Phys. 10(3), 552\u2013559 (2012)","journal-title":"Cent. Eur. J. Phys."},{"key":"1572_CR49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S0036141003426471","volume":"36","author":"R Alicandro","year":"2005","unstructured":"Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1\u201337 (2005)","journal-title":"SIAM J. Math. Anal."},{"key":"1572_CR50","series-title":"Lecture Notes of the Unione Matematica Italiana","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/978-3-540-36546-4_1","volume-title":"Topics on Concentration Phenomena and Problems with Multiple Scales","author":"A Braides","year":"2006","unstructured":"Braides, A., Gelli, M.S.: From discrete systems to continuous variational problems: an introduction. In: Braides, A., Chiad\u00f2, V. (eds.) Topics on Concentration Phenomena and Problems with Multiple Scales. Lecture Notes of the Unione Matematica Italiana, vol. 2, pp. 3\u201377. Springer, Berlin (2006)"},{"key":"1572_CR51","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1137\/050646251","volume":"5","author":"B Schmidt","year":"2006","unstructured":"Schmidt, B.: A derivation of continuum nonlinear plate theory from atomistic models. SIAM Multiscale Model. Simul. 5, 664\u2013694 (2006)","journal-title":"SIAM Multiscale Model. Simul."},{"key":"1572_CR52","doi-asserted-by":"crossref","first-page":"551","DOI":"10.3934\/nhm.2007.2.551","volume":"2","author":"A Braides","year":"2007","unstructured":"Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2, 551\u2013567 (2007)","journal-title":"Netw. Heterog. Media"},{"key":"1572_CR53","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1016\/j.jmps.2006.12.004","volume":"55","author":"GJ Rodin","year":"2007","unstructured":"Rodin, G.J.: Higher-order macroscopic measures. J. Mech. Phys. Solids 55, 1103\u20131119 (2007)","journal-title":"J. Mech. Phys. Solids"},{"key":"1572_CR54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00205-008-0138-0","volume":"190","author":"B Schmidt","year":"2008","unstructured":"Schmidt, B.: On the passage from atomic to continuum theory of thin films. Arch. Ration. Mech. Anal. 190, 1\u201355 (2008)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"1572_CR55","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/S0167-8442(00)00054-9","volume":"35","author":"F Fraternali","year":"2001","unstructured":"Fraternali, F.: Complementary energy variational approach for plane elastic problems with singularities. Theor. Appl. Fract. Mech. 35, 129\u2013135 (2001)","journal-title":"Theor. Appl. Fract. Mech."},{"issue":"4","key":"1572_CR56","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1080\/15376490600845587","volume":"14","author":"F Fraternali","year":"2007","unstructured":"Fraternali, F.: Error estimates for a lumped stress method for plane elastic problems. Mech. Adv. Mater. Struct. 14(4), 309\u2013320 (2007)","journal-title":"Mech. Adv. Mater. Struct."},{"key":"1572_CR57","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0370-2693(87)90541-7","volume":"194","author":"D Espriu","year":"1987","unstructured":"Espriu, D.: Triangulated random surfaces. Phys. Lett. B 194, 271\u2013276 (1987)","journal-title":"Phys. Lett. B"},{"key":"1572_CR58","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1142\/S021773239000192X","volume":"5","author":"CF Bailiie","year":"1990","unstructured":"Bailiie, C.F., Johnston, D.A., Williams, R.D.: Nonuniversality in dynamically triangulated random surfaces with extrinsic curvature. Mod. Phys. Lett. A 5, 1671\u20131683 (1990)","journal-title":"Mod. Phys. Lett. A"},{"key":"1572_CR59","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1051\/jp1:1996246","volume":"6","author":"G Gompper","year":"1996","unstructured":"Gompper, G., Kroll, D.M.: Random surface discretization and the renormalization of the bending rigidity. J. Phys. I Fr. 6, 1305\u20131320 (1996)","journal-title":"J. Phys. I Fr."},{"issue":"4","key":"1572_CR60","doi-asserted-by":"crossref","first-page":"4762","DOI":"10.1103\/PhysRevE.55.4762","volume":"55","author":"DE Discher","year":"1997","unstructured":"Discher, D.E., Boal, D.H., Boey, H.S.: Phase transitions and anisotropic responses of planar triangular nets under large deformation. Phys. Rev. E 55(4), 4762\u20134772 (1997)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"1572_CR61","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1260\/0266-3511.29.3.145","volume":"29","author":"F Fraternali","year":"2014","unstructured":"Fraternali, F., Carpentieri, G.: On the correspondence between 2D force networks and polyhedral stress functions. Int. J. Space Struct. 29(3), 145\u2013159 (2014)","journal-title":"Int. J. Space Struct."},{"key":"1572_CR62","doi-asserted-by":"crossref","first-page":"2276","DOI":"10.1016\/j.ijsolstr.2010.04.024","volume":"47","author":"A Baratta","year":"2010","unstructured":"Baratta, A., Corbi, O.: On the equilibrium and admissibility coupling in NT vaults of general shape. Int. J. Solids Struct. 47, 2276\u20132284 (2010)","journal-title":"Int. J. Solids Struct."},{"key":"1572_CR63","volume-title":"Elements of Algebraic Topology","author":"JR Munkres","year":"1984","unstructured":"Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Boston (1984)"},{"key":"1572_CR64","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1177\/1081286507086898","volume":"13","author":"JL Ericksen","year":"2008","unstructured":"Ericksen, J.L.: On the Cauchy\u2013Born rule. Math. Mech. Solids 13, 199\u2013220 (2008)","journal-title":"Math. Mech. Solids"},{"key":"1572_CR65","doi-asserted-by":"crossref","unstructured":"Gurtin, M.E.: The linear theory of elasticity. In: Handbuch der Physik (Encyclopedia of Physics), VIa\/2, pp. 1\u2013295 Springer-Verlag (1972)","DOI":"10.1007\/978-3-662-39776-3_1"},{"key":"1572_CR66","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams, R.A.: Sobolev Spaces. Academic Press, Cambridge (1975)"},{"key":"1572_CR67","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719208","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"2002","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM Editions, Philadelphia (2002)"},{"key":"1572_CR68","doi-asserted-by":"crossref","first-page":"1575","DOI":"10.1142\/S0218202505000893","volume":"15","author":"I Babuska","year":"2005","unstructured":"Babuska, I., Podnos, G., Rodin, G.J.: New fictitious domain methods: formulation and analysis. Math. Models Methods Appl. Sci. 15, 1575\u20131594 (2005)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1572_CR69","unstructured":"SURF, Visual Numerics, Inc.: Smooth bivariate interpolant to scattered data that is locally a quintic polynomial in two variables. IMSL$$^{\\textregistered }$$ Fortran Numerical Math Library, Chap. 3. http:\/\/www.vni.com (2007)"},{"key":"1572_CR70","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/15376494.2010.483319","volume":"18","author":"F Fraternali","year":"2011","unstructured":"Fraternali, F., Marino, A., El Sayed, T., Della Cioppa, A.: On the structural shape optimization through variational methods and evolutionary algorithms. Mech. Adv. Mater. Struct. 18, 225\u2013243 (2011)","journal-title":"Mech. Adv. Mater. Struct."},{"issue":"2","key":"1572_CR71","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s10957-018-1417-z","volume":"180","author":"AA Novotny","year":"2019","unstructured":"Novotny, A.A., Soko\u0142owski, J., \u017bochowski, A.: Topological derivatives of shape functionals. Part I: theory in singularly perturbed geometrical domains. J. Optim. Theory Appl. 180(2), 341\u2013373 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1572_CR72","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s10957-018-1419-x","volume":"180","author":"AA Novotny","year":"2019","unstructured":"Novotny, A.A., Soko\u0142owski, J., \u017bochowski, A.: Topological derivatives of shape functionals. Part II: first-order method and applications. J. Optim. Theory Appl. 180(3), 683\u2013710 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1572_CR73","doi-asserted-by":"crossref","unstructured":"Fraternali, F., Carpentieri, G., Montuori, R., Amendola, A., Benzoni, G.: On the use of mechanical metamaterials for innovative seismic isolations systems. In: Proceedings of the 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), pp. 349\u2013358 (2015)","DOI":"10.7712\/120115.3401.636"},{"key":"1572_CR74","doi-asserted-by":"publisher","unstructured":"Amendola, A., Fabbrocino, F., Feo, L., Auricchio, F., Fraternali, F.: Dependence of the mechanical properties of pentamode materials on the lattice microstructure. In: Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016), vol. 1, pp. 2134\u20132150 (2016). https:\/\/doi.org\/10.7712\/100016.1947.6004","DOI":"10.7712\/100016.1947.6004"},{"issue":"2","key":"1572_CR75","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s10957-011-9894-3","volume":"152","author":"W Saleem","year":"2012","unstructured":"Saleem, W., Khan, M.A., Ch, S.R.: Formulation and execution of structural topology optimization for practical design solutions. J. Optim. Theory Appl. 152(2), 517\u2013536 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1572_CR76","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1023\/A:1022637216104","volume":"98","author":"W Achtziger","year":"1998","unstructured":"Achtziger, W.: Multiple-load truss topology and sizing optimization: some properties of minimax compliance. J. Optim. Theory Appl. 98(2), 255\u2013280 (1998)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01572-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-019-01572-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01572-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,1]],"date-time":"2020-12-01T08:05:08Z","timestamp":1606809908000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-019-01572-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,13]]},"references-count":76,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["1572"],"URL":"https:\/\/doi.org\/10.1007\/s10957-019-01572-1","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2019,8,13]]},"assertion":[{"value":"24 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}