# zbMATH — the first resource for mathematics

Anisotropy and shape optimal design of shells by the polar-isogeometric approach. (English) Zbl 1439.49031
This paper is concerned with the optimal design of composite structures, in particular, of shells. The objective is to maximise the shell stiffness by optimising over both the shape, i.e., a domain $$\Omega \subset \mathbb{R}^3$$, and the elastic anisotropic properties that may vary pointwise on $$\Omega$$, i.e., an elastic tensor field $$\mathbb{E}:\Omega \to \bigotimes^4\mathbb{R}^3$$.
The model adopted in this paper is Naghdi’s shell model, and the (static) state equation is given via a variational formulation, namely the “virtual work principle”. This principle states that the virtual work of applied loads is equal to the strain energy. An optimisation problem is formulated to find the argmin of an objective function constructed based on the strain energy, subject to various constraints arising from the geometry of the problem.
One of the main features of this paper, apart from the aforementioned formulation of the optimal control problem, is the combination of the isogeometric approach for determining basis functions and the polar formalism for representing the components of elastic tensors. Several interesting numerical examples are studied thoroughly, and various open questions are discussed towards the end of the paper.
Reviewer: Siran Li (Houston)
##### MSC:
 49J53 Set-valued and variational analysis 49K99 Optimality conditions 74K25 Shells 49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
BIANCA
Full Text:
##### References:
 [1] Banichuk, Nv, Problems and Methods of Optimal Structural Design (1983), New York: Springer, New York [2] Allaire, G., Shape Optimization by the Homogenization Method (2002), New York: Springer, New York · Zbl 0990.35001 [3] Bendsøe, Mp; Sigmund, O., Topology Optimization (2004), Berlin: Springer, Berlin [4] Gurdal, Z.; Haftka, Rt; Hajela, P., Design and Optimization of Laminated Composite Materials (1999), New York: Wileys, New York [5] Vannucci, P., Anisotropic Elasticity (2018), Singapore: Springer, Singapore · Zbl 1401.74001 [6] Montemurro, M.; Vincenti, A.; Vannucci, P., Design of the elastic properties of laminates with a minimum number of plies, Mech. Compos. Mater., 48, 4, 369-390 (2012) [7] Vannucci, P., Designing the elastic properties of laminates as an optimisation problem: a unified approach based on polar tensor invariants, Struct. Multidiscip. Optim., 31, 5, 378-387 (2005) · Zbl 1245.74008 [8] Vannucci, P.; Vincenti, A., The design of laminates with given thermal/hygral expansion coefficients: a general approach based upon the polar-genetic method, Compos. Struct., 79, 3, 454-466 (2007) [9] Vannucci, P.; Barsotti, R.; Bennati, S., Exact optimal flexural design of laminates, Compos. Struct., 90, 3, 337-345 (2009) [10] Vincenti, A.; Desmorat, B., Optimal orthotropy for minimum elastic energy by the polar method, J. Elast., 102, 1, 55-78 (2010) · Zbl 1273.74109 [11] Vincenti, A.; Vannucci, P.; Ahmadian, Mr, Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy, Mech. Adv. Mater. Struct., 20, 3, 242-255 (2013) [12] Catapano, A.; Desmorat, B.; Vannucci, P., Stiffness and strength optimization of the anisotropy distribution for laminated structures, J. Optim. Theory Appl., 167, 1, 118-146 (2015) · Zbl 1327.74046 [13] Jibawy, A.; Julien, C.; Desmorat, B.; Vincenti, A.; Léné, F., Hierarchical structural optimization of laminated plates using polar representation, Int. J. Solids Struct., 48, 18, 2576-2584 (2011) [14] Montemurro, M.; Vincenti, A.; Vannucci, P., A two-level procedure for the global optimum design of composite modular structures—application to the design of an aircraft wing. Part 1: theoretical formulation, J. Optim. Theory Appl., 155, 1, 1-23 (2012) · Zbl 1255.90099 [15] Montemurro, M.; Vincenti, A.; Vannucci, P., A two-level procedure for the global optimum design of composite modular structures—application to the design of an aircraft wing. Part 2: numerical aspects and examples, J. Optim. Theory Appl., 155, 24-53 (2012) · Zbl 1255.90098 [16] Vannucci, P., Strange laminates, Math. Methods Appl. Sci., 35, 13, 1532-1546 (2012) · Zbl 1247.74054 [17] Hughes, T.; Cottrell, J.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194, 39-41, 4135-4195 (2005) · Zbl 1151.74419 [18] Bazilevs, Y.; Calo, Vm; Hughes, Tjr; Zhang, Y., Isogeometric fluid-structure interaction: theory, algorithms, and computations, Comput. Mech., 43, 1, 3-37 (2008) · Zbl 1169.74015 [19] Bazilevs, Y.; Calo, Vm; Zhang, Y.; Hughes, Tjr, Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput. Mech., 38, 4, 310-322 (2006) · Zbl 1161.74020 [20] Bazilevs, Y.; Hsu, Mc; Scott, M., Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines, Comput. Methods Appl. Mech. Eng., 249-252, 28-41 (2012) · Zbl 1348.74094 [21] Cho, S.; Ha, Sh, Isogeometric shape design optimization: exact geometry and enhanced sensitivity, Struct. Multidiscip. Optim., 38, 1, 53-70 (2008) · Zbl 1274.74221 [22] Qian, X., Full analytical sensitivities in NURBS based isogeometric shape optimization, Comput. Methods Appl. Mech. Eng., 199, 29-32, 2059-2071 (2010) · Zbl 1231.74352 [23] De Nazelle, P.: Paramétrage de formes surfaciques pour l’optimisation. Ph.D. thesis, Ecole Centrale Lyon (2013) [24] Julisson, S.: Shape optimization of thin shell structures for complex geometries. Ph.D. thesis, Paris-Saclay University. https://tel.archives-ouvertes.fr/tel-01503061 (2016) [25] Kpadonou, D.F.: Shape and anisotropy optimization by an isogeometric-polar approach. Ph.D. thesis, Paris-Saclay University (2017) [26] Frediani, Aldo; Mohammadi, Bijan; Pironneau, Olivier; Cipolla, Vittorio, Variational Analysis and Aerospace Engineering (2016), Cham: Springer International Publishing, Cham · Zbl 1355.93004 [27] Montemurro, M.; Catapano, A., On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates, Compos. Struct., 161, 145-159 (2017) [28] Morgan, K., The finite element method for elliptic problems, phillipe g. ciarlet, north-holland, amsterdam, 1978. no. of pages 530. price $${\$$ [29] Banichuk, Nv, Introduction to Optimization of Structures (1990), New York: Springer, New York [30] Ciarlet, Pg, An Introduction to Differential Geometry with Applications to Elasticity (2005), New York: Springer, New York [31] Love, Aeh, The small free vibrations and deformation of a thin elastic shell, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 179, 491-546 (1888) · JFM 20.1075.01 [32] Koiter, W.: Foundations and Basic Equations of Shell Theory: A Survey of Recent Progress. Afdeling der Werktuigbouwkunde: WTHD. Labor. voor Techn. Mechanica (1968) [33] Naghdi, P., Foundations of Elastic Shell Theory (1963), Amsterdam: North-Holland Publishing CO., Amsterdam [34] Reissner, E., On the theory of transverse bending of elastic plates, Int. J. Solids Struct., 12, 8, 545-554 (1976) · Zbl 0336.73026 [35] Bézier, P.: Essai de définition numérique des courbes et des surfaces expérimentales: contribution à l’étude des propriétés des courbes et des surfaces paramétriques polynomiales à coefficients vectoriels, vol. 1. Ph.D. thesis, University Paris 6 (1977) [36] Rogers, Df, An Introduction to NURBS with Historical Perspective (2001), Amsterdam: Elsevier, Amsterdam [37] De Boor, C., On the evaluation of box splines, Numer. Algorithms, 5, 1, 5-23 (1993) · Zbl 0798.65012 [38] Verchery, G.; Boehler, Jp, Les invariants des tenseurs d’ordre 4 du type de l’élasticité., Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, 93-104 (1982), Berlin: Springer, Berlin · Zbl 0516.73015 [39] Vannucci, P., Plane anisotropy by the polar method, Meccanica, 40, 437-454 (2005) · Zbl 1106.74016 [40] Vannucci, P., A note on the elastic and geometric bounds for composite laminates, J. Elast., 112, 199-215 (2013) · Zbl 1267.74023 [41] Vannucci, P.; Verchery, G., Anisotropy of plane complex elastic bodies, Int. J. Solids Struct., 47, 1154-1166 (2010) · Zbl 1193.74021 [42] Valot, E.; Vannucci, P., Some exact solutions for fully orthotropic laminates, Compos. Struct., 69, 2, 157-166 (2005) [43] Vannucci, P.; Pouget, J., Laminates with given piezoelectric expansion coefficients, Mech. Adv. Mater. Struct., 13, 5, 419-427 (2006) [44] Vannucci, P., Influence of invariant material parameters on the flexural optimal design of thin anisotropic laminates, Int. J. Mech. Sci., 51, 192-203 (2009) · Zbl 1264.74187 [45] Vannucci, P., A new general approach for optimizing the performances of smart laminates, Mech. Adv. Mater. Struct., 18, 7, 548-558 (2011) [46] Montemurro, M.; Koutsawa, Y.; Belouettar, S.; Vincenti, A.; Vannucci, P., Design of damping properties of hybrid laminates through a global optimisation strategy, Compos. Struct., 94, 11, 3309-3320 (2012) [47] Vannucci, P., The design of laminates as a global optimization problem, J. Optim. Theory Appl., 157, 2, 299-323 (2012) · Zbl 1267.90115 [48] Montemurro, M.; Vincenti, A.; Koutsawa, Y.; Vannucci, P., A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches, J. Vib. Control, 21, 9, 1778-1800 (2013) [49] Vannucci, P., The polar analysis of a third order piezoelectricity-like plane tensor, Int. J. Solids Struct., 44, 24, 7803-7815 (2007) · Zbl 1167.74409 [50] Vannucci, P., On special orthotropy of paper, J. Elast., 99, 1, 75-83 (2009) · Zbl 1188.74012 [51] Catapano, A.; Desmorat, B.; Vannucci, P., Invariant formulation of phenomenological failure criteria for orthotropic sheets and optimisation of their strength, Math. Methods Appl. Sci., 35, 15, 1842-1858 (2012) · Zbl 06108500 [52] Barsotti, R.; Vannucci, P., Wrinkling of orthotropic membranes: an analysis by the polar method, J. Elast., 113, 1, 5-26 (2012) · Zbl 1277.35030 [53] Vannucci, P., General theory of coupled thermally stable anisotropic laminates, J. Elast., 113, 2, 147-166 (2012) · Zbl 1336.74017 [54] Desmorat, B.; Vannucci, P., An alternative to the Kelvin decomposition for plane anisotropic elasticity, Math. Methods Appl. Sci., 38, 1, 164-175 (2013) · Zbl 1320.74030 [55] Vannucci, P.; Desmorat, B., Analytical bounds for damage induced planar anisotropy, Int. J. Solids Struct., 60-61, 96-106 (2015) [56] Vannucci, P., A note on the computation of the extrema of young’s modulus for hexagonal materials: an approach by planar tensor invariants, Appl. Math. Comput., 270, 124-129 (2015) · Zbl 1410.74007 [57] Vannucci, P.; Desmorat, B., Plane anisotropic rari-constant materials, Math. Methods Appl. Sci., 39, 12, 3271-3281 (2015) · Zbl 1343.74010 [58] Vannucci, P., A special planar orthotropic material, J. Elast., 67, 81-96 (2002) · Zbl 1089.74565 [59] Vannucci, P.; Verchery, G., Stiffness design of laminates using the polar method, Int. J. Solids Struct., 38, 50-51, 9281-9294 (2001) · Zbl 1016.74019 [60] De Boor, C., A Practical Guide to Splines. Applied Mathematical Sciences (2001), New York: Springer, New York · Zbl 0987.65015 [61] Patrikalakis, Nm; Maekawa, T., Shape Interrogation for Computer Aided Design and Manufacturing (2009), Berlin: Springer, Berlin [62] Hooke, R., A Description of Helioscopes, and Some Other Instruments (1675), London: John and Martyn Printer, London [63] Heyman, J., The Stone Skeleton (1995), Cambridge: Cambridge University Press, Cambridge [64] Cowan, Hj, The Masterbuilders (1977), New York: Wiley, New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.