Isogeometric Analysis for Topology Optimisation of Two Dimensional Planar and Laminated Composite Plate Continuum Structures

Subject Areas : Computational Mechanics

K.N.V Chandrasekhar ¹ , V Bhikshma ² , N Rakesh ³ , N Swapnareddy ⁴ , C Rakesh ⁵

1 - Department of Civil Engineering, CVR College of Engineering, Hyderabad, Telangana, India
2 - University College of Engineering, Osmania University, Hyderabad, Telangana, India
3 - Department of Civil Engineering, CVR College of Engineering, Hyderabad, Telangana, India
4 - Department of Civil Engineering, CVR College of Engineering, Hyderabad, Telangana, India
5 - Department of Civil Engineering, CVR College of Engineering, Hyderabad, Telangana, India

Received: 2021-03-13 Accepted : 2021-05-10 Published : 2021-06-30

Keywords: Topology, Isogeometric, Reinforced Concrete, Optimisation, Laminates,

Abstract :

Isogeometric analysis is the recent development in the field of engineering analysis with high performance computing and greater precision. This current research has opened a new door in the field of structural optimisation. The main focus of this research study is to perform topology optimisation of continuum structures in civil engineering using Isogeometric analysis. The continuum structures analysed here in this study are reinforced concrete, steel and laminated composite plates. Reinforced concrete is a rational union of concrete and steel. Topology optimisation of reinforced concrete structures is an emerging area of study to determine the optimal layout of material in the concrete domain. Laminated structures are made of several layers of material and bonded to achieve high stiffness and low weight to strength ratio. The deformed shape at the optimal state can be determined with topology optimisation of laminated composites. The formulation for composite plates is done using kirchoff thin plate theory without any shear contribution. B-splines are used to model the geometry. The objective is to optimise the energy of the structure and optimality criteria is used to calculate the newer values of relative densities. First order sensitivity analysis is performed to determine the newer values of objective function. The code is written in MatLab® and a few problems have been solved with different domains. The results are verified and have shown a good agreement with those in the literature.

References:

[1] Punmia B.C., Designs R.C.C., 2006, Laxmi Publications, ISBN: 9788131809426.
[2] Liang Q.Q., 1999, Topology optimisation of strut-and-tie models in non-flexural reinforced concrete members, International Conference on Mechanics of Structures, Materials and Systems, Wollongong, Australia.
[3] Leu L.-J., 2006, Strut and tie methodology for three dimensional reinforced concrete structures, Journal of Structural Engineering 132(6): 929-938.
[4] Michelle A.G.M., 1904, LVIII the limits of economy of material in frame-structures, Philosophical Magazine 8(47): 589-597.
[5] Khajehzadeh M., 2011, Modified particle swarm optimization for optimum design of spread footing and retaining wall, Journal of Zhejiang University-Science A 12(6): 415-427.
[6] James B., 2005, A Finite Element Approach to Reinforced Concrete Slab Design, Master of Science Thesis, Georgia Institute of Technology.
[7] Kwak H.G., Filippou F.C., 1992, Finite element analysis of reinforced concrete structures under monotonic and cyclic loads, Earthquake Engineering, Tenth World Conference, Rotterdam.
[8] Kaveh A., Behnam A.F., 2013, Design optimisation of reinforced concrete 3D structures considering frequency constraints via., a charged system search, Scientia Iranica A 20(3): 387-396.
[9] Kim H., Baker G., 2011, Topology Optimisation of Reinforced Concrete Structures, School of Engineering, University of Warwick.
[10] Oded A., 2013, A topology optimisation procedure for reinforced concrete structures, Computers & Structures 114-115: 46-85.
[11] Oded A., Michael B., 2010, Topology optimisation for conceptual design of reinforced concrete structures, 9th World Congress on Structural and Multidisciplinary Optimization, Shizuoka, Japan.
[12] Oded A., Ole S., 2013, Reinforcement layout design for concrete structures based on continuum damage and truss topology optimisation, Structural and Multidisciplinary Optimization 47: 157-174.
[13] Liang Q.Q., 2000, Topology optimisation of strut-and-tie models in reinforced concrete structures using an evolutionary procedure, ACI Structural Journal 97: 322-332.
[14] Sarma K., Adeli H., 1998, Cost optimisation of concrete structures, Journal of Structural Engineering 124(5): 570-578.
[15] Babiker S., 2012, Design optimisation of reinforced concrete beams using artificial neural network, International Journal of Engineering Inventions 1(8): 07-13.
[16] Guerra A., Kiousis P.D., 2006, Design optimisation of reinforced concrete structures, Computers and Concrete 3(5): 313-334.
[17] Singh B., Rai H.S., 2014, Optimisation of rcc beam, International Journal of Engineering, Business and Enterprise Applications 14: 21-34.
[18] Kaveh A., Sabzi O., 2012, Optimal design of reinforced concrete frames using big bang-big crunch algorithm, International Journal of Civil Engineering, Structural Engineering 10(3): 189-200.
[19] Yousif S.T., 2013, Optimum cost design of reinforced concrete continuous beams using genetic algorithms, International Journal of Applied Sciences and Engineering Research 12(4): 681-693.
[20] McCarthy T.J., McCluskey S., 2009, A particle swarm optimisation approach to reinforced concrete beam design according to AS360, Proceedings on the First International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering, Civil-Comp Press, Stirlingshire, UK.
[21] Karthiga N., 2014, Finite element analysis of reinforced concrete beams, International Journal of Innovative research in Advanced Engineering 1(8): 272-285.
[22] Chandrasekhar K.N.V., Sahithi N.S.S., 2018, A step by step illustrative procedure to perform isogeometric analysis and find the nodal displacements for a two dimensional plate structure, i-Managers Journal on Structural Engineering 6: 40-52.
[23] Chandrasekhar K.N.V., Sahithi N.S.S., 2018, A step by step procedure to perform isogeometric analysis of beam and bar problems in civil engineering including sizing optimisation of a beam, i-Managers Journal on Structural Engineering 7: 13-27.
[24] Sahithi N.S.S., Chandrasekhar K.N.V., 2019, Isogeometric topology optimisation of continuum structures using evolutionary algorithms, Journal of Applied Computational Mechanics 5: 414-440.
[25] Sairam K.S., Sinha P.K., 1991, Hygrothermal effects on the bending characteristics of laminated composite plates, Computers & Structures 40(4): 1009-1015.
[26] Kant T., Swaminathan K., 2001, Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory, Composite Structures 53: 73-85.
[27] Hughes T.J.R., 2010, Efficient quadrature for NURBS-based isogeometric analysis, Computer Methods in Applied Mechanical and Engineering 199: 301-313.
[28] Hassani B., 2012, An isogeomtrical approach to structural topology optimisation by optimality criteria, Structural and Multidisciplinary Optimization 45: 223-233.
[29] Benson D.J., 2011, A large deformation, rotation-free, isogeometric shell, Computer Methods in Applied Mechanical and Engineering 200: 1367-1378.
[30] Hoang X., Nguyena E.A., Nguyen-Xuan H., Thuc P.V., 2017, Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory, Computers and Structures 193: 110-127.
[31] Dufour J.-E., 2018, A cost-effective isogeometric approach for composite plates based on a stress recovery procedure, Composites Part B 138: 12-18.
[32] Valizadeh N., 2013, NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter, Composite Structures 99: 309-326.
[33] Wang X., 2015, Isogeometric finite element method for buckling analysis of generally laminated composite beams with different boundary conditions, International Journal of Mechanical Sciences 104: 190-199.
[34] Somireddy M., Rajagopal A., 2014, Meshless natural neighbor Galerkin method for the bending and vibration analysis of composite plates, Composite Structures 111: 138-146.
[35] Weeger O., 2014, Nonlinear frequency response analysis of structural vibrations, Computational Mechanics 54: 1477-1495.
[36] Patton A., 2019, Fast and accurate elastic analysis of laminated composite plates via. isogeometric collocation and an equilibrium-based stress recovery approach, Composite Structures 225: 111026.
[37] Shojaee S., 2012, Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method, Composite Structures 94: 1677-1693.
[38] Behshad A., Ghasemi M.R., 2014, NURBS-IGA modelling: analysis and optimisation of laminated plates, Tehnicki Vjesnik 21(4): 789-797.
[39] Hong Ch., 2013, A rotation-free isogeometric analysis for composite sandwich thin plates, International Journal of Composite Materials 3(6A): 10-18.
[40] Farshad R., Roger S., 2016, Isogeometric nonlinear shell elements for thin laminated composites based on analytical thickness integration, Journal of Micromechanics and Molecular Physics 01(03-04): 1640010.
[41] Shuang Q., 2018, Multilevel optimisation of laminated composite thin-walled structures using isogeometric analysis, 13th World Congress on Computational Mechanics (WCCM XIII), New York, NY, USA.
[42] Kapoor H., 2013, Isogeometric Finite Element Code Development for Analysis of Laminated Composite Structures, Ph.D. Thesis, Virginia Institute.
[43] Gupta A., Ghosh A., 2019, NURBS-based thermo-elastic analyses of laminated and sandwich composite plates, Sådhanå 44: 84.
[44] Cassanova C.F., 2014, Analysis of Composite Shells: Isogeometric Modelling and Damage Identification, Ph.D. Thesis, Universidad de Granada.
[45] Kiendl J.M., 2011, Isogeometric Analysis and Shape Optimal Design of Shell Structures, Ph.D. Thesis, Technische Universität München.
[46] Tran L.V., 2013, An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates, Finite Element Analysis and Design 70: 65-76.
[47] Tran L.V., 2013, Isogeometric Analysis for Functionally Graded Plates Using Higher Order Shear Deformation Theory, Universiteit Gent.
[48] Breitenberger M., 2013, Isogeometric layout optimisation of shell structures using trimmed NURBS surfaces, 10th World Congress on Structural Multidisciplinary Optimisation, Orlando, USA.
[49] Deng X., 2015, Isogeometric analysis of continuum damage in rotation-free composite shells, Computer Methods in Applied Mechanical and Engineering 284: 349-372.
[50] Kiani Y., 2018, NURBS-based isogeometric thermal postbuckling analysis of temperature dependent graphene reinforced composite laminated plates, Thin-Walled Structures 125: 211-219.
[51] PavanG.S., Nanjunda Rao K.S., 2016, Analysis of laminated composite plates using isogeometric collocation method, Eccomas Congress 2016, VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece.
[52] Liu H., 2018, Isogeometric analysis based topology optimization design with global stress constraint, Computer Methods in Applied Mechanical and Engineering 342: 625-652.
[53] Civalek O., Yavas A., 2006, Large deflection static analysis of rectuangular plates on two parameter elastic foundations, International Journal of Science & Technology 1(1): 43-50.
[54] Civalek O., Kiracioglu O., 2010, Free vibration analysis of Timoshenko beams by DSC method, International Journal for Numerical Methods in Biomedical Engineering 26: 1890-1898.
[55] Akgöz B., Civalek O., 2017, A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation, Composite Structures 176: 1028-1038.
[56] Wu H., Yang J., Kitipornchai S., 2017, Parametric instability of Thermo-mechanically loaded functionally graded graphene reinforced nanocomposite plates, International Journal of Mechanical Sciences 135: 431-440.
[57] Shen H-S., Xiang Y., Fan Y., Hui D., 2017, Nonlinear vibration of functionally graded graphene-reinforced composite laminated cylindrical panels resting on elastic foundations in thermal environments, Composites Part B 136: 177-186.
[58] Hsu M.H., Hsu Y.L., 2005, Generalization of two- And three-dimensional structural topology optimization, Engineering Optimization 37: 83-102.
[59] Hughes T.J.R., 2005, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanical and Engineering 194: 4135-4195.
[60] Liang Q.Q., 2002, Performance based optimisation for strut-and-tie modeling of concrete, Journal of Structural Engineering 128(6): 815.
[61] Behshad A., Ghasemi M.R., 2013, Isogeometric based modelling and analysis of laminated composite plates under transverse loading, Journal of Solid Mechanics 5(4): 380-390.
[62] Hong Ch., Pierre S., Bordas A., 2014, Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, European Journal of Mechanics / A solids 43: 89-108.
[63] Serdyuk D., 2012, Fundamentals of Optimization Problems, FAU Erlangen-Nürnberg, TU München, Uni Stuttgart.
[64] Tang P-S., Chang K-H., 2001, Integration of topology and shape optimisation for design of structural components, Structural and Multidisciplinary Optimization 22: 65-82.
[65] Chirehdast M., Gea H.Ch., 1992, Further advances in the integrated structural optimisation system, American Institute of Aeronautics and Astronautics 4817: 974-981.
[66] Hassani B., 2012, An isogeometrial approach to structural topology optimisation by optimality criteria, Structural and Multidisciplinary Optimization 45: 223-233.
[67] Hong Ch., Nguyen-Xuan H., Bordas S.P.A., Nguyen-Thanh N., Rabczuk T., 2015, Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory, Mechanics of Advanced Materials and Structures 22: 451-469.
[68] Liang Q.Q., 2007, Performance-based optimisation: A Review, Advances in Structural Engineering 109(6): 739-753.
[69] Austin J., Thomas C., Hughes J.R., Bazilevs Y., 2009, Isogeometric Analysis Toward Integration of CAD and FEA, John Wiley & Sons.
[70] Hervandil M., Anna S., Fonseca J.S.O., 2002, Topology optimisation of continuum two-dimensional structures under compliance and stress constraints, Mecanica Computacional XXI: 2732-2751.
[71] Abolbashari M.H., Keshavarzmanesh Sh., 2006, On various aspects of application of the evolutionary optimisation method for 2D and 3D continuum structures, Finite Elements in Analysis and Design 42: 478-491.
[72] Yang X.Y., Xie Y.M., Steven G.P., 2005, Evolutionary methods for topology optimisation of continuous structures with design dependent loads, Computers & Structures 83: 956-963.
[73] Kant T., Swaminathan K., 2002, Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures 56: 329-344.