Evolutionary Topology Optimization of Continuum Structures
Topology optimization has been an interesting area of research in recent years. The main focus of this paper is to use an evolutionary swarm intelligence algorithm to perform Isogeometric Topology optimization of continuum structures. A two-dimensional plate is analyzed statically and the nodal displacements are calculated. The nodal displacements using Isogeometric analysis are found to be in good agreement with the nodal displacements acquired by standard finite element analysis. The sizing
... lysis. The sizing optimization of the beam is then performed. In order to determine the stress at each point in the beam a formulation is presented. The optimal cross-section dimensions by performing Isogeometric analysis are acquired and verified with the cross-section dimensions achieved by hiring bending stress and shear stress criteria, as well. The topology optimization of a two-dimensional simply supported plate continuum and a problem on three-dimensional continuum are optimized and presented. The results show that the minimum weight which is found by applying Isogeometric topology optimization gives better results compared to the traditional finite element analysis. problems (Yang, 2010) . The main applications of firefly algorithm are digital image compression and processing, featured selection and fault detection, antenna design, structural design, scheduling, semantic web composition, chemical phase equilibrium, clustering, dynamic problems, and so on. (Archana, 2017) . In this study, Section 1.1 presents the objectives of the study to determine the optimal distribution of material having minimum weight using Isogeometric analysis and evolutionary algorithms. Section 2 discusses the literature review of the work done by several authors who applied Isogeometric analysis in structural mechanics. Section 3 will represent the theory behind the formulation of plate problems. The NURBS basis functions are clearly explained. Section 4 shed lights on the problem statement. In Section 5, a flowchart of the method used to perform topology optimization is presented and discussed. In Section 6, analysis of plate problems in two-dimensions is presented. In section 7, the sizing optimization of a beam is offered. In section 8, the topology optimization of a simply supported two-dimensional plate and a three-dimensional domain are presented. Section 9 briefly concludes the work done in this paper and sets a direction for future studies. 1.1 Objectives of the study a) Performing the Isogeometric topology optimization of continuum structures using an evolutionary algorithm. b) Determining the minimum weight distribution of material which is subjected to the constraints of stress and displacement. 1.2 Assumption of the study a) The material obeys Hooke's law. b) Buckling analysis is not included in this study. Literature review The concept of Isogeometric Analysis was first introduced by T.R.J Hughes (Hughes, 2005) and his colleagues at the University of Texas at Austin. The basic idea of IGA is to represent the geometry accurately. The initial work of Isogeometric analysis was first developed to fill the gap between FEA and CAD. The exact geometric model is generated with basis functions from NURBS (Non-Uniform Rational B-Splines) which is a new method for analysis of solids, fluids, structures and other types of problems which are governed by partial differential equations. This method, not only helps with generating the exact geometric model, but also assists with simplifying mesh refinement by eliminating the need for communication with the CAD geometry after the initial mesh is constructed (Attila, 2010). Nguyen, et al. (Nguyen, 2015) discussed the concepts of Isogeometric analysis in details. A brief introduction to NURBS functions is done. The preliminary concepts and implementation of IGA have been discussed. With the help of Matlab®, the difference between IGA and conventional FEA has been explained with an overview of recent developments and shortcomings of IGA. IGA is mostly helpful for the plate and shell type of problems as NURBS basis functions allows straightforward construction and it is particularly helpful for thin shells as it constructs a rotation free formulation. NURBS also provides advantages for structural vibration problems. IGA is much helpful in solving partial differential equations of fourth-order derivate. This paper also discussed the shortcomings of NURBS. NURBS always produces watertight geometries which always complicates mesh generation. They have also presented Matlab® implementation for one, two, and three-dimensional Isogeometric finite element analysis for structural and solid mechanics. Luis (Luis, 2011) applied the Isogeometric analysis to perform linear and non-linear analysis of a few basic structural problems in engineering. Mit (Mit, 2015a), in his paper, did a review on the application of Isogeometric analysis for problems in one-dimensional, two-dimensional, and three-dimensional cases. Gondegaon (Gondegaon, 2016) performed the static and modal analysis of few basic problems. Gondegaon (Gondegaon, 2014) applied IGA to model the geometry which exactly represents the domain. Nguyen (Nguyen, 2015) has presented the computer implementation aspects for several problems in engineering. Hartman (Hartman, 2011) applied Isogeometric analysis to LS-Dyna and concludes that the IGA gives better results over traditional methods with less computational effort. Mit (Mit, 2015b) in another paper, performed stress calculations using Isogeometric analysis. Nagy (Nagy, 2010) used variational formulation and applied Isogeometric analysis to solve the differential equations by Galerkin weak formulation. Mateus (Mateus, 2013) employed Isogeometric analysis to perform free vibrational analysis of bars. Laura (Laura, 2012) presented a few open issues in Civil Engineering which include interface modeling. Milos (Milos, 2016) discussed the free vibration analysis of beam structures using Isogeometric concept. Clough and Penzien (Clough, 1993) argued the formulation of stiffness matrix and mass matrix for beams and bars. In Civil engineering, beam elements are primary components in a structure. A free vibration analysis is required for a beam as it is always subjected to some dynamic excitation. The traditional method hired for this type of excitation is the Bernoulli-Euler beam. For thick beams, transverse shear deformation and rotatory inertia effect are included in Timoshenko beam as they give more adequate results. Milos (Milos, 2016) has used Isogeometric analysis approach for free vibration analysis beam and compared it with dynamic stiffness method (DSM) and conventional finite element method (FEM). Valizadeh (Valizadeh, 2011) employed an improved Isogeometric analysis by employing the Lagrange Multiplier method. For the purpose of designing the structural systems, earlier they used to plan a general layout which efficiently supports the predicted design loads. This prediction of the structural design systems is generally done by either fixed or variable topology shape optimization. There are two techniques which are helpful for optimizing the shape and topology of the system. Ohsaki (Ohsaki and Swan, 2002) mainly focused on the topology optimization of trusses and frames with a detailed explanation of the two techniques with some example problems on continuum optimization.