Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp. 397-409, Warsaw 2016 DOI: 10.15632/jtam-pl.54.2.397 MULTIOBJECTIVE AND MULTISCALE OPTIMIZATION OF COMPOSITE MATERIALS BY MEANS OF EVOLUTIONARY COMPUTATIONS Witold Beluch, Adam Długosz Silesian University of Technology, Institute of Computational Mechanics and Engineering, Gliwice, Poland e-mail: witold.beluch@polsl.pl; adam.dlugosz@polsl.pl The paper dealswith themultiobjective andmultiscale optimization of heterogeneous struc- tures bymeans of computational intelligencemethods. The aim of the paper is to find opti- mal properties of composite structures in a macro scale modifying their microstructure. At least two contradictory optimization criteria are considered simultaneously.A numerical ho- mogenization concept with a representative volume element is applied to obtain equivalent macro-scale elastic constants. An in-house multiobjective evolutionary algorithm MOOP- TIM is applied to solve the considered optimization tasks. The finite element method is used to solve the boundary-value problem in both scales. A numerical example is attached. Keywords: composite, numerical homogenization,multiobjective optimization, evolutionary algorithm 1. Introduction Composites are structural materials which are increasingly used and constantly gain in popula- rity due to their properties. In particular, the favourable strength/weight ratio causes them to displace traditional structural materials such as metals and their alloys in many areas of tech- nology. Their properties depend on such parameters as the properties of constituent materials, volume fraction of the constituents as well as shape and location of the reinforcement. Proper manipulation of suchparameters allows obtaining the desired behaviour of composite structures. In order to obtain the best (for given criteria) properties, optimization methods have to be applied.Since theapplication of conventional, typically gradient-based, optimizationmethods for composites may encounter difficulties due to multimodality and discontinuity of the objective function, it is reasonable to use global optimization methods, like bio-inspired optimization algorithms, e.g. evolutionary algorithms, artificial immune systems or particle swarm optimizers (Michalewicz and Fogel, 2004; De Castro and Timmis, 2002; Kennedy and Eberhart, 2001). In real optimizationproblems, it is veryoftennecessary to considermore thanone criterion at the same time. If the criteria are contradictory, the optimization task belongs to multiobjective ones.Thededicated implementations of bio-inspired global optimizationmethods can be applied to solvemultiobjective optimization tasks. A survey of the state of the art of themultiobjective evolutionary algorithms can be found in Zhoua et al. (2011). Proper determination of the effect of micro-structure of heterogeneous materials on their behaviour at the macro level allows the optimal design of heterogeneous materials. Multiscale optimization allows designingmaterials in one scale level to obtain the desired properties of the material ondifferent scale(s). Different homogenizationmethods are typically applied to perform calculations inmore than one scale in a reasonable time (Kouznetsova, 2002; Buryachenko, 2007; Zohdi andWriggers, 2005). There are numerous approaches to the multiscale modelling. Analytical or semi-analytical methods are typically used to determine the equivalentmaterial constants for inclusions or voids of regular shape, e.g. circular, elliptical or spherical (Eshelby, 1957; Bensoussan et al., 1978). 398 W. Beluch, A. Długosz The applied in the present paper attitude is based on numerical homogenization methods belonging to so-called upscalingmethods. Simulation in this group of homogenizationmethods is carried out hierarchically in different scales utilizing the representative volume element attitude. The computational intelligencemethodshave been successfully applied by the authors tomultio- bjective optimization problems of composite structures at the macro scale only, see e.g. Beluch et al. (2008). The application of computational intelligencemethods for the single-objectivemul- tiscale identification of material constants in heterogeneous materials was presented by Beluch and Burczyński (2014). 2. Multiscale modelling Many structuralmaterials like composites, porousmaterials or polycrystallinematerials are non- homogeneous on a certain observation level. In order to model such materials more precisely, considerations in a macro scale only may be insufficient. Taking into account different scales allows modelling different geometric and material properties of the structures. A macro-scale model may contain various types of external loads (mechanical, thermal, electrical, etc.). Meso andmicro scales make it possible to consider such elements as discontinuities or imperfections, like cracks, voids, inclusions or surface roughness (Nemat-Nasser and Hori, 1993; Vernerey and Kabiri, 2014). A nano-scale level includes e.g. crystal lattice defects while an atom-scale le- vel allows incorporating molecular mechanics effects (Burczyński et al., 2007). The number of considered scales depends on the required accuracy of the model (Ilic and Hackl, 2009). The proper determination of the influence of themicro-structure of heterogeneous materials on their behaviour at themacro level allows optimal designing of them.An appropriate selection of the component materials, geometry and volume ratio of constituents allows creating mate- rials with desired properties, including those which cannot be obtained with the application of homogeneous materials only (Takano and Zako, 2000). The direct application of more than one scale in numerical calculations by means of nu- merical methods such as the finite element method (FEM) (Zienkiewicz and Taylor, 2000) or boundary element method (BEM) (Brebbia and Dominiguez, 1989) leads to systems with such large numbers of degrees of freedom that they are very hard or even impossible to be solved. In order to overcome this problem, different homogenization techniques are employed. In the present paper, numerical homogenization techniques are applied to find the parameters of the equivalentmaterial for composite structures.The behaviour of heterogeneousmedia is described bydifferential equationswithdiscontinuous coefficients like elastic constants in linear-elastic pro- blems.Theaimof the numerical homogenization is to determine continuous, effective coefficients of differential equations which are applied to a higher scale. A typical attitude in the numeri- cal homogenization consists in the determination of constitutive relation between averaged field variables, like stresses and strains (Ptaszny and Fedeliński, 2011). 2.1. Numerical homogenization of heterogeneous materials Numerical homogenization techniques belong to upscaling methods which perform hierar- chical simulation in particular scales andmake use of the representative volume element (RVE) concept (Hill, 1963). They allow obtaining macroscopically homogeneous, equivalent materials which behave in the macro scale as microscopically heterogeneous ones. RVEs are used for globally or locally periodical structures. RVE represents the structure of the whole medium or its part, so it has to include all information required for a thorough description of the structure and properties of the medium (Hashin, 1964). Multiobjective and multiscale optimization of composite materials... 399 Numerical homogenization can be performed under certain conditions: a) The principle of the scales separation requires that RVE size lRVE must be significantly greater than the microstructure characteristic dimensions lmicro and considerably smaller than the characteristic dimensions lmacro in the macro scale (Zohdi andWriggers, 2005) lmicro ≪ lRVE ≪ lmacro (2.1) It is commonly assumed that the RVE is the smallest possible volume representing the entire medium or its part. RVE should meet two conflicting criteria: be large enough to be representative of the entire structure and as small and uncomplicated (geometrically and materially) as possible in order to carry out its precise numerical analysis (Madi et al., 2006). In the case of fully regular structures (commonly used for fiber-reinforced composites), RVE may contain only one centrally placed core. Such an RVE is called a unit cell. b) Averaging is performed according to the relation 〈·〉= 1 |V | ∫ V (·) dV (2.2) where 〈·〉 is the averaged value of the field under consideration, V – RVE volume. c) The condition specifying the equality of the average energy density in themicro scale and the macroscopic energy density at the point of macrostructure corresponding to the RVE (Hill condition) has the form (Kröner, 1972) 〈σijεij〉= 〈σij〉〈εij〉 (2.3) where: σij and εij are stress and strain tensors in the micro scale. d) Appropriate boundary conditions, e.g. periodic boundary conditions (Kouznetsova, 2002): periodic displacements and anti-periodic tractions on opposite faces of the RVE, as shown in Fig. 1, are u+i =u − i ∀r∈ ∂V : n + i =−n − i t + i =−t − i ∀r∈ ∂V : n + i =−n − i (2.4) Fig. 1. RVE boundaries for periodic boundary conditions In addition to the boundary conditions, strain boundary conditions from the higher scale are imposed on every RVE (localization). If FEM is applied to solve the boundary-value problem in both scales, the RVE is assigned to each integration point in the micro scale (Kuczma, 2014). Averaged stresses, calculated according to Eq. (2.2), are obtained as a result of numerical computations in the micro scale (homogenization). Averaged stresses are transferred to the 400 W. Beluch, A. Długosz higher scale in order to calculate homogenized material parameter values at the macro scale taking into account the constitutive equation for the homogenized material. Assuming that the considered composites can be treated as orthotropic materials in the plane strain state, the constitutive equation in the Voight notation has the form (Gibson, 2012)    〈σ11〉 〈σ22〉 〈σ12〉    =    Q11 Q12 0 Q22 0 · Q33       〈ε11〉 〈ε22〉 〈ε12〉    = E (1+ν)(1−2ν)    1−ν ν 0 1−ν 0 · 0.5−ν       〈ε11〉 〈ε22〉 〈ε12〉    (2.5) whereQij are the elements of the resultant elastic constants tensorQ, i,j =1,2,3. In the considered case, determination of the Q matrix elements requires performing of 3 independent analyses in themicro scale for eachRVE. If thematerial is linear and fully periodic, only one RVE has to be analysed for the whole structure. Having determined the components of theQmatrix, the elastic constants of the equivalent material are calculated according to Eq. (2.5). 3. Formulation of the optimization problem In many engineering optimization problems, more than one optimization criterion have to be taken into account simultaneously. Moreover, the considered criteria are often contradictory, which leads to multiobjective optimization (MOO) tasks. MOO results in a set of trade-off solutions instead of only one optimal solution in single-objective optimization tasks. The aim of the two-scale multiobjective optimization of composite structures is to find some of its properties in the micro scale (represented by the RVE) which optimize the behaviour of the structure in the macro scale. To solve the boundary value problem in the macro andmicro scales, the commercial FEM software MSCMarc and MSC Nastran has been applied. In order to combine MOOPTIM with FEM software, appropriate programming interfaces have been developed. The block diagram of themultiobjective andmultiscale evolutionary optimization is presented in Fig. 2. Fig. 2. A block diagram of the multiobjective andmultiscale evolutionary optimization Multiobjective and multiscale optimization of composite materials... 401 3.1. Definition of the multiobjective optimization task AMOOproblemcanbe treated as a search for a vectorx∈D, whereD is a set of admissible solutions being a subset of design space X (Deb, 2001) x= [x1,x2, . . . ,xn] T (3.1) which minimizes the vector of k objective functions f(x)= [f1(x),f2(x), . . . ,fk(x)] T (3.2) The vector x has to satisfym inequality constrains gi(x)­ 0, i=1,2, . . . ,m and p equality constrains hi(x)= 0, i=1,2, . . . ,p. There exist many attitudes to the multi-objective optimization problems (Laumann et al., 2004). A priori methods are based on the transformation of a multiobjective problem into a single-objective one (Collette and Siarry, 2003). Themost popularmethods from this group are: i) weighted sum method in which each criterion has its own weight value; and ii) ε-constraint method inwhich the optimization is performed for a chosen criterionwhile the remaining criteria are treated as constrains.Theadvantage of thea priorimethods is that single-objectivemethods canbeapplied,but thedrawback is that someveryoftenunrealistic assumptionsof theobjectives have to bemade before the optimization starts. The second group state interactivemethods,which demand an interactionwith the decision- maker (DM) during the optimization to achieve additional information (Luque et al., 2011). There existmanymulti-criteria decision-makingprinciples.For example, inPhelps andKoksalan (2003), a pair-wise comparison of solutions is used to include DM’s preference. The guided multiobjective evolutionary algorithm (G-MOEA) uses amodified definition of dominance (see: Section 3.2) which has beenmodified based upon theDM’s preference information (Branke and Deb, 2004). Both aforementioned groups of methods result in one solution of the optimization process. In the a posteriori methods, a set of compromise (trade-off) solutions is determined in the first step of the optimization procedure. The DM is required to choose the most preferred solution in the second step. 3.2. Pareto concept in multiobjective optimization of composites An attitude belonging to a posteriorimethods is employed in the present paper. Themultio- bjective optimization is performed using the Pareto concept of non-dominated solutions (Ehr- gott, 2005). If the minimization problem is considered, a solution x is strongly dominated by the solution x∗ if ∀i∈{1,2, . . . ,k} : fi(x ∗)