matlab and labview as an integrated tool to minimize ...

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

MATLAB AND LABVIEW AS AN INTEGRATED TOOL TO MINIMIZE THERMAL RESIDUAL STRESSES IN FUNCTIONALLY GRADED STRUCTURES Silva, T.A.N., Loja, M.A.R. ADEM/ISEL - Instituto Superior de Engenharia de Lisboa, Av. Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal. IDMEC/LAETA - Instituto de Engenharia Mecânica, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal. GI-MOSM/ISEL - Grupo de Investigação em Modelação e Optimização de Sistemas Multifuncionais E-mail of the corresponding author: [email protected]

SUMMARY: Thermal residual stresses often arise due to the manufacturing process, involving plastic deformation or abrupt temperature gradient, or to the different thermal expansion coefficients of structural components. This fact can thus result in stress states that remain within a structural member in the absence of external loads, being desirable to obtain, as long as possible, a minimum level of residual stresses as well as smoother stresses transitions in the materials interfaces. Dual-phase functionally graded materials provide superior thermal and mechanical performances when compared to the traditional laminated composites due to its continuous properties variation characteristic. This relevant fact contributes to the mitigation of stress concentrations by gradually varying the microstructure and/or composition of materials in a gradient structure. With the present work it is intended to demonstrate how student’s motivation and competences can be developed both in the fields of mechanical design and structural optimization, by taking as demonstration the problem of minimizing thermal residual stresses in functionally graded structures using differential evolution optimization technique. These goals are enabled by taking advantage of the friendly interaction between two commercial software packages, MATLAB and LabVIEW, which can provide an integrated tool for engineering education and research. In this work, the authors present a virtual platform that enables to students an easier perception and intermediate results tracking, concerning to the optimization process, as well as the influence of considering different temperature distribution profiles, layer thicknesses or volume fraction law exponents, among other optimization related parameters, on thermal residual stresses.

Keywords: MATLAB, LabVIEW, Functionally graded material, Thermal residual stresses, Differential Evolution.

1. INTRODUCTION With this work, it is intended to illustrate the combined use of numerical and graphical computation platforms to integrate different areas of knowledge, which beyond any other advantages, aims to achieve a faster and more effective perception of physical phenomena. Taking into account the base knowledge acquired in Mechanical Engineering BSc, within the fields of deformable bodies mechanics and composite laminates, and its further exploration at a deeper and more complex level at Mechanical Engineering MSc, it was understood to be of interest to develop an integrated tool that would allow to present to students new areas of knowledge, motivating them simultaneously to research. The tool here developed, with a didactical character, elects as fundamental purpose motivating the future Masters in Mechanical Engineering for a holistic view of Engineering, not confined to a specific course curricular content. In this sense, it was considered of interest to address emerging areas within the structural optimization field, choosing the application of differential evolution (DE) technique to structures made of functionally graded materials (FGM).

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Dual-phase FGM are a particular type of composite materials, whose properties are tailored to vary continuously, depending on the composition distribution of its two constituents, and which use is increasing on the most diverse application fields. These materials are known to provide superior thermal and mechanical performances when compared to the traditional laminated composites, because of its continuous properties variation characteristic, which enables, among other advantages, smoother stress distribution profiles. Moreover, it is well known that abrupt transitions in material properties within a composite structure can cause stress concentrations, which can be mitigated by gradually varying the microstructure and/or composition of materials in a gradient architecture (Surendranath et al., 2003). Since the concept of FGM emerged, one can conclude of a great research involvement on the study of these materials, namely for relaxation of thermal stresses (Koizumi, 1993). In fact, one of the significant capabilities of these materials is related to their behaviour in high temperature conditions, thus being initially thought as thermal barriers (Uemura, 2003). Among several other published works, Reddy (2000) proposed a theoretical formulation and finite element models based on the third-order shear deformation plate theory for the analysis of through-thickness functionally graded plates, assuming its modulus of elasticity to vary according to a power-law distribution in terms of the volume fractions of its constituents. In terms of optimization, the behaviour of systems that exhibit large oscillations and various local minima is result of the interaction of multiple design variables or parameters. Hence, optimization strategies must be able to deal with this non-linear and/or non-convex design spaces and still find the best global solution. Population-based evolutionary algorithms, such as genetic algorithm, particle swarm optimization, ant colony optimization and DE (Storn and Price, 1997), are nature inspired computing paradigms able to predict complex emergent behaviours, being suitable to be used in structural optimization problems. Several authors have explored the application of DE in this field and reported that DE proved to be very robust and suitable to be applied with advantages as a global optimizer (Kitayama et al., 2011). In the field of engineering education, several journals are published, by the global community of engineering education societies and associations all over the world (IFEES, 2012). Almarshoud (2011) reviewed the use of MATLAB’s graphical user interface (GUI), as well as, LabVIEW in electrical engineering education, namely to set up remote laboratories. The referred research activity has parallel in the mechanical engineering education, where we can find works from vehicle dynamics to solar radiation, just to mention a few areas. On the other hand, it is known that students learn in a more efficient way if their education is framed in a broader, multidisciplinary and hands-on curriculum (Du et al., 2009). Considering this background, the present work intends to give an overview of an educational platform built recurring to a visual programming language (LabVIEW 2009) in conjunction to a numerical computation commercial software (MATLAB - R2009b), in order to minimize thermal residual stresses in functionally graded structures (Silva and Loja, 2011). Therefore, the authors decided to organize this paper in such a way that is supposed to mimic the use of the proposed educational platform.

2. MATLAB AND LABVIEW AS AN INTEGRATED TOOL With a broad dissemination among the scientific community, MATLAB is a high-level numerical computing software for algorithm development and data manipulation. MATLAB can be used in a wide range of applications and it has the capability to be extended to deal with specific problems, by means of user defined functions. Moreover, MATLAB codes, known as .m files, can be integrated with or in other computational applications or software packages. Though MATLAB gives us tools for building custom GUIs, we think that this is not its stronger advantage, taking into account the fact that we still need to dig under the code to set up the interface, apart the essential code related to the application itself. On the opposite side, LabVIEW is a development environment object-oriented graphical programming language, named G. Initially oriented for data acquisition and control systems, LabVIEW has grown to be a general purpose programming environment. Each LabVIEW program, known as virtual instrument (VI), is composed by a front panel and a block diagram. Taking the analogy with a hypothetic electronic device, the front panel should mimic the real instrument with data inputs (controls) and outputs (indicators), either numerical or graphical; while the block diagram gathers all the program blocks, which are linked by data wires, as in any electronic device. Note that the term program block encompasses all the controls, indicators and program structures, such as for and while loops or case structures, in the block diagram (NI, 2009). Naturally, MATLAB and LabVIEW can be integrated in order to access to the advantages of both. For this purpose, LabVIEW provides two types of programming structures, which are able to interpret the already programmed .m files, namely MathScript Node and MATLAB Script. Both structures use code with MATLAB syntax and enable the use of LabVIEW’s variables as inputs to the .m code and the use of .m file’s outputs as

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

inputs to the front panel indicators, with some restraints related to the data type of both inputs and outputs of the .m code. The MathScript Node can execute scripts written in the MATLAB language syntax. However, this structure executes the scripts using a LabVIEW’s built-in interpreter, i.e., it is not required to have MATLAB installed in your computer. Therefore, there are some non-supported MATLAB functions, and the path to user defined functions must be declared in LabVIEW. On the other hand, the MATLAB Script fully supports MATLAB functions, because it calls MATLAB to execute the scripts, demanding for a licensed MATLAB software package installed on the computer and it is only available for Windows OS (NI, 2009). In the current work, every calculation is programmed in .m files and its input variables and results are manipulated and presented in the LabVIEW front panel. Implicitly, this paper aims to give a wider dimension to a huge amount of already available MATLAB code files by the use of a friendly and appellative graphical interface, considering the integrated use of both MATLAB and LabVIEW software packages.

3. EDUCATIONAL PLATFORM OVERVIEW The present work aims to present an educational platform intended to develop students’ motivation and competences both on mechanical design and structural optimization. For this purpose, the authors selected the problem of minimizing thermal residual stresses in a sandwich panel constituted by a metallic core and outer FGM layers using differential evolution optimization technique. Taking advantage of the friendly interaction between MATLAB and LabVIEW, it is proposed an integrated tool for engineering education and research with several valences. The educational platform here presented demands students to work with a specific kind of composite material, FGM, and to conclude about the influence of structural properties and temperature distribution along thickness on the thermal residual stresses of the structure. Furthermore, it is expected that students can gain competences on optimization processes through the use of DE, in order to minimize the referred residual stresses. Note that the presented platform basically only uses LabVIEW resources to set up the user interface, while calculations are performed numerically, making use of both MathScript Node and MATLAB Script. The following subsections present the capabilities and available options of the proposed educational platform, as well as, the equations behind the interface. The organization of the referred subsections respects the hierarchy of the platform, being thus: (i) the main VI front panel; (ii) a VI dedicated to the general use of DE strategy; (iii) a VI where the use of DE to minimize thermal residual stresses on FGM structures is explored; and (iv) a documentation repository VI on the approached subjects.

3.1. Base platform - Analysis The base platform or main VI is not only a dock for all the subroutines or subVIs but also an interface where the problem of thermal residual stresses generated in FGM structures is addressed, according to the formulation described on sections 3.1.1 and 3.1.2. Figure 1-7 illustrates the main interface and its options.

3.1.1. FGM sandwich structure modelling In this paper it is considered a dual-phase FGM constituted by metallic and ceramic particles, titanium and zirconia, respectively. Usually FGM structures are submitted to high temperature environments. Hence, it is necessary to take into account the effective properties of these materials, which depend not only on the position but also on the temperature (Reddy and Chin, 1998). In order to allow for a better prediction of the mechanical behaviour of these structures, we consider the effective material properties Peff using the rule of mixtures of Voigt (Li and Wang, 2005). Peff ( z , T ) = Pm (T )Vm ( z ) + Pc (T )Vc ( z )

(1)

where the material property P associated to a generic thickness coordinate z and a certain temperature T, depends on the corresponding metal and ceramic properties, Pm and Pc , weighted by the corresponding volume fractions, Vm and Vc . Several structural configurations can be selected, being the distribution of the metal volume fraction across the thickness z ∈ [ hle , hri ] given by Vm (Nguyen et al., 2008).

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

 z − h  p le   for hle ≤ z ≤ h1  h1 − hle   Vm ( z ) =  1 for h1 ≤ z ≤ h2 p    z − hri  for h ≤ z ≤ h 2 ri   h2 − hri  

(2)

where hle and hri are the coordinates of the left and right outer surfaces of the structure and h1 and h2 are respectively the coordinates between the left FGM layer and the core, and between the right FGM layer and the core, when applied. Note that p the exponent of the volume fraction power law (2).

3.1.2. Thermal Residual Stresses Thermal residual stresses often arise due to the manufacturing process, involving plastic deformation or abrupt temperature gradient, or to the different thermal expansion coefficients of structural components. This fact can thus result in stress states that remain within a structural member in the absence of external loads, being desirable to obtain, as long as possible, a minimum level of residual stresses as well as smoother stresses transitions in the materials interfaces. FGM are known to provide superior thermal and mechanical performances when compared to the traditional laminated composites. Due to its continuous properties variation characteristic, it is possible to obtain smoother stress distribution profiles, avoiding abrupt transitions in material properties within a composite structure. Since FGM emerged, its usage is connected to thermal stresses relaxation. In this study, as base structure, it is considered a sandwich FGM panel, infinitely long in the width direction and with the thickness limits already mentioned (Section 3.1.1). According to Ravichandran (1995), the thermal residual stresses in this type of structures can be described as

  A − A1 E (zE − E )  2   A1  2 E1 2  1 σ ( z ) = E ( z ) α ( z ) − +  ∆T 2 E1 E1 E3 − E 2    

(3)

where hsup

A1 =

∫ α ( z ) E ( z )dz ;

hsup

A2 =

hinf

∫ zα ( z ) E ( z )dz h inf

and hsup

E1 =

∫ E ( z )dz ; hinf

hsup

E2 =

∫ zE ( z )dz ; hinf

hsup

E3 =

∫z

2

E ( z )dz

hinf

being α ( z ) the thermal expansion coefficient and E ( z ) the elastic modulus. In the present work, it was neglected the temperature dependency of the material properties.

3.1.3. Main VI description Despite the several used examples, we do not intend to cover all the output possibilities, considering the combination of the input parameters. In Figures 1-7, we present different views of the main VI. These figures try to cover all the user controlled available options. Note that all calculations of this VI are performed by MathScript Nodes, which reuse previously programmed MATLAB codes. On Figure 1, several areas on the interface are enumerated. On the left end side, we can define several parameters related to the structure type (Figure 1, Item 1) and the temperature distribution (Figure 1, Item 2). The controls available on Item 3 (Figure 1) are the ones which are used to call the implemented subVIs, described on sections 3.2-3.4. The right end side of the interface is dedicated to the outputs. Temperature and metal volume fraction distributions (Figure 1, Item 5) and thermal residual stress profiles (Figure 1, Item 6) are presented. On this last item a zoom function is available.

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

1

4

5 1 - Structure definition options; 2 - Temperature distribution options; 3 - Access to subroutines; 6 4 - Schematic structure type definition;

2 3

5 - Plots for temperature and metal volume fraction distribution; 6 - Thermal residual stress distributions. Figure 1 – Analysis platform: content overview.

Figure 2 – Analysis platform: structure type selector (ex.: “Just one FGM layer”).

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Figure 3 – Analysis platform: temperature distribution selector (ex.: “Only FGM”).

On Figures 1-3 it is shown that the structure type image (Figure 1, Item 4) is driven by the structure type selector (Figure 1, Item 1). Note that, in the cases illustrated by Figures 2 and 3, some buttons are disabled. It is due to the selected structure type, for which theses ones does not make sense. Figure 3 highlights the temperature distribution type selector. Note the difference on the residual stress profiles shown on Figures 1-3. As expected, the stress profile varies with the selected structure type, being relevant the symmetrical character of it. This circumstance is patent when residual stress profiles of Figures 1Figure 1, 5-7 are compared. Note that, for symmetrical structures, in the presence of a symmetrical temperature distribution the moment equilibrant can be neglected. Figure 4 presents outputs for the case of a symmetrical structure submitted to a linear temperature distribution. Also on this figure, we found an example of a pop up description associated to a control.

Figure 4 – Analysis platform: pop up description (FGM button).

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Figure 5 – Analysis platform: example of an asymmetrical structure total thermal residual stress profile.

Figure 6 – Analysis platform: example of an asymmetrical structure thermal residual stress profile - force equilibrant.

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Figure 7 – Analysis platform: example of an asymmetrical structure thermal residual stress profile - moment equilibrant.

3.2. Base platform - Optimization This subVI is accessed from the main one by clicking on the Run Optimization button (Figure 1, Item 3). Here we can use several typical test functions in order to study the influence of DE’s parameters on the best solution. This is an adaptation of the application developed by F.J. Ahlers (Price and Storn, 2012).

3.2.1. Optimal Design Problem A generic minimization problem can be stated as: min Ω (b )

(3)

subject to: bilow ≤ bi ≤ biup , i = 1, K , ndv and Ψ j ( q, b) ≤ 0, j = 1,K, nbc

where Ω (b ) is the objective function, b is the design variables vector, which values are bounded by the lower and upper limits bilow and biup , respectively, and Ψ j (q, b) are the nbc inequality constraint equations. Note that the length of b is limited by the total number of design variables n dv . In this context, the design variables can be thicknesses and/or the exponents of the metal volume fraction law that dictate how the material combination will be characterized across the thickness coordinate of the FGM layers. The limits imposed to the design variables are used as boundary constraints of the optimization process. The objective function considered in this study is the thermal residual stress distribution function associated to a thin or moderately thin FGM sandwich panel with metallic core (Silva and Loja, 2011).

3.2.2. Differential Evolution description Developed by Price and Storn (1997), DE is a population-based approach to function optimization that generates trial individuals by calculating vector differences between other randomly selected members of the population. Given an objective function to be minimized, DE begins by randomly generating population

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

vectors of n members per design variable. These vectors, also known as individuals, will evolve over the progression of the optimization procedure, simulating an evolving population. For each generation or iteration, the objective function is evaluated n times, trying to find the best population member until a termination criterion is achieved. A flowchart of DE algorithm is presented on Figure 8. It is worth to mention that DE congregates characteristics from global optimization and meta-heuristics algorithms. Although it is a global search technique, DE preserves a search direction vector, which gives an important descent property, and the randomness of populations, which improves the method's robustness (Kitayama et al., 2011).

Figure 8 – Differential Evolution strategy general flowchart.

3.2.3. Differential Evolution VI description In Figures 9 and 10, two optimization examples of well known test functions are presented, the Rosenbrock and Griewangk functions, respectively. In Figure 9, several areas on the interface are enumerated.

3

1

1 - Test function and design parameters options; 2 - DE algorithm options; 2

Figure 9 – Optimization platform: Rosenbrock function.

3 - Progress of DE algorithm and outputs;

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Figure 10 – Optimization platform: Griewangk function.

By the available examples it is possible to study the influence of several parameters on the best solution. Several user controlled variables are available on this interface, which allow students to explore in depth the optimization strategy. Moreover, it is possible to set up numerous test functions. In Figure 9, Item 1 congregates all test function and design parameters controls. Note that it is prepared to set up constrained optimization. DE’s parameters can be defined, as well as, termination criterions for the method (Figure 9, Item 2). Regarding the optimization strategy, F is the selection weighting factor and CR is the binomial crossover probability constant. On the same item can be found information about DE strategies through the icon ?, in front of the strategy selector. The region 3 of Figure 9 is dedicated to the outputs. Note that the user can select which pair of design parameters wants to display. Here, it is also shown the best solution, as well as, the total number of functions evaluations (for each generated DE member) and generations.

3.3. Application platform - FGM structure optimization test case The Application platform is dedicated to a FGM structure optimization example and is a step forward, on the educational point of view, of the work previously published by Silva and Loja (2011). With the test case presented in this section, the authors aim to merge the previously addressed contents, through the use of DE to minimize the thermal residual stresses in a FGM structure. Figures 11 and 12 are outputs of the present subVI, which call MATLAB for all the calculations. In this specific case a MATLAB Script is used in order to run a .m file, previously programmed. Note that the user can include more variables and/or change the current ones by offline editing the referred .m file. This subVI can be opened from the main one by clicking on the FGM button (Figure 1, Item 3).

3.3.1. FGM structure optimization example VI description The examples here presented were obtained using F=0.5, CR=1 and DE/best/1/bin optimization strategy, tailored for small population sizes and fast convergence, where best specifies the vector of lowest cost from the current population to be mutated, 1 the number of difference vectors used for calculation, and bin a binomial crossover scheme. Note that F is the selection weighting factor and CR is the binomial crossover probability constant. On Figure 11, the progress through the optimization iterations of design parameters and optimal solution can be tracked (Items 1 and 2). There are also available the values of the optimal solution and parameters set (Item 4) and the comparison between optimal and reference stress distributions (Item 5). Note that the

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

1

3

1 - Design parameters evolution;

4

2

5

2 - Optimum value evolution; 3 - Example general information; 4 - Optimum design parameters; 5 - Residual stress profiles (opt. vs. ref.).

Figure 11 – Application platform: content overview (10 generations).

reference stress distribution were obtained considering p = 10 and hFGM = 0.005m . Figures 11 and 12 differ by the number of considered iterations. On this interface, it is also presented the time spent to perform the optimization task.

Figure 12 – Application platform: example for 50 generations.

CSEI2012 – Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação Lisboa, 2-3 Abril de 2012

Figure 13 – Documentation platform: selector example.

3.4. Documentation platform The so called documentation platform supports the entire educational application and is available from the main VI by clicking on the Documentation button (Figure 1, Item 3). Here it is possible to get access to several references, from scientific papers to general information on the internet. Figure 13 shows how we can select a document from Wikipedia, although there are other www sites available and a few selected papers on the addressed subjects.

4. CONCLUSIONS In this work, the authors present an educational platform that enables to students an easier perception and intermediate results tracking, concerning to the optimization process, as well as the influence of considering different temperature distribution profiles, layer thicknesses or volume fraction law exponents on thermal residual stress distributions. It is intended to demonstrate how student’s motivation and competences can be developed both in the fields of mechanical design and structural optimization, by taking as demonstration the problem of minimizing thermal residual stresses in a sandwich panel constituted by an aluminium core and outer FGM layers, using differential evolution optimization technique. These goals are reached by taking advantage on the friendly interaction MATLAB and LabVIEW, which can provide an integrated tool for engineering education and research. By this, the educational potential of MATLAB .m files is clearly increased. Besides reusing code, it is easier and quicker to develop graphical interfaces. The combined use of both numerical and graphical software packages promote a more user friendly educational environment for both students and scholars, overtaking some drawbacks of each programming environment, i.e., it becomes easier to set up a graphical user interface avoiding flags or calls and, on the other hand, we can reuse numerical code just by copy and paste. Unlike MATLAB’s GUI, LabVIEW’s VIs are almost code free, in terms of setting up the front panel. This fact decreases drastically the time to the final output, promoting the enrolment of students on more advanced tasks and the interest on their own codes, and, on the scholar side, a more effective time management, regarding the preparation of classes and the approach to students. Moreover, it is possible to generate executable applications to distribute among students or give them remote access to the platform over the internet, which does not demand students to have powerful personal computers or to install all the software packages used by their professors. It is worth to mention that both referred options to provide the platform to students do not require any kind of programming, they are available in LabVIEW.

ACKNOWLEDGEMENTS This work was partially supported by Fundação para a Ciência e a Tecnologia (FCT) through the Project PTDC/EME-PME/120830/2010 and the PhD grant SFRH/BD/44696/2008.

REFERENCES Almarshoud, A.F., “The advancement in using remote laboratories in electrical engineering education: a review”, European Journal of Engineering Education, 36 (5), 425-433, 2011.

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Becker Jr., T.L., Cannon, R.M., Ritchie, R.O., “An approximate method for residual stress calculation in functionally graded materials”, Mechanics of Materials, 32, 85-97, 2000. Du, X., de Graaff, E., Kolmos,A., “Research on PBL Practice in Engineering Education.”, Sense Publishers, Rotterdam, 2009. IFEES - International Federation of Engineering Education Societies, http://www.ifees.net/, Jan 2012. Kitayama S, Arakawa M, Yamazaki K, “Differential evolution as the global optimization technique and its application to structural optimization”, Applied Soft Computing, 11(4), 3792-3803, 2011. Koizumi, M., “The concept of FGM,” Ceramic Transactions: Functionally Gradient Materials, 34, 3-10, 1993. Li, L., Wang, T., “A unified approach to predict overall properties of composite materials”, Materials Characterization, 54 (1), 49-62, 2005. National Instruments (NI) Corporation, LabVIEW 2009 Help, 2009. Nguyen, T.-K., Sab, K., Bonnet, G., “First-order shear deformation plate models for functionally graded materials”, Composite Structures, 83 (1), 25-36, 2008. Ravichandran, K.S. “Thermal residual stresses in a functionally graded material system”, Materials Science & Engineering: A, 201, 269-276, 1995. Reddy, J.N., “Analysis of functionally graded plates”, International Journal for Numerical Methods in Engineering, 47, 663-684, 2000. Reddy, J.N., Chin, C.D., “Thermo-mechanical analysis of functionally graded cylinders and plates”, Journal of Thermal Stresses, 21 (6), 593-626, 1998. Silva, T.A.N., Loja, M.A.R., “Minimization of Thermal Residual Stresses on Functionally Graded Sandwich Structures Using Differential Evolution”, In Proc. of Int. Symp. on Computational Intelligence for Engineering Systems (ISCIES'2011), Portugal, 2011. Storn , R. and Price, K., “Differential Evolution - a simple and efficient heuristic for global optimization over continuous spaces”, Journal of Global Optimization, 11, 341-359, 1997. Storn , R. and Price, K., Differential Evolution Homepage, http://www.icsi.berkeley.edu/~storn/code.html, Jan 2012. Surendranath, H., Bruck, H. A. and Gowrisankaran, S., “Enhancing the optimization of material distributions in composite structures using gradient architectures”, International Journal of Solids and Structures, 40 (12), 2999-3020, 2003. Uemura S., “The activities of FGM on new applications”, Materials Science Forum, 423-425, 1-10, 2003.