Multi-objective optimization of laser-welded steel sandwich panels for static loads using a genetic algorithm

Engineering Structures 49 (2013) 508–524

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Multi-objective optimization of laser-welded steel sandwich panels for static loads using a genetic algorithm Jeffrey D. Poirier, Senthil S. Vel ⇑, Vincent Caccese Mechanical Engineering, University of Maine, Orono, ME 04469, USA

a r t i c l e

i n f o

Article history: Received 3 November 2011 Revised 1 August 2012 Accepted 19 October 2012 Available online 5 January 2013 Keywords: Structural optimization Laser welding Steel sandwich panels Finite element analysis Genetic algorithms Evolutionary optimization

a b s t r a c t We present a methodology for the multi-objective optimization of steel sandwich panels for prescribed quasi-static loads. The steel sandwich panels consist of prismatic V-cores that are bonded to the facings using laser stake welds. Candidate sandwich panel designs are analyzed using geometrically nonlinear finite element analysis. The finite element model is validated by comparing the deflection and stresses for a representative sandwich panel with published experimental and numerical results. Sandwich panels are optimized for multiple, conflicting objectives using an integer-coded non-dominated sorting genetic algorithm. The methodology is illustrated through two optimization case studies. In the first study, we consider a rectangular steel sandwich panel configuration in which the facing segments are bonded to the core segments using double welds and optimize the panel geometry to minimize its deflection and mass. The second optimization study concerns a square steel sandwich panel in which the facings are bonded to the core segments using a single weld. The results demonstrate that the proposed methodology can be used to design lightweight laser-welded steel sandwich panels with superior structural performance. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction It has long been known that sandwich structures composed of stiff outer layers connected by a relatively low-density core result in high specific strength and stiffness, leading to substantial design advantages. Properly designed steel sandwich panels offer substantial resistance to static and dynamic loads due to their high relative stiffness and inherent energy absorbing capacity. To that end, steel sandwich construction has great potential for use in ships, building and bridge structures, especially for hazard reduction in situations of high wind, storm surge, earthquakes or accidental blast. Lok and Cheng [1] listed several other advantages of steel sandwich construction including the simplification of traditional connection processes, accurate construction, less surface distortion, better retention of pressure and low water leakage, greater flexibility for designers to create elegant curves, and ease of material transportation. They also noted that difficulty in fabrication and reliability of the face-sheet/core connection has been a continual problem in the widespread use of steel sandwich panels. Laser welding techniques, especially where laser and gas metal arc welding are combined into a hybrid welding process, hold great promise for overcoming the manufacturing impediments that have stood in the way of steel panel fabrication [2]. Past fabrication ⇑ Corresponding author. Tel.: +1 207 581 2777. E-mail address: [email protected] (S.S. Vel). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.10.033

methods for steel sandwich panels have relied upon fastening methods such as using periodically spaced screws, bolts or rivets [3], adhesive bonding [4], resistance welding [5], or brazing [6]. In comparison, laser welding of the core to the face-sheets in a steel sandwich panel system results in a robust, reliable and environmentally resistant connection that can be expected to have many years of service. In this process, the metal core is bonded directly to the metal face-sheet using a through-thickness stake weld as shown in Fig. 1 to create a continuous and reliable attachment. Laser welding can be performed at much greater speeds with production rates of 5–10 times that of conventional welding [7]. Good control over weld quality and weld profile has been demonstrated along with reduced residual stresses when compared to conventional welding [8]. Some other advantages of laser welding include ease of process automation, high productivity, increased process reliability, low distortion of the finished part and no requirement for filler materials. The core, which is an essential element in sandwich panel construction, is used predominately to resist transverse shear force. Core designs for sandwich panels can take on many forms and shapes depending upon the intended end use [9]. Prismatic cores are preferred in sandwich construction because they are simple to manufacture and their high longitudinal stiffness makes them ideal in cases where orthotropic plate action is preferred. The configuration used in steel sandwich panels typically results in a highly orthotropic structure where it is necessary to consider the

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Fig. 1. Fabrication of laser welded steel sandwich panels, (a) laser stake welding of prismatic cores to facings and, (b) post production laser welded sandwich panel (Applied Thermal Sciences Inc., Sanford, Maine, USA).

effects of shear deformations even at large length to depth ratios due to the low core transverse shear rigidity [10–14]. Tan et al. [15] performed an important experimental study of a spot-welded V-core type sandwich panel and found good correlation with finite element results. The effect of a discrete face-sheet/core connection in a C-core type sandwich panel was studied by Fung et al. [16] for use in building structures. The C shaped core material they analyzed was connected to the face-sheets using screws. They modeled this connection as a line of contact and developed a mathematical formulation for the panel response including the weak axis shear stiffness, which considers the local response of the core and the face-sheet/core connection. Lok and Cheng [1] developed a mathematical formulation for truss-core type sandwich panels. They developed expressions to predict the orthotropic stiffness and quantified the effect of the core angle on the response. Chang et al. [17] presented a closed-form solution based on the Reissner–Mindlin plate theory for the bending of corrugated-core sandwich panels. The dynamic response of steel sandwich panels to intense pressure impulses has been studied by Xue and Hutchinson [18]. Fleck and Deshpande [19] investigated the blast response of steel sandwich beams and showed an order of magnitude improvement in blast resistance over monolithic beams for the case of water loads. The advances in laser welding have made it possible to produce steel sandwich panels of different sizes and core geometries. When designing steel sandwich panels, the designer is faced with the daunting task of having to select a set of geometric parameters, such as the number of prismatic cores, thicknesses of the plate members and the corrugation angle, from a myriad of potential choices. Since the structural response and weight of a sandwich panel is highly dependent on the choice of geometric parameters, it is important to develop a robust methodology for the optimization of laser welded steel sandwich panels. Classical optimization methods, such as the gradient-based methods, suffer from certain inher-

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ent disadvantages such as a tendency to get stuck in local optima. In comparison, genetic algorithms, which belong to a class of search and optimization methods that mimic evolution through natural selection of ‘‘genetic’’ information, are better at finding global solutions and are easy to parallelize [20,21]. Genetic algorithms have found applications in many areas of engineering [22–26]. Klanac and Kujala [27] utilized a genetic algorithm to optimize the design of a stiffened steel cardeck. They considered two different optimization criteria, namely weight or cost of production, and obtained optimum structures for each objective separately. The design of practical laser welded sandwich panels will require the simultaneous maximization or minimization of multiple objectives. For example, we may want to minimize the weight while maximizing the stiffness. First attempts at trying to solve multiobjective optimization problems involved scalarizing the multiple objectives into a single objective using a weighted sum approach [28]. In those cases, the obtained solution is highly sensitive to the weights used in the scalarization process. However, the principles of true multi-objective optimization, where the design objectives are considered independently and simultaneously, give rise to a set of equally optimal solutions, known as Pareto-optimal or noninferior solutions, instead of a single optimal solution [29]. Upon completion of the optimization procedure, the designer can view the manner by which the Pareto-optimal solutions are distributed in the performance space, perform trade-off studies and choose the most suitable solution based on higher level information (e.g. see Pelletier and Vel [24]). The objective of this paper is to present a methodology for the multi-objective optimization of steel sandwich panels under quasi-static loading. Candidate designs are analyzed using a geometrically nonlinear finite element analyses via the general-purpose code ABAQUS [30]. An integer-coded genetic algorithm, based on the elitist non-dominated sorting genetic algorithm (NSGA-II), is implemented to obtain Pareto-optimal designs for multiple objectives. The finite element model is validated against experimental results presented by Tan et al. [15]. The optimization methodology is illustrated through two model problems having various conflicting objectives. In the first model problem, a 6 m  2.1 m rectangular sandwich panel is optimized simultaneously for mass and deflection with constraints on the yield safety factor. In the second model problem, a 2 m  2 m square panel is optimized for different loads, core heights and varying number of prismatic cores. The results demonstrate that it is possible to obtain lighter and stiffer panels through systematic optimization of geometric parameters and plate thicknesses.

2. Steel sandwich panel geometry In the present analysis, we consider discontinuous V-core sandwich panels such as the one shown in Fig. 2a. It is more computationally efficient to model thin members of the sandwich panel as plates or shells rather than treating them as three dimensional bodies. Accordingly, the facings and prismatic core segments are modeled as thin plates about their respective mid-surfaces as depicted in Fig. 2b. The stake welds are modeled as vertical plate segments that connect the midsurfaces of the facings and core segments. The length, width and total height of the sandwich panel are denoted by L, W and H, respectively. The sandwich panel geometry is described in terms of a representative unit cell with the core segments bonded to the facings using either one or two stake welds as shown in Fig. 3. The number of prismatic cores, which is also the number of repeating unit cells across the plate width, is denoted by Nc. The thickness of the top and bottom facings are denoted by t1 and t2, respectively. The angle of the inclined segment of the core to the horizontal is denoted by a and the core

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(a)

L/2 H CL

W/2 Laser welds Midsurfaces of plate members

CL

(b)

h CL CL Fig. 2. Quarter model of discontinuous V-core sandwich panel depicted as (a) full 3-dimensional sandwich panel geometry, (b) simplified panel geometry with prismatic core segments and facings modeled as thin plates.

t1

(a) w1

w1 tc

h α

w2

t2

t1

(b) β w1

γ w1 w1 tc λw2

h w2

α t2

Fig. 3. Representative unit cell with (a) single stake weld connecting facing and core segments, (b) two stake welds connecting facings and core segments.

plate thickness by tc. The widths of the horizontal segments of the core are w1 and w2. The distance between the midsurfaces of the facings, denoted by h, is related to the total depth of the sandwich panel and thicknesses of the facings as h = H  t1/2  t2/2. The stake welds are located symmetrically about the vertical axis that passes through the center of the unit cell. In the single weld configuration, the welds are positioned at the middle of the horizontal core segments as shown in Fig. 3a. In the double weld configuration, the welds are spaced at a distance of k  w2 from the end of the horizontal core segment of width w2, where k < 0.5. The welds on the horizontal core segments of width w1 are spaced at distances of b  w1 and c  w1 from the two ends as depicted in Fig. 3b, where b and c are both less than 0.5. End plates of thickness

tb are welded on all four boundaries of the sandwich panel. The edges of the sandwich panel are assumed to be simply supported and a uniform distributed load is applied on either the top or bottom facings of the panel.

3. Analysis of steel sandwich panels Several studies have been performed on the response of corrugated core sandwich panels to static loads. Libove and Hubka [31] developed a small deflection theory for the analysis of corrugated core sandwich panels and presented analytical expressions for the transverse shear and bending rigidities. Chang et al. [17] ana-

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lyzed simply supported continuous V-core steel sandwich panels using the Reissner–Mindlin plate theory. In their model, the sandwich panel is treated as a thick orthotropic plate and the effects of transverse shear deformation are included. The elastic rigidities of steel sandwich panels were calculated using the expressions of Libove and Hubka [31] and the deflections obtained using a Naviertype solution. Subsequently, parametric studies were performed for varying corrugation angles, core to face sheet thickness ratio and pitch to core depth ratio [17]. Cheng et al. [32] described a general numerical analysis approach for evaluating elastic constants in web core type sandwich structures. They used a series of shell type finite element models where representative boundary conditions were applied on a unit cell of the sandwich structure being analyzed. Recently, Yorulmaz et al. [33] employed frame analysis to evaluate the transverse shear stiffness of sandwich panels and calculated the deflection using the Reissner–Mindlin plate theory. While the orthotropic plate analysis reviewed in the previous paragraph is useful for evaluating the deflection of sandwich panels, calculation of the stresses in the facings and core segments requires a refined analysis that takes into consideration the local effects within each cell. In the present work, the deflection and stresses in sandwich panels under quasi-static loads are obtained through numerical simulations using the general-purpose finite element package ABAQUS [30]. The facings, prismatic cores and boundary plate members are discretized using eight-node thick shell elements (ABAQUS S8R elements). This type of a quadrilateral shell element has eight nodes on the element boundary with three rotational and three displacement degrees of freedom at each node. The reduced integration option is selected with four quadrature points on the mid-section and five integration points through the thickness. The vertical weld links that connect the facings and core are also modeled using shell elements. Simply supported boundary conditions are enforced on the edges by restraining the vertical displacement of the boundary plates and the sandwich panel is subjected to a uniform distributed load acting on its top or bottom surfaces. Only a quarter of the steel sandwich panel is modeled in ABAQUS due to the symmetry of the loading and boundary conditions. Depending on the geometry and magnitude of the distributed load, the facings and core segments may be subjected to large displacements. Therefore, a geometrically nonlinear analysis is performed to accurately capture the deflection and stresses of sandwich panels. Contact conditions are enforced between the facing and core segments to prevent material interpenetration. To define contact between surfaces within ABAQUS standard, contact pair definitions are created for specific surfaces. Contact pairs are defined for surface pairs with the potential to come in contact during quasi-static loading. Surface-to-surface discretization is used for master and slave surface designations because it is less sensitive to the chosen surface designation than node-to-surface discretization. Surface-to-surface contact conditions are enforced in an average sense over the slave surface as opposed to discrete points in node-to-surface contact since, in general, the former provides more accurate results for the stresses and contact pressure [30]. The contact interaction between pairs is modeled as ‘‘hard’’ contact whereby surfaces are allowed to separate after contact initialization as the load increases. The analysis maintains a minimum distance between the core and facings segments equal to the initial separation between those surfaces. Friction between contact surfaces is considered negligible and not used in our contact interaction model. The material is modeled as elastic with Young’s modulus E and Poisson’s ratio m. The displacements and stresses are calculated at key locations of interest in the sandwich panel and the von Mises stress is compared with the yield strength of the material to check for plastic yielding. In order to minimize the influence of stress concentrations due to the boundary condi-

tions, elements which have nodes that lie on the boundaries are excluded when checking for yielding. It is noted that we do not assess the integrity of the laser welds. The stress field around the welds requires a detailed three dimensional stress analysis which is beyond the scope of the present optimization study. The reader is referred to the work by Singh [34] on the analysis of laser welded connections. 4. Formulation of the optimization problem A sandwich panel design, D, is represented by a real-valued array consisting of the parameters that fully define the geometry of the panel. For the single-weld configuration shown in Fig. 3a, the design variables are the number of cells, Nc, corrugation thickness, tc, top facing plate thickness, t1, bottom facing plate thickness, t2, corrugation segment widths, w1 and w2, and core angle, a. Two possible loading cases are considered. The first is when the sandwich panel is loaded on the top facing as depicted in Fig. 2. In the second case, the sandwich panel is flipped over prior to applying the loading. This is equivalent to applying the distributed load on the bottom facing of the original sandwich panel geometry. Since the widths and thickness of the facing segments between the prismatic cores are different for the top and bottom facings, the local deflections due to the distributed load will also be different depending on the surface to which the load is applied. We introduce a parameter S which is defined as 1 if the load is applied on the top facing and 0 if it is applied on the bottom facing. The outer dimensions of the sandwich panel, namely L, W and H, which are prescribed based on the intended application, are held constant during the optimization procedure. The thickness of the boundary plate, tb, is also held constant. In the single-weld configuration, the complete set of parameters that define the sandwich panel geometry and loading direction are

D ¼ fN c ; t 1 ; t2 ; t c ; w1 ; w2 ; a; Sg:

ð1Þ

Thus, there are eight decision variables in the single-weld configuration. In the double-weld configuration, there are three additional parameters, namely the weld spacing parameters k, b and c. Thus, the set of parameters that completely specify the geometry of a panel in the case of double-welds are

D ¼ fN c ; t 1 ; t2 ; t c ; w1 ; w2 ; a; k; b; c; Sg:

ð2Þ

The general form for a multi-objective optimization problem with a number of objective functions to be minimized can be represented in the form:

Find D Minimize F j ðDÞ;

j ¼ 1; 2; . . . ; J;

Subject to g k ðDÞ 6 0;

k ¼ 1; 2; . . . ; K;

hm ðDÞ ¼ 0;

ð3Þ

m ¼ 1; 2; . . . ; M:

where Fj(D) are the objective functions that need to be minimized, gk(D) is the kth inequality constraint and hm(D) is the mth equality constraint. In problems that involve one or more objective functions that need to be maximized, the duality principle [29] states that a maximization problem can be converted to a minimization problem by multiplying the corresponding objective function by 1. Specific forms of objective functions and constraints are considered in Section 8. 5. Multi-objective optimization using genetic algorithms In general, a multi-objective optimization algorithm yields a set of optimal solutions, instead of a single optimal solution [29]. The reason for the optimality of many solutions is that no one solution can be considered better than any other with respect to all objec-

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tive functions. A single solution is said to be Pareto-optimal if improvement in one objective is only achieved at the expense of another objective. The set of such optimal solutions are known as Pareto-optimal or non-inferior solutions [29,35]. The primary goals of a multi-criteria optimization algorithm are to guide the search towards the global Pareto-optimal region and to maintain population diversity in the Pareto-optimal front. We now describe the multi-objective genetic algorithm used in this study. The objective of this algorithm is to search through the entire feasible solution space and identify a set of Pareto-optimal solutions. Our implementation utilizes an integer-coded version of the non-dominated sorting genetic algorithm developed by Deb [29] (NSGA-II) to obtain a set of Pareto-optimal designs. The NSGA-II algorithm is modified to include an archive of the historically non-dominated designs [24]. Genetic algorithms belong to a class of search and optimization techniques known as evolutionary algorithms. They work with a population of candidate solutions, and directly manipulate the unique strings of decision variables that define the solutions. For an individual solution, each decision variable is referred to as a gene, whereas the sequence of decision variables is known collectively as an individual’s chromosome. Genetic operators, such as gene cross-over and mutation, act on parent chromosomes to generate similar yet novel gene sequences, i.e. children. It is common to use binary coding to represent the genes that make up an individual’s chromosome [21]. However, binary representation is not suitable for representing standard plate thicknesses and geometric dimensions used in sandwich panel designs. To overcome this limitation, the genes for candidate sandwich panels are represented using an integer coding scheme. In this method, all possible choices for a design variable, the alleles, are grouped into an array. For example, the array tc lists the possible choices for the core plate thickness,

  ð2Þ ðmÞ ðN tc Þ t c ¼ t ð1Þ : c ; tc ; . . . ; tc ; . . . ; tc

ð4Þ

A specific core plate thickness t ðmÞ is identified by its index m. In c this coding scheme, a particular single-weld sandwich panel den o ðjÞ ðkÞ ðmÞ ðnÞ ðpÞ ðrÞ ðqÞ sign D ¼ N ðiÞ is represented in our c ; t 1 ; t 2 ; t c ; w1 ; w2 ; a ; S implementation of the NSGA-II by the corresponding integercoded chromosome, C ¼ fi; j; k; m; n; p; q; rg. Each design in the population is assigned a fitness for each of the objective functions to be minimized, as well as a measure of the constraint violation. To do this, the geometric parameters for an individual are read and a model of the sandwich panel is created in ABAQUS. Subsequently, a finite element analysis is performed to determine quantities of interest such as the maximum deflection and peak von Mises stress for prescribed loading and boundary conditions (see Section 3). These outputs are then utilized to create and assign the values of the objective functions, Fj(D), and constraint values, gk(D) and hm(D), to the individual design. In the case of multiple constraint violations, we combine all the constraint violations into a single constraint value, c(D), as follows [29] K M X X cðDÞ ¼ Hðg k ðDÞÞg k ðDÞ þ jhm ðDÞj; k¼1

ð5Þ

m¼1

ð1Þ DðiÞ and DðjÞ are feasible; with ðaÞ DðiÞ is no worse than DðjÞ in all objectives; and ðbÞ DðiÞ is strictly better than DðjÞ in at least one objective: ð2Þ DðiÞ is feasible while individual DðjÞ is not: ð3Þ DðiÞ and DðjÞ are both infeasible; but DðiÞ has a smaller constraint violation: ð6Þ (i)

(j)

If each of these is false, then designs D and D are said to be non-dominated. Note the constraint violation of an infeasible individual D(j) is defined by the function c(D(j)). The concept of constrain-domination enables us to compare two individuals in problems that have multiple objectives and constraints, since if D(i) constrain-dominates D(j), then D(i) is better than D(j). Perhaps most easily visualized in the case of two objective functions, Fig. 4 provides a graphical depiction of the dominance relation where two objectives are to be minimized. If we compare individuals D(1) and D(4), we note that D(4) has lower objective function values in both objectives. Therefore D(4) is said to dominate D(1). Similarly, D(4) dominates D(5) since D(4) clearly has a smaller value for the first objective although both have an identical value for the second objective. Upon comparing the solutions marked D(4) and D(3) in Fig. 4, we see that while one of these solutions is better in one objective, the other solution is better with respect to the other objective. This scenario violates the first condition of (6), and thus, D(4) and D(3) are said to be non-dominated. Lastly, we compare D(4) and D(6). Although, D(6) has lower objective function values in both objectives, it is an infeasible solution and therefore D(4) is said to dominate solution D(6). Thus, D(4) is not dominated by any other solution and it is said to be a non-constrain-dominated solution. Similarly, D(2) and D(3) are also non-dominated solutions. Based on these observations, we conclude that the individuals D(2), D(3) and D(4) are the non-constrain-dominated individuals and the individuals D(1), D(5) and D(6) are dominated. The non-constrain-dominated set consisting of the individuals D(2), D(3) and D(4) is designated to be of rank 1. The rank 1 individuals are then temporarily disregarded from the population and the non-constraindominated solutions of the remaining population are found and designated as the non-constrain-dominated set of rank 2. This procedure is continued until the entire population is classified into various subpopulations r(q) of rank q. Infeasible solutions are ranked according to the magnitude of their constraint violation. For example, in Fig. 4, the ranked subpopulations are r(1) = {D(2), D(3), D(4)}, r(2) = {D(1), D(5)} and r(3) = {D(6)}.

Feasible Designs Infeasible Designs

D(2)

15

F2 (minimize)

512

D(1)

10

D(5)

D(4) 5

where HðÞ is the Heaviside step function which equals unity if the operand is positive and zero otherwise. In order to perform multi-objective optimization, it is necessary to first understand the concept of constrain-domination that is used to compare individual designs within a population. Following the definition by Deb [29], an individual D(i) is said to constraindominate an individual D(j), if any of the following conditions are true:

D(6) D(3) 0

0

1

2

3

F1 (minimize) Fig. 4. Graphical depiction of the constrain dominance relation for the simultaneous minimization of two objectives.

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NSGA-II Evolutionary Search Cycle

P

g

M

P g

g

}

Controlled Elitist selection of parents }

} }

} }

Updated parent population

V

r1 r2 r3 r4

Crowded distance sort

V

Non-constraindominated Sort

V

Mating pool via tournament selection

VV V V

Parent population

P

g+1

r5

Q

Q

g

g

} }

Children via crossover and mutation

Combined population R

g

Update historical archive

H

Combine r1 with historical archive

g

H

g+1

Determine non-constraindominated set of (Hg U rg)

Fig. 5. Schematic of the controlled elitist non-dominated sorting multi-objective genetic algorithm showing the process by which the parent population, Pg, and historical archive of non-dominated solutions, Hg, are updated from generation g to g + 1.

(a)

(b)

Fig. 6. Crossover and mutation genetic algorithm operators, (a) example integer-based uniform crossover interchange of decision variables to generate two offspring from parents A and B, (b) example integer coded mutation operation on offspring A.

5.1. Evolutionary search process The genetic algorithm used in this study involves seven main operators. They are, (i) fitness evaluation, (ii) non-dominated sort and rank, (iii) tournament selection, (iv) crossover and mutation, (v) crowded distance sort, (vi) controlled elitist selection, and

(vii) historical archive construction. The process begins with a population of randomly created designs P0. A schematic of the process that is used to update the parent population, Pg, and historical archive of non-dominated solutions, Hg, from generation g to g + 1 is shown in Fig. 5. It is assumed that the objective function and constraint values have been previously evaluated for all N individuals

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in the parent population, Pg. The parent population is ranked into various subpopulations, r(q), of rank q based on constrain-domination. The first step in creating an offspring population for a multiobjective GA is to construct a mating pool using tournament selection. Two designs from the parent population are selected at random to compete in a tournament. The winner of the tournament, which is chosen based on a tournament selection operator [29], has a copy of itself placed in the mating pool. This process is repeated until each solution in the parent population has competed twice and the N spots in the mating pool have been filled. Once completed, the resulting mating pool will contain more copies of the more desirable individuals and fewer of the less desirable individuals from the parent population. Next, an offspring population, Qg, consisting of N individuals, is created from the parent population, Pg, through uniform crossover and mutation. The integerbased uniform crossover operator takes two distinct parent individuals and interchanges each decision variable, as shown in Fig. 6a, with a probability, 0 < pc 6 0.5. The crossover probability controls the degree to which children are similar to their parents. Following crossover, the mutation operator changes each of the children’s integer coded decision variable with a mutation probability, pm, from its current value to a random integer (see Fig. 6b). Next, the objective functions and constraint violations of each individual in Qg are computed and the parent population, Pg, and offspring population, Qg, are combined to create Rg = Pg [ Qg. Subsequently, the individuals in Rg are non-dominated sorted and ranked. The archive of non-dominated solutions, Hg, is updated to include the better ranked individuals in the combined population Rg. In order to ensure a population that is uniformly distributed in the objective space, preference is given to designs that are less crowded by sorting the combined population, Rg, from largest to smallest crowding distance within each rank [24]. Once individuals of equal rank are sorted in descending order of crowding distance, a controlled elitism mechanism is used to select N designs to form the updated parent population, Pg+1, from the combined population Rg [38]. The process is iterated for a fixed number of generations, G, or until a specific termination criterion is satisfied. In the end, the set of historical non-dominated designs, HG, is reported as Pareto-optimal.

6. Routine for automated analysis of candidate designs The genetic algorithm described in Section 5 generates new candidate chromosomes at each generation whose fitness must be evaluated. Each chromosome is translated to a corresponding set of design variables that completely define the sandwich panel geometry, plate thicknesses and loading surface. Prior to creating a finite element model for a given chromosome it is important to ensure that its geometric parameters lead to a feasible sandwich panel design. In order for the prismatic core to fit within a unit cell, the combined width of the horizontal core segment, w2, and the inclined portion of the facing needs to be less than the width of the representative unit cell, i.e., w2 + 2hc/tan a 6W/Nc, where hc = H  t1  t2  tc is the vertical distance between the core segments w1 and w2. Designs that fail this geometric feasibility condition are assigned a constraint value and sent back to the optimization algorithm with no need for a finite element analysis. If the geometric feasibility condition is satisfied, then the total width of the prismatic core, including the horizontal segment width of w1, is compared with the width of the unit cell. If the total width of the core is less than the width of the unit cell, the design corresponds to a sandwich panel with discontinuous prismatic cores. If the total width of the core exceeds the cell width, w1 is truncated to form a continuous corrugated core. In the latter case of a continuous core, the locations of the stake welds are modified to reflect the contin-

uous nature of the horizontal segments. In the case of a single weld, the location of the stake weld is shifted to the center of the newly created continuous horizontal core segment of width 2w1. A similar process is used for double welds whereby the stake welds are placed at a distance of 2cw1 from the edges of the core segment of width 2w1. The primary method of constructing a finite element model in ABAQUS is through the use of the Graphical User Interface (GUI) which allows the user to create parts and finite element models manually. Since the optimization of laser welded sandwich panels typically calls for the analyses of several thousand different candidate designs, it is impractical for the designer to create finite element models manually. Therefore, a custom interfacing routine has been developed for the automated analysis of candidate designs. The interface routine, which utilizes the ABAQUS Scripting Interface [36,37], allows for the finite element analysis of sandwich panels without needing to invoke the GUI. After reading the chromosome information for each candidate design, it generates a finite element model and submits it for analysis in ABAQUS. Once the analysis is complete, it reads the output file, extracts the displacements and stresses and calculates the objective functions for that sandwich panel design. A schematic of the interfacing routine is shown in Fig. 7. Once a candidate design passes the geometric feasibility check, the corresponding design parameters are written to a formatted text file. This text file is then catenated with a generic finite element modeling script and an additional script that retrieves the results from the ABAQUS output file upon completion of the analysis. The catenated script allows for the creation of a finite element model for any feasible design presented by the optimization algorithm. The modeling script uses the design parameters to generate the construction points for a unit cell. The script then loops through the number of cells defined and appends additional construction points for successive unit cells. The construction points are appropriately connected using straight lines to create a 2D sketch of the panel’s side profile. Subsequently, the 2D sketch of the sandwich panel is extruded out of plane to form a 3D model represented by planar surfaces. The extrusion of the sandwich panel profile includes the side plate that runs parallel to the extrusion process to simplify the model construction. Next, the other side plate is created as a separate part because its surface is perpendicular to the extrusion process. It should be noted that during the creation of the 2D sketch, ABAQUS uses a right hand rule to designate surface normal directions. The order in which two construction points are connected by a line determines the corresponding normal vector of the resulting extruded surface. This is important to take into consideration as the effects of surface normals can influence certain types of contact interactions and loading directions. Following geometry construction, the script adds material definitions and assigns section properties for the top facing, bottom facing, core, welds, and side plates. The section properties define the corresponding thicknesses for each section type, number of through-thickness integration points and the reference surfaces. Our modeling script selects the midplanes as the reference surfaces by default. Once the section properties have been assigned, the created parts are instanced and assembled together. The side plate is ‘‘welded’’ to the top and bottom facings through the use of tie constraints. ABAQUS uses ‘steps’ to specify the type and sequence of analyses to be performed. The analysis step is defined prior to the application of boundary conditions and loading since complex finite element model analyses can have multiple steps to which different boundary conditions and loading are applied. Since the goal of this study is to conduct an analysis for quasi-static loading, the modeling script specifies a general static step with geometric nonlinearity. Next the contact interaction properties are specified between

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515

Fig. 7. Schematic of the routine that interfaces the elitist non-dominated sorting multi-objective genetic algorithm to ABAQUS.

the horizontal segments of the facings and core. The simply supported boundary conditions are applied to the side plates and symmetric boundary conditions are enforced about the centerlines. The quantities of interest, such as displacements, strains and von Mises stress, are specified as the field outputs ABAQUS must save to its Output Data Base (ODB) file. An approximate element size is specified for use by ABAQUS to mesh the model geometry. ABAQUS will automatically reduce the element size in specific locations depending on the local geometry. Once this last stage of the modeling is completed, an ABAQUS input file is written. The interface routine then submits the input file for analysis to ABAQUS Standard. Once the analysis is complete the script accesses the ODB file, extracts the displacements and stresses, and saves the results to a separate text (output) file. The interface routine then uses the information in the output file to evaluate the objective functions and constraint values. These are then returned to the NSGA-II genetic algorithm. 7. Posteriori buckling analysis Some of the candidate designs generated by the genetic algorithm during the optimization process may exhibit either local or global buckling. It is difficult and computationally prohibitive to perform an automatic buckling analysis in addition to a stress analysis for each candidate design. Therefore, the sandwich panels are optimized for stiffness or yield strength using stress analysis. Once the Pareto-optimal set of design solutions has been identified, detailed analyses are performed for specific designs of interest to assess their buckling load. This provides ‘‘higher-level’’ information that one can incorporate for better trade-off comparisons in the objective space. Eigenvalues found from a linear buckling analysis of sandwich panels may be inaccurate or incorrect due to geometric and contact non-linearities. In an attempt to account for the non-linearities, a geometrically non-linear static analysis is performed using a small

preload, Pi. Subsequently a linear eigenvalue buckling analysis is performed using Lanczos eigensolver with a uniform unit pressure load. The smallest positive eigenvalue obtained for the buckling analysis is denoted by kb. Adding the initial preload to the computed eigenvalue buckling load gives the predicted buckling load, Pb, for the structure, i.e. Pb = Pi + kb. The predicted buckling load is used as the preload in the next step and another two-step analysis is performed to obtain a refined estimate for the buckling load. While some designs were seen to exhibit a convergence of the predicted buckling load as the preload was increased, other designs had predicted buckling modes that changed with preload and did not always converge. Therefore, the preceding two-step formulation was used to obtain a general estimate of the buckling load. To investigate the specific buckling phenomenon more closely, a quasi-static geometrically nonlinear analysis was performed with a very fine mesh and small load increments in an attempt to manually resolve the buckling load and mode shape. The buckling modes observed in the two-step analysis were used as a guide to identifying buckling in the quasi-static general analysis. Buckling is usually characterized by the sudden drop in stiffness of a particular region of the panel. Accordingly, we plot the global load vs. local deflection curve for different locations on the sandwich panel and look for evidence by an abrupt change in slope in the load– deflection curve. The corresponding load at which the slope changes significantly is considered to be the buckling load of the sandwich panel. 8. Numerical results and discussion 8.1. Validation of finite element model In this section, the finite element model is validated by comparing the deflections with experimental, numerical, and analytical results for the steel sandwich panel configuration shown in Fig. 8a

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(a)

(b)

Uniform distributed load p

165

y

182.5

115

x

2.5

L=5996 mm

z

107.5

2.5 2.5

H= 110 mm

365 82.5

W=2120 mm

415

(d) 25

(c) 8

20

6

Stress σx (Mpa)

Deflection uz (mm)

7

5 4 3

Present FE Tan FE study Tan Exp. study

2 1 0

0

0.5

1

15

10

Present FE Tan FE study Tan Exp. study

5

1.5

2

2.5

0

3

0

0.5

1

Distance x (m)

1.5

2

2.5

3

Distance x (m)

Fig. 8. Sandwich panel studied by Tan et al. [15] and comparison of results with present FE model, (a) steel sandwich panel geometry and quasi-static loading, (b) unit cell configuration (dimensions in mm), (c) comparison of centerline deflection of the bottom surface, (d) comparison of centerline longitudinal stress, rx, of the bottom surface.

top surface of the panel. The Young’s modulus and Poisson’s ratio of steel are taken to be 209 GPa and 0.3, respectively [15]. A convergence study is performed to determine the number of S8R elements required to obtain accurate results. It is found that roughly 15,000 elements are required to obtain converged values for the displacements and stresses. A total of 26,000 elements are used in the validation study to ensure accurate results. The deflections and stresses from the present analysis are compared with previously published results in Table 1. In their experimental study, Tan et al. [15] applied a uniform distributed load on the top surface using air bags and measured the centerline deflection on the bottom surface. They also performed a FE analysis using approximately 120 shell elements. The experimental and finite element deflections of the bottom surface, which were obtained from the deflection curves presented in Fig. 8 of their paper [15], are 7.6 mm and 6.0 mm, respectively. Lok and Cheng [1] developed an analytical model for steel sandwich panels and compared their results with finite element analyses. Their analytical and finite element deflections for the representative sandwich panel, which we presume are the average deflections, are 6.86 mm and 6.78 mm,

and b which was originally studied by Tan et al. [15]. It is considered as the representative sandwich panel and used as a baseline for comparison. The representative panel consists of a continuous corrugated steel core with each horizontal core segment bonded to the facing using two rows of spot welds. In the present work, the spot welds have been replaced by continuous laser welds. The sandwich panel is of length L = 5996 mm, width W = 2120 mm and total depth H = 110 mm. The panel geometry consists of 4 cells across the panel width. The facings and corrugated core sheet are of identical thickness t1 = t2 = tc = 2.5 mm. The width of the core segments are w1 = 82.5 mm and w2 = 165 mm, the vertical distance between the midsurfaces of the top and bottom facings is h = 107.5 mm and the corrugation angle a = tan1(102.5/100) = 45.7°. The weld location parameter (k) for core segment w2 is 0.1515. Due to the continuous corrugation, there is only a single weld for the core segment w1 within each cell, the location of which is defined by c = 0.303 (see Fig. 3). The thicknesses of the vertical weld elements and boundary plates are assumed to be 2.5 mm and 12 mm, respectively. A uniform distributed load of magnitude p = 5.5 kPa is applied on the

Table 1 Comparison of deflections and stresses for the sandwich panel geometry studied by Tan et al. [15]. The deflections are for a uniform distributed load of 5.5 kPa whereas the stresses are for a uniform load of 6.9 kPa. Top CL uz (mm)

Bottom CL uz (mm)

Average CL uz (mm)

Bottom CL rx (MPa)

Results from previously published studies Tan et al. [15]: Experimental study Tan et al. [15]: FE study Lok and Cheng [1,15]: Analytical Lok and Cheng [1,15]: FE study Chang et al. [15,17]: Analytical

– – – – –

7.6 6.0 – – –

– – 6.86 6.78 6.75

23.8 18.8 – – –

Present finite element analysis (a) w/o Nonlinear strains, w/o contact (b) w/ Nonlinear strains, w/o contact (c) w/ Nonlinear strains, w/ contact

7.97 7.90 7.14

5.99 6.02 5.63

6.98 6.96 6.39

17.47 18.59 18.77

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nate in Fig. 8c. The experimentally measured deflection is higher than the finite element value obtained by Tan et al. [15] as well as the predictions of our finite element model that includes contact conditions and geometric nonlinearity. The corresponding variation of the longitudinal stress rx is depicted in Fig. 8d. There is good agreement between our finite element results and those of Tan et al. [15], while the experimentally obtained stresses are larger. The stresses rx and ry at the center of the bottom facing are listed in Table 1. The experimentally obtained longitudinal stress, rx, on the bottom surface is 23.8 MPa [15]. Our finite element model predicts a longitudinal stress, rx, of 18.77 MPa on the bottom surface which is in good agreement with the value of 18.8 MPa obtained by Tan et al. [15] from their finite element model.

Table 2 Decision variables for optimization study I. Decision variable

Design choices

Nc t1 (mm) t2 (mm) tc (mm) w1 (mm) w2 (mm)

2, 3, 4, . . . , 20 0.5, 0.625, 0.75, . . . , 5.0 0.5, 0.625, 0.75, . . . , 5.0 0.5, 0.625, 0.75, . . . , 5.0 2.5, 5.0, 7.5, . . . , 100.0 5, 10, 15, . . . , 200.0 15, 20, 25, . . . , 90 0.1, 0.2, 0.3, 0.4 0.1, 0.2, 0.3, 0.4 0.1, 0.2, 0.3, 0.4 0, 1

a k b

c S

8.2. Optimization case study I: Double-weld configuration respectively. Chang [17] modeled the representative sandwich panel using the Reissner–Mindlin plate theory and obtained an average deflection of 6.75 mm. Three different finite element models are analyzed in the present validation study to assess the influence of geometric nonlinearity (large strains, deflections and rotations) and contact conditions (see Section 3) on the displacements and stresses. In the first case, an analysis is performed without contact conditions and geometric nonlinearity. The corresponding centerline deflection on the top and bottom surfaces are 7.97 mm and 5.99 mm, respectively. Inclusion of geometric nonlinearity in the present analysis causes the deflection of the top surface to reduce slightly from 7.97 mm to 7.90 mm (Table 1). The corresponding deflection on the bottom surface is 6.02 mm which is close to the deflection of 6.0 mm obtained by Tan et al. in their finite element analysis [15]. However, when contact conditions are enforced in addition to geometric nonlinearity, the centerline deflection of the top and bottom surfaces reduces to 7.14 mm and 5.63 mm, respectively. The corresponding average centerline deflection of 6.39 mm is slightly lower than the results presented by Lok and Cheng [1] and Chang et al. [17]. The centerline deflection of the bottom surface is shown as a function of the longitudinal coordi-

Maximum deflection uz (mm)

Generation 1

Generation 2

Generation 5

15

15

15

10

10

10

5

5

5

0

0

20

40

60

80 100 120 140 160 180

Areal density

0

0

20

(kg/m2)

40

60

80 100 120 140 160 180

Areal density

Generation 10

Maximum deflection uz (mm)

In this first model problem, we perform a multi-objective optimization of laser-welded steel sandwich panels that have the same outer dimensions and loading as the representative rectangular sandwich panel considered by Tan et al. [15] in their experimental and finite element studies. The goal of the optimization study is to minimize the mass and deflection of the panels while ensuring that the optimized designs are strong enough to preclude plastic yielding. The length, width and height of the panels are held fixed at L = 5996 mm, W = 2120 mm and H = 110 mm, respectively. The panel is subjected to a uniform distributed load of p = 5.5 kPa on its top surface. The approximate element size used to generate the finite element mesh is 35 mm. The facings and core are connected together using double welds as depicted in Fig. 3b. The number of cores Nc, facing plate thicknesses t1, t2, core plate thickness tc, core angle a, width of core segments w1, w2, weld location parameters k, b, c and panel loading parameter S are treated as optimization variables. Geometric nonlinearity and contact conditions are included in the finite element models when analyzing candidate designs. The yield strength, Sy, of the steel is assumed to be 310 MPa.

0

10

10

10

5

5

5

40

60

80 100 120 140 160 180

Areal density (kg/m2)

0

0

20

40

60

80 100 120 140 160 180

Areal density (kg/m2)

60

80 100 120 140 160 180

Generation 100 15

20

40

Generation 50 15

0

20

Areal density (kg/m2)

15

0

0

(kg/m2)

0

0

20

40

60

80 100 120 140 160 180

Areal density (kg/m2)

Fig. 9. Time lapse sequence showing the evolution of the parent population, Pg, at 1, 2, 5, 10, 50 and 100 generations.

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15

Deflection uz (mm)

A

10

Representative sandwich panel (Tan et al. [15])

5 B C

0 0

20

40

E

D

60

80

100

120

140

160

180

Areal density (kg/m2) Fig. 10. Final Pareto-optimal designs obtained from the historical archive, HG, for minimum displacement and areal density as discussed in optimization case study I.

The two objective functions that we seek to minimize in the present study are the areal mass density and peak transverse deflection of the top surface. The designs are subject to four constraints listed in Eq. (7). Given a candidate design D, the first inequality constraint g1(D) ensures that the width of an individual prismatic core is less than the width of the unit cell, i.e., w2 + 2hc/ tan a 6W/Nc. This constraint is enforced by the normalized geometric feasibility criterion g1(D) 6 0. If the candidate design satisfies this criterion, a geometrically nonlinear finite element analysis is performed using an incremental approach where the distributed load is gradually increased until the prescribed load, p, is reached. In certain cases, the finite element analysis may fail at a load of pmax < p due to excessively large deformation of the panel. This typically happens when the thicknesses of the facings and core are insufficient to support the distributed load. Such designs are assigned a constraint violation proportional to the difference p  pmax via the normalized equality constraint h1(D) = 0. If the finite element analysis runs to completion, the deflection and stresses are computed for the entire sandwich panel. Two additional constraints are introduced on the deflection and stresses to preclude excessive deformation and plastic yielding. The peak deflec-

Table 3 Selected Pareto-optimal designs for optimization study I. Design

Nc

t1

t2

tc

w1

w2

a

k

b

c

S

Continuous corrugation

Areal density (kg/m2)

Deflection (mm)

Yield factor Sy/r0

Tan’s panel A B C D E

4 17 15 16 16 17

2.5 .625 1.25 1.875 3.125 5

2.5 .5 .5 1.5 3.0 4.875

2.5 .5 .625 .625 .875 1.625

82.5 9.32 2.5 2.55 4.28 6.88

165 5 5 5 5 5

45.7 65 60 60 60 60

.15 0.2 0.4 0.1 0.1 0.1

0.3 0.3 0.3 0.1 0.1 0.2

0.3 0.3 0.2 0.2 0.2 0.4

0 1 1 1 1 0

Y Y N Y Y Y

62.3 17.2 23.0 36.3 62.3 105.6

7.2 13.8 3.0 0.8 0.4 0.2

3.2 1.1 2.9 10.8 17.9 51.7

Tan’s Case

Design A

Nc = 4 , Areal density = 62.3 kg/m2, Max Deflection = 7.2 mm σ'

97 MPa

Design B

Nc = 17 , Areal density = 17.2 kg/m2, Max Deflection = 13.8 mm σ'

0 MPa

Design C

Nc = 16 , Areal density = 36.3 kg/m2, Max Deflection = 0.8 mm

282 MPa

Nc = 16 , Areal density = 62.3 kg/m2, Max Deflection = 0.4 mm 29 MPa

0 MPa

100 MPa

0 MPa

0 MPa

Design D

σ'

Nc = 15 , Areal density = 23.0 kg/m2, Max Deflection = 3.0 mm σ'

Design E

σ'

13 MPa

0 MPa

Nc = 17 , Areal density = 105.6 kg/m2, σ' Max Deflection = 0.2 mm

6 MPa

0 MPa

Fig. 11. Depiction of core geometry and von Mises stress distributions of selected designs on the Pareto-optimal front for case study I along with the corresponding results for the reference panel geometry studied by Tan et al. [15]. The deformation is amplified by a factor of 10 for the sake of clarity.

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Find D Minimize F 1 ðDÞ ¼ qareal ðDÞ; F 2 ðDÞ ¼ maxuz ðD; xÞ; x2X

w2 þ 2h= tan a  1 6 0; Subject to g 1 ðDÞ ¼ W=N c maxx2X uz ðD; xÞ g 2 ðDÞ ¼  1 6 0; dlimit h1 ðDÞ ¼ pmax =p  1 ¼ 0;  0  Z 1 r ðD; xÞ H  1 dV ¼ 0; h2 ðDÞ ¼ V x2X Sy

ð7Þ

where the areal density qareal(D) = Mass (D)/(L  W). It is noted that the mass of the vertical boundary plates are not included when computing the areal density. The range of values for each decision variable in the optimization study are listed in Table 2. The NSGA-II optimization is performed using a population size of N = 150 individuals, gene crossover probability pc = 0.4, gene mutation probability pm = 0.1 and controlled elitism parameter value of 0.1 [38]. The maximum deflection of feasible designs is set at dlimit = 15 mm. The algorithm is terminated after G = 100 generations. Fig. 9 depicts the evolution of the parent population. It shows the objective function values of the parent population, Pg, at generations g = 1, 2, 5, 10, 50 and 100. As is evident from the figure, the algorithm progresses from the random designs in generation 1 to a fairly well defined front of non-dominated designs by generation 50. There appears to be only a slight improvement in the performance over the next 50 generations. The final Pareto-optimal designs obtained from the historical archive, HG, are shown in Fig. 10. Included in the figure is the performance of the representative sandwich panel studied by Tan et al. [15] which has an areal density of 62.3 kg/m2 and exhibits a maximum deflection of 7.14 mm. Four of the designs on the Pareto-optimal front, labeled A through E, are singled out for further investigation. The design variables and corresponding objective function values for the chosen designs are listed in Table 3. Design A is the lightest with an areal density of 17.2 kg/m2 and a deflection of 13.8 mm. Design C exhibits a 41.7% reduction in areal density and a 88.9% reduction in deflection compared to Tan’s geometry. It should be noted that the focus of Tan et al.’s work was on the comparison of theoretical predictions with experimental results and not on the optimization of steel sandwich panels [15]. Design D has nearly the same mass as Tan’s geometry but its deflection is 94.4% smaller. Design E is one of the heavier designs with an areal density of 105.6 kg/m2 and a deflection of 0.2 mm. The Pareto-optimal front exhibits a knee between designs B and D. Designs to the left of B exhibit a significantly higher deflection for only a slight decrease in areal density. Similarly, designs to the right of D exhibit a significant increase in areal density for just a slight decrease in deflection. Pareto-optimal designs be-

tween B and D, such as design C, offer a good compromise between weight and stiffness. The core geometry and von Mises stress distribution for the selected designs on the Pareto-optimal front are shown in Fig. 11 along with the corresponding results for Tan’s geometry. The thicknesses of the facings and core elements are represented by scaled line thicknesses in the cell geometry. The deformation is amplified by a factor of 10 for the sake of clarity in the plots. The selected Pareto-optimal designs have anywhere from 16 to 18 core elements compared to the four core elements used by Tan et al. [15]. The corresponding core plate thicknesses, tc, of the selected designs are smaller than Tan’s geometry. The increased number of core members with smaller core plate thicknesses lead to lightweight designs with higher transverse shear rigidity, and hence smaller deflection. However, the core plate members cannot be too thin since they may buckle thereby leading to higher deflection. The von Mises stress distribution is plotted for the selected Pareto-optimal designs in Fig. 11. The peak von Mises stress of Design A, which is the lightest design, is 282 MPa. Design D exhibits a peak von Mises of 17 MPa which is significantly smaller than the peak von Mises stress of 97 MPa for the reference panel geometry although both have nearly the same areal density. The yield safety factor for a sandwich panel is defined as the ratio of the yield strength to the maximum von Mises stress Sy/r0 over the entire panel. The yield safety factor at any location of the sandwich panel will be greater than or equal to the yield safety factor obtained for the entire panel. The yield safety factors are listed in Table 3

(a) 5.5 5.0 4.5

Distributed load (kPa)

tion for the entire panel maxx2Xuz(D; x) is subjected to the constraint that it be less than a prescribed deflection of dlimit. The peak deflection constraint is enforced through the normalized inequality constraint g2(D) 6 0. Finally, we compare the von Mises stress, r0 , with the yield strength, Sy, of the material to check for plastic yielding of the candidate design. Since the peak von Mises stress is sensitive to the finite element mesh and abrupt changes in geometry, we characterize the extent of plastic yielding by the sandwich panel volume that experiences stress levels greater than the yield strength. This is done by computing the total volume of all elements in the finite element mesh where r0 P Sy. Designs that exhibit plastic yielding are penalized in proportion to the volume of material that has plastically yielded through the normalized equality constraint h2(D) = 0. The optimization problem is formally stated as follows:

4.0 3.5 3.0 2.5 2.0 1.5

point a point b point c

1.0 0.5 0 0

2

4

6

8

10

12

14

Magnitude of displacement (mm) Top facing

(b)

a

b

Local buckling

Core

c

Bottom facing Fig. 12. Buckling of design A from optimization case study I showing (a) load– deflection curves at three different points on the panel, (b) buckled shape at a load of 5.5 kPa and the location of the load–deflection points shown in (a).

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(b) 20

80

18 16

70

Number of cells

Corrugation angle (degrees)

(a) 90

60 50 40 30

12 10 8 6

20

4

10

2

0

0

20

40

60

80

100

Areal density

120

140

160

0

180

0

20

40

(kg/m2)

60

80

100

120

140

160

180

140

160

180

Areal density (kg/m2)

(c) 6

(d) 100 90

tc t1 t2

5

80 70

4

w2 (mm)

Thickness (mm)

14

3 2

60 50 40 30 20

1

10 0

0

20

40

60

80

100

120

140

160

0

180

0

20

40

60

80

100

120

Areal density (kg/m2)

Areal density (kg/m2)

Fig. 13. Geometric parameters for the Pareto-optimal designs plotted as a function of areal density, (a) corrugation angle, (b) number of cells, (c) thicknesses of facings and core, (d) width of the horizontal core segment, w2.

for designs A to E along with the corresponding value for Tan’s panel. Design A has the lowest yield safety factor of 1.1 and Design E has a relatively high value yield safety factor of 51.7. A refined finite element analysis is performed on the selected Pareto-optimal designs to investigate if any of them have buckled. The distributed load on the top facing is gradually increased and the displacement is plotted as a function of the applied load for several different points on the panel. Fig. 12a shows the load– deflection curves for Design A at three different locations labeled a, b and c. Point a is located at the center of the top facing, point b lies on the core and point c is on the bottom facing as depicted in Fig. 12b. The load–deflection curve corresponding to point c in Fig. 12a shows a sharp change at a load of approximately 2.3 kPa. The slope decreases rather abruptly indicating a sudden loss in stiffness at that load which may be due to buckling. The displacement magnitude at the prescribed load of 5.5 kPa is shown on an exploded view of the panel in Fig. 12b. The bottom facing exhibits local wrinkling near point c as evidenced by the undulating displacement field with a characteristic wavelength. The geometric parameters for the Pareto-optimal designs are plotted as a function of the areal density in Fig. 13. The vast majority of optimal designs have a core angle, a, of 60° (see Fig. 13a) which is larger than the 45.7° core angle of Tan’s geometry. A few designs with low areal density have a slightly higher core angle of 65°. The number of cells in the optimized designs are in the range of 15–18 with a majority of designs having 16 cells. The thicknesses of the facings and cores are plotted in Fig. 13c. In order to obtain lightweight but stiff designs, the genetic algorithm keeps the core plate thicknesses small and gradually increases the thickness of the top and bottom facings with increasing mass until they reach their maximum thickness of 5 mm. Subsequently, the algo-

rithm increases the core plate thickness to provide additional stiffness. This transition occurs at an areal density of about 90 kg/m2. The increase in core plate thickness beyond this point does not lead to a substantial reduction in deflection although the mass increases significantly as is evident from Fig. 10. 8.3. Optimization case study II: Single-weld configuration In this next study, we perform a multi-objective optimization of square sandwich panels made of Weldox 700E steel. We had considered a double-weld configuration in the first optimization study to compare the optimized designs with Tan’s geometry. In this second optimization study, we consider the single-weld configuration shown in Fig. 3a since it is more economical compared to the double-weld configuration. Fig. 2b shows one such steel sandwich panel in which the core segments are connected to the facings using a single laser stake weld. The length, width and height of the sandwich panel are L = W = 2 m and H = 100 mm, respectively. The

Table 4 Decision variables for optimization study II. Decision variable

Design choices

Nc t1 (mm) t2 (mm) tc (mm) w1 (mm) w2 (mm)

2, 3, 4, . . . , 20 0.5, 0.625, 0.75, . . . , 5.0 0.5, 0.625, 0.75, . . . , 5.0 0.5, 0.625, 0.75, . . . , 5.0 2.5, 5.0, 7.5, . . . , 100.0 5, 10, 15, . . . , 200.0 15, 17.5, 20, . . . , 90 0, 1

a S

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(a) 50 45

Generation 1 Generation 5 Generation 10 Generation 25 Generation 50

Deflection uz (mm)

40 35 30 25 20 15 10 5 0 0

20

40

60

80

100

120

140

160

180

200

Areal density (kg/m2)

Deflection u z (mm)

(b) 15

A

10

5

B C 0

0

20

40

60

80

100

120

D

140

160

180

Areal density (kg/m2) Fig. 14. Results for case study II with 80 kPa loading optimized for minimum areal density and deflection depicting (a) evolution of parent population, and (b) final set of Pareto-optimal designs from the historical archive.

thicknesses of the vertical weld elements and boundary plates are held constant at 2.5 mm and 10 mm, respectively. The mass of the vertical boundary plates is 62.8 kg. The number of cores Nc, facing plate thicknesses t1, t2, core plate thickness tc, core angle a, width of core segments w1,w2 and panel loading parameter S are treated as optimization variables. Weldox 700E is a high strength steel with a yield strength of Sy = 859 MPa [39]. The objective functions and constraints are the same as in the previous optimization study. The approximate element size used to generate the finite element mesh is 25 mm. The sandwich panel is subjected to a uniform distributed load of p = 80 kPa. As done in the previous study, the maximum deflection of feasible designs is set at dlimit = 15 mm. The range of values for each decision variable in the optimization study are listed in Table 4. The genetic algorithm is run with a population size of

150 individuals. The evolution of the parent population is shown in Fig. 14a. The algorithm quickly progresses from the random designs of generation 1 to a well defined front of designs in generation 50. The algorithm was terminated at 50 generations since the front did not advance significantly with each passing generation. The deflection and areal density of the final set of Pareto-optimal designs in the historical archive is shown in Fig. 14b. The Pareto-optimal designs vary in areal density from 24.2 kg/m2 to 160.8 kg/m2. We select four Pareto-optimal designs, labeled A through D, for further study. The geometric parameters, areal density and deflection for each of the four designs are listed in Table 5. Design A, which is the lightest design, has an areal density of 24.2 kg/m2 and exhibits a deflection of 14.9 mm. Design D is the heaviest design with an areal density of 160.8 kg/m2 and deflection of 1.0 mm. The core geometry and von Mises stress of all four designs are shown in Fig. 15. Design A has 10 prismatic cores whereas Designs B, C and D have 20 each. The core angle, a, for Designs A, B and C is 65°. Design D has a slightly smaller core angle of 62.5°. Design B, which lies near the knee of the Pareto-optimal front, offers a good compromise between weight and stiffness. Design A is 54.8% lighter but its deflection is 496% higher than B. Likewise, Design D is 200.6% heavier but its deflection is only 60% lower than Design B. As in the previous optimization study, a detailed buckling analysis was performed for the selected designs. It is found that buckling initiates at a load of approximately 46 kPa for Design A. The buckling load of the other three designs are higher than the design load of 80 kPa. Optimization studies are performed for two additional design loads of 40 kPa and 160 kPa and the corresponding Pareto-optimal fronts are shown in Fig. 16a along with the front for the 80 kPa load. The Pareto-optimal fronts look qualitatively similar although the front moves away from the origin as the magnitude of the load is increased. This is due to the fact that for a design of given mass the deflection increases with load. To assess the performance of the optimized designs to different loads, the designs that were optimized for the 40 kPa load are subjected to a load of 160 kPa. Their performance is then compared with the designs that were specifically optimized for a load of 160 kPa. The results of the comparison are shown in Fig. 16b. It is found that most of the 40 kPa designs either suffer plastic yielding or excessive deflection (greater than the constraint value of 15 mm) in addition to plastic yielding. Only 36 of the 106 designs were able to withstand a load of 160 kPa without plastic yielding. This indicates that steel sandwich panels need to be optimized for prescribed design loads. If the design load changes, it is necessary to rerun the algorithm to obtain designs that are optimized for that specific load. The sandwich panel designs described earlier in the present work are optimized for a variable number of prismatic cores up to a maximum of 20. As the number of cores increases, the fabrication cost also increases due to the higher number of stake welds that need to be performed. It is therefore important to keep the number of cores to a minimum. Accordingly, we investigate the influence of the number of cores on the performance of the sandwich panels by conducting optimization studies in which the number of prismatic cores is fixed. Three optimization studies are performed for a distributed load of 80 kPa with the number of

Table 5 Selected Pareto-optimal designs for optimization study II. Design

Nc

t1

t2

tc

w1

w2

a

S

Continuous corrugation

Areal density (kg/m2)

Deflection (mm)

Yield factor Sy/r0

A B C D

10 20 20 20

.875 2 4.75 5

1.125 2.625 4.375 4.875

1 1 2.25 5

2.5 2.5 2.5 2.5

5 5 5 5

65 65 65 62.5

1 0 1 1

N N N N

24.2 53.5 110.2 160.8

14.9 2.5 1.2 1.0

1.3 1.8 9.0 21.4

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Design B

Design A

Nc = 10 , Areal density = 24.2 kg/m2, Max Deflection = 14.9 mm

σ'

500 MPa

Nc = 20 , Areal density = 53.5 kg/m2, Max Deflection = 2.5 mm

σ'

150 MPa

0 MPa

0 MPa

Design C

Nc = 20 , Areal density = 110.2 kg/m2, Max Deflection = 1.2 mm

Design D

σ'

50 MPa

Nc = 20 , Areal density = 160.8 kg/m2, Max Deflection = 1.0 mm

0 MPa

σ'

45 MPa

0 MPa

Fig. 15. Depiction of core geometry and von Mises stress distributions of selected designs on the Pareto-optimal front for case study II with 80 kPa loading optimized for minimum areal density and deflection. The deformation is amplified by a factor of 10 for the sake of clarity.

cores fixed at 4, 8 or 16. The corresponding Pareto-optimal designs are shown in Fig. 17. The Pareto-optimal front for 16 cores is nearly identical to the one obtained using a variable number of cores. However, the fronts for the 4 and 8 cores are significantly different. For example, Pareto-optimal designs that have an areal density of 60 kg/m2, have deflections of roughly 6.6 mm, 3.4 mm and 2.3 mm for 4, 8 and 16 cores, respectively. In other words, when compared to a panel optimized for four cores, the deflection of the optimized 8 and 16 core designs are smaller by 50% and 66.18%, respectively. Similarly, for a given deflection, the areal density of the optimized panels increase as the number of cores is reduced. It is therefore important to weigh the increase in performance with the potential increase in fabrication cost as the number of cores is increased. It is possible to include cost (material as well as fabrication costs) as an additional objective function during the optimization process to ensure that the designs are as cost effective as possible, which is very important if the system is to be implemented on a widespread basis. It is noted that the majority of optimal designs, especially those near the knees of the Pareto-optimal curves, have a core angle, a, that is approximately 42.5°, 40° and 60° for 4, 8 and 16 cores, respectively, compared to 65° for the case of variable number of cores. An optimization study was also performed for a distributed load of 80 kPa for a single weld configuration with the objective functions set to maximize the yield factor of safety and minimize the areal density. The optimization variables were kept the same as those used in previous optimizations in order to compare the results with the 80 kPa optimization for minimal deflection and minimal areal density. Fig. 18a shows the displacement vs. areal density for the two different objective function optimizations described previously. As expected, the designs specifically optimized for minimal displacement have a smaller displacement than those designs that were optimized for maximum yield safety factor. It is noted that the algorithm identifies fewer designs in the Pareto-

optimal front when optimizing the panels for yield safety factor, especially designs of higher mass, when compared with the optimization for minimal displacement. It is likely that more designs would have been identified if the algorithm had been run for longer. Fig. 18b depicts the yield factor of safety vs. areal density for the same designs shown in Fig. 18a. The horizontal dashed line in Fig. 18b represents a yield factor of safety of 1. All optimized designs fall above the dashed line since yielding was considered a constraint violation in both optimization studies. The results shown in Fig. 18a and Fig. 18b show that optimization based on different objective functions can lead to designs with different structural responses that are dependent on the objectives specified and the constraints imposed. It should be noted that it is possible to account for multiple loading cases although we have considered only a single load case in the present work. This can be done by analyzing the response of each candidate design to multiple loading conditions with the worst-case deflection or factor of safety used as an objective in the optimization study.

9. Conclusions A methodology has been presented for the multi-objective optimization of laser-welded steel sandwich panels under quasi-static loads. Candidate designs are analyzed using a geometrically nonlinear finite element analysis via ABAQUS [30]. The finite element analysis is validated by comparing the displacements and stresses for the sandwich panel geometry studied by Tan et al. [15] against their experimental and numerical results. The geometry and plate thicknesses of the core and facings are optimized using an integercoded version of the elitist non-domination sorting genetic algorithm. The genetic algorithm is interfaced with ABAQUS using a custom interfacing routine.

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(a) 15

(a) 15 Optimized for yield safety factor Optimized for displacement

Displacement uz (mm)

Deflection uz (mm)

40 kPa 80 kPa 160 kPa 10

5

0 0

10

5

0 20

40

60

80

100

120

140

160

180

0

20

40

(b) 30

80

100

120

140

160

180

140

160

180

(b) 25 Yield factor of safety (Sy /σ')

160 kPa designs 40 kPa designs Yield Yield and excess deflection

25

Deflection uz (mm)

60

Areal density (kg/m2)

Areal mass (kg/m2)

20

15

10

Optimized for yield safety factor Optimized for displacement

20

15

10

5

5

0

0

20

40

60

80

Areal mass

100

120

140

160

180

15

Deflection uz (mm)

Variable number of cores 4 cores 8 cores 16 cores

10

5

0

20

40

60

80

100

0

20

40

60

80

100

120

Areal density (kg/m2)

(kg/m2)

Fig. 16. Comparison of results for case study II showing (a) Pareto-optimal designs corresponding to three different design loads of 40 kPa, 80 kPa, and 160 kPa, (b) Pareto-optimal designs optimized for a load of 160 kPa along with the performance of Pareto-optimal designs that are optimized for 40 kPa when subjected to a load of 160 kPa.

0

1 0

120

140

160

180

Areal density (kg/m2) Fig. 17. Comparison of Pareto-optimal designs optimized for a load of 80 kPa with 4, 8 or 16 prismatic cores.

Fig. 18. Comparisons of results from case study II for a load of 80 kPa depicting (a) the displacement vs. areal density and (b) the yield factor of safety vs. areal density of Pareto-optimal designs optimized for minimum areal density and minimum deflection shown along side the Pareto-optimal designs optimized for minimum areal density and maximum yield factor of safety.

The proposed methodology is illustrated through two multiobjective optimization studies. In the first study, we consider rectangular steel sandwich panels that have the same overall length, width and height as the reference sandwich panel analyzed by Tan et al. [15]. Each segment of the core is bonded to the facings through a pair of laser welds similar to the double spot weld configuration used by Tan et al. [15]. The goal of the study is to obtain Pareto-optimal designs that minimize the deflection and areal density of the panel under prescribed loads. Constraints are set on the peak von Mises stress to ensure that the resulting sandwich panels do not exhibit plastic yielding. The geometry, thicknesses of the facings and core plate elements, number of prismatic cores, weld locations and the variable that specifies the loading surface are treated as optimization parameters. The algorithm generates a large number of Pareto-optimal designs that the designer can choose from ranging in areal mass density from 17.2 kg/m2 to 154.9 kg/m2. The Pareto-optimal front exhibits a distinct knee in the objective space. The designs close to the knee of the Paretooptimal curve offer a good balance between stiffness and mass. In the second study, we optimize 2 m  2 m square sandwich panels for mass and deflection. The optimized designs, which converge to a clearly defined Pareto-optimal front within 50 generations, range in mass from 24.2 kg/m2 to 160.8 kg/m2 for a design

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load of 80 kPa. Optimization studies are also performed for 40 kPa and 160 kPa loads. It is shown that the designs obtained for a particular load are not necessarily optimal designs for other loads. In other words, sandwich panels need to be optimized for a specific design load. When the number of prismatic cores is treated as an optimization variable, the resulting optimized sandwich panel designs are essentially truss core sandwich panels with continuous corrugations and core angle, a, that lies in the range of 60–65°. However, in the case of fixed number of cores, the optimal core angle depends on the number of cores. Acknowledgements The authors are grateful to Andrew J. Goupee for his help with the implementation of the NSGA-II algorithm and Jacob L. Pelletier for his constructive comments that helped improve the presentation of this work. References [1] Lok T-S, Cheng Q-H. Elastic stiffness properties and behavior of truss-core sandwich panel. ASCE J Struct Eng 2000;126:552–9. [2] Roland F, Manzon L, Kujala P, Brede M, Weitzenbock J. Advanced joining techniques in European shipbuilding. J Ship Prod 2004;20:200–10. [3] Fung TC, Tan KH. Shear stiffness for Z-core sandwich panels. ASCE J Struct Eng 1998;124:809–16. [4] Kennedy SJ, Murry TM. Ultimate strength of an SPS bridge – the Shenley Bridge, Québec, Canada. In: 2004 Annual conf of the transp assn of CA, Québec City, Québec; 2004. [5] Astech Inc. Innovative structural materials for the marine industry, Santa Ana, CA 92705, USA; 2005. [6] Cote F, Deshpande VS, Fleck NA, Evans AG. The out of plane compressive behavior of metallic honeycombs. Mater Sci Eng A 2004;V380:272–80. [7] Blomquist PA, Orozco N, Patch D. Laser fabricated structural shapes for new construction, overhaul and repair. J Ship Prod 2004;20:114–21. [8] Caccese V, Berube K, Blomquist PA, Webber SR, Orozco NJ. Effect of weld geometric profile on fatigue life of cruciform welds made by laser/GMAW processes. Mar Struct J 2006;19:1–22. [9] Kujala P, Romanoff J, Tabri K, Ehlers S. All steel sandwich panels – design challenges for practical applications on ships. PRADS 2004, LubeckTravemunde, 12–17 September, vol. 2; 2004. p. 915–22. [10] Plantema FJ. Sandwich construction. John Wiley & Sons Inc.; 1966. [11] Zenkert D. An introduction to sandwich construction. London: Engineering Materials Advisory Services; 1995. [12] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancaster (PA): Technomic Publishing Co. Inc; 1999. [13] Allen HG. Analysis and design of structural sandwich panels. Oxford, New York: Pergamon Press; 1969. [14] Kardomateas GA. An elasticity solution for the global buckling of sandwich beams/wide panels with orthotropic phases. J Appl Mech Trans ASME 2010;77:1–7.

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