Multilevel Design of Sandwich Composite Armors for Blast Mitigation using Bayesian Optimization and Non-Uniform Rational B-Splines

I n regions at war, the increasing use of improvised explosive devices (IEDs) is the main threat against military vehicles. Large cabin”s penetrations and high gross accelerations are primary threats against the occupants” survivability. The occupants” survivability under an IED event largely depends on the design of the vehicle armor. Under a blast load, a vehicle armor should maintain its structural integrity while providing low cabin penetrations and low gross accelerations. This investigation employs Bayesian global optimization (BGO) and non-uniform rational B-splines (NURBS) to design sandwich composite armors that simultaneously mitigate the cabin”s penetrations and the reaction force at the armor”s supports. The armors are made of four layers: steel, carbon fiber reinforced polymer (CFRP), aluminum honeycomb, and CFRP. BGO is a methodology to solve optimization problems that require the evaluation of expensive black-box functions such as the finite element (FE) simulations of the vehicle armor under a blast event. BGO has two main components: the surrogate model of the black-box function and the acquisition function that guides the optimization. In this study, the surrogate models are Gaussian processes and the acquisition function is the multi-objective expected improvement function. NURBS generate the armor”s shape. The numerical examples show three alternatives to optimize the armor at two levels: (1) thicknesses of the sandwich”s layers and (2) the armor”s shape. The three design alternatives differ in the number of optimized levels and the optimization approach (sequential or simultaneous). The results show that the simultaneous optimization of the thicknesses of the sandwich”s layers and the armor”s shape is the most effective approach to design vehicle armors for blast mitigation.


Introduction
T he most dangerous threats for American military personnel in modern war sites are the improvised explosive devices (IEDs).IEDs have caused closely half of the fatalities (4536 deaths from 2006 to 2019 [1]) and injured thousands of officers (20,000 officers until 2010 [2]).To combat these statistics, blast-protective vehicles such as the Mine Resistant Ambush Protected Vehicle (MRAPV) have been designed.One of the factors that contributes to the blastworthiness of the MRAPV is its v-shaped underbody [3].The ability of the v-shaped underbody to deflect the blast energy has shown that better armor designs can increase occupant survivability.
The aircraft industry has broad experience in studying acceleration injuries.The dynamic response index (DRI), originally developed to study spinal injuries due to high accelerations during the ejection seat, is an occupant injury metric that can be used to evaluate the blast mitigation characteristics of a vehicle [4].DRI"s studies have shown that the risk of serious injuries is directly related to the ramp up acceleration experienced by the occupants [2].In addition to large accelerations, other factors that cause significant occupants" injuries are fragments entering the cabin and large cabin"s penetrations [5].Therefore, vehicle armors need to be designed to mitigate both large accelerations and cabin"s penetrations.
High-performance vehicle armors have been designed using surrogate-based optimization and generative design methods.In a single objective framework, the response surface method and gradient-based optimization designed sandwich composite armors.It was observed that optimal armor"s shapes prevent critical deflections and proper sizing of the armors" layers mitigate large accelerations [6].Artificial neural networks and genetic algorithms solved two-objective optimization problems to understand the effect of the armor"s curvature on the blast response of the sandwich structure.It was observed that the minimization of the deflection, maximization of the energy absorption, and minimization of the armor"s weight are competing objectives [7].This observation highlights the importance of addressing the design problem in a multi-objective framework, which captures the interactions between the objectives.In the case of generative design methods, the hybrid cellular automaton (HCA) method designed armors made of steel and aluminum.The HCA method distributed the materials through the armor"s volume in order to mitigate the displacement and acceleration caused by a blast load [2].
In the last years, sandwich composites panels have found increasing use over monolithic panels in blast mitigation applications.The stiffness and strength of steel layers prevents cabin"s perforation by fragments generated during the blast [2,8].Cellular material cores, frequently in the form of metallic foams or honeycombs (HCs), lead to lightweight designs capable of absorbing large amounts of energy [8].Several studies have pursued the use of functionally graded foams to design high performance armors [9].CFRP laminates have high strength-to-weight ratio, durability, and adaptability to curved shapes.Additionally, their failure mechanisms (delamination, matrix cracking, and fiber breakage) dissipate energy [10].CFRP laminates are often placed as reinforcements of existing structures to enhance blast resistance [11].
The design of sandwich composite armors for blast mitigation involves expensive simulations that prevent the use of traditional optimization techniques.BGO is an alternative to handle expensive simulations.It is a gradient-free optimization methodology to solve black-box function problems [12].A black-box function is an entity that does not have a closed form, but that can be evaluated at any point of the design domain, e.g., the FE simulation of an armor under a blast load.BGO has two main components: a surrogate model of the black-box function and an acquisition function.The surrogate model is a probabilistic predictive model of the black-box function.The acquisition function employs the probabilistic predictions of the surrogate model to search for optimal designs.BGO has solved design problems in multiple domains including control engineering, manufacturing, structural optimization, and blast mitigation [13,14,15,16].
This paper employs BGO to design sandwich composite armors for blast mitigation at two levels: (1) the thicknesses of the sandwich"s layers and (2) the armor"s shape.The armor"s shape is determined by a NURBS surface.The position of the control points of the NURBs surface are optimized in order to generate optimal shapes for the composite armor.
The numerical examples section presents three alternatives for the design of blast mitigation armors.These design alternatives differ in the number of optimized levels and the optimization approach (sequential or simultaneous).For comparison purposes, all the alternatives execute 100 iterations of the BGO algorithm.BGO minimizes two objectives: the armor"s penetration and the reaction force at the armor"s supports.The design variables are the thicknesses of the sandwich layers and the control points of the NURBS.The optimization problem includes constraints over the total thickness of the sandwich composite and the armor"s height.The explicit FE analysis code LS-DYNA simulates the blast event, and MATLAB performs the optimization.
The results show that simultaneous optimization of the thicknesses of the sandwich layers and the armor"s shape is the most effective approach to sandwich design composite armors for blast mitigation.

Sequential Design of Experiments
Design of computer experiments generates space-filling sampling plans that convey the maximum amount of information about a computer simulation [17,18].Traditional experimental design (ED) techniques include fractional designs, orthogonal arrays, and Latin hypercube sampling (LHS) [19,20].Traditional techniques present two major problems when facing constrained problems: sampling of unfeasible designs, and oversample or undersample of the simulation code [21].
In contrast to traditional techniques, sequential design (SD) methods can produce space-filling sampling plans for constrained problems (Figure 1).SD methods generate feasible sampling plans by considering the distribution of the samples in the design domain and the results of previous evaluations of the simulation code [22,23].This paper employs the Sequential Experimental Design (SED) toolbox presented in [21,22,23].

Gaussian Process
A stochastic process is a collection of indexed random variables.A Gaussian process (GP) is a stochastic process in which every finite collection of its random variables has a multivariate normal distribution [13].This particular feature places the GP as the most popular surrogate model in BGO.GP regression and kriging metamodeling are common names referring to the use of GPs as surrogate models of black-box functions [24,25,26].
The GP regression model Y(x) of a black-box function y(x) is defined as FIGURE 1 SD sampling plan of a problem with three design variables that are constrained to lie in the sphere with center at (3, 0, 0) and radius 2. The sampling plan has 120 designs and where f(x) is a regression function, Z(x) is a GP with zero mean and correlation matrix σ2 ψ, and x is a d-dimensional vector.
The goal of the regression function f(x) is to capture the trend of y(x) [12,26].In general form, f(x) can be defined as where b i (x) are k fixed basis functions and a i are the regression coefficients.
The GP Z(x) refines the predictions of the GP regression model Y(x) by using the residual information left by f(x) [12,26].The kernel function Ψ(x, x ' ) produces the correlation matrix ψ of Z(x).Valid kernel functions generate positive semi-definite correlation matrices regardless of the chosen pair of points (x,x ' ) [12,26].For example, where θ i and p i are parameters that control the characteristics of the regression model, such as the smoothness of the predictive model or the importance of each component of the design vector.
The regression coefficients a i , the GP variance s 2 , and the parameters θ i and p i are known as the hyperparameters of the GP model [27].Popular techniques to find the hyperparameters of a GP model include the maximum likelihood estimation, the maximum a posteriori, and the fully Bayesian approach [12].
After the training process, the GP regression model provides a mean predictive function ŷ x and a variance function ŝ2 x .ŷ x predicts the response of the black-box function y(x) at the design point x and ŝ2 x estimates the accuracy of the prediction (Figure 2).Given the effectiveness of Z(x) to refine the predictions of the GP model Y(x), it has been observed that a constant regression function ( ) = x f µ is enough to produce satisfactory predictive models [27].For a regression function ( ) = x f µ and a set of n samples X={x 1 ,..., x n }, with responses y={y 1 ,..., y n }, the GP predictor and the variance of the prediction at a point x are ( ) ( ) ( ) and where r(x) is the correlation vector between the samples X and the predicting point x, Ψ is the correlation matrix of the samples, and 1 is a n-vector of ones [27].MATLAB implementations of GP regression can be found in [20,26].

Cross-validation
There are several regression and correlation functions to build a GP model.Examples of regression functions include zero, first and second order polynomials.Examples of correlation functions are Gaussian, spherical, and spline kernels [20].The accuracy of a GP model depends on the chosen regression and correlation functions.The validation of the surrogate is the process that determines the most appropriate regression and correlation functions for the model.
One validation approach is to simulate additional points and compare their response with the predictions of the surrogate model.The best regression and correlation functions correspond to the pair that produces the lowest prediction errors.This validation approach is infeasible when dealing with expensive simulation codes, in which additional simulations cannot be afforded.
A validation approach that does not require additional simulations is leave-one-out cross-validation [28].Leaveone-out cross-validation consists in removing one sample, named y(x i ), from the n samples.Then, a n-1 surrogate is created with the remaining samples.The error between the true response of the removed sample and the prediction of the n-1 surrogate is calculated, e y y i n x x i i ˆ1 (Figure 3).This process is repeated for all the samples, and the root-meansquare error (RMSE) of the surrogate is calculated.We employ the RMSE to select the regression and correlation functions for the GP surrogate models knowing that high accuracy models present low RMSE values [28,29,30]

Expected Improvement
In BGO, the expected improvement (EI) function is the most popular acquisition function to solve single-objective optimization problems [12].The EI function determines the amount of improvement that a new sample x will produce in the current data.For a minimization problem, the EI function is defined as where y pbs = min{y 1 ,..., is the GP mean prediction at the design point x, and ŝ2 x is the variance of the prediction.
The closed-form of the EI function is where u y y s pbs x x x ˆ/ .ϕ( .) and ϕ( .) are the Gaussian probability and cumulative density functions [30].
During the solution of the optimization problem, the EI function is maximized to determine the next design to be sampled (simulated) or to finalize the optimization.If the maximum EI is larger than a threshold value, the design that maximizes the EI function is simulated (black-box function evaluation) and the GP surrogate is updated.Otherwise, the optimization stops.
Maximization of the EI function balances exploitation of the surrogate model and exploration of the design domain.The process identifies designs with promising predictions of the black-box function and\or high uncertainty in their predictions (Figure 4) [13].

Multi-objective Expected Improvement
The multi-objective expected improvement (MEI) function is the extension of the EI function to solve multi-objective problems.Considering an optimization problem with two objective functions, y 1 (x) and y 2 (x), to be minimized, the evaluation of the two functions using a sampling plan X produces an initial Pareto front with m designs where y y The MEI function is defined as where P[I(x)] is the probability of improving both functions y 1 (x) and y 2 (x) if the design x is sampled, and min(d 1 ,..., d m ) is the minimum Euclidean distance between the centroid of the probability of improvement and the current Pareto front [30] (Figure 5).
The probability of improvement is ) where x .The centroid of the probability of improvement where As in the single objective case, the MEI function is maximized to determine the next design to be simulated or to stop the optimization.The EI and MEI functions can be extended to solve constrained optimization problems [30,31].

Non-Uniform Rational B-Spline
NURBSs are the industry standard in the generation and representation of geometries for computer-aided manufacturing (CAM) and computer-aided design (CAD) systems.Other fields that employ NURBS include robotics, self-driving cars, virtual reality, and isogeometric analysis [32,33,34].
A NURBS surface is a bivariate vector-valued function of the form and where P i, j is a control point with weight w i, j .The NURBS surface has n and m control points in the u and v directions that define a control net (Figure 6).The degree of the NURBS surface in the u and v directions is p and q, respectively.{N i, p (u)} and {N j, q (v)} are non-uniform rational B-spline basis functions.The characteristics of {N i, p (u)} and {N j, q (v)} depends on the knot vectors U and V [33].For example, uniform knot vectors produce periodic basis functions.The knot vectors are defined as , , , , , ,, , ... ... ... and , , , , , , , , ... ... ... , where r=n + p + 1 and s = m + q + 1.
The knot vectors also influence other features of the NURBS surfaces, e.g., knot vectors with repeated elements at the beginning or at the end produces surfaces with clamped edges [32].In addition to the knot vectors, the position and weights of the control points determine the shape of the NURBS surface [35].
For a given knot vector, either the positions or weights of the control points can be used to modify the NURBS shape [35].In this study, the BGO algorithm finds the z-coordinates of the control points of NURBS surfaces that produce optimal armor"s shapes.The NURBS surfaces are generated using the toolbox presented in [36].

Optimization Methodology
This study employs SD of experiments, GP regression, leaveone-out cross-validation, MEI function, and NURBS in the design optimization of sandwich composite armors for blast mitigation (Figure 7).
SD of experiments generates sampling plans that satisfy the constraints over the bounds of the design variables, the maximum thickness of the sandwich composite, and the armor"s height.The sampling plan is simulated and GP surrogate models are trained using zero, first and second order regression functions, and Gaussian and exponential correlations.Leave-one-out cross-validation determines the most suitable regression and correlation functions.After selecting the initial surrogates, the genetic algorithms implementation of MATLAB finds the design that maximizes the MEI function.This design is simulated and the GP models are updated.Commonly, this process is repeated until the maximization of the MEI reaches a threshold value, which indicates the convergence of the optimization algorithm.In this study, we run 100 iterations of the BGO algorithm for all the design approaches in order to compare their performance.

Numerical Model
The baseline geometry of the armor is a 1x1 m 2 plate with clamped edges.A spherical blast load of 5 kg of TNT is located at 0.4 m above the center of the armor (Figure 8).The top surface of the armor is set as the loading surface using the *BLAST_SEGMENT_SET card.The blast load is calculated using the spherical free-air burst option of the *LOAD_BLAST_ENHANCED card.This card employs the empirical equations of the CONWEP (Conventional Weapons Effects Program) model [37].
The armor is made of four layers: steel, CFRP, aluminum HC and CFRP.The armor"s FE model employs shell elements to model the CFRP layers and solid elements to model the steel and HC layers (Figure 9).All the layers have 50x50 elements in the x-y plane.The solid elements use formulation type 2 and the shell elements use formulation type 16 [38].
The steel layer is modeled using the *MAT_PIECEWISE_ LINEAR_ PLASTICITY card.The properties for the material model are included in Table 1.The thickness of the steel layer is fixed to 10 mm in order to represent a vehicle"s underbody panel that is reinforced by the CFRP and HC layers.
Numerically, the layup of the CFRP laminate is represented using a twenty-four through thickness integration rule.The effective properties of the CFRP laminate is calculated using laminated shell theory.This theory is activated using the *CONTROL_SHELL card (LAMSHT = 1).The *CONTROL_ ACCURACY (INN = 2) card is used to minimize the sensitivity of the material coordinate system to in-plane shearing, which preserves the accuracy of the orthotropic shell elements [38].
The HC layer are made of aluminum foil AL5052 with density ?f and yield stress S y equal to 2680 (kg/ m 3 ) and 300 (MPa), respectively.The HC layer is made of regular hexagonal cells (Figure 10).The HC layer is modeled using the *MAT_ CRUSHABLE_FOAM card.Table 3 contains the homogenized properties of the HC expressed as functions of the aspect ratio t/D, where t is the wall thickness and D is the cell width.These expressions come from the non-linear homogenization implementation presented in [6]

Numerical Examples
This section presents three alternatives to design sandwich composite armors for blast mitigation.The armors employs two CFRP layers and an aluminum HC layer to reinforce of a 10-mm steel panel.In all the cases, the thickness of the composite armor is restricted to be less than or equal to 40 mm.The objectives of the optimization are to minimize the armor"s penetration and the reaction force at the supports.The armor"s penetration is defined as the maximum displacement of the armor"s nodes along the z-coordinate.In the three cases, 100 iterations of the BGO algorithm are performed.As suggested in [27], the sampling plans for the initial GP surrogates have 10d samples, where d is the number of design variables.The unreinforced 10-mm steel panel has a mass of 80 kg.This panel has a penetration of 102 mm and reaction force of 2740 kN under the blast event.

Size Optimization (SO)
The first design alternative is the size optimization (SO) of the thickness of the CFRP and aluminum HC layers where x = [x 1 x 2 x 3 ] and x i are the thickness of the reinforcing layers (Figure 11).The upper and lower bounds of the design variables are included in Table 4.The lower bounds ensure a mesh quality that prevents numerical instabilities.The upper bounds define a design domain that reduces the existence of infeasible designs.
Figure 12 shows the 30 samples generated by SD of experiments that considers the linear constraint over the armor"s thickness.The most appropriate surrogate model for the penetration has a first-order regression function and Gaussian     correlation (RMSE P = 1.7 mm), and the surrogate model for the reaction force has a zero-order regression function and Gaussian correlation (RMSE F = 211.2kN).
Figure 13 shows, the response of the 10-mm steel panel, the distribution of the training samples and the SO Pareto front.The CFRP and HC layers have increased the stiffness of the armor, which reduces the armor"s penetration but increases the reaction force.None of the SO designs dominates the 10-mm steel plate.
The optimal sandwich configurations start with designs made of thick CFRP layers (stiff designs), and end in designs with thick HCs (compliant designs) (Figure 14).Interestingly, the total thickness of the last set of optimal designs is less than 40 mm, which indicates that a design does not need to use all the available material to be optimal.
With respect to the 10-mm steel plate, the heaviest optimal design increases the weight of the armor in 47%, increases the reaction force in 53%, and decreases the penetration in 60%.The lightest optimal design increases the weight of the armor in 12.5%, increases the reaction force in 25%, and reduces the penetration in 27%.

Shape Optimization (SHO)
A sequential design approach for the vehicle armor can involve optimizing first the sandwich composite and then the armor"s shape.This example shows the shape optimization (SHO) of an armor with a predefined sandwich configuration.The sandwich configuration is one of the optimal designs of the SO example.
The selected sandwich configuration is the SO-Pareto design 12, x = [6.316 2] mm (Figure 13).This design has a mass, reaction force, and penetration of 95.8 kg, 3799 kN and 54.9 mm, respectively.The armor"s shape is generated by a NURBS surface with twenty-five control points (Figure 15).Given the symmetry of the problem and the boundary  The distribution of the SO and SHO Pareto fronts suggest that SO tends to reduce the penetration while SHO is capable of reducing both the penetration and the reaction force (Figure 16).Some SHO-optimal designs dominate the baseline design (SO-Pareto design 12).
Some training samples are dominated by the baseline design, which indicates that some shapes decrease the performance of the armor even for an optimal sandwich configuration.
Although none of the SHO-optimal designs dominates the 10-mm steel panel, some of the new designs substantially reduce the penetration (from 102 mm to less than 70 mm) while presenting similar reaction forces (2740 kN).
With respect to the 10-mm steel plate, the design approach produces a maximum increment in weight of 21% and a maximum increment in reaction force of 39%.The maximum reduction in penetration is 53%.The maximum increment in weight with respect to the SO-Pareto design 12 is less than 2%.These performance metrics suggest that shape optimization is an excellent alternative to generate lightweight armors for blast mitigation.Three optimal armor"s shapes are shown in Figure 17.

Size and Shape Optimization (SSHO)
In the third design alternative, the sandwich composite and the shape of the armor are optimized simultaneously.The constraints over the maximum thickness of the sandwich and the height of the armor are the same as the constraints used in the two previous examples.The problem formulation is find x ∈ R 3 minimize f 1 (x): Penetration minimize f 2 (x): Reaction force subject to: x 1 + x 2 + x 3 = 30 mm min(z armor (x)) = -25 mm max(z armor (x)) = 45 mm  For the generation of the initial surrogates, SD of experiments generates 70 samples that satisfy one linear and two non-linear constraints.The most appropriate GP surrogate models for the penetration (RMSE P = 3.4 mm) and reaction force (RMSE F = 445.4kN) have a first-order regression function and Gaussian correlation.SSHO leads to better results than SO and SHO.
The simultaneous optimization of the sandwich composite and armor"s shape produces a Pareto front that dominates both the SO and the SHO Pareto fronts (Figure 19).Additionally, some SSHO designs dominate the 10-mm steel plate.
With respect to the 10-mm steel plate, the design alternative produces a maximum increment in weight of 44%, a maximum increment in reaction force of 28%.The maximum reduction in penetration is 69%.
The thickness of most of the optimal sandwich configurations is less than 40 mm (Figure 20), and many of the design variables that control the armor"s shape have bound values (Figure 21).This indicates two aspects about the design of armors for blast mitigation: (1) the shape is the main contributor to the blastworthiness of the armor, and (2) the FIGURE 18 Size and shape optimization -Three design variables define the sandwich configuration and four design variables define the armor"s shape configuration of the sandwich improves the blastworthiness produced by the armor"s shape.
The three designs of Figure 22 show the evolution in the armor"s shape to produce low penetrations and/or low reaction forces.

Conclusion
This work presents three alternatives for the design optimization of sandwich composite armors for blast mitigation.These alternatives optimize the armor at two levels: sandwich configuration and armor"s shape.The design alternatives differ in the number of optimized levels and the optimization approach (sequential or simultaneous).
The three design alternatives are size optimization (SO) of the sandwich composite, shape optimization (SHO) of the armor, and simultaneous size and shape optimization (SSHO).All the alternatives employ SD of experiments and BGO.
SD of experiments generates space-filling sampling plans that satisfy constraints over the sandwich thickness and armor"s shape.The BGO framework uses GP regression and MEI as the surrogate and acquisition functions, respectively.The objectives of the optimization are the minimization of the armors" penetration and the minimization of the reaction force at the supports of the armor.For the sake of comparison, all the design alternatives execute 100 iterations of the optimization algorithm.The design alternatives that optimize the armor"s shape use NURBS.Optimization of the z-coordinates of the control points of the NURBS generates the optimal armor"s shapes.
The results show that the most effective approach to design vehicle armors for blast mitigation is the simultaneous optimization of the sandwich composite and the armor"s shape.The shape is the main contributor to the blastworthiness of the armor.The sandwich configuration enhances the blast mitigation capabilities produced by the armor"s shape.
Designers can use several aspects of this study to solve other optimization problems.Given that for the BGO framework the FE simulations are black-box functions, the presented design methodologies can produce optimal composite parts for other applications that require an FE code, e.g., crashworthiness.
Ongoing research aims to the development of other acquisition functions for BGO, multi-fidelity optimization, and design under uncertainty.

FIGURE 2 GP
FIGURE 2 GP regression model.The black line is the GP predictive mean ( ) ŷ x and the gray area denote the 95% confidence interval defined by the variance function ( ) (i) is a design of the current Pareto front.

FIGURE 4
FIGURE 4 Illustration of Bayesian optimization procedure using EI as the acquisition function for a minimization problem.The left scale is for the objective function and the right scale is for the EI function.The EI function is high where the GP model predicts low values of f(x) (exploitation at iteration 1) and where the uncertainty of the prediction is high (exploration at iteration 2)

FIGURE 5 .
FIGURE 5 Illustration of the components of the multiobjective expected improvement.The black dots denote the current Pareto front.The design x has predictions ( ) 1 ŷ x . The numerical examples use regular hexagonal cells of width D = 40 mm and wall thickness t = 1.84 mm.T he * CON TAC T_T I E D_ SH E L L _ E D GE _TO_ SURFACE_BEAM_ OFFSET card models the contacts

FIGURE 8
FIGURE 8 Baseline geometry, boundary and loading conditions of the sandwich composite armor

FIGURE 10
FIGURE 10 Regular hexagonal honeycomb geometry.The example problems use cells of width D = 40 mm and wall thickness t = 1.84 mm

FIGURE 11
FIGURE 11 Size optimization -Design variables

FIGURE 12
FIGURE 12   Size optimization -SD of experiments generates 30 samples that satisfy the constraint over the maximum thickness of the sandwich composite, which must be less than or equal to 40 mm

FIGURE 15
FIGURE 15 Shape optimization -Twenty-five control point NURBS that describes the shape of the armor and resulting armor"s design.Four design variables rule the z-coordinates of the nine unclamped control points

FIGURE 16
FIGURE 16 Shape optimization -SHO training samples, SO and SHO Pareto fronts, baseline design for SHO and response of the 10-mm steel panel

FIGURE 17
FIGURE 17 Shape optimization -z-coordinate contour plot (in m) of three armor"s shapes that produce low penetration and/or reaction force

FIGURE 22
FIGURE 22 Size and shape optimization -z-coordinate contour plot (in m) of three SSHO designs that produce low penetration and/or reaction force

TABLE 2
Material properties of carbon fiber unidirectional tape T700GF 12k/2510 © The Authors.

TABLE 3 Homogenized
material properties of regular hexagonal honeycomb

TABLE 4 Size
optimization -Lower and upper bounds of the design variables