It is important to understand the effect of curvature on the blast response of curved structures so as to seek the optimal configurations of such structures with improved blast resistance. In this study, the dynamic response and protective performance of a type of curved metallic sandwich panel subjected to air blast loading were examined using LSDYNA. The numerical methods were validated using experimental data in the literature. The curved panel consisted of an aluminum alloy outer face and a rolled homogeneous armour (RHA) steel inner face in addition to a closedcell aluminum foam core. The results showed that the configuration of a “soft” outer face and a “hard” inner face worked well for the curved sandwich panel against air blast loading in terms of maximum deflection (MaxD) and energy absorption. The panel curvature was found to have a monotonic effect on the specific energy absorption (SEA) and a nonmonotonic effect on the MaxD of the panel. Based on artificial neural network (ANN) metamodels, multiobjective optimization designs of the panel were carried out. The optimization results revealed the tradeoff relationships between the blastresistant and the lightweight objectives and showed the great use of Pareto front in such design circumstances.
The increasing threats of unexpected explosions on the battlefield and the terrorist actions threatening the public security have stimulated much interest in the development of more effective blastresistant materials and structures. The traditional blastresistant structures are usually designed in a bulky and solid way, which leads to poor operational performance and high costs [
In recent times, interest in curved sandwich panels for blast mitigation has grown by virtue of their spatial curvatures which provide additional stiffness against impulsive loads [
In addition to performance prediction, several pioneer works in seeking the optimal configurations of blast loaded sandwich structures have been reported. Liang et al. [
In this study, the dynamic response and protection performance of a type of curved metallic sandwich panel subjected to air blast loading were numerically examined and optimized by means of explicit nonlinear finite element (FE) simulations coupled with a metamodel based optimization procedure. The paper is structured as follows: Section
The curved sandwich panel considered in this work is shown in Figure
Baseline dimensions of curved sandwich panel.
The panel has the following baseline geometries: radius of curvature
The loading and boundary conditions are schematically shown in Figure
Loading and boundary conditions and meshes for 1/4 model of the curved sandwich panel.
The blastresistant performance of a structure can be evaluated by several indices. First of all, the maximum resultant deflection (MaxD) of the structure under shock loads needs to be reduced or confined to survivable levels. For instance, large deflection of an infantry vehicle floor under a landmine attack may result in the injury or even fatality of the occupant [
The nonlinear explicit FE programme LSDYNA 971 was used for the simulation. Only a quarter of the panel was modeled to shorten the simulation time due to the symmetric nature of the problem. Corresponding constraints were defined on the two symmetric planes, while the other two edges were fully clamped (Figure
The
The constitutive property of the RHA steel was represented by the JohnsonCook (JC) model [
Mechanical properties, JC model constants, and failure parameters for the facesheet materials of the curved sandwich panel.
Facesheet material  Density 
Young’s modulus 
Poisson’s ratio 
Yield stress 
Tangent modulus 
JC material constants and failure parameters  




 



 
RHA steel [ 
8000  210  0.28  —  —  950 
560 
0.014 
0.26 
Al2024 T3 [ 
2680  72  0.33  318  0.737  —  —  —  — 
Aluminum alloy Al2024 T3 was chosen for the outer face of the sandwich panel in view of its light weight. The constitutive behavior of the material is based upon the piecewise linear plasticity material model, Mat 24, in LSDYNA [
The closedcell aluminum foam is a lightweight material with excellent plastic energy absorbing characteristics and has been used as sandwich cores against blast loads [
The yield stress
Material constants for aluminum foam [







0  0  0  0.22 

590  140  40  320 

2.21  0.45  1.4  4.66 
Failure (element erosion) occurs when the plastic volumetric strain
The CONWEP (conventional weapons effects program) empirical model [
The CONWEP model implemented as the Load_Blast function [
Overpressure evolution of surface blast with 1 kg TNT charge and SoD = 350 mm.
To validate the numerical methods in use, the blast responses of curved sandwich panels tested by Jing et al. [
Specifications of curved sandwich panels and experimental [
Number of specimens  Radius (mm)  Facesheet thickness (mm)  Core thickness (mm)  Foam relative density (%)  Mass of TNT charge (g)  Inner face central deflection (mm)  

Test  FE  
1  250  0.8  10  15  20  17.80  17.55 
2  250  0.8  10  15  30  34.90  31.99 
3  500  1.0  10  15  20  10.72  12.53 
4  250  0.5  10  15  20  25.34  24.90 
5  250  1.0  10  15  20  12.66  14.22 
6  500  0.8  10  15  20  24.28  21.06 
A correlation plot between the experimental and numerical inner face central deflection is shown in Figure
(a) Comparison of numerical and test results of inner face central deflections and (b) tested and simulated panel deformation and simulated resultant displacement contour of specimen 1 in Table
The dynamic response of the curved sandwich panel with baseline geometries to blast loading is presented here. Figure
Simulated deformation and resultant displacement contours on (a) the sandwich panel, (b) the circumferential symmetrical plane, and (c) the longitudinal symmetrical plane at typical times after detonation.
Figure
Time histories of (a) central deformation of facesheets and foam core crushing and (b) energy absorption of the curved sandwich panel under blast loading.
The above results show that the strategy of using a “soft” outer face and a “hard” inner face in constructing a blastworthy flat sandwich panel [
It is of special interest in this research to analyze the effect of curvature on the blast resistance of the metallic sandwich panel. For this purpose, a group of sandwich panels with radii of curvature ranging from 250 to 1000 mm were investigated, all other dimensions (
Effect of curvature radius on the blastresistant indices of curved sandwich panel.
Figure
Simulated deformation and resultant displacement contours on the Cplanes of curved sandwich panels with different curvatures: (a)
In this study, an optimization problem was formulated and solved to find the optimal solutions for achieving multiple objectives in developing high performance cylindrical aluminum foam sandwich panels for lightweight applications, for example, vehicle armor structures. These objectives included mass and inner face deflection minimization as well as blast energy absorption maximization. Three individual case studies emphasizing on different objectives were considered. The multiobjective optimization problem is formulated as follows.
Find
which satisfy
and minimize
[
[
[
As the side length
Among the three design cases, Case Study 1 is focused on designing the curved panels whose deflection and mass both need to be minimized. As these two objectives are generally competing with each other, a set of best tradeoff solutions to the optimization problem are to be identified. Next, Case Study 2 aims at seeking an optimal set of panel designs with minimum deflection and maximum energy absorption, while panel mass is not the primary consideration in this case. For consistency, the original maximizing objective is converted to a minimizing objective of minus EA. Lastly the objective in Case Study 3 is to attain a set of panel designs that have the maximum energy absorption per unit structural mass, that is, SEA, and the minimum inner facesheet deflection under blast loading.
Of the three objectives, the panel mass can be easily calculated as an explicit function of the design variables. However, the MaxD and energy absorption of the panel are hardly computed explicitly and need to be numerically determined. To expedite the optimization process, the expensive FE analyses were replaced by metamodel predictions for objective function evaluations, which are discussed next.
Engineering optimizations generally require a large amount of performance evaluations to formulate objective and constraint functions. In this context, metamodels are extensively used instead of expensive FE analyses for fast iteration. In this study, the complex physical relationships between the blastresistant performance functions and the design variables of the curved panels are approximated by the artificial neural networks (ANNs) metamodels, which have been proven effective and efficient in the design of composite structures [
Based on performance of the biological neural system for data and information processing so as to learn and create knowledge, ANNs are formed of a diagram of simple processing elements called neurons that work together to solve problems. Each neuron returns an output signal when the weighed sum of the inputs exceeds an activation value. The output value is computed by the activation functions according to the inputs. The ANNs typically have three layers of neurons: neurons at the input and output layers represent the variables and responses of a system, respectively, while a hidden layer in between is composed of nonlinear activation functions to introduce flexibility into the modeling.
To model the complex blast responses of the curved panels while avoiding possible occurrence of overfitting problems, the ANNs used in this study are classical feedforward models with only one hidden layer as shown in Figure
Architecture of the threelayer feedforward neural network.
The ANNs were trained using the FE results of 200 design of experiments (DoEs) points, which were generated using the Sobol deterministic algorithm filling in a uniform manner the design space by maximally avoiding the design points of each other.
Regression plots, showing correlations between the FE values (inner facesheet MaxD and energy absorption of the panel) and the network’s outputs, are in Figure
Error analysis results of ANN metamodels for MaxD and energy absorption.
ANN metamodels  MAX  MAPE (%) 


MaxD  1.0748 mm  4.2005  0.9996 
Energy absorption  0.2686 kJ  0.4743  0.9999 
Regression plots of ANN metamodels (a) MaxD of inner facesheet and (b) energy absorption of panel.
Solutions to a constrained multiobjective optimization problem as in the present study are a group of best tradeoff designs, called “Pareto front” in the feasible domain where all the constraints are satisfied. The final design can be chosen from the Pareto front later on. Here, we used the nondominated sorting genetic algorithm (NSGAII) for the defined optimization problem. NSGAII features two effective sorting principals, that is, the elitist nondominated sorting and crowding distance sorting. The algorithm has proven rather effective for various multiobjective engineering optimization problems including composite designs [
Optimization procedure for curved sandwich panel blastresistant design.
To investigate the effects of design variables on the antiblast objective functions of the curved sandwich panel, a correlation analysis was performed using the DoE results. The correlation coefficient
Correlation matrix between objective functions and design variables.
Based on the ANN metamodel of MaxD, we employed the NSGAII to solve the optimization problem with objectives defined in Case Study 1. The 200 Sobol DoE designs were used as the first generation and the genetic algorithm iterated for 50 generations and was considered to converge.
Figure
Optimum panel designs for Case Study 1.
Panel 





Mass (kg)  MaxD (mm)  

ANN  FE  
P1  998.10  1.00  1.73  40.38  135  4.55  99.55  104.87 
P1′  993.46  1.00  1.96  40.45  135  4.72  94.51  99.53 
P2  427.15  3.22  3.99  58.66  355  14.99  10.41  11.65 
P3  351.81  1.00  3.98  59.31  289  9.99  28.57  29.26 
Note: P1 is the solution of minimum mass, P1′ is the solution of minimum mass with MaxD constraint of 95 mm, P2 is the solution of minimum MaxD, and P3 is the solution of minimum MaxD with mass constraint of 10 kg.
Optimization design results of Case Study 1.
Metamodel accuracy does affect not only the performance of the solution but also the design constraints. In Table
From Table
Figure
Optimum panel designs for Case Study 2.
Panel 





Mass (kg)  MaxD (mm)  EA (kJ)  

ANN  FE  ANN  FE  
P4  417.27  3.48  3.34  58.37  353  14.99  11.62  11.35  39.51  39.58 
P5  357.80  3.99  1.00  59.99  264  13.31  29.97  32.50  93.27  92.42 
P6  999.97  3.99  1.81  59.99  137  11.21  38.49  47.21  122.14  121.63 
P7  999.99  3.99  1.00  59.98  135  10.62  42.15  54.28  173.34  169.41 
Note: P4 is the solution of minimum MaxD, P5 is the solution of maximum EA with MaxD constraint of 30 mm, P6 is the solution of minimum MaxD with EA constraint of 120 kJ, and P7 is the solution of maximum EA.
Optimization results of Case Study 2.
It is seen from Table
The optimization results for Case Study 3 are shown in Figure
Optimum panel designs for Case Study 3.
Panel 





Mass (kg)  MaxD (mm)  SEA (kJ/kg)  

ANN  FE  ANN  FE  
P8  371.07  3.45  2.71  59.84  356  14.97  12.06  11.73  2.89  2.91 
P9  342.49  3.97  1.02  59.81  258  13.24  29.94  32.76  7.13  6.91 
P10  994.08  2.86  1.01  59.89  137  8.43  56.09  64.37  20.01  20.02 
P11  992.60  1.15  1.01  50.04  137  4.70  99.95  105.23  34.03  35.52 
P11′  993.90  1.28  1.01  50.91  137  4.98  94.24  99.56  32.41  33.39 
Note: P8 is the solution of minimum MaxD, P9 is the solution of maximum SEA with MaxD constraint of less than 30 mm, P10 is the solution of minimum MaxD with SEA constraint of larger than 20 kJ/kg, P11 is the solution of maximum SEA, and P11′ is the solution of maximum SEA with MaxD constraint of 95 mm.
Optimization results of Case Study 3.
The dynamic response of cylindrical curved sandwich panel to air blast loading was numerically studied. Three distinctive stages of panel deformation, that is, aluminum alloy front face deformation, RHA steel inner face deformation, and structural vibration, were identified. The results prove the effectiveness of using a “soft” material for outer facesheet and a “hard” material for inner facesheet in constructing the curved sandwich panel against blast loading. It was also found that the panel curvature has a monotonic effect on the SEA of the curved panel under investigation. However, MaxD of the panel was nonmonotonically affected by the panel curvature and this justifies the application of optimization techniques in the blastresistant design of such panels.
Three individual case studies each with two objective functions, that is, MaxD and mass, MaxD and EA, and MaxD and SEA, were presented in the design of different blastworthy curved sandwich panels, along with MaxD and mass constraints. Geometric parameters and foam core density were used as design variables. An ANN metamodel and NSGAII based multiobjective optimization procedure was proposed to solve the formulated design problems. The optimization results show that the two objectives in each design case conflict with each other, generally preventing simultaneous optimums from being reached. The identified Pareto front, however, provided a foundation for further decision making according to real applications of the curved panels. Metamodel approximation error may result in infeasible designs of the panel, and this could be overcome by assigning more stringent constraints on the violated boundaries.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (nos. 50905024, 51105053), the Liaoning Provincial Natural Science Foundation of China (no. 20102026), the Research Fund for the Doctoral Program of Higher Education of China (nos. 20090041120032, 20110041120022), and the Fundamental Research Funds for the Central Universities (nos. DUT13LK47, DUT14ZD212).