Isaac Bastos
Correia Reis
Optimization of Layer Configurations for Ballistic
Impact on Light-Weight Armour Plates
Otimizac¸ ˜ao da Configurac¸ ˜ao de Camadas para
Protec¸ ˜oes Bal´ısticas de Baixo Peso
Isaac Bastos
Correia Reis
Optimization of Layer Configurations for Ballistic
Impact on Light-Weight Armour Plates
Otimizac¸ ˜ao da Configurac¸ ˜ao de Camadas para
Protec¸ ˜oes Bal´ısticas de Baixo Peso
Dissertac¸ ˜ao apresentada `a Universidade de Aveiro para cumprimento dos requisitos necess ´arios `a obtenc¸ ˜ao do grau de Mestre em Engen-haria Mec ˆanica, realizada sob orientac¸ ˜ao cient´ıfica de Jo ˜ao Alexandre Dias de Oliveira, Professor Auxiliar e de Filipe Teixeira-Dias, Reader do School of Engineering da The University of Edinburgh (United King-dom).
Esta dissertac¸˜ao teve o apoio dos projetos UID/EMS/00481/2019FCT -FCT - Fundac¸˜ao para a Ciˆencia e a Tecnologia; e
CENTRO010145FEDER022083 -Programa Operacional Regional do Centro (Centro2020), atrav´es do Portugal 2020 e do Fundo Europeu de Desenvolvimento Regional
presidente / president Prof. Doutor Ant ´onio Gil D’Orey de Andrade Campos Professor Auxiliar, Universidade de Aveiro
Doutor Jo ˜ao Filipe Moreira Caseiro
Investigador, CENTIMFE - Centro Tecnol ´ogico da Ind ´ustria de Moldes, Ferra-mentas Especiais e Pl ´asticos
Prof. Doutor Jo ˜ao Alexandre Dias de Oliveira Professor Auxiliar, Universidade de Aveiro (orientador)
longo do semestre tornaram-se essenciais e imprescind´ıveis. Tamb ´em pela motivac¸ ˜ao e o gosto pela otimizac¸ ˜ao transmitido.
Ao Professor Doutor Filipe Teixeira-Dias pelo apoio dado sempre que necess ´ario ao longo do semestre.
A todos os meus amigos que, de uma forma ou outra, influencia-ram o meu percurso acad ´emico. Sem eles a conclus ˜ao desta etapa de 5 anos n ˜ao teria sido carregada de t ˜ao bons momentos e de tantas aprendizagens.
Um enorme agradecimento `a minha fam´ılia que sempre me apoiaram e continuar ˜ao a apoiar a todos os n´ıveis e sem eles seria imposs´ıvel completar esta jornada.
Por ´ultimo, a todos aqueles que influenciaram o meu percurso acad ´emico e, mais importante, o meu desenvolvimento pessoal.
tion
abstract The broad field of engineering is facing a paradigm shift where advanced optimization methods and techniques are more often used to solve complex problems. Most of these problems either require the analysis of a large amount of data or the solving of complex calculations, or even both. This dissertation aims to develop an understanding of non-linear optimization algorithms applied to a complex engineering design problem: a multi-layer plate under a ballistic impact.
To solve a complex design engineering problem, the most efficient way is to combine non-linear optimization algorithms with a software capable of simulating the model and event. Accordingly, the first part of this document focuses on developing a Python script of the simulation model system using Abaqus API. The usage of an Abaqus Python script to simulate the event allows to generate specific variables and post-processing outputs essential to its posterior integration with optimization algorithms. Nevertheless, the development of a model that simulates a ballistic impact is complex and, thus, a sounding understanding on the physics and mechanics behind such an event are properly discussed. These insights are then used to validate the dynamic response and equilibrium of the simulated model. Furthermore, several modeling strategies are considered and analyzed throughout the first part of this document.
The second part of this dissertation aims to acquire a comprehensive understanding of three optimization algorithms: Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Simulated Annealing (SA). The perfor-mance and efficiency of each algorithm, as well as numerous programming and optimization strategies, are tested in four different benchmarks. Each benchmark increases in complexity regarding its precedent and they all use the Abaqus Python script previously developed.
This dissertation culminates in a multi-objective optimization procedure that uses the most efficient algorithm out of the three algorithms tested in the previous benchmarks. This multi-objective procedure uses every single-objective formulation, variables and constraints from the previous benchmarks which results in a highly non-linear problem. The results from this complex optimization problem are analyzed using and discussed.
Simulado (SA), Otimizac¸ ˜ao Multi-Objetivo
resumo O amplo ramo da engenharia enfrenta uma mudanc¸a de paradigma, na qual m ´etodos e t ´ecnicas avanc¸ados de otimizac¸ ˜ao s ˜ao cada vez mais usados para resolver problemas complexos. A maioria desses problemas requer a an ´alise de uma grande quantidade de dados ou requer a resoluc¸ ˜ao de c ´alculos complexos, ou at ´e mesmo ambos. Esta dissertac¸ ˜ao tem como objetivo desenvolver um estudo compreensivo de algoritmos de otimizac¸ ˜ao, aplicados a um problema complexo de projeto de engenharia: uma placa com multiplas camada sob um impacto bal´ıstico.
Para resolver um problema complexo de engenharia de projeto, a forma mais eficiente consiste em combinar algoritmos de otimizac¸ ˜ao n ˜ao-linear com um software capaz de simular o modelo e o evento. Assim, a primeira parte deste documento ´e focada no desenvolvimento de um c ´odigo em Python do modelo de simulac¸ ˜ao atrav ´es da API do Abaqus. O uso de um c ´odigo Python para simular o evento permite gerar vari ´aveis espec´ıficas e resultados de p ´os-processamento que s ˜ao essenciais para sua posterior integrac¸ ˜ao com algoritmos de otimizac¸ ˜ao. No entanto, o desenvolvimento de um modelo que simule um impacto bal´ıstico ´e complexo e, portanto, uma compreens ˜ao intr´ınseca sobre a f´ısica e a mec ˆanica de tal evento ´e discutido adequadamente. Esses conhecimentos adquiridos s ˜ao posteriormente usados para validar a resposta din ˆamica e o equil´ıbrio do modelo simulado. Al ´em disso, v ´arias estrat ´egias de modelagem s ˜ao consideradas e analisadas ao longo da primeira parte deste documento.
A segunda parte desta dissertac¸ ˜ao visa adquirir uma compreens ˜ao abrangente de tr ˆes algoritmos de otimizac¸ ˜ao n ˜ao-lineares: otimizac¸ ˜ao por enxame de part´ıculas (PSO), algoritmo gen ´etico (GA) e recozedura simulada (SA). O desempenho e a efici ˆencia de cada algoritmo, bem como numerosas estrat ´egias de programac¸ ˜ao e otimizac¸ ˜ao, s ˜ao testados em quatro benchmarks. Cada benchmark aumenta em complexidade em relac¸ ˜ao ao seu precedente e todos usam o c ´odigo Python do modelo em Abaqus previamente desenvolvido.
Esta dissertac¸ ˜ao culmina num processo de otimizac¸ ˜ao multi-objetivo que utiliza o algoritmo mais eficiente dos tr ˆes algoritmos testados nos benchmarks anteriores. Este procedimento multi-objetivo utiliza todas as formulac¸ ˜oes, vari ´aveis e restric¸ ˜oes das formulac¸ ˜oes dos benchmarks anteriores, o que resulta num problema altamente n ˜ao linear. Os resultados desse complexo problema de otimizac¸ ˜ao s ˜ao analisados e discutidos.
1 Introduction and Background 1
1.1 Armour Materials and Structures . . . 1
1.2 Optimization Studies . . . 2
1.3 Objectives and Scope . . . 3
1.4 Motivation . . . 3
1.5 Dissertation Structure . . . 4
2 Physics of Wave Propagation 5 3 Model Discussion and Validation 9 3.1 Abaqus Scripting and Python Programming . . . 9
3.2 Model Design . . . 11
3.2.1 Impact Region Analysis . . . 12
3.2.2 Boundary Conditions . . . 15
3.2.3 Mesh Convergence . . . 19
3.2.4 Symmetry Validation . . . 22
3.2.5 Dynamic Response of the System . . . 25
3.2.6 Elastic Wave Velocity Analysis . . . 25
3.2.7 Contact Type . . . 29
3.2.8 Configuration Study . . . 31
3.3 Interlayer Parameters Analysis . . . 33
3.4 Base Model Script Final Remarks . . . 36
4 Optimization Introduction and General Concepts 37 4.1 Fundamentals of Optimization . . . 37
4.1.1 Standard Mathematical Formulation . . . 37
4.1.2 Exterior Penalty Function Method . . . 37
4.2 Non-Linear optimization Algorithms . . . 38
4.2.1 Genetic Algorithm . . . 38
4.2.2 Particle Swarm Optimization . . . 42
4.2.3 Simulated Annealing . . . 44
5 Implementation and Methodologies 47 5.1 Benchmark I (Continuous Variable — Minimize weight subject to stress constraint) 48 5.1.1 Benchmark I — PSO . . . 49
5.2.2 Benchmark II — GA . . . 61
5.2.3 Benchmark II — SA . . . 63
5.2.4 Benchmark II — Summary . . . 64
5.3 Benchmark III (Discrete Variable — Minimize weight subject to stress constraint) 64 5.3.1 Benchmark III — PSO . . . 66
5.3.2 Benchmark III — GA . . . 68
5.3.3 Benchmark III — SA . . . 70
5.3.4 Benchmark III — Summary . . . 72
5.4 Benchmark IV (Discrete Variable — Minimize the rear stress subject to weight constraint) . . . 72 5.4.1 Benchmark IV — PSO . . . 72 5.4.2 Benchmark IV — GA . . . 74 5.4.3 Benchmark IV — SA . . . 76 5.4.4 Benchmark IV — Summary . . . 77 5.5 Alternative Approach . . . 77 6 Multi-Objective Optimization 81 6.1 Fundamentals and Methodologies . . . 81
6.2 Pareto Optimality and Utopia Point Concept . . . 82
6.3 Formulation and Strategies . . . 82
6.3.1 Normalization of the Objective Functions . . . 83
6.3.2 PSO Strategies and Operational Parameters . . . 84
6.4 Results and Analysis . . . 85
7 Final Remarks 89 7.1 Conclusion . . . 89
7.2 Further Work . . . 90
Bibliography 90 A Multi-objective Optimization Python Code 95 B Additional Graphics 113 B.1 Benchmark I . . . 113 B.1.1 PSO . . . 113 B.1.2 GA . . . 114 B.1.3 SA . . . 115 B.2 Benchmark II . . . 116 B.2.1 PSO . . . 116
B.3.1 PSO . . . 121 B.3.2 GA . . . 122 B.3.3 SA . . . 123 B.4 Benchmark IV . . . 125 B.4.1 PSO . . . 125 B.4.2 GA . . . 125 B.4.3 SA . . . 127
3.1 Results from the 3 boundary condition scenarios. . . 17
3.2 Evaluated variables in each step of the mesh optimization. . . 20
3.3 Results of a full sized model and a quarter model simulation. . . 23
3.4 Comparison between the velocity obtained from the simulation and from the theoretical formulation. . . 27
3.5 Description of each plate configuration scenario. . . 31
3.6 Transmission ratio between layers for each configuration. . . 33
3.7 Mechanical properties for each interlayer studied. . . 33
3.8 Comparison of the accuracy of results and the computational cost between the original and the optimized model. . . 36
5.1 PSO operational parameters. . . 49
5.2 Genetic algorithm operational parameters for benchmark one. . . 51
5.3 Simulated Annealing operational parameters. . . 54
5.4 Benchmark I final resutls. . . 58
5.5 Benchmark I final results after running five times for each algorithm. . . 58
5.6 Benchmark II final resutls. . . 64
5.7 Material’s database and correspondent indices. . . 65
5.8 PSO operational parameters for Benchmark III. . . 66
5.9 Genetic algorithm operational parameters for benchmark four. . . 68
5.10 Simulated Annealing operational parameters for benchmark three. . . 70
5.11 Final results of benchmark number three. . . 72
5.12 Final results of benchmark number four. . . 77
6.1 Normalization parameters used for the multi-objective optimization. . . 84
6.2 PSO operational parameters used in the multi-objective optimization problem. . 85
2.1 Transmission and reflection of an elastic wave at an interface. . . 6
3.1 Flowchart of how Abaqus scripting interface interacts with Abaqus Kernel. [Inc 2004] . . . 10
3.2 Model of the ballistic system. a) configuration nomenclature; b) ballistic impact model. . . 11
3.3 Front and rear impact region sets. . . 12
3.4 a) Stress from all nodes at the front impact region (graph extracted from Abaqus) and b) average stress at every time increment using a Python script. . . 13
3.5 a) Stress from all nodes at the rear impact region (graph extracted from Abaqus) and b) average stress at every time increment using a Python script. . . 14
3.6 Total force due to contact pressure at every time increment. . . 15
3.7 Three boundary condition studies: (a) plate clamped by the side surfaces of the plate, (b) plate pinned at the side edges of the rear surface of the plate, (c) plate clamped at the rear surface of the plate. . . 16
3.8 Evolution of σzfrom all nodes at the rear surface for a) boundary condition case 1, b) boundary condition case 2 and c) boundary condition case 3. . . 18
3.9 Number of elements against the time required to complete the simulation and the peak stress at the rear surface for case 1; the parameters chosen are marked with the dashed line. . . 20
3.10 Number of elements against the time required to complete the simulation and the peak stress at the rear surface for case 2; the parameters chosen are marked with the dashed line. . . 21
3.11 Details of the mesh. . . 22
3.12 Front view of the full plate. . . 23
3.13 Stress from all nodes of the rear surface of the plate: a) full model; b) quarter model. . . 24
3.14 Identification of the nodes previously selected at the plate’s mesh. . . 27
3.15 Displacement against time for each node being analyzed. . . 27
3.16 Distance between each node against the instant when they suffered the first wave impact. Calculation of the slope which represents the wave velocity. . . 28
3.17 Snapshots of the elastic wave propagation and interference. . . 28
3.18 a) Contact type 1. b) Contact type 2. c) Contact type 3. d) Contact type 4. . . . 30
3.19 Graph combining the four contact type results. . . 31
3.20 Effect of the base model material configuration on the rear stress. . . 32
3.21 Impedance of Aluminium and Steel. . . 32
4.2 Single-point crossover illustration. . . 40
4.3 Double-point crossover illustration. . . 40
5.1 Flowchart of the optimization process. . . 47
5.2 Weight evolution, for the PSO in Benchmark I. . . 50
5.3 Evolution of the best, worst and mean values of the objective function with the number of iterations, for the PSO in Benchmark I. . . 50
5.4 Evolution of the best, worst and mean values of the objective function without penalization against the number of iterations, for the PSO in Benchmark I. . . . 50
5.5 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the PSO in Benchmark I. . . 51
5.6 Weight evolution for the GA in Benchmark I. . . 52
5.7 Evolution of the best, worst and mean solutions of the objective function with penalization against the number of iterations, for the GA in Benchmark I. . . . 53
5.8 Evolution of the best, worst and mean solutions of the objective function without penalization against the number of iterations, for the GA in Benchmark I. . . . 53
5.9 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the GA in Benchmark I. . . 54
5.10 Weight evolution, for the SA in Benchmark I. . . 55
5.11 Evolution of the best, worst and mean values of the objective function with pe-nalization against the number of iterations, for the SA in Benchmark I. . . 56
5.12 Evolution of the best, worst and mean values of the objective function without penalization against the number of iterations, for the SA in Benchmark I. . . 56
5.13 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the SA in Benchmark I. . . 57
5.14 Evolution of the best, worst and mean values of the objective function against the number of iterations, for the PSO in Benchmark II. . . 59
5.15 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the PSO in Benchmark II. . . 60
5.16 Comparison of the position at each evaluation with the overall best position at each iteration, for the PSO in Benchmark II. . . 60
5.17 Evolution of the best, worst and mean values of the objective function against the number of iterations, for the GA in Benchmark II. . . 61
5.18 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the GA in Benchmark II. . . 62
5.19 Comparison of the position at each evaluation with the overall best position at each iteration, for the GA in Benchmark II. . . 62
5.20 Stress evolution, for the SA in Benchmark II. . . 63 5.21 Evolution of the best, worst and mean values of the objective function against
5.23 Comparison of the solution at each evaluation without penalization with the overall best solution at each iteration, for the PSO in Benchmark III. . . 67 5.24 Combination of materials’ indexes for the PSO in Benchmark III. . . 68 5.25 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the GA in Benchmark III. . . 69 5.26 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration, for the GA in Benchmark III. . . 69 5.27 Combination of materials’ indexes for the GA in Benchmark III. . . 70 5.28 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration, for the SA in Benchmark III. . . 71 5.29 Combination of materials’ indexes for the SA in Benchmark III. . . 71 5.30 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the PSO in Benchmark IV. . . 73 5.31 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration. for the PSO in Benchmark IV. . . 73 5.32 Combination of materials’ indexes for the PSO in Benchmark IV. . . 74 5.33 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the GA in Benchmark IV. . . 74 5.34 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration, for the GA in Benchmark IV. . . 75 5.35 Combination of materials’ indexes for the GA in Benchmark IV. . . 75 5.36 Stress evolution, for the SA in Benchmark IV. . . 76 5.37 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the SA in Benchmark IV. . . 76 5.38 Combination of materials’ indexes for the SA in Benchmark IV. . . 77 5.39 Weight evolution for the alternative approach to the Benchmark III. . . 79 5.40 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration, for the alternative approach to the Bench-mark III. . . 80 5.41 Combination of materials’ indexes for the alternative approach to the Benchmark
III. . . 80 6.1 Pareto curve of the multi-objective optimization. . . 86 6.2 Results obtained from the multi-objective optimization when weight y = 0. . . 86 6.3 Results obtained from the multi-objective optimization when weight y = 0.5. . 87 6.4 Results obtained from the multi-objective optimization when weight y = 1. . . 88 B.1 Thickness evolution for the PSO in benchmark I. . . 113 B.2 Comparison of the position at each evaluation with the overall best position at
each iteration, for the PSO in benchmark I. . . 114 B.3 Thickness evolution for the GA in benchmark I. . . 114 B.4 Comparison of the position at each evaluation with the overall best position at
each iteration, for the GA in benchmark I. . . 115 B.5 Thickness evolution for the SA in benchmark I. . . 115
B.10 Rear stress evolution for the GA in benchmark II. . . 118 B.11 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the GA in benchmark II. . . . 118 B.12 Thickness evolution for the GA in benchmark II. . . 119 B.13 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the SA in benchmark II. . . . 119 B.14 Comparison of the solution at each evaluation with the overall best position at
each iteration, for the SA in benchmark II. . . 120 B.15 Thickness evolution, for the SA in benchmark II. . . 120 B.16 Comparison of the position at each evaluation with the overall best position at
each iteration, for the SA in benchmark II. . . 121 B.17 Weight evolution for the PSO in benchmark III. . . 121 B.18 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the PSO in benchmark III. . . 122 B.19 Weight evolution for the GA in benchmark III. . . 122 B.20 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the GA in benchmark III. . . . 123 B.21 Weight evolution for the SA in benchmark III. . . 123 B.22 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the SA in benchmark III. . . 124 B.23 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the SA in benchmark III. . . . 124 B.24 Rear stress evolution for the PSO in benchmark IV. . . 125 B.25 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the PSO in benchmark IV. . . 125 B.26 Rear stress evolution for the GA in benchmark IV. . . 126 B.27 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the GA in benchmark IV. . . . 126 B.28 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the SA in benchmark IV. . . . 127 B.29 Comparison of the solution at each evaluation without penalization with the
overall best solution at each iteration, for the SA in benchmark IV. . . 127 B.30 Evolution of the best, worst and mean values of the objective function against
the number of iterations, for the alternative approach to the benchmark III. . . . 128 B.31 Evolution of the best, worst and mean values of the objective function without
penalization against the number of iterations, for the alternative approach to the benchmark III. . . 128
Introduction and Background
The field of armour systems, and in particular armour plates, whether for defence or protective systems, is facing a paradigm shift. If the focus, until approximately 30 years ago, was primar-ily to build armour systems that could withstand determined impact loads, now the focus has a multi-objective stand point. Not only it is necessary to fulfill ballistic impact or blast require-ments but it is also necessary to fulfill optimal design requirerequire-ments often related to production costs and the weight of the structure. Although the need and demand for such lightweight ar-mour systems is increasing at a noticeable pace, there are yet limited literature and progress related with weight and/or cost optimization in such structures. Besides, most of the improve-ments are experimentally determined, relying in an empirical and intuitive perspective. Thus, conclusions are locally optimal and specific to the test circumstances. There are several reasons that one can point for the lack of progress in the wider context of impact optimization such as the computational cost associated with optimization studies and user experience in modelling ballistic impacts, as suggested by [Chen 2001]. The prevailing reason for this high complexity is that most impact events have a nonlinear and transient nature.
1.1
Armour Materials and Structures
A prevalent need for a continuous research on lightweight armour systems had emerged since centuries ago. However, the application of metals had prevailed on most armour structures until the 1960s. Although it is possible to name four materials (steel, aluminum, magnesium and titanium) that are practical and functional as contenders for armour applications, only steel and aluminum are currently used in a wide scale due to raw material costs and ability to be worked and welded. These two metals have been widely used as they are good all-rounders, offering reasonable hardness with good ductility and toughness [Hazell 2015]. Although their usage as protective systems is becoming obsolete as there are light-weight non-metallic options available such as ceramics and high-strength fibers, it still is likely that metals will be carried on for some time. This is due to developments in porous or micro-architecture structures resulting in promis-ing designs such as cermets (hybrid armour systems made of a ceramic-metal combination).
In 1992, one of the first trials [Hetherington 1992] has been carried to prove that a two-layer design combining ceramic with aluminum presents reasonable mechanic properties while being lighter than a single layer metal plate. Since then, a two-layer design configuration has been widely used for modern armour structures. The principle in this design is to have a front plate made of a hard and high-purity ceramic, while the rear plate is made of a ductile material such
reduction of wave propagation (and damage) velocity.
Hybrid materials are combinations of two or more materials assembled in such ways as to have attributes not offered by either one alone [Ashby 2005]. Most often, hybrid materials are obtained by filling the gaps present within it’s volume with other materials. However, they can also be sandwich structures, foams and more. Hybrid material’s enhanced properties are be-coming better understood and one example is the usage of ceramic inclusions to cause projectile deflection, self-sealing of the hole and forced shear localization [Gu and Nesterenko 2007].
High-performance fibres (woven fabrics) make use of the high tensile strength of the fibres (∼ 2−3 GPa) and their low densities (∼ 1000−1500 kg m3). Because they have reasonable strains to failure (∼ 3% − 6%) and they have excellent energy absorbing abilities (depicted by the area under the stress-strain curve) [Hazell 2015].
1.2
Optimization Studies
Despite an increasing demand in lightweight armour structures, there is still little literature and progress in this field. The authors [Park et al. 2005] made one of the first experiments regarding the design optimization of a multi-layered plate under ballistic impact by numerical simulation. The goal of this study was to find the minimum weight of the target plate by adjusting its layer’s thicknesses while being able to withstand a high velocity impact without allowing penetration and while retaining a specific volume. For the simulations, NET2D (a Lagrangian explicit time integration finite element code developed by two of the authors) was used. The Johnson-Cook constitutive model was implemented due to the high-strain-rate plastic deformations involved. It was verified that the objective to minimize strain energy in order to maximize the strength of the target can also be expressed by minimizing the average temperature or average equivalent plastic strain (EQPS). As such, two case studies were presented. The first where the objec-tive was to minimize the average temperature, and the second were the objecobjec-tive-function was to minimize the EQPS. The restrictions for both cases were identical, concerning about layer thicknesses and limiting the maximum EQPS at the critical element. At the end, both objective-functions presented similar results. The optimization algorithm used was a Response Surface Method (RSM). The optimal results from the optimization algorithms were in conformity with the results from the NET2D code. Although being one of the first studies on the field of structure optimization in transient events, it is evident the need and the potential for future work: the ma-terial properties need proper validation and the optimization algorithm is not the most accurate and had only two variables.
Authors [Yong et al. 2008] presented a study on the application of genetic algorithms for optimizing composites against impact loadings. Two optimization scenarios were presented, a low velocity impact of a slender laminated strip and a high velocity impact of a rectangular plate by a spherical impactor. The goal was to minimize the peak deflection in the plate under impact and to minimize the penetration velocity or maximize the rebound velocity, respectively. The variables in both cases were the ply angle configurations, but it is worth noting that in the for-mer scenario the ideal stacking sequence was known, which was 0° for all plies. Thus, the low velocity case study worked as a benchmark for the algorithm techniques used. A genetic
al-the impact analysis. A comparison with a commercial optimization package that included opti-mization methods such as the Monte Carlo and Neural Network techniques, LS OPT, was also made. For the low velocity benchmark, both Monte Carlo and Neural Network techniques re-sults were particularly good. It was verified that the Monte Carlo method performed better for smaller test sets while the neural network had better results for larger sets. The genetic algorithm adopted performed better results for the high velocity scenario where the search space is larger and non-linear. A study carried by the same author [Yong et al. 2010], focused on applying the same genetic algorithm coupled to LS DYNA to optimise hybrid multilayered plates subject to ballistic impacts. In this study, low cost but sufficiently accurate models were generated so they could be later used by the optimization algorithm to generate new hybrid offsprings. The goal was to minimize weight and design costs from a selection of isotropic metals, polymers and orthotropic fibre-reinforced laminates. Both the number of plies and their material’s mechanical properties were variable while maintaining their thickness constant. Experimental validation of the optimal designs identified were successfully carried out using a single stage gas gun. Nev-ertheless, the authors are clear that further work needs to be done with sufficient computational hardware and using finer meshes, equations of state and sophisticated material models so that hybrid systems can be identified from a wide range of materials, designs and threads.
1.3
Objectives and Scope
The aim of this dissertation is to develop an understanding of non linear optimization algorithms applied to a complex engineering design problem: a multi-layer plate under a ballistic impact. Furthermore, this dissertation can be divided into three interdependent objectives:
1. Develop a sound understanding of the physics and mechanics involved in a transient event such as that of a ballistic impact on a multi-layer plate. These insights can be used in Abaqus Python scripting to construct and validate a proper simulation model of such event and, ultimately, prepare this script to be fully integrated in future optimization algorithms. 2. Acquire a comprehensive knowledge on three metaheuristic algorithms: Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Simulated Annealing (SA). Evaluate the performance of each algorithm in four different benchmarks that gradually increase in complexity. These benchmarks are intended to test not only the algorithms, but their integration with the Python simulation model built previously. Accordingly, different pro-gramming strategies, algorithm parameters, problem formulations and constraints are to be discussed.
3. Perform a multi-objective optimization using the most efficient algorithm selected from the previous benchamrks. This final optimization procedure is intended to minimize the weight of the multi-layer plate and, also, minimize the stress at the rear side of the plate. This multi-objective problem uses advanced techniques and methods such as the normal-ization of each single-function and the weighted sum method. Besides, it integrates both continuos and discrete variables.
1.4
Motivation
The objectives for this dissertation completely fulfill four major branches of the mechanical engineering field which involves programming, optimization, structural and simulation skills.
basic concepts are just an extension of what optimization algorithms are intrinsically aimed to do. Furthermore, this dissertation was selected with the ambition of developing a thorough understanding of optimization algorithms and consolidate programming and simulation skills.
1.5
Dissertation Structure
This document is structured into 8 different chapters, that are organized as follows:
• Chapter 1: Introduction and framework on the filed of design optimization, ballistic im-pacts and armour materials and structures. The objectives and motivation for this disser-tation are set;
• Chapter 2: Introductory concepts on the physics and mechanics of wave propagation; • Chapter 3: Incremental building of a generic Abaqus Python script model of the
sys-tem. Development of variables of study. Validation of the model in terms of its dynamic response integrity and wave velocity;
• Chapter 4: General concepts on the fundamentals of engineering optimization; Introduc-tion and general concepts of three metaheuristic algorithms;
• Chapter 5: Development of four optimization benchmarks that aimed test the three al-gorithms, different problem formulation, constraints, variables of study and programming strategies. The previously generated Abaqus Python script is integrated in the optimiza-tion formulaoptimiza-tions and, thus, also tested. At the end, the overall most efficient algorithm is selected;
• Chapter 6: Implementation of a multi-objective optimization procedure using the se-lected algorithm from the previous benchmarks. A proper Pareto’s curve is generated and explained;
• Chapter 7: Final remarks, which include the major conclusions of this dissertation and proposals of future work on this field.
Physics of Wave Propagation
As one of the major objectives of this dissertation implies the study, development and analysis of ballistic model designs (which imply a projectile impact upon a target), the sound understanding of how waves propagate and behave through such system is required. For the scope of this chapter, it is important to note that the majority of the concepts here presented were based on [Hazell 2015].
Stress waves arise every time there is contact between a moving object and a stationary object. Such waves will propagate from the point of impact and move into the projectile and target simultaneously [Zukas and Scheffler 2001]. On low velocity collision scenarios, the wave is expected to be elastic in nature. By increasing the projectile’s velocity, the resultant wave can be inelastic. Nevertheless, this dissertation will mainly focus on impacts that result only in elastic waves.
The phenomenon of wave propagation is nothing more than the manifestation of a syn-chronous movement of the object’s constitutive particles as they are affected by the wave and through every direction starting from the impact region. The velocity uP by which these parti-cles move when affected by the wave depends on the object materials’ mechanical properties, namely its elastic modulus, E, and density, ρ0, can be stated as
uP= √σ ρ0E
, (2.1)
where σ is the stress induced by the impact. It is important to note, though, that particles located at free edges will have twice this velocity due to the phenomena of rarefaction.
A ballistic impact model is essentially composed of a target and an impactor and, both of these objects are simultaneously affected by a wave that emanates through them after the impact, as explained before. However, this wave velocity, c0, will vary according to the material in study. For an elastic wave travelling in a bounded medium, such as a bullet, its wave velocity corresponds to
c0= s
E
ρ0. (2.2)
For an elastic wave travelling within the target (semi-infinite medium), its velocity will be slightly higher than the value calculated in 2.2 and will be function of the Poisson’s ratio, ν, as
c0=
ρ0(1 + ν)(1 − 2ν). (2.3)
The transmission of an elastic wave is influenced by the composition of its medium. For instance, in a target composed by two or more layers of different materials, when an elastic wave of intensity σI arrives at those layers’ interfaces there will be a portion that will be transmitted σTinto the next layer whilst a portion will be reflected σRback. This event, which is depicted in Figure 2.1 is of a great importance for further studies in this dissertation as the target modeled in every simulation will be composed of three layers. The proportion of the elastic wave which is reflected and transmitted is directly dependent on the relative mechanical impedance of the materials. The elastic impedance, which has units of kg/m2s, is
Z =pEρ0 = ρ0c0. (2.4)
Figure 2.1: Transmission and reflection of an elastic wave at an interface. The interface between two materials A and B is at a state of equilibrium if
σI= σT+ σR, (2.5)
where the subscripts I, T and R stand for the incident, transmitted and reflected wave, respec-tively. The continuity at the interface can be stated as
uI= uT+ uR, (2.6)
where u represent the particle velocity. Ultimately, from equations 2.5 and 2.6, it is possible to derive equations 2.7 and 2.8. These equations are intended to measure the intensity of stress that is transmitted and reflected and they essentially depend on each individual material impedance, as σT σ = 2 √ EBρB √ √ ! , (2.7)
σR σI = 2 √ EBρB− √ EAρA √ EAρA+ √ EBρB ! . (2.8)
From equations 2.7 and 2.8 the transmitted wave will always be positive (compressive) whilst the reflected wave can either be positive or negative (compressive or tensile, respectively) depending on the properties of the materials. If material B has a lower impedance than material A, the reflected wave has a tensile nature whilst if the material B has a higher impedance than material A, the reflected wave will be compressive. If a plate isn’t backed by anything (material B is air), the stress wave is completely reflected back.
Model Discussion and Validation
This chapter aims to develop and validate a fully operational Python script for a ballistic simu-lation model to be later integrated in optimization algorithms (Chapters 5 and 6). In order to do so, a thorough analysis and testing of different methods and criteria will be discussed throughout this chapter.
As a means of improving the time required for the simulation procedures and to automate repetitive tasks, a parametrization approach was adopted whenever possible. Parametric studies allow to generate, execute and gather results of multiple analyses that differ only in the values of some of the parameters used in place of input quantities [Simulia 2014].
3.1
Abaqus Scripting and Python Programming
The Abaqus scripting interface, which is an extension of the Python object-oriented program-ming language, is an application programprogram-ming interface (API) [Inc 2004]. This scripting in-terface, which are Python scripts, are intrinsically powerful and dynamic as they can be used to create and modify a model, submit a job and read and write from an output database. Ulti-mately, the Abaqus scripting interface is a crucial tool not only to develop parametric studies but also to create scripts that can be integrated in other program subroutines, such an optimization algorithm (Chapters 5 and 6). The Abaqus scripting interface also allows to bypass the Abaqus graphic user interface (GUI) and comunicate directly with the kernel. Figure 3.1 illustrates how Abaqus scripting interface interacts with Abaqus’ Kernel.
Besides being the standard programming language used in the Abaqus scripting interface and, thus, used to build every simulation model present in this dissertation, Python was also the chosen language to program the optimization algorithms in both chapters 5 and 6. Python is an open-source programming language designed to help programmers write clear and logical programs [Altom and Chapman 1999]. It is an interpreted programming language, dynamically typed and supports multiple paradigms, including procedural, object-oriented and functional programming. It is also supported in multiple operating systems, which led to a continuously growing global community.
Figure 3.1: Flowchart of how Abaqus scripting interface interacts with Abaqus Kernel. [Inc 2004]
3.2
Model Design
The geometry and dimensions of the ballistic system that are consistently used throughout this dissertation, consist of a three layer plate and a cylindrical projectile which adopted nomencla-ture is illustrated in Figure 3.2a. This model geometry and dimensions arise as a starting point from a previous study on the field of ballistic impacts [Pittman 2017] and in the context of a partnership between the University of Aveiro and the University of Edinburgh. This study was conducted using LS Dyna as a software to simulate the ballistic impact system and some of it’s results and conclusions will be used as a standpoint for comparison and further development throughout this chapter.
(a)
(b)
Figure 3.2: Model of the ballistic system. a) configuration nomenclature; b) ballistic impact model.
conve-As noticeable in Figures 3.2, each layer of the plate is divided into several partitions. These partitions are larger, in area, as they are more distant from the impact region. In fact, at the impact region itself, it was created a partition that has the exact same area as the projectile’s base. Such partitions were programmed into the Python script of the model for several purposes: being able to optimize the mesh density of the plate, as explained in Section 3.2.3, and to generate outputs at specific important areas of plate, as elucidated in Section 3.2.1.
3.2.1 Impact Region Analysis
This section is intended to validate two variables of study generated and calculated using Abaqus Python scripting. These variables will be constantly used throughout this dissertation, with special emphasis in optimization procedures. Accordingly, it is imperative to have a sound understanding on the principles used to generate these outputs.
As every simulation model is created using Abaqus Python scripting, it is possible to take advantage of Abaqus parametric input which allows greater flexibility in building and manip-ulating models [Simulia 2014]. Accordingly, when programming the script for this model, the plate was set in such way that every one of its layers automatically has a specified number of features assigned, such as sets, surfaces and partitions. These features are independent of the number of layers a plate may be programmed to have. So if a plate has 4 layers, each one has assigned features. An example of a design feature that is programmed in every layer is a geo-metric set at the front impact region and another at the rear impact region as illustrated in Figure 3.3. The creation of such sets is important, as they can be used to generate specific outputs, such as the average stress within that set’s area. Besides, they were created due to the necessity of having reliable and good quality variables of study that can be applied, for example, in future optimization problems as in Chapter 5. As such, these sets will, essentially, be used as a means to generate two important study variables: the peak stress from the average of all nodes at the impact region set and the total force due to contact.
Peak stress from the average of all nodes at the impact region set
Graphs from Figures 3.4 and 3.5 illustrate the process of averaging the stress, σz, of all nodes within the impact region at every time increment for both the front and rear impact region sets, respectively.
After executing the simulation in Abaqus, the data related with the σz of every node within both the front and rear impact regions is stored in predefined variables in the Python script that are then used to calculate the average at every time instant. The ultimate interest in doing so is to obtain the maximum average stress at the impact region. This process allows the generation of a useful and reliable criteria variable, although not being a direct output form the simulation.
(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ·10−4 −1.5 −1 −0.5 0 ·108 Time [s] Stress [P a]
Stress at the front IR
(b)
Figure 3.4: a) Stress from all nodes at the front impact region (graph extracted from Abaqus) and b) average stress at every time increment using a Python script.
(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ·10−4 −6 −4 −2 0 2 ·107 Time [s] Stress [P a]
Stress at the rear IR
(b)
Figure 3.5: a) Stress from all nodes at the rear impact region (graph extracted from Abaqus) and b) average stress at every time increment using a Python script.
Total Force due to Contact
Since the impact region’s set corresponds to the area of contact between the projectile and the front surface of the plate, it can also be used to generate the total force due to contact pressure at every instant, as is depicted in figure 3.6. This generated output is used in Section 3.2.5 to calculate and validate the system’s equilibrium and dynamic response.
From graph 3.6, it is perceptible that the impact occurs at ∼ 3 µs, leading to a gradual in-crease of the contact force until it reaches its maximum value of 22.1 KN, at ∼ 20 µs. This point of maximum contact force corresponds to the instant at which the projectile changes direction to its opposite due to the elastic effect of the plate. At ∼ 36 µs, the projectile ceases its contact with the plate.
0 1 2 3 4 5 6 7 8 ·10−5 0 0.5 1 1.5 2 2.5 ·104 Time [s] F orce [N] Contact Force
Figure 3.6: Total force due to contact pressure at every time increment.
3.2.2 Boundary Conditions
Besides the usage of symmetry boundary conditions (that will be validated in Section 3.2.4) and a predefined projectile velocity, it was also established a boundary condition regarding how the plate is supported. Accordingly, three different scenarios for this boundary condition were studied: the first where the plate is clamped by the side surfaces of the plate (Figure 3.7a), the second where the plate is only pinned at the side edges of the rear surface of the plate (Figure 3.7b) and the third where the plate is clamped at the rear surface of the plate (Figure 3.7c).
From Figure 3.8 it is possible to compare the stress at the rear surface of the plate at each time increment for each one of the three scenarios studied. From this analysis, the graph from Figure 3.8c (third scenario) depicts a more uniform behaviour comparatively to the other two graphs. Table 3.1 presents the results obtained for each scenario and it is perceptible that both scenarios one and two have very close values for the stress at the front and bottom impact region and, also, the contact force. These results do not seem congruent with graphs 3.8a and 3.8b at the first sight. However, the results are correct as they only refer to the average stress calculated at the impact region.
It is indisputable that the boundary condition where the plate is clamped at the rear surface present more uniform results throughout the entire impact time period. This may be explained
(a)
(b)
(c)
Figure 3.7: Three boundary condition studies: (a) plate clamped by the side surfaces of the plate, (b) plate pinned at the side edges of the rear surface of the plate, (c) plate clamped at the rear surface of the plate.
as the particular configuration of this boundary condition minimizes the propagation of waves throughout the OX and OY axis. Accordingly, waves propagating in the OZ direction suffer significantly less interference when comparing to the results from the other boundary conditions. Thus, these results are more reliable either for interpretation or to measure the average peak stress at the impact region as explained in Section 3.2.1 or even to perform a mesh study as in Section 3.2.3. Furthermore, this boundary condition was elected to be implemented in further simulations throughout this dissertation.
Table 3.1: Results from the 3 boundary condition scenarios. Boundary Condition Scenario Max. σz[MPa]
Front Impact Region
Max. σz[MPa]
Rear Impact Region Contact Force [N]
1. Encastered at the sides 29.024 0.4885 10800.85
2. Pinned 29.025 0.4885 10800.6
(a)
(b)
(c)
Figure 3.8: Evolution of σz from all nodes at the rear surface for a) boundary condition case 1, b) boundary condition case 2 and c) boundary condition case 3.
3.2.3 Mesh Convergence
The mesh configuration has a direct correlation with the quality of the results being measured and with the computational cost of each simulation. As the main purpose of this chapter is to create and validate a generic model script to be integrated in non-linear optimization algorithms that require hundreds of simulations, it is mandatory to find an optimal mesh size that requires the shortest time possible to run without compromising the results.
During the ballistic impact phenomenon, there are stress waves that propagate throughout the plate’s horizontal plane (transverse waves) and through the thickness direction (longitudinal waves). Section 3.2.6 goes into more detail about the propagation of elastic waves and how it is strictly related with the mesh quality. Thus, this mesh convergence study was divided into two major steps: convergence study along the plate’s horizontal plane and a convergence study through the plate’s thickness direction. The decisive criterion in both cases was the peak stress measured at the rear plate’s impact region (explained in Section 3.2.1). For the purpose of this study, the model was a plate of only one layer. The element type used was a eight-node brick element with reduced integration (C3D8R) and with hourglass control for the plate. The projectile, as a rigid body, uses a four-node bi-linear rigid quadrilateral (R3D4).
Mesh Convergence Through the Horizontal Plane of the Plate
In this first mesh study, the mesh size of each partition of the plate varied while keeping the mesh size through the thickness direction always constant and equal to 3 mm. Hence, a para-metric study was performed, where the mesh size in each partition varied within the interval: [0.05, 0.001] m, divided in 11 increments.
Table 3.2 presents not only the average peak stress at the rear surface of the plate but also the final velocity of the projectile and its acceleration, and the average peak stress at the front surface. Furthermore, it is perceptible an identical convergence ratio for each one of those result as the mesh density increases. Moreover, although not being a direct output from the simulation as explained in chapter 3.2.1, it is legitimate to use the average peak stress at the rear impact region as the criteria for choosing the best mesh configuration.
The graph in Figure 3.9 illustrates the evolution of the average peak stress at the plate’s rear impact region and the time required to complete the simulation as the number of elements of the plate increases. As shown in the graph, the time required to perform the simulation increases slightly more that ten times (∼ ×10) from 80304 to 564604 elements while having an increase of ∼ ×1.15 in the peak stress at the rear impact region. For that reason, it is reasonable to chose the mesh configuration that resulted in a total number of elements of 80304 (as marked in dashed lines in Figure 3.9) to be integrated in future simulations. It is important to note, though, that this choice also takes into consideration future optimization procedures where the mesh density plays an important role for the computational cost of the model.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 ·105 2 3 4 5 ·107 Number of Elements Stress [P a] 0 200 400 600 800 1,000 1,200 T ime [s]
Rear Peak Stress Execution Time
Figure 3.9: Number of elements against the time required to complete the simulation and the peak stress at the rear surface for case 1; the parameters chosen are marked with the dashed line.
Table 3.2: Evaluated variables in each step of the mesh optimization.
Step No. Elements
Final projectile vel. [m/s]
acceleration [m/s2]
max. σz
at front Imp. Region [Pa]
max. σz
at rear Imp. Region [Pa] run time [s] 1 2128 -1.4142 -94818.7216 53565663.57 38863771.87 39.19 2 5120 -1.4104 -94880.0602 60870498.80 39627614.36 39.45 3 11350 -1.4138 -90992.7863 59642913.13 38680990.07 40.29 4 27885 -1.4167 -95154.8419 60002882.43 40075827.26 43.16 5 34164 -1.4153 -91244.8496 60296438.85 40368236.26 55.41 6 47008 -1.4127 -94930.1962 60462640.23 41294198.13 77.38 7 61488 -1.4133 -95110.4932 60924852.43 41390959.97 89.63 8 80304 -1.4134 -95030.9502 60761544.75 41025602.84 108.24 9 129640 -1.4093 -94882.9273 60569767.40 42470707.77 237.79 10 266120 -1.4122 -95028.9499 62651536.78 45119206.56 457.53 11 564604 -1.4100 -94918.3998 63673923.88 47164009.64 1130.97
Mesh Convergence Through the Plate’s Thickness Direction
The second study intends to optimize the mesh density along the thickness direction of the plate while maintaining the mesh density at the plate’s surface constant. For this reason, the mesh results obtained from the previous study are used and kept constant throughout the current mesh study.
Following the same reasoning as in the previous study, the mesh size along the thickness of the plate varied in the given interval: [0.01, 0.0001], divided by 11 increments. As shown in graph 3.10, the time required to execute the simulation increases significantly (∼ ×3.7) from 26880 to 77500 elements while the average peak stress at the rear impact region only increases ∼ ×1.024. Thus, the mesh configuration parameters that resulted in a number of 26880 elements (as marked in dashed lines in Figure 3.10) will be used in further simulations.
0 1 2 3 4 5 6 7 8 9 ·104 2 3 4 5 ·107 Number of Elements Stress [P a] 0 50 100 150 200 T ime [s]
Rear Peak Stress Execution Time
Figure 3.10: Number of elements against the time required to complete the simulation and the peak stress at the rear surface for case 2; the parameters chosen are marked with the dashed line.
Mesh Optimization Results
The mesh configuration results are graphically illustrated in figure 3.11. This mesh is structured and the number of elements intentionally increases as the distance to the impact region decreases. By doing that, the total number of elements can be drastically reduced while not compromising stress propagation, especially thought the z direction (thickness direction). Hence, the mini-mum execution time can be guaranteed allowing a better performance of the simulation model when integration in future optimization procedures. Nevertheless, as every simulation in this dissertation is made using Python scripts, these parameters can be easily replaced if necessary.
Figure 3.11: Details of the mesh.
3.2.4 Symmetry Validation
The use of symmetry boundary conditions can drastically reduce computing time and data stor-age space. Accordingly, only one quarter models were built and used for the scope of this dissertation, as suggested in Chapter 3.2.
To validate the use of symmetries in the model, two identical simulations were replicated: a full size model (Figure 3.12) and a quarter size model (Figure 3.11). The evaluation criteria was the average peak stress at the rear and front impact region, the final velocity of the projectile, maximum contact force and the resultant stress. The resultant stress is calculated as follows
Rs= Fc Ac
N, (3.1)
where Rsis the resultant force, Fc the maximum contact force and Acis the area of contact. A comparison of the stress measured at the rear impact region of the plate using both full and quar-ter sized models is illustrated in Figure 3.18. Table 3.3 depicts the results from both simulations and it is perceptible that the percentage difference is under 1% in every parameter evaluated except for the stress at the front impact region which is 2.3%. Besides, there was a decrease of ∼ 68.9% in execution time when using the quarter sized model. Thus, for the purpose of this dissertation, a quarter model of the system is arguably validated as equivalent to the whole model.
Figure 3.12: Front view of the full plate.
Table 3.3: Results of a full sized model and a quarter model simulation.
Whole model A quarter of the model Deviation [%] Peak σzat Rear Imp. Region [Pa] 45245582.23 45517851.77 0.6018 Peak σzat Front Imp. Region [Pa] 64204347.27 62726039.52 2.3025
Final Velocity [m/s] -1.415 -1.412 0.2279
Maximum Contact Force [N] 88636 4 ×22102 0.2572
Resultant Stress [Pa] 70352969 70533921 0.2572
(a)
(b)
Figure 3.13: Stress from all nodes of the rear surface of the plate: a) full model; b) quarter model.
3.2.5 Dynamic Response of the System
When modeling and simulating transient events it is important to verify and validate the dynamic forces in the system. Hence, the following method was followed:
1. Contact force analysis
A comparison was made between the force calculated due to the acceleration induced in the projectile and the average contact force from Figure 3.6 during the contact period. Thus, the acceleration of the projectile during the contact period can be calculated as
a = vf− vi tf− ti = −1.41 − 1.5 3.61 × 10−5− 3.38 × 10−6 = 88 993.59 m s−2, (3.2) where a is the acceleration, v is the velocity, t is the time instant and the subscripts f and i correspond to the final and initial instants, respectively.
Knowing the mass of the projectile, calculating the force induced during the contact period is straight forward, as
F = m × a = 0.147025 × 88993.59 = 13 084.28 N, (3.3) where F is the force in Newton and m is the mass of the projectile in kilograms. Comparing the results from Equation 3.3 to the values from Figure 3.6, which resulted in an average force during the contact period of 13074.83 N, the final deviation is only of 0.072%. Accordingly, the contact force of the interaction between the projectile and the plate is approved.
2. Resultant Stress Analysis
From the resultant force in Equation 3.3 and knowing the contact area between the projectile and the plate, it is possible to calculate the resultant stress from Equation 3.1 as:
Rs = Fc Ac
= 13084.28
(π × 0.022)/4 = 41.65 MPa. (3.4) The average stress from the impact region (Figure 3.4b) during the contact period is 37.07 MPa. Comparing with the result from Equation 3.4, the deviation is ∼ 11.63%. This deviation can be considered tolerable since the graph from Figure 3.4b itself, is already the result of a process of averaging as explained in Chapter 3.2.1. Thus, two processes of averaging are conducted in order to obtain the above result of 37.07 MPa.
3.2.6 Elastic Wave Velocity Analysis
An important component of this dissertation involves calculations and analyses related with the mechanics of wave propagation in a transient event. Consequently, a proper validation of this phenomena in Abaqus was carried out. To ensure that the elastic wave is measured with enough resolution and that the results are accurate, three rules must be followed [Graff 1991]:
1. Time increment size should be low enough to capture the smallest natural period of inter-est;
alloy (E= 70GPa, υ= 0.33, ρ = 2700 kg m ) was used. Furthermore, it is possible to calculate the theoretical elastic wave speed at the plate by applying Equation 2.3, which results is c= 6197.824 m/s. Thus, its impulse wavelength Iwcan be calculated as [Graff 1991]:
Iw= 2 × c × dt = 2 × 6197.824 × 2.0 × 10−6 = 2.48 × 10−2m, (3.5) where c is the theoretical wave speed. Therefore, a mesh size of h = 1 × 10−3 m was used in the simulation model as it is more than enough to capture the propagation of the elastic wave (h < Iw).
The critical time increment T can be calculated as: T = h c = 1 × 10−3 6197.824 = 1.61 × 10 −7 s. (3.6)
Accordingly, a maximum time increment of 1.0 × 10−7 s was used for the scope of this study. Finally, by using a small time increment (T = 1.0 × 10−7) s and a sufficiently small mesh size (h = 1 × 10−3m), the Courant–Friedrichs–Lewy condition (CFL condition) is met in order to maintain the accuracy of solutions [Gnedin et al. 2018, Pleˇsek et al. 2012].
Having defined the required size of the mesh and the maximum step increment, it is now necessary to perform the simulation such that the necessary output is generated and identified. The adopted method to calculate and validate the elastic wave velocity is enumerated as follows:
1. Chose the plane where the elastic wave will be evaluated;
2. Identify and select a path of nodes that follow the same direction and are in the previously defined plane;
3. Calculate the linear distance between each selected node;
4. After completing the simulation, obtain the time instant by which each previously selected node was hit by the elastic wave;
5. Calculate the slope between the selected nodes and compare the results with the theoretical formulation in Equation 2.3.
Following the method explained above, three nodes were selected as shown in figure 3.14. After completing the simulation, a graph of the displacement in the OX direction against time was generated for each node (Figure 3.15), allowing to get the very first instant when each node suffered a displacement (which is the equivalent to the instant where each respective node suf-fered the first elastic wave impact). Having the distance between each node and their respective time attached, it was then possible to calculate the slope between nodes which corresponds to the elastic wave velocity. As can be seen in the graph from Figure 3.16, the calculated slope, 6133.6 m/s, is very close to the theoretical value which is 6197.82 m/s (Equation 2.3). The deviation is close to 1% as shown in Table 3.4.
Figure 3.17 clearly depicts how elastic waves propagate through the horizontal plane of the plate. It starts with a snapshot close to the instant of the impact at 54 µs and it shows how waves disperse in same proportion in both OX and OY directions. At instants t = 92 µs waves are already being reflected thought the length of the plate due to its short distance to the impact. At t = 98 µs is shown reflection through the plate’s width and after t = 128 µs reflected waves are
Figure 3.14: Identification of the nodes previously selected at the plate’s mesh. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ·10−4 −0.5 0 0.5 1 ·10−7 P1 P2 P3 Time [s] Displacement [m] P1 P2 P3
Figure 3.15: Displacement against time for each node being analyzed.
Table 3.4: Comparison between the velocity obtained from the simulation and from the theoret-ical formulation.
Slope [m/s] Theoretical Value [m/s] Deviation [%]
1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 ·10−5 0 5 · 10−2 0.1 0.15 0.2 Time [s] Distance [m] y = 6133.6x - 0.0342
Figure 3.16: Distance between each node against the instant when they suffered the first wave impact. Calculation of the slope which represents the wave velocity.
3.2.7 Contact Type
When simulating events that involve contact and interaction between different bodies, a proper contact interaction model must be defined according to the particular features that characterize the event to be modeled. Abaqus provides two different algorithms for modeling contacts [Inc 2004]:
• 1. General contact
– This algorithm allows the contact definition between many or all regions of a model with a single interaction. It typically includes all type of bodies in the model, the sur-face definition is automatic and, besides, it has very few restrictions on the sursur-faces involved. The contact constraint is the penalty method. General contact algorithm is more suited to models with multiple components and complex topology.
• 2. Contact pair
– This algorithm requires the specification of each of the individual surface pairs that can interact with one another. It has more restrictions on the types of surfaces in-volved and has two types of contact constraint: kinematic compliance and penalty method. Penalty method is more appropriate for interactions in which at least one of the bodies is rigid and, as such, it was chosen when using the contact pair algorithm. For the purpose of this study, a plate of three layers was modeled and four contact type scenarios were studied:
– Case 1:
A general contact interaction algorithm was used in the model. – Case 2:
A contact pair algorithm (surface to surface) was used between the projectile and the front surface of the plate and also between layers.
– Case 3:
A contact pair algorithm (surface to surface) was used between the projectile and the front surface. A Tie constraint1was used between each layer.
– Case 4:
A general contact interaction algorithm was used in the model. A Tie constraint was used between each layer.
1
same applies for cases 3 and 4 except that the model here used has a tie constraint between layers. The purpose was to begin by using a general model and then compare the results with a more specific one. As depicted in figure 3.18, the stress at the rear surface of the plate followed the same behavior between contact types 1 and 2 and between contact types 3 and 4. The same behaviour is illustrated in graph from Figure 3.19 when comparing the average peak stress at the rear impact region for all cases.
The contact types 3 and 4 slightly outperformed contact types 1 and 2 in terms of execution time. This may be explained due to the usage of a tie constraint algorithm in the former cases, which is less broaden than the latter cases’ algorithms in terms of applicability and, thus, results in less computational processing. Because the main focus of this chapter is to create a simulation model to be implemented in optimization studies, it is of best interest to opt for a model that requires the less computational processing whenever justifiable. Thus, because contact 2 do not allow abaqus to calculate the total force due to contact pressure at the impact region (as explained in chapter 3.2.1), the contact type number 3 will be constantly used in further simulations.
(a) (b)
(c) (d)
1 2 3 4 1.3 1.4 1.5 1.6 1.7 ·107 1.66 · 107 1.67 · 107 1.43 · 107 1.45 · 10 7 Contact Type Peak Stress [P a]
Figure 3.19: Graph combining the four contact type results.
3.2.8 Configuration Study
A configuration study of the model was carried in order to find the critical combination that would result in the highest peak stress at the rear surface. Hence, only two layers were modeled in this study, the front and bottom layers. The purpose of this study is to find a controlled configuration for the front and bottom layers so that further studies on interlayer parameters (as in Section 3.3) may be also controlled and, thus, lead to reliable conclusions. Ultimately, the base model that results from this study will be useful to begin testing different non linear optimization algorithms as the number of variables can be reduced to only properties related with the interlayer, such as its mechanical properties or thickness. In order to begin this configuration test, four scenarios were considered as shown in Table 3.5:
Configuration Front Layer Rear Layer
AA Aluminium Aluminium
AS Aluminium Steel
SA Steel Aluminium
SS Steel Steel
Table 3.5: Description of each plate configuration scenario.
As illustrated in the graph from Figure 3.20, the configuration that culminates in the highest or critical stress at the rear surface is when both the front and rear layer have steel mechanical properties attached. Figure 3.21 shows the impedance of both aluminium and steel materi-als. These values were calculated using Equation 2.4 and it is evident that steel has a larger impedance comparing with that of aluminium. These results are expected as steel has a much larger Young’s module and density than aluminium. The transmission ratio σT
σI (from Equation 2.7) was calculated for every configuration and is presented in Table 3.6. The transmission ratio is equal to one when the front and rear layers are made of the same material, resulting in a total continuity of stress between both layers. Configuration AS has a transmission ratio above one, which means that the stress wave is amplified at the interface. However, this amplification was
AA AS SA SS 0 1 2 3 1.99 · 108 2.72 · 108 2.23 · 108 Configuration Peak Stress [P a]
Figure 3.20: Effect of the base model material configuration on the rear stress.
not enough to result in a rear stress superior to the one measured in configuration SS.
This study leads to conclude that configuration SS resulted in the larger stress at the rear impact region. However, configuration AS was chosen to be adopted as the base configuration for the front and rear layers in next section.
Aluminium Steel 0 1 2 3 4 ·107 1.37 · 107 3.96 · 107 Material Impedance [kg/m 2.s]
Configuration Transmission Ratio
AA 1
AS 1.4848
SA 0.5151
SS 1
Table 3.6: Transmission ratio between layers for each configuration.
3.3
Interlayer Parameters Analysis
Thus far, the script model has been validated in terms of its mechanical and physical integrity. However, this section aims to start testing the behaviour of certain parameters. These studies are important regarding the optimization procedures that will be described further on this document. For the scope of this study, a parametric study was carried in order to speed up the study and, also, to test the Python model script thus far developed. As such, to begin with simple tests, three different parameters were varied: the interlayer thickness, the interlayer material and the projectile velocity. The material’s mechanical properties used as the interlayer are shown in Table 3.7. The graph in Figure 3.22 depicts the impedance for each of the materials in Table 3.7. It is evident the contrast in these values between the different materials.
Table 3.7: Mechanical properties for each interlayer studied.
Material Elastic Modulus, MPa Density, kg m−3 Yield Strength, MPa Poisson’s Ratio
Nylon-6 3000 1140 82 0.35
EPDM 2.5 960 16.8 0.499
Cork 9000 293 1 0.3
Aluminium Foam 103.08 410 1.24 0.05
Nylon-6 EPDM Cork Aluminium Foam
0 0.5 1 1.5 2 ·106 1.85 · 106 48,989.79 1.62 · 106 2.06 · 105 Material Impedance [kg/m 2.s]
Figure 3.22: Impedance of the materials used as interlayer.
The graph from Figure 3.23a displays the correlation between the peak stress at the rear impact region of the plate, for four different interlayer materials and a constant projectile velocity
