My PhD thesis is on the "Optimisation of the impact response of composite materials using genetic algorithms." It was done at the Department of Aeronautics, Imperial College. My PhD/DPhil was awarded in 2009. My PhD thesis can be found in the Senate House Libraries, see Senate House Library link
Genetic algorithms are programs that optimise problems by emulating the behaviour of populations undergoing Darwinian evolution. Composite materials are materials with two or more heterogeneous phases, such as carbon-fibre reinforced plastics.
The following section discusses work done during my PhD and is organised as a narrative of successfully optimised problems.
The work presented is, to the author's knowledge, the first known attempt to automate the material selection and design process to develop a composite plate capable of defeating a given ballistic threat. Partial automation of the material modelling procedure was also achieved. Seven different materials were selected from, while attempting to keep cost and/or weight minimum. However, the technique developed is able to choose from an arbitrary number of materials against an arbitrary number of goals. The material selection pool consisted of aluminium alloys, monolithic plastics (polycarbonate and ultra-high molecular weight polyethylene), mild cold-rolled steel, glass/epoxy and carbon/epoxy plates; see below:
Optimisation proceeded at two levels here. The first was to use the optimisation algorithms to improve experimental-FE correlation, by tuning the finite element erosion strains. Element erosion is an accepted evil in modelling high velocity impact events with Lagrangian meshes, which involve severe element distortions. Without element erosion, the time step for explicit calculations would diminish to near zero values, resulting in the computation grinding to a halt. The movie below shows a ballistic impact test on a monolithic Lexan polycarbonate sheet (click to download), filmed using a Phantom v5.1 high speed camera, at 20000 pictures/second and an exposure time of 2Ás. Impact velocity by a 0.89g 6mm diameter stainless steel sphere is 266m/s, exit velocity 137m/s.
From the tests, an experimental ballistic curve is generated. The goal is to get the finite element simulation to replicate this curve, so that points outside our experimental data set may be reproduced computationally. The animation below shows the results of a FE simulation (click to download). The computed exit velocity is 123m/s, within experimental variability. Simulation time took 37s on a commodity overclocked Pentium 4 Intel D 805 2x3.33GHz CPU (tested with Prime95 for errors).
Despite the coarse mesh and computationally inexpensive elastic-plastic material model, relatively good experimental-FE correlation was achieved by the evolutionary algorithm; see ballistic curves below.
The resulting material models were used to develop minimum cost, weight and cost & weight designs that met the design ballistic threat of a 0.89g 6mm diameter stainless steel sphere moving at 350m/s. The designs identified by the algorithm were:
A high speed video clip of a high velocity impact on a PC-PC-Al2014 system is shown below. Impact velocity is 365m/s and the projectile is arrested after perforating the first two PC plates. Note the gross deformation of the final Al plate, which failed by global plastic dishing rather than localised shear plugging. The former damage mode is a more efficient energy absorber than the latter one. The FE simulation results are presented after that.
A pdf version of this aspect of my research can be downloaded from here.
From the results presented here, we have seen the optimisation algorithms developed used successfully in both improving the material modelling process, and then the material selection & design process.
Low velocity impact on composite structures are a menace because even relatively low energy impact events such as dropped tools can cause significant reductions in strengths and stiffness, with compressive strength losses of up to 70% possible. These significant reductions are due to the lack of plasticity as a means of absorbing energy in thermoset-resin composites, which absorb energy by creating fracture surfaces instead. Furthermore, the damage is often not very visible to the naked eye, and is thus termed Barely Visible Impact Damage (BVID).
The goal of this portion of the research was to develop tools that can reduce the energy absorbed by the composite during an impact event. This would translate to less energy spent creating new fractures within the composite. The variables in this design case were the lamination angles and material (GFRP or CFRP). Based on the numerical optimisation results, a GFRP laminate with a [▒15░, ▒60░]s layup was manufactured and compared against a baseline crossply stacking sequence. The initial and final kinetic energies were calculated using high speed camera footage (see below) and integration of the force-time history of a piezoelectric force transducer.
The optimal layup absorbed about 8.3% less energy than the crossply layup, at the same impact velocity. The force-displacement histories are shown below. From the force-displacement plot, it is clear that the area enclosed by the graph (energy absorbed by the composite) was more for the baseline crossply layup in comparison to the optimal one.
The optimisation algorithms developed were thus able to improve the low velocity impact performance of the composites tested.
In general, it is relatively straightforward to determine the natural frequency of rectangular plates given their dimensions and composition. It is less easy to determine the dimensions and composition of such plates given the frequency desired. Work was done to develop tools capable of:
ESDU item A8336, 83035 and 83036 were used here to evaluate the natural frequecies, while an optimisation algorithm was successfully used to solve both scenarios above. Variables included ply angles (if orthotropic), ply materials and ply thickness.
For the first problem, the graph below shows the possible range attainable, given 10 different materials (e.g. CFRP, GFRP, Aluminium, Titanium, Tungsten, Steel, Brass, Magnesium alloys) and a maximum ply number of 10. Where the plotted points line up with a line of gradient 1, the attained frequency matches the target frequency. Given the material selection pool, this range is 206Hz - 38.5 kHz. Thus a designer can know a priori if they need to expand their search space to accommodate or avoid certain modal frequency requirements.
For the second problem, suppose our target natural frequency is 2000Hz. The graph below shows that given the materials, ply angles and thickness range available, this target is achievable by ply counts ranging from 5-13 plies. Furthermore, as a secondary objective, the optimiser was tasked with minimising the spread between the first three modes of vibration. This is measured is the Standard Deviation and is kept as low as possible (it linearly increases with number of plies).
Obtaining a robust optimisation technique for finding the global optimum of a wide variety of mathematical functions is of interest not only to mathematicians but also to the engineering community. This is because many complex real-world problems can be idealised as a mathematical function. Here, the optimisation techniques developed were successfully tested against the mathematical functions tabulated below. The optimiser successfully located the exact global optimum in all cases or arrived within a very close vicinity of it. All tests were re-run with ten random seeds to ensure the reproducibility of the results.
The first five functions are from De Jong's 1975 PhD thesis, "An Analysis of the Behaviour of a Class of Genetic Adaptive Systems" (University of Michigan). The sixth function is one of my own concoction, meant to challenge the optimisation algorithm with multiple local optima, a huge search space and noise.
Some features of the functions successfully optimised by my techniques:
Visualisations of the mathematical test functions are shown below.
Last updated: 2009/05/13.