Surface imaging and oxidative etching of macro-crystalline ...



A comparison of topology optimization and genetic algorithms for the optimization of thermal energy storage compositesAbstract The structure of high thermal conductivity matrices have been optimized to improve the discharge behaviour of thermal energy stores. Topology optimization and genetic algorithms were validated by verifying their ability to converge to the global optimum for small grid sizes. Genetic algorithms are prone to converge to local maxima while requiring significantly longer convergence times compared to topology optimization. This results in tree structures which consistently outperform a standard rectangular fin by a margin, which increases with grid refinement. Store discharge after ~2 hrs can be improved from 70% to 90% at a loading of 10 vol%. Topology optimization resulted in structures representing parallel sheets, which are as thin as the grid allows. These configurations can maintain the maximum surface area between the low and high conductivity materials at high refinement, resulting in the best performance. Time required for 99% store discharge is decreased by 70% using a 50x50 optimization grid at a loading of 10 vol%.Keywords: thermal energy storage; transient conduction; genetic algorithm; topology optimization1. IntroductionThermal energy storage (TES) systems are used to accumulate energy for use in both heating and cooling applications. Interest in TES is growing rapidly for a variety of reasons, including the incorporation of highly variable, renewable resources into the energy grid [1-3]. These systems are mainly used to mitigate differences between the energy supply and demand. In this capacity, the stores must be rapidly charged and discharged. However, frequently the best storage medium, in terms of energy density, exhibits low thermal conductivity leading to unacceptably low heat transfer rates [4-6]. Even in phase transition systems, conduction through the solid is usually the rate limiting process [7, 8]. Thus, solid composites are used to enhance the transient behaviour.These composites can vary from dispersed particles to randomly structured low-density foams [9-12]. Structured materials offer an interconnected, high conductivity pathway for conduction and as such do not suffer from the high interface resistances found in particulate systems. Unfortunately, these interlinked materials have higher production costs compared to simple additives [9]. In an effort to decrease the overall cost and enhance the efficiency of the composite, the shape or topology of these structures should be optimized. This involves finding the structure, which maximizes the heat transfer at low mass fractions.This is an optimization problem frequently encountered in heat dissipation applications [13-15]. However, a widely accepted and systematically verified methodology to solve these problems is not currently established. Shape or topology optimization has been studied for many years [16], specifically as applied to solid mechanics. A standardized framework for its application to heat transfer however is not yet formalized [17]. Work in this area is comparatively new and quite disperse [13]. For example, since the approach solves an approximation of the problem, uncertainty remains regarding the optimality of the obtained solutions [17, 18]. Despite these concerns, significant improvements have been realized in terms of heat transfer efficiency [19-21].The advantage of this approach is that no a-priori structure is imposed or assumed. An alternative approach is to propose an arbitrary composite structure. A few basic parameters are then selected, such as length, thickness or branching dimensions [22]. Any conventional optimization algorithm can be used to search the parametric space for an optimum of a given objective function [23-25]. Genetic algorithms have been applied to these optimization problems with varying degrees of success [26-31]. However, very few studies have focused on the direct application of genetic algorithms to the unstructured shape optimization problem [32].In this context, both topology optimization and genetic algorithms have been predominantly applied to the optimization of steady state heat transfer problems [13, 19-21, 26-32]. Only recently have transient studies become more prevalent [33-35]. In addition, topology optimization has been mostly applied to finite element models, as these are preferred in solid mechanics. Only limited applications have used finite volume models [36], which are more dominant in heat transfer modelling. The application of these two approaches to transient conduction problems using the finite volume approach are of direct relevance to the energy storage problem.The aim of this study is to compare the performance of topology optimization and genetic algorithms applied to transient conduction. The structural design aims to maximize the objective function, which is defined as the cumulative energy discharged over a fixed time. This is calculated as a percentage of the total energy stored. The performance is compared in terms of the required computational time and the maximum achieved objective function value. In addition, for small problems the algorithms are validated against the true global optimum. The global optimum is found by enumerating all possible configurations. The influences of the problem size (grid dimensions) and the composite mass loading are also examined. The developed structures are analysed to determine if generalized insights into the optimal configuration of energy storage composites can be ascertained. The structures are evaluated in the context of practical feasibility and relative cost savings. 2. Background and Methods2.1. Finite Volume SimulationIn topology optimization, a standard test configuration is known as the “volume-to-point” problem [37]. This involves the two-dimensional shape optimization of a heat sink placed in a square cavity where heat generation takes place at every location. Every point in the volume is made of either a high or low conductivity material. The boundaries are insulated everywhere, except at the base of the heat sink as demonstrated in Figure 1 A. Fig. 1: A) Volume to point problem; B) Energy storage problemThis simplified configuration enables rapid testing of different algorithms on a standardized problem. However, this is a steady state formulation and does not incorporate transient behaviour, as the objective is typically to minimize the steady state temperature distribution. Instead, a slightly modified arrangement, as shown in Figure 1B, has been utilized in this investigation. In this case three boundaries are considered insulated whereas the fourth is exposed to a convective boundary condition (T∞ = 0 ?C). The entire cavity is placed at an arbitrary hot starting temperature (T0 = 100 ?C) and allowed to cool.Using the accepted standard methodology for the finite volume method, as detailed in engineering heat transfer textbooks [38, 39], the domain is subdivided into a number of sub-elements to yield a grid. In the current case, square subunits are used, making it equivalent to the finite difference method. The two dimensional, transient conduction equation is discretized and applied to each subunit to generate the set of implicit difference equations. The resulting matrix equation is solved simultaneously using the LAPACK routine “_gesv”, found in the Python package NumPy. This procedure generates the temperature distribution at each grid point within the energy store over time. These temperatures can be used to calculate the heat flux at any point within the store, at any instant in time. The chosen performance measure is the total heat output across the face exposed to convection over a chosen discharge period. This value is calculated by adding the heat flux across each convective element to account for the full face and integrating this value over time. The end of the discharge period is chosen to be 8000 s (~2 hrs), which in the best performing systems results in up to ~90% of the total energy stored within the square having been removed by convection. Using a set time period fixes the simulation time required, as opposed to a fixed discharge fraction, which would require a variable timeframe and the need for iteration. 2.2. Topology optimizationThis technique has been extensively applied to solid mechanics to find structures, which can withstand a given force using the minimum amount of material [40]. The current mathematical formulation of the topology optimization problem may be stated as follows [17]: find the distribution of material that minimizes an objective function while maintaining the overall material density at or below a specific value. Since the volume of interest is subdivided into a grid, each grid point can either be occupied by a high or low conductivity material. The design variable, representing which material is present, takes on a value of either one or zero respectively. This type of optimization problem is commonly known as a mixed integer (binary) non-linear program (MINLP) [41]. Such problems become very difficult to solve when the number of variables, i.e. in this case the size of the grid, becomes very large. In an effort to circumvent this, a standard practise is to solve the problem by relaxing the integer values into continuous variables [41]. Then the problem can be solved using conventional optimization algorithms, suited to large, nonlinear problems. In topology optimization, the preferred algorithm is the so-called “method of moving asymptotes” that applies conservative, convex, separable approximations to solve these problems using the gradients of the objective function and the inequality constraint (maximum material fraction) [42].Finally, a method must be found to convert the obtained continuous solution to a discrete form. It is an established fact that in doing so, the optimality of the obtained solution will be less than the continuous version [41]. In topology optimization, this is achieved by imposing a penalty function on the calculation of the material thermal conductivity. For a continuously distributed medium, the thermal conductivity may be calculated as the volume (or mass) average of the individual thermal conductivities: k= klow+khigh-klowxP (1)Where x represents the volume fraction of the high conductivity material and the penalty parameter, P = 1. This initial, fully continuous solution is used as input into further iterations. By gradually increasing the penalty parameter, solution values between 0 and 1 become increasingly less optimal. In essence, any value other than 1 results in a low thermal conductivity, thereby moving the objective function away from optimality. A typical progression is shown in Figure 2 for the “volume-to-point” problem [21]. A maximum value of 5 is conventionally used [17,21]. Fig. 2: Typical topology optimization progression for increasing penalty [21]This constitutes the bare minimum for the implementation of topology optimization. Wide varieties of modifications exist, for example speeding up the approach by using the adjoint method to calculate derivatives or using filtering to avoid mesh refinement issues [17]. To facilitate a direct comparison, only the most basic versions of both algorithms have been used.The gradient is calculated by perturbation: each design variable is increased by a small amount and the change in the objective function is calculated. This method is simple and unambiguous. In the current problem, the constraint is simply a sum of all the design variables divided by the total number of grid points. Thus, the change in this value for a given change in any design variable is a constant: one divided by the total number of grid points. The starting point is a homogenous composite with each grid point set to the final target value for the overall loading.2.3. Genetic algorithmGenetic algorithms (GAs) are a specific family of problem solving methods within the broader field of evolutionary computing [43]. They are derived from an analogy loosely based on the evolution of natural systems and consequently a large number of variations exist, as detailed in several textbooks on the subject [43-45]. In short, an environment is filled with a population of individuals (randomly generated), in this case possible structural configurations. The forces of survival, mutation and reproduction act on the population, gradually driving it towards the solutions, which have a better objective function value, or fitness. In essence it is a stochastic trial-and-error (or generate-and-test) problem solving process.A “steady-state” algorithm maintains the size of the population by creating only enough new solutions, or children, to replace a fixed number of individuals (Replacement %) each cycle. To select the configurations for replacement, a tournament approach can be followed. In this case a small number of individuals are selected at random (Tournament size %), their fitness is evaluated and the weakest participants are discarded. This method has the added advantage that weaker individuals may persist in the population, potentially leading to advantageous future combinations. The opposite situation is a problem commonly encountered in genetic algorithms known as premature convergence [45]. This is the tendency of the system to converge the entire population rapidly to the best solution found at an early stage. Conceptually it is equivalent to an inability to maintain genetic diversity in the population. Maintaining diversity ensures a more thorough search of the solution space. In optimization terms, it promotes the search for a global optimum rather than convergence to local optima. The number of solutions selected for the tournament, or tournament size, drives what is known as the selection pressure. The selection pressure reflects the rate at which more fit designs dominate over less fit solutions. It creates a trade-off between faster convergence and the likelihood of finding the global optimum.For a binary representation, mutation is introduced through bit flipping. A random number generator is used to produce a number between zero and one, if the number is less than the “mutation probability” the design variable is flipped, i.e. 0 to 1 or 1 to 0. Mutation introduces variation, thereby curtailing premature convergence but at the same time potentially disrupting good solutions. Recombination is achieved by uniform crossover, which avoids positional bias [44]. For this, two parents are selected at random. For each grid point a random number between zero and one is generated, if the number is greater than 0.5 the mother’s value is inherited, else the father’s. A second child is generated by inverting this mapping. An example of the process applied to a 5x5, ~10% fin matrix is shown in Figure 2. Fig. 3: Schematic representation of uniform recombinationIn order to maintain the vol% of high conductivity material constant it is necessary to check that the arrays were combined in manner that preserved this value in each of the children. If not, the recombination procedure is repeated until this criterion is satisfied. The termination condition is usually based on the stagnation of the population fitness level. GAs invariably follow a typical progression [43]: an initial period of rapid increase in the population fitness followed by a pronounced slowing and eventual stagnation. It is not sensible to continue cycling once this convergence has taken place within the population, since further iteration does not significantly improve the objective function value.2.4. ComputationThe coding platform used for simulation and optimization was Python 3.6.4. The “method of moving asymptotes” optimizer was implemented using the NLopt open-source library for nonlinear optimization [46]. The architecture is an Intel Xeon? CPU E5-1607 operating at 3.1 GHz and equipped with 32 GB of RAM. The encoded algorithms are summarized in the following figures with full code provided in the Supporting Material.Fig. 4: Finite volume simulation flowchartFig. 5: Topology optimization flowchartFig. 6: Genetic algorithm flowchartTo find the global optimum all possible design configurations must be evaluated. From a mathematical perspective, this is a purely combinatorial problem: how to select a number of locations to place high conductivity material, from all possible locations. For example, a 5x5 system (25 elements in total) with 10 vol% of fin material (rounded up to 3 elements) is represented by the number of ways in which three numbers can be selected from a list of 1 to 25 where order doesn’t matter and replacement is not allowed. Arithmetically this is equivalent to calculating combinations:nk=n!k!(n-k)!=253=2300 (2)All possible combinations can be evaluated by generating the complete set of combinations. This is achieved using the module itertools from the Python Standard Library. Each configuration is evaluated using the finite volume simulation to calculate its achieved objective function value.3. Results and discussion3.1. ValidationThe first step is to validate the finite volume simulation, which is used to calculate the objective function value. This is done by comparing the results of the custom coded finite volume model to the established commercial package ANSYS Fluent?. Both techniques are used to simulate the discharge of a 10x10 cm, a two dimensional heat store, which is subdivided into 25x25 grid points. A traditional straight fin of size 3x15 (~7 vol%) is added to the centre of the store to enhance heat transfer, as shown schematically in Figure 7A. Grid points are indexed starting with (0, 0) in the top left corner.Fig. 7: (A) Storage grid schematic (B) Results comparison: markers = ANSYS Fluent?, lines = Custom codeThe storage medium used is a typical wax with properties: k = 0.4 W.m-1.K-1, Cp = 2800 J.kg-1.K-1 and ρ = 990 kg.m-3. While the fin is aluminium with properties: k = 202 W.m-1.K-1, Cp = 870 J.kg-1.K-1 and ρ = 2720 kg.m-3. The store is placed at an initial temperature of 398 K and cooled using a fluid at 298 K with a convective heat transfer coefficient of 500 W.m-2.K-1. For validation, the simulation was run over 500 time steps of 10 s each. The store temperature distributions after the elapsed time (83.3 min) are given in Figure 8 for both techniques.Fig. 8: Simulation results (A) Custom code (B) ANSYS Fluent?The qualitative agreement is excellent and a more quantitative indication of the relative performance is given in Figure 7B. Here the temperatures at the four points indicated on Figure 3A are compared over time. The markers represent the Fluent? results and the black lines are the custom code. The profiles correspond well, with a maximum deviation of around 1%. This minor deviation may be caused by a difference in the method used to calculate the average thermal conductivity across material interfaces; a harmonic average was used in the custom code. When the grid subdivision was further increased by a factor of 10, no difference was observed in the profiles, indicating that convergence is already achieved. Unfortunately, a widely accepted validation test case for the simplified algorithms presented here is not available, mainly due to the wide range of possible variations. Instead, the algorithms are validated by considering their performance relative to the true global optimum for simple problems. The number of combinations increases with an increase in the grid size as well as the loading. The maximum values examined were a 7x7 grid with loadings of up to 20 vol%. The enumeration procedure, at an execution frequency of 340 μs per evaluation, still took ~1 month to complete, as there are more than 8 billion possible combinations (8.22x109). To visualize the results they are normalized, with a value of 100 assigned to the best performing design (highest objective function) and 0 to the worst. These values are sorted from low to high and plotted in Figure 9 for a 5x5 grid, with the total number of solutions scaled from 0 to 100. Fig. 9: Solution enumeration on a 5x5 grid as a function of loading.As the loading percentage is increased, the curve tends towards an approximately linear relationship across the possible solutions. This is to be expected: as the number of high conductivity elements is increased, the number of closely related structures increases: many minor variants are possible around a “core” design. Thereby a more gradual transition from low performing to high enhancement configurations occurs. However, in the case of interest, i.e. low loading, the behaviour is very non-linear. This is further demonstrated in Figure 10 where the solution space found for a 7x7 grid (10 % loading) is displayed, with focus on the uppermost 0.5% of solutions. Fig. 10: Uppermost 0.5% of solution space on a 7x7 grid (10% loading)In this case, 50% of the benefit gained by structuring of the high conductivity material can only be found in the top 0.25% of all possible configurations. This still amounts to around ~5000 designs, where large portions are simple variants of a few principal configurations. Nonetheless, there is a large potential for loss in performance if a configuration is selected outside of these “core designs”. For further clarification, consider the top 9 solutions of the full solution set, i.e. the true global optima, as depicted in Figure 11. It should be noted that the data set of all possible solutions include solutions where the fin is not anchored to the convective boundary, since no heuristics have been applied to limit the search space. Fig. 11: Top 9 globally optimal solutions (7x7 grid, 10% loading)The solutions in Figure 11 are arranged in order of decreasing optimality starting from top left, moving across then down. The discussion assumes the most optimal design is first followed by the rest in in order of decreasing optimality. The first and second designs are distinct where the third is a mirror image of the second. The objective values for the second and third designs are identical. The fourth and fifth designs are symmetrical variations on the first: the base of the fin is located centrally with a single, furthermost element relocated. The same is true for the sixth and seventh designs, which are non-symmetrical variants of the second and third configurations. Finally, designs eight and nine are distinct but symmetrical, representing the third “core design”. Moving further along the set of all solutions multiple elements will be relocated, resulting in a wider range of variants and branches. As stated, this effect will be compounded by increasing the loading and increasing the grid refinement. Topology optimization is a gradient-based method and is deterministic. Hence for a specific progression of the penalty parameter, the final result will always be the same. In general, it was found that a slow progression, increments of 0.5 as suggested in literature [21], provided the best results. For this case, the optimization correctly converged to the global optimum for a 7x7 grid at 10% loading. The genetic algorithm on the other hand is stochastic and as such randomly converges to any of the “core designs”. This is demonstrated by Figure 12, which plots the performance of the algorithm solution, scaled to the global optimum, for 100 consecutive runs. Fig. 12: Algorithm outcomes for 100 runs of a 7x7 grid (10% loading)These results reveal that the algorithm converges to the global optimum, roughly 25% of the time (also attained for 1000 runs). This indicates that there are in fact four “core designs” which have high levels of optimality. The first set constitutes a base (top node) at the central node, the second set has a node on either side of the central node and the third two has nodes away from the central node. Thus from Figure 11 it can be argued that the final set of “core designs” is one where the fin is located at the outer edge (left or right) of the grid but still anchored to the convective boundary. It may be concluded that the algorithm is likely to find a design related to the globally optimal “core design”, roughly once in every x runs, where x = half the number of columns (due to symmetry). The algorithm does however find a solution in the top 0.25% of all configurations 100% of the time, as indicated by the performance in Figure 12, which is always within 50% of the top design. Since mutation only modifies one or two elements at a time, once a “core design” has taken hold in the population, it is very difficult for the optimizer to cross over to a more optimal version. This would require the entire base of the structure to be moved; hence, the algorithm tends to get stuck in these local maxima. However, by randomly seeding the population the global optimum is periodically found. Since it cannot be predicted exactly when the optimum is found, all genetic algorithm tests were repeated at least 3x times to maximize the likelihood of at least one run converging to the global optimum. Despite this drawback, the results clearly demonstrate both algorithms are capable of converging to the global optimum for small grids.3.2. PerformanceFor larger grids, it is not feasible to enumerate all possible combinations with the current computational capability in a reasonable amount of time. Instead, to assess the performance of the designs found by each algorithm they are compared to a base case. For this, material arranged in a standard, single rectangular fin with a depth of around two thirds of the store was selected. The depth was chosen arbitrarily to constrain the base case to a rectangle and it is possible that a slight improvement in performance could be found by optimizing the fin length. The final objective function values, i.e. the % energy discharged in 8000 s, are given in Table 1 for all the designs, as a function of increasing grid size. The largest grid size was determined by a maximum convergence time of one month for the genetic algorithm. Table 1: Objective function values (10% loading)Grid sizeNo FinRandom DistributionStandard Rectangular FinTopology OptimizationGenetic Algorithm256 (162)39.1%41.1%70.3%68.5%79.9%529 (232)39.1%41.9%70.1%84.6%85.2%2500 (502)39.1%42.4%69.5%91.1%90.4%The lowest discharge is value is found when no fin is present. An additional case has been included where the material is arranged in a totally random fashion. It is notable that in this case there is no substantial improvement over the performance with no fin. Thus, for additives which tend to clump and fail to achieve a fully homogenous distribution, very little additional benefit is realised. This is in agreement with recent findings regarding energy storage composites [9]. The standard rectangular fin improves discharge from 39% to ~70% compared to no fin. Interestingly, at the smallest grid size considered, the topology optimization result performs worse than the standard fin while the genetic algorithm design improves on it by ~10%. As the grid is increased, the values for both algorithms become virtually the same, representing a 15% increase above the standard fin. Finally, at the maximum grid refinement considered, the topology optimization approach performs better than the genetic algorithm design. The reasons behind these shifts will be discussed in the next section. Irrespective, at high refinement both algorithms achieve a substantial 20% increase in discharge above the standard fin design. The data indicates that upon further refinement, the results would improve even more, but since the discharge approaches 100% asymptotically, these improvements would level off at the current loading. The times required for convergence at different grid sizes are demonstrated in Figure 13. Fig. 13: Convergence time as a function of grid size (10% loading)As the grid size is increased, both algorithms show a roughly second order based increase in the required computational time to achieve a converged solution. With both in their most basic formulations, the topology optimization approach requires significantly less time than the genetic algorithm, roughly one order of magnitude. The two primary reasons for this are the stochastic search, which requires multiple executions to move beyond local maxima and the random search pattern, which requires numerous iterations to converge. Based on literature recommendations [43] the starting values for the genetic algorithm (GA) parameters were chosen as given in Table 2. Table 2: GA starting parameter valuesParameterValuePopulation10Mutation probability0.5Replacement %10Tournament size %20All of these values were varied to explore the impact on the convergence behaviour. None were found to improve the convergence time meaningfully. The execution time of the topology optimization could be further decreased using the adjoint method to speed up gradient calculations [47]. Thus, it is clear that even with potential heuristic improvements to the genetic algorithm, for example searching only interconnected structures, it is highly unlikely that it will ever reach the computational speed of the topology optimization route.3.3. Structural analysisAs can be seen from Table 1, the performance levels achieved by the algorithms are clearly a function of the grid refinement. Furthermore, the topology optimization designs begin to outperform the genetic algorithm at high refinement. The reasons behind these observations are evident when the design structures displayed in Figure 14 are evaluated. Fig. 14: Designs as a function of refinement (10% loading)The topology optimization designs are qualitatively consistent, several straight structures which increase in number and depth as the grid is refined. As the temperature gradient and hence the heat flow is unidirectional in this case, i.e. upward, it seems that the designs are a direct, consistent consequence of this. It is presumed that at very high numerical precision the lines would be straight and evenly spaced. The structures created by the genetic algorithm on the other hand clearly contain an element of chaotic randomness. The most refined configuration is however very reminiscent of the tree structures found by topology optimization for the steady state “volume-to-point” problem [21] as seen in Figure 2. As the grid is refined, the resulting structures in all cases are thinner and longer, thereby increasing the surface area for heat transfer between the high and low conductivity media. This accounts for the increase in performance of both approaches with increased grid refinement. However, this is also the reason why the topology optimization designs eventually outperform the genetic algorithm. As the tree structure is refined, additional branches are added which collect more heat that must be conducted out of the store. This is achieved via the central “trunk” structure. However, as the branches are subdivided a point is reached where they require a thicker “trunk” structure to accommodate the high heat flow. This phenomenon is just starting to manifest in the genetic algorithm design with the highest refinement, where regions of double element thickness are observable along the central “trunk” structure. At this point, the surface area for exchange between high and low conductivity material is no longer maximized. This is not the case for the topology optimization based design, which leads to its superior performance at high refinement. Because of the incremental nature of the genetic algorithm, search procedure it will always favour tree structures as it initially randomly selects a base structure or “core design”, which gives the maximum initial objective function gain, followed by branching and enhancement.A recent steady state study has analytically proven the non-optimality of tree structures, for specific heat conduction problems [37]. It makes sense that further refinement will simply lead to additional surface area creation and further enhanced performance for the topology optimization design. For a hypothetical steady state configuration with pure conduction the maximum thermal conductivity (i.e. optimal solution) is known from analytical considerations as the pure parallel model [9]. In this case the arrangement would constitute continuous fins spanning the store from top to bottom, very similar to the topology optimization sheet arrangement. However, from a steady state perspective the overall thermal conductivity is unchanged for a single, thick fin as opposed to multiple, thinner fins. However, for the transient case, maximisation of the surface area for exchange has a substantial impact on performance.From the perspective of optimally designing energy storage composites, the results indicate that the best option is to use high conductivity sheets, which are as thin as possible. The both optimization results indicate that randomly structured materials such as foams [4] and additives [5] are suboptimal. Ultimately though, cost and other practical considerations will determine the minimum thickness, which can be achieved for these thin sheet composites.The obtained structures may not signify the true global optimum, but they do represent a significant improvement above the standard rectangular fin design for a given storage volume. The enhancement in the dynamic performance of an optimized store is demonstrated in Figure 15 for a grid size of 50x50 at a loading of 10%. Fig. 15: Dynamic performance comparison 50x50 grid (10% loading)The time to fully discharge (99%) the store using a standard rectangular fin is reduced by two thirds from 505 minutes to 151 by the topology optimization design (70% reduction). In addition, the peak (initial) discharge rate is almost doubled. The structures developed by the genetic algorithm do not only represent inferior designs at high refinement, but they would also be very difficult to construct practically. The topology optimization design on the other hand simply requires thin, flat sheets or foils. Based on the outcomes of this investigation there is little to no value in progressing the designs to higher refinement by using more computing power or enhanced efficiency techniques such as parallel processing. 4. ConclusionsTopology optimization and genetic algorithms provide a platform for the shape optimization of energy storage composites. Matrix structures offer higher thermal conductivity than additives due to the decreased effect of interface resistance. However, these materials tend to be more expensive, thus by optimizing their structure to gain maximum benefit in terms of dynamic performance, the overall cost of the energy store can be decreased. The aim of this study was to compare the structures developed by these two techniques, in terms of computational time required and reduction in energy store discharge time. To validate the methodologies, their ability to find the global optimum was evaluated for small systems. The following can be concluded regarding algorithm performance:The system behaviour was found to depend heavily on the loading. At low fractions of high thermal conductivity material, the behaviour tends to be highly non-linear. In the case of a 7x7 grid with 10% loading, 50% of the enhancement achievable by structural optimization was encompassed in only the top 0.25% of all possible designs. Furthermore, it was found that for the most part, these designs represented subtle variations of a few “core designs”. These designs are dictated by the grid size and nature of the discharge problem.Both algorithms were found capable of converging to the global optimum for small grids; however, the genetic algorithm requires approximately an order of magnitude more computational time than the topology optimization approach. For larger grids the global optimum is unknown and performance was compared to a standard rectangular fin. In this case, the genetic algorithm consistently provided an improved design. The relative improvement of the design was found to increase with larger grid sizes. Initially the topology optimization designs underperformed slightly; however, as the grid refinement was increased this approach eventually outperformed even the genetic algorithm.The topology optimization substantially outperforms the genetic algorithm in terms of computational time. It was found that parameter tuning had very little impact on improving the convergence time of the genetic algorithm. While the algorithm could be improved by heuristics, which limit the search space it is unlikely that it will surpass the topology optimization approach, which can be significantly improved using the adjoint method for gradient calculations.A structural analysis indicated that the two algorithms converge to fundamentally different designs. The genetic algorithm favours a tree structure, whereas the topology optimization results in progressively thinner sheet arrangements. In the case of the former, a point is reached where the heat gathered by the branch structures cannot be conducted out via a thin trunk. At this stage, the trunk thickens and the surface area for heat exchange between the high and low thermal conductivity materials is no longer maximized. The topology optimization on the other hand can indefinitely reduce the thickness of the sheets to maintain a maximum surface area. The incremental nature of the genetic algorithm means it will always favour a given base structure early on, as this maximizes the objective function gain. Thus, the algorithm is prone to converge to these local maxima.At the highest refinement considered, a 50x50 grid, the designs delivered by both of the algorithms represent a significant improvement over a standard rectangular fin. The energy discharged after a ~2 hours can be increased to 90% by the algorithms compared to 70% for a rectangular fin and 40% for no fin. The time required to reach 99% energy discharge can be improved to 151 minutes by the topology optimization design compared to 505 minutes for a standard fin. The work demonstrates that optimal performance in energy storage composites can be achieved by using high conductivity sheets, which are as thin as possible. 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