Application of layout optimization to the design of additively manufactured metallic components
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Abstract
Additive manufacturing (‘3D printing’) techniques provide engineers with unprecedented design freedoms, opening up the possibility for stronger and lighter component designs. In this paper ‘layout optimization’ is used to provide a reference volume and to identify potential design topologies for a given component, providing a useful alternative to continuum based topology optimization approaches (which normally require labour intensive postprocessing in order to realise a practical component). Here simple rules are used to automatically transform a line structure layout into a 3D continuum. Two examples are considered: (i) a simple beam component subject to threepoint bending; (ii) a more complex airbrake hinge component, designed for the Bloodhound supersonic car. These components were successfully additively manufactured using titanium Ti6Al4V, using the Electron Beam Melting (EBM) process. Also, to verify the efficacy of the process and the mechanical performance of the fabricated specimens, a total of 12 beam samples were load tested to failure, demonstrating that the target design load could successfully be met.
Keywords
Layout optimization Topology optimization Additive manufacture EBM 3D Printing1 Introduction
Additive manufacturing (‘3D printing’) techniques have matured rapidly in recent years, and are now starting to deliver on their promise of providing engineers with unprecedented design freedoms. However, to date there has been a limited range of tools available to engineers wishing to exploit these freedoms, with the result that many components produced using additive manufacturing techniques have had similar forms to those produced using conventional manufacturing methods (e.g. casting or machining). When strong and light components are required, structural optimization techniques have the potential to address this.
To date the predominant structural optimization technique applied to component design has been continuum topology optimization (see e.g. Deaton and Grandhi (2014)). For example, the Solid Isotropic Material with Penalization (SIMP) method has been implemented in a number of commercial FEbased packages, e.g. OptiStruct, Ansys, etc. (Rozvany 2009). As a result SIMP is now used as a design tool in many industries, particularly the aerospace and automotive industries, for example by Ford (Wieloch and Taslim 2004) and EADS (Tomlin and Meyer 2011). Although the application of continuum topology optimization techniques to the design of additively manufactured components is described in a number of papers in the literature, frequently researchers have stopped short of fabricating the designs (e.g. Brackett et al. 2011, Aremu et al. 2010), or, if fabricated, have not undertaken load testing (e.g. Razaie et al. 2013). However, in the study described by Cansizoglu et al. (2008), additively manufactured designs were load tested, though it was found that the load test results did not completely match predictions. Another important issue is that although the solutions obtained using SIMP and other comparable continuum topology optimization techniques comprise continua, as pointed out by Deaton and Grandhi (2014), the solutions obtained are often only suitable for conceptual design purposes, and significant manual postprocessing of the final solution is normally required in order to obtain a viable design solution; to address this new formulations capable of obtaining solutions which are closer to the final design are being actively investigated (e.g. Guo et al. 2014). Here the potential for ‘layout optimization’ to be used instead of continuum based methods is explored. A workflow is proposed which includes automatic transformation of the normal ‘line structure’ output from the layout optimization procedure into a continuum.
Numerical layout optimization, first proposed by Dorn et al. (1964), can now be used to obtain extremely accurate solutions, which can be, in the case of 2D problems, almost identical to exact analytical solutions (e.g. see Gilbert and Tyas 2003; Sokół and Lewiński 2010; Pichugin et al. 2012). These can serve as reference solutions for use in later stages of the design process. However, the forms of the truss structures which are identified can appear very impractical. This is because these typically contain numerous joints and interconnecting elements, which would be prohibitively expensive to fabricate using conventional manufacturing techniques. However, the practicality of the solutions obtained using layout optimization can be viewed in a new light when using additive manufacturing, because complex forms can often be fabricated without difficulty. Additionally, a layout optimization based approach can be expected to be particularly useful for scenarios where the final component occupies only a small percentage of the permitted design space, situations where continuum based alternatives appear less wellsuited (e.g. 10 % is the minimum volume fraction allowed in the continuum based optimization software recently described by Aage et al. (2013)).
 1.
Determine a reference volume, V _{0}, for the component in question by performing one or more high resolution layout optimization runs, taking account of the extent of the design domain and the loads and boundary conditions, but no account of ‘practical’ constraints.
 2.
Determine a practical layout by performing further layout optimization runs (these may involve runs with various practical constraints included, or simply the use of lower nodal resolutions if fewer elements are required in the final design).
 3.
Perform postprocessing steps as necessary (e.g. impose minimum area or buckling / overall stability constraints, if not explicitly enforced in step 2), to obtain an updated design and associated volume V.
 4.
Check structural efficiency e = V _{0}/V. Repeat from step 2 if efficiency e is below an acceptable threshold.
 5.
Obtain a continuum model by converting the linestructure using geometrical rules. (Recheck structural efficiency if desired.)
2 Layout optimization
2.1 Basic formulation
Numerical layout optimization provides a powerful and efficient means of identifying the optimum topology of discrete truss structures. For a given set of load cases and support conditions, layout optimization can be used to determine a minimum volume truss topology; here the objective will be to optimize components for strength. With the basic single load case ‘plastic’ formulation (Dorn et al. 1964) the resulting structure will be fully stressed when the design load is applied; the design solution obtained will also be the same as the topology derived using the minimum compliance truss formulation. However, for practical problems the use of multiple load cases is often necessary to ensure robustness.
Where σ _{0} is the limiting material stress. The virtual strain can easily be calculated from nodal values in the dual LP problem and, to ensure the problem size does not grow too rapidly, only members most violating the above criteria are added initially in the procedure (Gilbert and Tyas 2003). The procedure terminates when no potential members violate the criteria, ensuring that the solution obtained is the same as would have been obtained using a fully connected ground structure. The procedure was extended to allow multiple load cases and unequal limiting tensile and compressive stresses to be handled by Pritchard et al. (2005). The latter formulation has been programmed in a C++ based optimization tool developed at the University of Sheffield and the Mosek (2014) interior point linear programming library is used to obtain solutions.
2.2 Including rigid shell structures
In addition to truss bar members, rigid shell structures can also be included to facilitate the modelling of more realistic scenarios. For example, a component to be optimized will often have to integrate into a larger assembly of components, and to model component interfaces rigid shell structures can conveniently be employed (e.g. in the case of the simple beam problem that will be considered in this paper, a rigid shell structure is used to model the region where the load is to be applied by the universal testing machine).
Where B _{ s } is the equilibrium matrix collecting terms for each rigid shell structure, and where f _{ s } collects external forces (F _{ s }) and moments (M _{ s }) applied to each rigid shell structure.
2.3 Determining a reference truss volume and a practical layout
It has been found that modern numerical layout optimization techniques can provide very good estimates of the exact analytical solution, often well within 1 % of the latter in the case of twodimensional problems (e.g. Darwich et al 2010, Sokół and Lewiński 2010). Although the corresponding truss layouts cannot be manufactured directly, due to the high number of members that are usually involved, they do provide a useful reference volume. When attempting to yield a practical solution various rationalization techniques can be applied (e.g. He and Gilbert 2015), or alternatively, the optimization can simply be performed with a coarse nodal discretization and, provided the layout obtained is practical, and the associated volume lies within an acceptable percentage of the reference volume, then it can be deemed acceptable. For this initial investigation any solution that fell within 20 % of the reference volume was deemed to be potentially acceptable.
3 Transformation of line structure to a continuum
The output from the layout optimization procedure will be a frame comprising a series of onedimensional line elements. In order for this to be realized physically, it must be transformed into a threedimensional continuum. Various means of achieving this are possible; here the emphasis has been on simplicity and ease of implementation. Should the basic approach prove to be attractive then clearly the level of sophistication can in due course be increased.
Thus, for sake of simplicity, it was decided that each line element would be transformed into a solid circular element (i.e a simple cylinder). Steps must be taken to address situations where two or more members, once expanded, occupy some of the same volume. This can occur in the vicinity of joints and in regions where members overlap. Also, at this stage, members which are too thin to be effectively fabricated using the chosen additive manufacturing process, or which are susceptible to buckling, need to be resized. These stages, together with the final form generation, constitute steps in the postprocessing workflow, which will now be outlined in sequence.
3.1 Modify layout
3.1.1 Collinearity
3.1.2 Overlapping at nodes
Considering the example shown in Fig. 3c, i and j would be the vectors of elements AC and AB respectively, which have radii r _{ i } and r _{ j }.
3.2 Create nodes at intersections
Layout optimization will frequently generate topologies that include members that crossover one another, presenting two issues: (i) because there is no explicit node at a crossover location (which now forms a joint) the simple buckling analysis that will be presented in Section 3.4 will not provide accurate results; (ii) the joint expansion routine (detailed in Section 3.5) will not be applied at all. The simplest way of addressing both these issues is to find all crossover locations and to then add explicit nodes at these locations, thereby splitting adjoining elements at the newly created node.
3.3 Sizing optimization
After modifying the layout of members, a secondary sizing optimization can be performed to update the member crosssectional areas. The same layout optimization formulation and design parameters are used, but now using the updated layout as the ground structure. Because the problem size is very small this optimization step adds an insignificant amount of time to the overall postprocessing workflow.
3.4 Resize member areas
Any members that violate this criterion are resized. The choice of a suitable effective length factor (k _{ e f f }) is here investigated experimentally  see Section 4.3.2.
3.5 Volume expansion at joints
Given that numerous elements will often converge at a node in a layout optimization solution, there will often be a significant amount of overlapping in the vicinity of a joint. To avoid high localized stresses at such locations additional material must be introduced. Equation (1la) is evaluated at each node to calculate the radius of the joint in proportion to the area and the number and orientation of the elements connected to the node.
Note that because additional material is introduced at joints, the volumes of proposed layouts which include large numbers of joints may be significantly increased in this step, potentially rendering the design unviable. In other words, the proposed procedure relies on the initial layout optimization solution being practical, and amenable to postprocessing using the procedure described. If this is not the case the postprocessed volume may significantly exceed the reference volume.
3.6 Solid model generation
3.7 Output CAD file
A STereoLithography (STL) file can then be derived in preparation for additive manufacture and a STEP, IGES or other standard CAD file can be created for input into a commercial finite element analysis package. Here the NURBS based geometries were created using Rhinoceros V5.0 and the Grasshopper plugin.
4 Component design examples
The layout optimization workflow outlined in Sections 2 and 3 was applied to two case study problems: a single point loaded beam and an airbrake hinge for the Bloodhound Supersonic Car (SSC) project, with the former being fabricated and load tested.
4.1 Choice of additive manufacturing process
Although there are many categories of additive manufacture, the most prevalent type for fabricating geometrically complex components involve the use of a powderbed. Parts are built from the ground up by depositing thin layers of fine powder (<100μm thick) and using a laser or electron beam to melt the 2D cross section in each layer. Generally laser systems have a higher resolution, and yield parts with a better surface finish (Rafi et al. 2013), but suffer from residual stresses as a result of the high cooling rates (Vrancken et al. 2013). Electron beam melting (EBM) on the other hand alleviates these residual stresses by preheating each layer before melting. Also, when melting Titanium Ti6Al4V, the anisotropic mechanical properties associated with the growth of columnar grains during solidification, as seen with other metals when additively manufactured, is largely avoided. This is because Ti6Al4V experiences a phase change upon cooling (at 882.5 ^{∘}C), where the large vertically aligned columnar grains transform into a fine, basketweave like microstructure, known as the Widmanstatten microstructure (AlBermani et al. 2010). Ti6Al4V also has a near perfectly plastic response (Rafi et al. 2013), making it very suitable for use in conjunction with the suggested plastic optimization formulation (see Section 2.1). Polymer materials such as Nylon12 can also be used but the material properties are very sensitive to process parameters and can be inconsistent between builds and machines (Vasquez et al. 2011; Zarringhalam et al. 2006).
As Ti6Al4V parts made from EBM seem to have consistent and near isotropic properties it would appear to be the ideal material to be used for fabricating optimized components, which can then be load tested.
4.2 Material properties
Ultimate Tensile Strength (UTS) data of Ti6Al4V ELI using the EBM process (Khalid Rafi et al. 2012)
Build orientation  Finished state  UTS (MPa)  

Mean  SD  
Vertical  Machined  928  9.8 
As built  842  13.8  
Horizontal  Machined  978  3.2 
As built  917  30.5 
4.3 Example 1: beam subject to point load
4.3.1 Problem definition
The problem definition shown in Fig. 7 consists of a cuboidal domain with four support locations at the corners of the base. A halfcylindrical rigid shell structure (radius = 7.5mm and length = 20mm) discretized using 266 triangular elements is located centrally in the design domain but with the flat surface coplanar with the top surface of the design domain. Each of these triangular elements could accept multiple truss connections. The 100kN load was applied as a pressure on the flat surface of the rigid shell structure.
This arrangement meant that, even when a single load case was involved, the probability of encountering a solution that was in unstable equilibrium with the applied loading was low. This avoided the need to consider multiple load cases (or the use of a stability formulation, such as the one described by Tyas et al. (2005)) in this preliminary study.
Nodes used for the layout optimization were generated spatially in the design domain using cubic grids of various densities. These nodes, together with those positioned on each triangle forming the rigid shell structure, were used to create the ground structure for each optimization.
4.3.2 Design candidates
Example 1: Volumes of line models resulting from layout optimization with differing nodal densities. (Volumes after the members have been resized to account for buckling are shown for different assumed end conditions)
Nodal spacing  No. of nodes *  No. of potential members  Vol.  Vol. after member resizing  

A: FixedFixed  B: FixedPinned  C: PinnedPinned  
(cm ^{3})  Δ %  (cm ^{3})  Δ %  (cm ^{3})  Δ %  (cm ^{3})  Δ %  
−  8  68,724  38.1  15.2  38.1  15.2  38.1  15.2  38.76  17.2 
20  81  89,096  35.5  7.4  35.6  7.7  36.07  9.1  36.98  11.9 
8  756  499,213  34.17  3.4  37.9  14.6  39.55  19.6  43.48  31.5 
4  4961  11,541,432  33.61  1.7  
2  35721  530,708,402  33.32  0.8  not resized  
\(\infty \)  −  −  33.06 ^{∗∗}  0.0 
4.3.3 Final designs
A total of nine examples of the 20mm nodal spacing design (shown in Fig. 8a) were fabricated; three identical specimens (for the purposes of repeatability) for each assumed member end condition shown in Table 2. The main difference between each group is the crosssectional area of the innermost inclined compressive members, which were deemed susceptible to buckling for all three assumed joint conditions.
Additionally, the simple six element truss design shown in Fig. 8a was also designed to act as a benchmark (first entry in Table 2). Again three specimens of this design were fabricated. Note that for the problem considered the benchmark is actually quite competitive, being less than 10 percent heavier than the optimized design. However, for more complex design problems this difference can be expected to be greater.
4.4 Finite element analysis verification
4.5 Example 2: Bloodhound SSC airbrake hinge
4.5.1 Problem definition
Example 2: Load cases considered for Bloodhound SSC airbrake hinge (prior to FoS of 2.4 being applied)
Load case  F _{ x } (N)  F _{ y } (N) 

1  7550  5500 
2  −2981  4171 
3  765  3675 
4  6399  2440 
5  12272  4545 
The Bloodhound team require the hinge to resist the loads shown in Table 3 without yielding or failure. A factor of safety (FoS) of 2.4 was therefore specified, to be applied to the loads listed in Table 3.
The initial optimizations were performed using a regular cubic nodal grid, as used for Example 1. Potential members created between these nodes that crossed any of the design domain boundaries (Fig. 10b) were omitted from the ground structure. Nodes shown on the shaded plane in Fig. 10b were subject to a displacement constraint in all three Cartesian directions. A rigid shell structure composed of 78 triangular elements was placed at the tip to represent the connector ring (outer diameter = 50mm; inner diameter = 24.5mm). Loads were applied directly to the shell structure, on the shell elements shown in Fig. 10b.
From a practical design and manufacturing perspective, the use of a cubic nodal grid proved to be problematic. This is because the resulting topologies generally had numerous members, many of which were very thin. To determine a more appropriate nodal distribution a parametric model was created that would allow nodal locations to be modified in a straightforward manner. The parametric model creates splines that follow the profile of the design domain (illustrated in Fig. 10b). The number and relative spacing of these splines could be defined across the breath and height of the design domain. Nodes could then be created along these splines at a specified spacing. Although this approach is devised based on engineering judgement, a safeguard is that the resulting volume can always be compared with the reference volume.
Example 2: optimization performed using cubic nodal grids of varying densities. The entry marked with a (*) is the final design produced using a nodal distribution created with the parametric model. A fixedpinned end condition assumption was used for the buckling analysis during postprocessing
Nodal spacing  No. of nodes  No. of potential members  Vol.  Vol. after member resizing  

(cm ^{3})  Δ %  (cm ^{3})  Δ %  
Parametric*  157  13,994  34.00  13.8  35.21  17.9 
12.5  340  36,824  33.64  12.6  34.81  16.5 
5  1022  2,988,133  31.37  5.0  
3.5  8420  21,720,412  30.80  3.1  not resized  
\(\infty \)  −  −  29.87 ^{∗∗}  0.0 
5 Additive manufacture and load testing of components
5.1 Manufacture
All specimens were manufactured from gas atomized Ti6Al4V powder using an Arcam EBM S12 system. Layers were deposited at a thickness 70 μm and the standard Arcam Ti6Al4V melt, wafer support and preheat themes for were used. In addition to the standard process themes a modified pin support theme based on the standard ‘Nett’ theme was used to produce porous pin supports.
5.1.1 Initial build study
Example 1: design and measured masses of the benchmark and ‘fixedfixed’ optimized designs
Sample  STL mass (g)  Measured mass (g)  Δ % 

Benchmark  284.5  269.2  5.4 
Optimized  273.6  259.8  5.0 
5.2 Example 1: beam subject to point load
5.2.1 Fabrication
Standard scaling factors were also applied to the whole sample geometry, as prescribed by the manufacturer Arcam (1.0068 in x and ydirections and 1.0093 in the zdirection). These account for shrinkage during cooling.
5.2.2 Load testing arrangement
5.2.3 Load testing results
Example 1: volume, mass and load test results for all the fabricated beam specimens
Specimen  Line model  Solid model  Fabricated  Ultimate  Strength to  

(resized members)  Solid model  (total mass ^{∗})  Mass  load  weight ^{∗∗}  
volume  (truss only)  (g)  (g)  (kN)  (kN/g)  
(cm ^{3})  Volume (cm ^{3})  Mass (g)  
O1  284.6  85.9  
O2  38.10  41.15  182.3  284.5  283.2  91.6  0.499 
O3  281.8  95.3  
A1  273.4  81.6  
A2  35.60  38.06  168.6  273.6  268.8  80.3  0.477 
A3  269.4  79.4  
B1  260.2  93.2  
B2  36.07  38.24  169.4  266.8  258.8  102.4  0.583 
B3  258.6  100.9  
C1  260.6  101.7  
C2  36.98  39.20  173.7  270.0  262.8  98.6  0.590 
C3  262.4  107.2 
It is evident that most of the specimens designated B and C were able to carry the design load of 100kN, whereas, as expected, specimens designated A did not.
5.3 Example 2: Bloodhound SSC airbrake hinge
5.3.1 Fabrication of Bloodhound SSC airbrake hinge
To demonstrate the design could be realised physically, the Bloodhound SSC airbrake hinge was also fabricated, using the same approach as used for the beam specimens. The fabricated design is shown in Fig. 11b.
Example 2: Volume and mass data of final Bloodhound SSC airbrake hinge design
Line model  Solid model  Solid model  Fabricated  

(resized members)  (truss only)  (total ^{∗})  mass  
volume  volume  mass  mass  (g) 
(cm ^{3})  (cm ^{3})  (g)  (g)  
35.21  35.27  156.25  726.09  704.4 
6 Discussion
6.1 Optimization methodology
This study has demonstrated that layout optimization can be used as a part of a workflow to automatically produce 3D CAD models, ready for additive manufacture. Physical load testing of beam specimens designed using this workflow, and then fabricated using the EBM process, generally met or slightly exceeded the required load capacity. The beam problem was selected for its simplicity for this exploratory study and as such the potential for mass reduction was limited (approx. 7 % compared with the benchmark design in this case). The difference in the measured strengthtoweight ratio of the benchmark and optimized specimens was a more significant 18 % increase in the case of the latter. For the more complex Bloodhound SSC airbrake hinge problem the volume reduction was however much more significant (approx. 69 %).
It is clearly imperative to properly account for buckling in the proposed workflow. When checking the optimized designs, the specimens designed assuming ‘fixedpinned’ and ‘pinnedpinned’ end support conditions in the buckling calculations generally met or slightly exceeded the design load. Those designed using the ‘fixedfixed’ assumption (samples A13) failed through buckling at around 80 % of the design load, with the buckling failure of specimen A2 shown in Fig. 14. This might suggest that at least the ‘fixedpinned’ end support condition should be used in future when considering buckling. However, all three of the benchmark specimens (i.e. samples O13) which were analysed using the ‘fixedpinned’ end conditions failed through buckling at approx. 91 % of the design load. This indicates that more a rigorous frame buckling instability analysis should in future be undertaken.
When increasingly fine nodal discretizations are used, layout optimization can be used to provide an estimate of the likely mathematical optimum solution, with this then providing a reference volume (V _{0}) for future design studies. However, there is a need to identify design solutions which are more practical (i.e. which do not contain numerous thin members, which are difficult to manufacture using additive manufacturing techniques and/or which are susceptible to buckling). This can be achieved by ensuring the optimization formulation includes appropriate manufacturability and/or buckling constraints. Alternatively, as in this study, coarse nodal discretizations can be used to obtain simpler design solutions, with checks then performed retrospectively. Provided the volume of the resulting component is within an acceptable of margin of the reference volume then this, or indeed any other strategy, can be justified. The main drawback of the strategy adopted here is that some iterations were required in order to obtain a viable design solution. However, the need for iteration could potentially be reduced through the use of a geometry optimization rationalization step (this involves adjusting the positions of nodes, which in turn leads to removal of many thin elements (He and Gilbert 2015). This could potentially replace the sizing optimization step described in Section 3.3.
The present study made use of solid circular crosssections to keep the solid model generation and fabrication stages as straightforward as possible. However, in future the use of more structurally efficient crosssections could be used (e.g. cruciform sections, which are more buckling resistant). This will often mean that the volume does not increase following the main optimization.
The finite element analysis highlighted that stress concentrations will occur at the joints if material is not added to compensate for overlapping elements, as expected. The simple volume expansion algorithm used in this study did successfully address this problem, though a more rigorous volume conservation algorithm could be applied at the joints, potentially reducing the volume of material added in this step.
Although the workflow presented here is comparatively simple, the results from the beam load tests, and the significant mass reduction achieved in the case of the Bloodhound SSC airbrake hinge, serve to demonstrate the potential of the method. A key potential benefit of the method is that the process of transforming the optimization result into a feasible CAD model is simpler than when using many existing continuum based methods (these methods often require significant manual interpretation and/or postprocessing to produce a feasible design, with smooth and well defined surfaces).
6.2 Additive manufacturing
As mentioned earlier, a scaling factor of 1.05 was applied to all element areas of each specimen to compensate for an issue with the EBM process which led to many of the truss members in the fabricated specimens being undersized. This scaling factor was, however, applied globally, whereas the degree to which the truss members were undersized was found to vary locally. For example, only truss members not aligned to the build direction (i.e. not vertical) appeared to be affected and the general trend was that members with a greater angle to the build direction were affected more. However, rather than developing more specific scaling factors, it is probably more worthwhile to develop more robust EBM process themes, tailored for truss type structures.
7 Conclusions

A workflow in which layout optimization is used to automatically design components suitable for fabrication via additive manufacture has been developed.

Beam specimens designed using the proposed workflow, and fabricated using titanium Ti6Al4V, generally met or slightly exceeded the target design load capacity.

A reference volume, derived by performing a series of increasingly fine resolution layout optimization runs, was used to quantify the structural efficiency of the components produced. The two designed components were at least 80 % efficient (i.e. were within 20 % of their respective reference volumes).

The volumes of the designed components were also compared with those of simpler benchmark components, of the sort that would be designed by manual means. Although the volume of the optimized beam was just 7 % lower than the benchmark, the optimized Bloodhound SSC airbrake hinge component was a far more significant 69 % lower than the benchmark. This highlights the potential for applying the proposed workflow to realworld engineering problems.

The workflow lays a foundation on which further developments can be made. Potential extensions, which should result in greater mass reduction, and/or less need for manual intervention by the user, have been outlined.

Whilst the Electron Beam Melting (EBM) additive manufacture process employed was successfully able to fabricate all the designs described, various dimensional accuracy issues were encountered which need to be addressed in the future.
Notes
Acknowledgments
The first author acknowledges the sponsorship of the Advanced Metallic Systems Centre for Doctoral Training, an EPSRC funded centre involving the Universities of Sheffield and Manchester.
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