Application of layout optimization to the design of additively manufactured metallic components

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.

Layout optimization is based on the ground structure approach, in which nodes distributed across the design space are interconnected by potential members (or ‘truss bars’). Various formulations are possible; here each member is assumed to have two variables associated with it: force and cross-sectional area. The objective of the optimization is to minimize the total structural volume, subject to equilibrium constraints and constraints which ensure that the cross-sectional area of each member is sufficient, given the forces across all load cases to be sustained and the tensile and compressive strength of the material: subject to where V is the total volume of the structure; l is a vector of individual member lengths, l ^T={l ₁,l ₂,...l _m }, where l _i is the length of member i (i=1,...,m), a is a vector of member cross-sectional areas, a ^T={a ₁,a ₂,...,a _m }, where a _i is the cross-sectional area of member i (i=1,...,m). Also, B is a suitable (3n×m) equilibrium matrix containing direction cosines; q is a vector of member axial forces, q ^T={q ₁,q ₂,...,q _m }, where q _i is the force in member i (i=1,...,m); f ^α is a vector of applied loads and load cases, where α represents an individual load case (α=1,...,p) and \(\mathbf {f}^{\alpha \mathrm {T}} = \{{f_{1}^{x}},{f_{1}^{y}},{f_{1}^{z}},{f_{2}^{x}},{f_{2}^{y}},{f_{2}^{z}},...,,{f_{m}^{z}}\}^{\alpha }\)where \({f_{j}^{x}}\), \({f_{j}^{y}}\), \({f_{j}^{z}}\) are the x, y and z direction components of the live load applied to node j (j=1,...,n). Finally, σ ⁺ and σ ⁻ are respectively the limiting tensile and compressive stresses that can be sustained by the material. This type of problem can be solved using linear programming (LP). Note that kinematic compatibility constraints are not required in the plastic formulation, even when multiple load cases are involved. This is because plastic yielding of members can occur, allowing forces within the structure to redistribute as necessary. Theoretical issues associated with the plastic multiple load case formulation are considered in more depth by Rozvany et al. (2014).

However, the use of a fully connected ground structure becomes computationally expensive for large-scale problems. This is because as the number of nodes n in a problem grows, the number of potential members quickly becomes large (since there will be n(n−1)/2 connections in a fully connected ground structure). As the majority of potential members will not be represented in the final design (i.e. will have zero area), representing all potential members becomes rather inefficient. To address this, the method originally developed by Gilbert and Tyas (2003) can be used. This method involves starting with a comparatively sparse ground structure (e.g. with only adjacent nodes interconnected) and then adaptively adding additional potential members in an iterative solution procedure. The Michell-Hemp optimality criteria (which imposes limits on the virtual strain 𝜖 _i which can be experienced by each potential member i), is used to judge whether to admit additional members to the ground structure:

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.

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).

Triangular elements are used to form the rigid shell structure in this formulation, with the locations of these prescribed prior to the optimization (Fig. 2). For simplicity, a single connection point is created at the centroid of each triangular element forming the rigid shell structure, which can accept multiple truss connections as well as the application of external forces. Equilibrium needs only to be considered at a single master node, with the equilibrium matrix for a single element ‘i’ shown in (1e). Here v _i is the vector [v _i,x ,v _i,y ,v _i,z ] of element ‘i’, d _i is the vector between the connection node of element ‘i’ to the shell and the master node, and l _i is the length of element ‘i’.

The global equilibrium of each rigid shell structure is then enforced, so that equilibrium equation (1b) is rewritten as:

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.

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 two-dimensional 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.

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.

Determining which members intersect with one another and the coordinates of the intersection can readily be found. Thus, considering the example depicted in Fig. 4, (1ia) and (1ib) can be evaluated: where n _A and n _C are the (x,y,z) coordinates of nodes A and C and where i and j are the vectors of elements AB and CD. Also, v _{C A} is the vector interconnecting nodes C and A and n _{E i} and n _{E j} are the coordinates of node E. If these two coordinates are the same and lie in between nodes A and B for member i and nodes C and D for member j then the members are deemed to crossover. A new node should be created at this location. Calculations of this type should be undertaken for each pair of members in the frame.

After modifying the layout of members, a secondary sizing optimization can be performed to update the member cross-sectional 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 post-processing workflow.

The buckling response of solid circular bars is intrinsically non-linear and thus it is convenient to consider buckling as a post-processing step, rather than during the main optimization phase. The relevant Euler buckling relationship, which must be satisfied for all compressive members, is as follows: where a _i , \(q_{i}^{-}\) and l _i are respectively the area, compressive force and length of member i and k _{e f f} is the effective length factor. Finally, E is the elastic modulus of the material.

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.

A simple minimum area constraint (1k) must also be enforced during the post-processing workflow. This is because many additive manufacturing processes will not effectively capture truss bars below a given diameter. where a _{m i n} is the minimum permitted cross-sectional area.

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.

Where the curly brackets in equation (1la) are Macaulay brackets, and where \(\hat {i}\) and a _i is the unit vector and area of the master member respectively, and \(\hat {j_{k}} = \{j_{1}, j_{2},...j_{m}\}\) and a _j,k ={a _j,1,a _j,2,...a _j,m } are the unit vectors and areas of all the other members connected to the node. This equation is evaluated with every member as the ‘master’ member, with the largest value for the radius being selected.

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 post-processing using the procedure described. If this is not the case the post-processed volume may significantly exceed the reference volume.

The final form of the component is conveniently generated using Non-Uniform Rational B-Splines (NURBS), which provide a mathematical representation of analytical and free-form geometries and are the industry standard in CAD exchange formats such as STEP and IGES (Piegl and Tiller 1995). Each member is created by extruding a circular cross-section along the length of the line element, with the area being set to that determined from the layout optimization process. At each joint a sphere is created with a radius calculated using (1la). At a set distance from the joint node, each truss member is expanded using a linear loft operation so that the end radius is equal to that of the sphere at the joint. This is shown schematically by the dotted line in Fig. 5. Boolean operations are then performed to create a single ‘watertight’ surface for the whole geometry.

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 plug-in.

Figure 6 shows graphically the main steps required to transform a line model into an STL mesh model ready for additive manufacture.

Although there are many categories of additive manufacture, the most prevalent type for fabricating geometrically complex components involve the use of a powder-bed. 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 Ti-6Al-4V, 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 Ti-6Al-4V experiences a phase change upon cooling (at 882.5 ^∘C), where the large vertically aligned columnar grains transform into a fine, basket-weave like microstructure, known as the Widmanstatten microstructure (Al-Bermani et al. 2010). Ti-6Al-4V 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 Nylon-12 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 Ti-6Al-4V 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.

As all specimens were to be tested in their as-built state (i.e. with no surface finishing), it is important to distinguish between the mechanical properties of as-built and machine finished Ti-6Al-4V from the EBM process. Ti-6Al-4V ‘extra low interstitials’ (ELI) powder was used for fabricating the specimens. Much of the data available from the machine manufacturer (Arcam) and from the literature for this powder was obtained using specimens that have been machined prior to testing. However one study compared the mechanical properties of machined and as-built tensile specimens and found there to be a significant difference in the ultimate tensile strength (UTS) (Khalid Rafi et al. 2012). These values are shown in Table 1. As all specimens in this study were to be built vertically a value of 842 MPa was selected as the limiting stress in the optimizations. An elastic modulus of 113.8 GPa (standard for Ti-6Al-4V) was used in the subsequent buckling analyses.

The Messerschmitt-Bölkow-Blohm (MBB) beam is a 2D benchmark optimization problem often referred to in the literature, consisting of two simple supports and an elevated load applied at a location midway between the supports. To better assess the capabilities of the layout optimization method, this standard problem has been adapted to 3D, as illustrated in Fig. 7.

A simplified elasto-plastic finite element analysis was performed for the ‘fixed-pinned’ optimized geometry with and without the joints expanded (see Section 3.5). The specimens with and without expansion at the joints were respectively meshed using 89138 and 104494 10-noded tetrahedral elements, as shown in Fig. 9a and b; in both cases quarter symmetry was assumed. At the time of the study strain hardening data was not available for as-built titanium manufactured using the EBM process. Thus an elastic-perfectly plastic model was assumed, with an elastic modulus of 113.8 GPa, yield stress of 842 MPa and Poisson’s ratio of 0.342. It is evident from Fig. 9a and b that stress concentrations at the joints were reduced by using the joint expansion algorithm, though not entirely eliminated.

The Bloodhound project aims to build a supersonic car that will break the land speed record, and also reach the landmark top speed of 1000 mph (1600 km/h). The predominant braking system is in the form of two deployable air-brakes positioned on each side of the car. Each air-brake is attached to the car with four hinges, as shown in Fig. 10a. The load exerted on each hinge is related to the aerodynamic drag experienced by the air-brake during deployment. Naturally the Bloodhound team wish to keep the structural weight of the car to a minimum and are thus investigating the use of optimization to reduce the mass of this and other components. This provides an opportunity to apply layout optimization to a practical engineering problem.

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 strength-to-weight 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 ‘fixed-pinned’ and ‘pinned-pinned’ end support conditions in the buckling calculations generally met or slightly exceeded the design load. Those designed using the ‘fixed-fixed’ assumption (samples A1-3) 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 ‘fixed-pinned’ end support condition should be used in future when considering buckling. However, all three of the benchmark specimens (i.e. samples O1-3) which were analysed using the ‘fixed-pinned’ 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 cross-sections to keep the solid model generation and fabrication stages as straightforward as possible. However, in future the use of more structurally efficient cross-sections 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 post-processing to produce a feasible design, with smooth and well defined surfaces).

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.