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Integrated Shape and Topology Optimization - Applications in Automotive Design and Manufacturing.


In structural mechanics, the requirements for structural elements and systems are highly complex. This is especially true for the very competitive automotive industry, where engineers are expected to shorten development and test time, while also cutting material and production costs. Moreover, in the quest for reducing fuel consumption and C[O.sub.2] emissions, light-weighting of structures is essential for more energy efficient transportation systems.

When assessing body structures of today's automobiles, the market is predominantly made up of thin walled pressed and welded steel structures. This solution has several advantages. Steel is an inexpensive and very versatile material, the forming processes and joining technologies (mostly spot welding) are well understood, and the employment of high strength steels has facilitated a significant improvement in crash safety performance while limiting the mass penalty.

Regardless of the strength of this approach, the use of materials such as aluminum, magnesium and fiber reinforced composites promises even further reduction in weight while also offering new design possibilities. Examples include highly complex shapes of castings, or fiber reinforced composite materials that are tailored to meet specific loading conditions. Furthermore, the emergence of large scale metal additive manufacturing in the aerospace sector may see this process transfer to the automotive sector, with an associated increase in the geometric complexity of future automotive bulk part designs.

Nevertheless, before such materials can be used in large volume production, there are a variety of issues to overcome, such as higher material cost, longer production cycle time (e.g. compared to sheet metal stamping), and the development of appropriate joining technologies.


In structural design, simulation and corresponding optimization methods have become indispensable. An important tool for the design of load bearing parts has become the topology optimization method [1,2]. Homogenization based approaches have been widely adopted. Here, the engineer starts with a feasible design space and applies loads and constraints. The topology optimization algorithm returns a structural layout describing main load paths in the form of a density distribution on the design domain [3,4,5,6].

In more detail, the designer starts with an initially solid volume describing the permissible design region - the "package space" -divided into a discrete mesh of finite elements. After applying loads and constraints the optimization algorithm then determines in an iterative process whether a region in the design space should contain material or not. In practice, most algorithms also assign intermediate densities, such as 'a design region is present to a level of 50%'. This results in a density distribution [rho]: [OMEGA] [right arrow] [0,1] on the package space [OMEGA]. Thus, the topology optimization algorithm can be viewed as a material distribution method.

There are two main approaches to define the objective function for the topology optimization. The first aims to find a density distribution [rho]: [OMEGA] [right arrow] [0,1] on the design space [OMEGA] that minimizes a certain functional such as the global compliance: Find [rho] such that

f=[1/2][u.sup.T]Ku is minimal, while the amount of used material equals a fixed percentage portion of the original material ('mass fraction approach').

Another approach is to find a density distribution with minimum overall mass, such that a set of constraint (e.g the displacement at some specific points) is minimized: Find

[mathematical expression not reproducible]

Once the algorithm has derived a suitable density [rho], the next step is to either fine-tune the resulting shape or else translate it into multiple manufacturable parts. Topology methods are predominantly useful for single parts, for example, cast aluminum suspension components. For complex assemblies, the topology results require considerable interpretation and rework by the designer. In this context, shape optimization methods are applied only after the topology optimization to reduce local stresses and to incorporate detailed manufacturability requirements [7,8].

Although topology optimization methods can generate load path proposals for the design of complex assemblies such as automobile body structures, they cannot produce the detail required to properly describe cross-section, or the intricate structure of a multi-layered panel with specific fiber orientations. Moreover, dynamic crash load cases have to be approximated by linear static equivalent loads. So, for a detailed CAE simulation, a more realistic finite element (FE) model representing the real geometry is needed. To achieve this level of detail, typically a CAD geometry is created, which is then translated into a FE simulation model, and CAE analysis with respect to stiffness, strength, crash performance and other aspects can be carried out. A specialized approach in this concept finding stage is offered by the parametric geometry tool SFE CONCEPT, which is designed to create finite element models of complex thin walled sheet metal structures. It is especially useful in the early design stage, where detailed geometry may not yet be available, or where the designer still has the freedom to modify the structural layout [9,10]. SFE CONCEPT provides an implicit parametric geometry description, where intuitive shape design variables can be defined and result in consistent complex geometry [11,12]. After a geometry update in batch-mode, the integrated shell mesher generates a consistent finite element model, which is particularly well suited for 2D elements. Today, SFE CONCEPT is mainly used in the automobile industry for early concept development of body structures. Crash related load cases are of particular importance [13,14]. One of the advantages of SFE CONCEPT is the ability to quickly assess large scale parametric shape modifications of a complete body structure including connections such as spot welds and adhesives [15,16]. Despite its many strengths, SFE CONCEPT's ability to generate load path information is limited to exchanging predefined structural members [9].

In this paper, we present a number of classes of structural optimization problems. The use of the Integrated Shape and Topology method to combine parametric geometry and topology methods is explored. Of particular interest is the application for complex automobile body structure, where the combination of parametric shape variation and the topology optimization for cast joints is examined.


The IST approach enables the integration of Solid Isotropic Material with Penalty (SIMP) topology methods with large scale parametric shape modification. Here we do not consider shape design performed as an additional step after topology optimization, which is a field of research in its own right [17,18]. Rather, we show that parametric shape modification before a topology algorithm adds a new dimension to the topology method. While a traditional topology optimization process starts with a fixed input design space, IST includes a parametrically variable structure as well as an adaptable design space for the SIMP algorithm (see Figure 1).

An introduction of the implementation of the IST process is given in [19]. We provide a short summary here. Applying IST, the engineer generally starts with the manual creation of a parametric geometry model, including a set of shape design variables x =([x.sub.1],..., [x.sub.n]) [epsilon] [R.sup.n]. The geometry consists of a part [[OMEGA].sub.x] that will be subjected to topology optimization, while [[LAMBDA].sub.x] is not. Importantly, the geometry [[OMEGA].sub.x] [union] [[LAMBDA].sub.x] is not fixed. Rather, for each set of design variables x, a different geometry model

X [right arrow] [[OMEGA].sub.x], [[LAMBDA].sub.x]

will be created.

Next either a Design of Experiments (DOE) or an optimization loop is conducted. The examples use Altair Hyperstudy and Dakota [20] to perform the optimization study. A single iteration in this loop consists of a geometry model update, the assembly of the analysis model, running of a SIMP algorithm, and finally the extraction of analysis results. Additional plug-in functions can be defined and integrated. IST provides a flexible scripting language to describe the process flow. An important step is the automated assembly of the finite element analysis model that needs to happen for every iteration. To facilitate this process, IST reads finite element data, creates connections between parts, applies loads and boundary conditions, defines specific grids for measurement of displacements, as well as offering many more useful features. This results in a finite element analysis model [[OMEGA].sub.x] [union] [[LAMBDA].sub.x] [union] [[GAMMA].sub.x] where [[GAMMA].sub.x] stands for the connections and boundary conditions. The next step is to initiate a SIMP topology optimization, for which Altair Optistruct is used [21]. Considering constraints [g.sub.j](x), a density distribution [[delta].sub.x] on [[OMEGA].sub.x] is determined together with some performance measure, which in many cases will be the overall mass M:

[mathematical expression not reproducible]

Depending on the objective function values determined by the finite element solvers, an outer optimization loop designates new values for the shape design variables x. This closes the geometry update loop.

The presented method allows two major new approaches: First, the initial design space that goes into the topology optimization algorithm can be subject to large scale parametric shape variation. Second, the topology design volume may be part of a complex system, which itself is subject to parametric optimization. This approach adds a geometric dimension to the design space for topology optimization and to the system it is embedded in, and largely increases the size of the optimization space open for exploration not possible in existing tools, since they do not offer a parametric description of geometry and package space.


In the following sections, three examples are presented, where the application of IST combines parametric shape modification with subsequent topology optimization.

The specific examples have been selected to demonstrate the relevance of this approach to the automotive industry. They highlight the versatility of IST in applications ranging from single load bearing parts or multi-piece assemblies to highly complex systems such as entire automobile body structures. For solid components (as opposed to thin walled stamped parts), manufacturing considerations are included to ensure that components can be either die cast or extruded.

In the first example, we look at a cast aluminum bracket mounted to an engine block, used to attach an air conditioner pump. IST tools are applied to simultaneously find load path layouts and optimal locations for the attachment bolts. With 11 geometry design parameters, genetic optimization proves to be effective, and the potential mass saving by combining the optimization for attachment point positions and topology is shown to be high. In a more general sense, IST allows us to combine standard topology optimization methods with the optimization of boundary conditions.

In the second example, a two-piece clamp, made from extruded aluminum, is optimized. The geometry of the components is optimized, and topology methods are used to minimize mass. In this type of application, where manufacturing requires extruded parts, IST is able to satisfy the extrusion constraint and optimize the shape of the components.

The third example showcases the parametric shape variation for an automobile body structure, combined with the topology optimization of cast connection joints. This class of application integrates complex parametric shape variation with the topology design of cast joints, and may become particularly useful for new approaches in the design and manufacturing methods of automobile body structures, where relatively simple extruded beam members are combined with complex shapes of cast joints. Here IST integrates complex thin walled sheet metal structure, extruded beam members and complex cast joints. It can determine a trade-off between necessary cross-section dimensions and size/topology of cast structures.

Application 1: Integrated topology optimization of a solid bracket while optimizing the positioning of attachment bolts

Problem Setup

In this example, we demonstrate the application of IST when determining a topology optimized solid bracket design in conjunction with the optimization of the positions of attachment bolts. Figure 2 shows the design space for a bracket (in light blue), attached to an engine block (dark blue) with four bolts (red). An air conditioner pump (yellow) is bolted to the bracket with three bolts (green).

It is standard practice to employ topology optimization to determine a concept load path design for the bracket. The design target in this case is to minimize the mass. As a critical constraint we require that the first Eigenfrequency of the system is not lower than 950 Hertz, so as to avoid possible resonant frequencies at low engine revolutions. A draw constraint is applied, to make sure that the part can be manufactured in a die cast process using a two piece die.

The exact positions of the bolts can make a significant difference to the static and dynamic performance of the design. In this study the engineer has the design freedom to modify the attachment location of the bracket, and we allow the location of the connection bolts to vary. Figure 3 shows the schematic setup: Bolts 1, 2, 5 and 6 can move in x- and in z-direction. Due to package constraints, the lower bolts 3, 4 and 7 can only move in x-direction.

Using the IST tool set, an analysis loop has been set up that allows us to assess the performance of different design configurations: First, a finite element mesh of the individual components is created manually as a one-off step. In an automated loop, the position of the seven bolts are then varied, connections between the bolts and the structure are created, and a topology optimization on the bracket is performed. Finally, the resulting mass is fed back to the optimizer, and a set of new design variable values is chosen.

It is well understood that any density distribution returned by the topology optimization algorithm requires manual rework in order to reduce local stress concentrations and to create a final design for a manufacturable part. Nevertheless, we can show that the mass obtained for different bolt locations is indicative of designs with high potential for a lowest mass solution. To demonstrate this, a full factorial DOE has been conducted for one of the bolts (bolt 2 in Figure 3), varying within 20mm in x-direction, and 40mm in z-direction. The graph in Figure 4 shows the resulting bracket mass, which varies significantly for different positions of this one bolt; for the variation in vertical direction the masses after topology optimization lie between approximately 350 and 550 grams. Figure 5 shows selected topology optimization results obtained when moving bolt 2 (shown in red) in z-direction. In the next section, we show that this result can be improved by allowing all 7 bolts to move.

Genetic Optimization with Eleven Design Parameters

In a more comprehensive optimization study, 11 geometric design parameters are introduced. They describe the location of the seven bolts as shown in Figure 3.

The response surface shown in Figure 4 (only one bolt moves) suggests the existence of a unique global minimum for that simplified case. However, for the optimization task with 11 design parameters, the response surface cannot be expected to be convex. Due to alternating characteristic load path solutions, the objective may even be discontinuous, so a gradient based algorithm is not suited here. This is why a genetic algorithm has been employed.

Also, it should be noted that the objective function values generated by the topology optimization algorithm are fuzzy to some degree. This means that a small change in the resulting mass does not necessarily translate into a different mass for the final design, since the density result returned by the SIMP algorithm requires interpretation before being translated into a manufacturable design, and the mass figures can only be indicative. This fuzziness is the reason why the design parameters are allowed only discrete values, in steps of 5 mm. The bolts can vary by 15 to 35 mm in x- and z-direction, within the limits shown in Table 1. Another advantage of discrete design parameter values is an increase in convergence speed, since previously calculated results can more readily be reused than for continuous design variables. The parameter limits are due to geometrical restrictions, and the values referred to as "0'' correspond to the initial design (see also Figure 8 for a schematic view of the parameter space).

As in the DOE analysis discussed in previous section, the objective of the optimization is to minimize the system mass. Constraints are the first Eigenfrequency, which is required to be above 950 Hz, and a draw constraint to assert manufacturability. Two static load cases represent forces of the driving belt ([F.sub.y] and [F.sub.z] in Figure 3), with displacement constraints in place for both loads.

Outer optimization terminates when no significantly lighter individuals are found. More precisely, the convergence criterion is to terminate when the best (=lightest) 20 individuals are within a range of only 2% difference in mass.


The actual genetic algorithm was provided by Dakota. With a population size of 30 individuals per generation and a survival rate of 10% percent, the Dakota optimization evaluated 290 individual designs within 200 hours on an Intel i7 1.6 GHz processor before converging due to small (less than 2%) changes in the minimum mass of the lightest 20 individuals.

After 100 iterations, the objective function declines - on average -steadily (Figure 6). Optimal location for the bolts can be seen in Figure 7 (design variable history for some of the parameters), and in Figure 8, where the bolt locations for the five lightest designs are shown.

The design with lowest mass show very similar load path configurations, all weighing around 380 grams (Figure 9). It can be concluded that the integrated geometry variation has significantly narrowed the design space for feasible low mass solutions. Even though the mass figures will change once an actual manufacturing design has been created, the potential to make a meaningful reduction in mass by simultaneously varying geometry parameters in an automated fashion is promising.

Application 2: Combined shape- and topology optimization for extruded two-piece assembly

In this example, we show the integrated optimization of both geometry and topology of an extruded assembly. Figure 10 shows the package space for a clamp that is used to attach a steel canopy to the tray of a pick-up truck. It consists of two main parts that are held together by a bolt (diameter = 8 mm). When tightened, the bolt generates the necessary clamping force.

To keep manufacturing cost low, the clamp is to be manufactured from extruded aluminum profiles, which are simply cut to length. The only necessary additional process steps are to drill the bolt holes and to deburr and round off sharp edges.

Optimization Objective

The objective here is to find optimal cross-sections and width for each of the two parts of the clamp, integrating parametric shape modification and topology optimization. To that end, three geometric design parameters have been introduced as shown in Figure 11 and Table 3. The first parameter [x.sub.1] describes the width of the parts as cut from the extruded profiles. Parameter [x.sub.2] varies the y-position of the split line between the two parts, while parameter [x.sub.3] modifies the location of the gap in z-direction.

The process works as follows: As an initial manual step, an SFE CONCEPT model representing the geometry has been created. Within this parametric model, the geometry design variables have been defined. Now new values for the variables can simply be input into SFE in batch mode, where the geometry is updated accordingly and a finite element mesh is created. The design variables x = ([x.sub.1], [x.sub.2], [x.sub.3]) describe the variation in millimeters with respect to some (arbitrary) initial configuration, referred to as x = (0,0,0).

Now IST controls the automated loop that starts with a file containing values for the three design parameters. This file is converted into a SFE batch file. Then SFE is run in batch mode, reading the design variables and updating the geometry accordingly. A SFE macro then creates and exports a finite element hexa-dominant volume mesh. At a mesh size of 2 mm, the analysis model consists of around 20000 elements.

Next, the IST tools create the necessary connections and boundary conditions: this is shown schematically in Figure 10, the lower side of the two clamping faces is constrained with single point constraints in vertical direction (SPC). For the facing side, a force F = 1000N is applied, representing the clamping force when the bolt is tightened. The bolt itself is modelled as a pivot point P using a Rigid Body Element (RBE2) allowing rotation around the x-axis only, the bending stiffness of the bolt is represented by a rotational spring. Finally, rigid body elements are created between the two sliding surfaces (RBE2's that allow the relative movement of the two parts in z-direction while restricting the movement in x- and y-direction). This completes the build of the analysis model.

As a next step, a topology optimization is performed. The objective hereby is to minimize the mass. The displacement of the load point F is limited to a maximum of 0.2 mm. At the same time, an extrusion constraint ensures that material is taken away homogeneously in the x-direction.

Full Factorial DOE

The computation time for one topology optimization step including model update, mesh generation, and compilation of the analysis model is about 10 minutes on an Intel i7 1.6 GHz processor (Table 2). This is relatively inexpensive and allows a comprehensive scan of the geometry parameter design space. A full factorial DOE for the three design parameters with 96 different shapes has been conducted (Table 3 shows the specific values).

The topology optimization performed on the different geometries leads to a distinct mass figure for each design. Figure 12 shows cuts through the three-dimensional design space for fixed coordinates [x.sub.1], [x.sub.2], [x.sub.3], respectively. The response surfaces show a convex characteristic, indicating that the problem could be well suited to a gradient based algorithm. We have tried this, and the optimization converged after 13 objective function evaluations, close to the optimum found by the full factorial DOE.

Looking at the DOE result, [x.sub.1] and [x.sub.2] have a big impact on the objective function, while the mass is relatively constant for changes in [x.sub.3]. Lowest mass solutions are achieved for small [x.sub.1], and [x.sub.2] around zero. A number of selected designs are shown in Figure 13.

Compared to the initial arrangement (x = (0,0,0, )), the mass has been reduced by about 14% to 218 grams. It is worthwhile to restate that the optimization result achieved by IST does not represent a final design and that the minimum mass figures will change, once a production design has been developed. Of course, this is the case whenever a topology optimization algorithm is utilized, and the IST optimization provides the engineer with a good starting point for a final design. Another aspect in this example is how IST is able to capture the extrusion constraint for the topology optimization while varying the length of the extruded part.

Application 3: Parametric shape variation of an automobile body structure combined with topology optimization of cast aluminum connection joints

With demands to reduce vehicle weight, new body structure design architectures are being investigated. A common theme is the application of light weight alloy extrusions combined with cast joints to form the vehicles space frame. Unlike the common steel monocoque, the space-frame architecture separates the structural requirements of the vehicle body from its styling form. This segregation in functional requirements can lead to the implementation of simpler structural geometry and the introduction of low formability materials such as Aluminum, Magnesium and Ultra High Strength Steels (UHSS). Low volume vehicle manufacturers such as Aston Martin and Lotus have employed these techniques and achieved significant weight savings, and recent trends appear to show more main-stream vehicle manufacturers adopting these strategies.

A recent research project involving some of the authors adopted this design philosophy to propose a high-level conceptual design for a lightweight modular vehicle platform. The body structure is based on a modular design employing simple constant cross-section structural members. A mix of materials and joining technologies has been explored. The architecture is representative of a space-frame, as shown in Figure 14.

Shape Modification Combined with Topology Optimization

One of the research objectives was to investigate design strategies of cast joints that serve as complexly shaped components to connect extruded beam members. As an example, in the present paper we focus on the two joints that connect the front rails with the first lower cross member. In this design task, typically topology optimization methods are applied for the castings.

In addition, we parametrically modify the geometry of the beams that connect into these joints. For any given geometry, a topology optimization on the cast connection joint is performed. Here the shape of the design volume that goes into the topology optimization

algorithm changes with the geometry, and the aim is to find a solution that combines both the shape of the beams and the joint topologies with highest stiffness.

To this end, a parametric SFE CONCEPT geometry model of the body structure is created. The geometry contains all relevant structural members; connections between parts include spot welds, laser welds, and adhesives (Figure 14). Two geometry design parameters have been defined within the concept model: The first parameter ([x.sub.1]) varies the width of the front rails on the inboard side, the second parameter ([x.sub.2]) the width of the first cross member on the rear facing side (Figure 15).

Analysis Loop using IST Tools

In order to realize an automated closed analysis loop, a number of steps have to be performed in batch mode. First, the geometry model is updated by SFE CONCEPT reading a text file that contains values for the design parameters. Then a finite element mesh is generated and exported.

Second, this FE mesh is pre-processed before it can be passed to the solver. One important step is to create the connection elements between the solid mesh and the shell mesh. By specifying a single command in the IST process setup script, the parts which are to be connected, as well as the type of connections, are defined. In the example at hand, Rigid Body Elements (RBE2's) with a single dependent node between the shell elements and the solids are used.

Third, loads and boundary conditions have to be applied suitable to changing geometry.

Fourth, a general problem when applying a combined shape/topology approach as described in this paper needs to be addressed: the resulting structures for varying initial geometries may have different masses as well as different performance results, and this creates a ranking problem of how to determine which design is to be called better. A number of approaches have been discussed in [19]. For the current example we use the following solution: for every specific geometry model, we measure the mass before a topology optimization is performed. Then the volume fraction value for the SIMP algorithm is adjusted such that the overall resulting mass is always identical. A simple user instruction realizes this step. The

optimization objective is then to minimize the weighted compliance for all static load cases. Essentially, we set out to find the stiffest solution for a given amount of mass.

Finally, the analysis results are extracted from the solver output files.

All these steps are defined in a simple command file. IST interprets this file and performs the necessary steps.

Parameter Study for Two Geometry Design Parameters

Figure 14 shows the three load cases used for this study. Besides bending (1) and torsional loads (2), longitudinal forces into the front rails are applied, simulating a static equivalent load for a full frontal impact (3). Since the torsional load case is unsymmetrical, no symmetry conditions could be applied. Therefore, both solid joints undergo topology optimization.

With respect to the initial position of ([x.sub.1], [x.sub.2]) = (0,0) the shape parameters vary for the rail width ([x.sub.1]) between -30 and 30 millimeters, and for the lower cross member ([x.sub.2]) between -20 and 40 millimeters. In 58 hours, a parameter study has been conducted using the same Intel i7 1.6 GHz processor as for the other examples, scanning the design space at 35 points, which results in the interpolated response surface shown in Figure 16.

It can be seen that there is a trade-off between the geometry of the beam members and the resulting mass of the topology optimized joints. The response surface suggests that a rail width of 95 mm and a cross member width of 85 mm is the best starting point for a refined design (corresponding to [x.sub.1] [approximately equal to] -5, [x.sub.2] [approximately equal to] -15). The resulting topologies are consistent in their geometrical characteristics, i.e. different geometries result in similar loadpaths. Figure 17 shows some topology optimization results for the left hand side joint (the results for the right hand side joint are symmetrical).


This study does not claim to be representative of a production design. Nevertheless, it highlights the potential of IST to simultaneously vary complex thin walled structures and cast connection joints. As an outlook, it may be beneficial to incorporate more load cases and to increase the number of geometry design parameters. For instance, the length of the extruded beams, and the cut angles could be added for both beams. Also, it will be interesting to include simple sizing parameters varying the gauges of the beam members.

More examples to explore in this context could be suspension components that combine castings with constant cross section beam members, or an instrument panel beam consisting of multiple extruded parts.


The IST approach is computationally costly, since for each set of geometry design variables, a topology optimization run is performed. In one such iteration step, typically the geometry update and meshing is fast, while the topology optimization part requires 90% of the computing time. For example, the last case study shown comprises roughly 140,000 elements, 30,000 of which are volume elements belonging to the topology design region. Here, one topology optimization run takes 100 minutes on an Intel i7 1.6 GHz machine, adding up to 60 hours of computing time for the 35 evaluations performed. On the other hand, structural problems with several design variables can be tackled using genetic algorithms (as shown in the first example). Thus, IST lends itself to the application of bionic algorithms which have the inherent advantage that they can easily be parallelized, so computing time for real life IST problems should be manageable.


Several applications for an integrated shape and topology optimization method (IST) have been presented. The authors used parametric shape variation in conjunction with topology optimization to explore new approaches to the structural concept layout. The examples discussed are applicable to the automobile industry, and range from single parts up to a highly complex automobile body structure. They demonstrate the versatility of the approach, and in particular, the following benefits of IST have been demonstrated:

* IST optimizes boundary conditions in addition to a load path layout design. This can lead to concept design proposals with the potential for significant mass saving.

* IST can vary the initial shapes of individual members of multi-piece assemblies. Simultaneously, it allows the topology of the parts to be optimized for lowest mass or highest stiffness. By parametrically modifying the design space, IST finds lighter solutions than topology methods alone do.

* The IST tool set is capable of assessing complex cast connections and their implications for beam structures made from extruded profiles.

* The integration of the parametric geometry software SFE CONCEPT with topology methods enables the shape optimization of highly complex automobile body structures together with the design of cast load bearing joints. This may prove to be of particular importance when new manufacturing approaches in automotive design become widespread.


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Klaus Fiedler

Klaus Fiedler


Bernard F. Rolfe

Deakin University

Timothy De Souza

Quickstep Automotive

Table 1. Design Variables for optimization using a genetic algorithm.
The parameters are allowed discrete values in steps of 5 mm.

Bolt 1   x-direction   [x.sub.1]     -5     20
         z-direction   [x.sub.2]    -20      0
Bolt 2   x-direction   [x.sub.3]    -15      5
         z-direction   [x.sub.4]    -10     30
Bolt 3   x-direction   [x.sub.5]    -10     10
Bolt 4   x-direction   [x.sub.6]    -15      5
Bolt 5   x-direction   [x.sub.7]     -5     15
         z-direction   [x.sub.8]      0     20
Bolt 6   x-direction   [x.sub.9]    -10      5
         z-direction   [x.sub.10]   -20      5
Bolt 7   x-direction   [x.sub.11]   -30     30

Table 2. Approximate computing time for one geometry/topology
optimization step.

Action                                         Computing time

SFE CONCEPT model update                       0.5
Finite element mesh generation                 0.5
Generate boundary conditions and constraints   0.5
Topology optimization                          9.0

Table 3. Geometry design variables for clamp optimization (in
millimetres). A value of 0 refers to the initial design.

Description                DV          Min   Max   Values

Width of clamp             [x.sub.1]   -6    18    -6,2,10,18
y-location of split line   [x.sub.2]   -8     4    -8,-4,0,4
z-location of split line   [x.sub.3]   -10   30    -10,-4,2,8,14,20
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Author:Fiedler, Klaus; Rolfe, Bernard F.; De Souza, Timothy
Publication:SAE International Journal of Materials and Manufacturing
Article Type:Technical report
Date:Jul 1, 2017
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