Application of manufacturing constraints to structural ...

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Application of manufacturing constraints to structural optimization of thin-walled structures T. Kuczek

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Institute of Rail Vehicles, Kraków University of Technology, Kraków, Poland Published online: 13 Mar 2015.

Click for updates To cite this article: T. Kuczek (2015): Application of manufacturing constraints to structural optimization of thin-walled structures, Engineering Optimization, DOI: 10.1080/0305215X.2015.1017350 To link to this article: http://dx.doi.org/10.1080/0305215X.2015.1017350

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Engineering Optimization, 2015 http://dx.doi.org/10.1080/0305215X.2015.1017350

Application of manufacturing constraints to structural optimization of thin-walled structures T. Kuczek ∗ Institute of Rail Vehicles, Kraków University of Technology, Kraków, Poland (Received 18 September 2014; accepted 31 January 2015) Topology optimization can be a very useful tool for creating conceptual designs for vehicles. Structures suggested by topology optimization often turn out to be difficult to implement in manufacturing processes. Presently, rail vehicle structures are made by welding sheet metal parts. This leads to many complications and increased weight of the vehicle. This article presents a new design concept for modern rail vehicle structures made of standardized, thin-walled, closed, steel profiles that fulfil the stress and manufacturing requirements. For this purpose, standard software for topology optimization was used with a new way of preprocessing the design space. The design methodology is illustrated by an example of the topology optimization of a freight railcar. It is shown that the methodology turns out to be a useful tool for obtaining optimal structure design that fulfils the assumed manufacturing constraints. Keywords: topology optimization; thin-walled structures; manufacturing constraints; rail vehicles

1.

Introduction

In the last 20 years, topology optimization has been characterized by dynamic development, both in research and broad industrial applications. This development has led to a steady increase in the applications of optimization technology in the past decade (Zhou et al., 2004). This has been especially evident since Bendsoe and Kikuchi’s studies caused a significant interest in this topic (Bendsoe and Kikuchi, 1988). The power of this technology lies in the initial impact in the design process. Currently, topology optimization has developed significantly and with the development of topology theory and an increased number of software tools it has become a strong method in the design of new products. As shown in the literature, topology optimization can generate efficient designs in engineering fields where products are already considered highly engineered. Commercial software plays a significant role in transforming an emerging technology from research to industry (Zhou et al., 2011). However, a range of problems, including porosity, mesh dependency, and local minimum results, often lead to a non-manufacturable topology that cannot be accepted later in the production process (Kong-Tian et al., 2006). It has been shown that topology optimization can help to create highly efficient design concepts. This very often leads to much more significant design improvement compared to sizing and shape optimization that can be only applied to a structure with a given layout (Zhou et al., 2002). At *Email: [email protected] © 2015 Taylor & Francis


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the initial stage of the design, fulfilment of the manufacturing requirements is one of the fundamental demands. So, the manufacturing constraints are crucial for the realization of the design concept. Typical manufacturing constraints described in the literature include casting, extrusion and stamping (Zhou et al., 2011). In the case of these constraints, the design symmetries and design patterns are included (Zhou, Shyy, and Thomas, 2001). Minimum member size control was the first manufacturing constraint introduced by the Altair Optistruct software released in 1999 (Zhou et al., 2011). A three-phase process was developed in which a density slope constraint corresponding to a given minimum member dimension is enforced during the first two phases, while during the third phase the density slope on the boundary is relaxed to eliminate the transition layer (Zhou, Shyy, and Thomas, 2001). In casting manufacturing constraints, the mesh is perfectly lined up in the draw direction to prevent the formation of cavities. The stamping manufacturing constraint attempts to enforce constant thickness in the design. With the sliding direction representing the die punch direction, the requirement for stamping feasibility is to keep the thickness constant in the stamp direction (Zhou et al., 2011). Presently, rail vehicle structures are made by welding sheet metal parts. This production method leads to many complications and increased weight of the vehicle. A large vehicle mass reduces the weight of transported cargo due to restrictions on the axle loading. This article presents a new design concept for modern rail vehicle structures made of standardized thinwalled, closed, steel profiles that fulfil the stress and manufacturing requirements. The light weight of vehicles is also one of the important factors for saving energy and improving performance. The resulting design should be made of standardized, commercially available, steel construction profiles (Kuczek, 2014). For this purpose, Altair Optistruct software for topology optimization will be used. However, for preprocessing the topology optimization design space, a new manufacturing constraints procedure will be used. The preprocessing procedure consists of preparating an equivalent design space (EDS). The EDS is discretized by using a specific size of finite elements and equivalent material properties. This article also presents the first use of EDS in a commercial software program. The new design methodology will be illustrated by an example of railcar structure optimization. The design obtained should fulfil all the requirements of current rail standards for freight cars (CEN, 2002). Topology optimization has so far been rarely used in the design of large vehicles. However, for fairly simple vehicle structures some successful tests have been carried out by Mrzyglod (2010) and Fredricson (2005).

2.

Manufacturing constraints procedure

Topology optimization of thin-walled structures leads to solutions that can cause problems in practical implementation. When the design space in topology optimization is discretized using finite elements of small dimensions, the optimization algorithm usually forms the final solution – thin rod-like structures with low usability in the production process (Thomas, Zhou, and Schramm, 2002). Therefore, a new design space preprocessing methodology has been presented. The manufacturing constraints procedure in topology optimization consists of discretization of the design space by solid-type finite elements of a hexahedron shape and equal dimensions. Furthermore, the equivalent stiffness of the design space is assigned to the finite elements. In Figure 1 the procedure of preprocessing of the design space is presented. The element stiffness of a solid cube is to be reduced to the stiffness of a thin-walled cube of similar dimensions. This must be done because the manufacturing requirements assume making railcar design with thin-walled profiles. The thin-walled structure has the same dimensions as the dimensions of finite elements of the design space. This allows to impose the manufacturing constraints on the final solution of topology optimization. The stiffness of a thin-walled cube was found (e.g.


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Figure 1. Example of the numerical procedure for design space preparation: (a) and (b) the first finite element (FE) model and (c) the value of the linear deformation for a thin-walled cube; (d) and (e) the second FE model; and the results of linear deformation tuning before (f) and after (g) modification of the stiffness.

dimensions: 180 × 180 × 180 × 5 mm) by application 100 N force (Figures 1(a) and 1(b)) and thereby obtaining the value of linear deformation (Figure 1(c)). For the thin-walled cube, 100 N force was applied to four edges of the cube (shown in Figure 1(b) as a single force vector). The Young’s modulus of the thin-walled cube was E0 = 2.1E + 011 N/m2 and the maximum linear deformation of the cube was d0 = 2.48E − 005 mm. The dimensions of the cube were selected on the basis of the cross-sectional dimensions of a standard profile to be used for the railcar design. For finding the equivalent stiffness of the EDS design space, the second FE model was prepared. The stiffness of a solid cube was found in the same way by application of 100 N force (Figure 1(d) and 1(e)), thereby obtaining the value of the linear deformation (Figure 1(f)). For the solid cube, 100 N force was applied to an entire side surface (shown in Figure 1(e) as four force vectors). In the illustration, the first case has four force vectors and the second case has one force vector, but this is only a graphical way to display the loads on the FE model system and their quantity is irrelevant. The Young’s modulus for the solid cube was E1 = E0 and maximum linear deformation of cube was d1 = 2.65E − 006 mm. The stiffness of this model was tuned to reach the value of linear deformation equal to the first model (Mrzyglod and Kuczek, 2014). After the Young’s modulus was reduced to a value of E2 = 2.3E + 010 N/m2 the maximum linear deformation of the cube increased to a value of d2 = 2.42E − 005 mm. The results of stiffness tuning are presented in Figures 1(f) and 1(g). The stiffness of the solid model after stiffness tuning will be

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(a)

(b)

10 kN

10 kN


Figure 2. Comparison of two steel beams (in cross section): (a) made of identical cubes with internal walls; (b) without internal walls.

Table 1. Comparison of von Mises stress and maximum displacement values for two steel beams.

von Mises stress Maximum displacement

Beam (a)

Beam (b)

Difference

1.19e7 N/m2 0.136 mm

1.16e7 N/m2 0.137 mm

2.53% 0.73%

Figure 3. An example of a simple cantilever beam structure problem: (a) simple cantilever beam; (b) and (d) the result of topological optimization without using design space constraints; (c) the result of topology optimization with using design space constraints.

used for topology optimization in the example. For the manufacturing constraints procedure presented, the FE model should be orthotropic. However, for simplification, the above methodology has been adopted because the differences between the orthotropic and isotropic model are minor, as illustrated by the example in Figure 2 and Table 1. In Figure 2(a), a beam made of identical cubes with internal walls is presented, and in Figure 2(b) a thin-walled beam is presented. The von Mises stress and maximum displacement differences in these two cases are less than 3%. In such circumstances, the author assumed that the isotropic model will be sufficient. The design methodology presented in the article assumes that the designed vehicle will be made of closed, thin-walled steel profiles. Therefore, due to manufacturing constraints it should be expected that a structure similar in shape to a set of connected, closed, thin-walled steel profiles of similar strength will result. In order to present the concept of design space manufacturing constraints, an example of a simple cantilever beam structure subjected to two asynchronous 100 N loads was used (see Figure 3(a)). This example is a modified version of the example shown in the articles authored by Rozvany (1998) and Mrzyglod (2012). The objective function in this case is the minimization of the mass of the structure constrained by the manufacturing



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and von Mises stress limits . The simple cantilever beam model was solved twice for two finite element mesh sizes. The FE mesh size for the model shown in Figure 3(c) was 10 mm and for the model presented in Figures 3(b) and 3(d) was 2.5 mm. The result of the topological optimization of the design space is presented in Figure 3. It is clear that the optimum structure in Figures 3(b) and 3(d) has a layout unsuitable for use in the case of a structure made of thin-walled profiles. The layout is not appropriate because the resulting structure obtained from the optimization should meet von Mises stress constraints and its shape should be similar to a set of closed, thin-walled steel profiles. The results presented in Figures 3(b) and 3(d) are similar to shell structures due to the small size of the mesh. The results presented in Figure 3(c) are similar to steel structures made of thin-walled profiles. Therefore, for topology optimization of the vehicle, a finite element mesh with dimensions similar to the dimensions of the steel profiles should be taken, which will be used for construction of the vehicle.

3.

An example of freight railcar structure optimization

As an example of the optimization methodology, a freight railcar structure was selected (Figure 4). It is a freight railcar designed in the sixties of the last century to transport loose bulk commodities such as coal, ore or grain. The main parameters of the railcar are: payload 57,000 kg, length 10.6 m, width 2.6 m, height 3.3 m, capacity 25 m3 . The optimization problem was to find the minimum mass of the vehicle structure subject to static von Mises stress limits and manufacturing constraints. The general optimization problem can be formulated as follows: min f (ρ) m

subject to g1 (ρ) ≤ gu MPa, g2 (ρ) = sa mm, g3 (ρ) = sb mm, 0 ≤ ρi ≤ 1,

Figure 4.

Freight railcar structure (409V-type).

i = 1, . . . , n,


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Figure 5.

T. Kuczek

Design space of the railcar structure for topology optimization.

where f (ρ) represents the objective function, g1 (ρ) represent the von Mises stress constraint response, gu is the g1 (ρ) upper bound, g2 (ρ) and g3 (ρ) represent the cross-sectional dimensions of the thin-walled steel profiles that are going to be used in the vehicle structure, ρi is the normalized material density of the ith element, and the total number of design variables is n. It was assumed that the structure of the vehicle would be made of steel with a yield strength of gu = 350 MPa and cross-sectional dimensions sa = sb = 180 mm. The assumed von Mises strength limit is in accordance with the CEN standards for rail vehicles (CEN, 2002). For the calculations, a minimum value of the yield strength of structural steel should be taken into account. In the FE model analysis, homogeneous isotropic linear-elastic material and solid type finite elements of hexahedron shape were used. For the implementation of the example (Figure 3) as well as the topology optimization of the railcar structure (Figure 4), Altair OptiStruct software was used (Altair, 2012). OptiStruct solves topological optimization problems using the solid isotropic material with penalization (SIMP) method. In the SIMP method, a pseudo material density is the design variable. The material density varies continuously between zero and one, with zero representing the void state and one the solid state. The SIMP method applies a power-law penalization for the stiffness–density relationship in order to push density towards the zero/one (void/solid) distribution: K(ρ) = ρ p K, where K is the penalized stiffness matrix of an element, K is the real stiffness matrix of an element, ρ is the density, and p is the penalization factor (which is always greater than one, with default penalty at 3.0) (Altair, 2012). The topology optimization was stopped when all constraints were satisfied (HMH stresses ≤ 350 MPa) and the decrease in the mass of the structure in subsequent iterations was less than 0.5%. The volume of a freight railcar body was assumed as the design space for topology optimization (Figure 5). To speed up the calculations, only one-quarter of the design space for topology optimization was considered with symmetrical boundary conditions. The manufacturing constraints procedure was used in the topology optimization.



Figure 6.

The meshed design space of the railcar structure.

Figure 7.

The loads and boundary conditions of the railcar structure.

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The design space was discretized by finite elements of dimensions equal to those of a cross section of the standard thin-walled profiles used in modern rail vehicle design (i.e. 180 × 180 × 180 × 5 mm) (Figure 6). Moreover, the EDS tuning procedure was used. The optimization model contains about 1400 hexagonal elements. To fulfil the structural requirements of railway standards, the vehicle structure has to be characterized by suitable static load endurance (CEN, 2002). In Figure 7, the boundary conditions and loads of analysis and optimization are presented. One-quarter of the vehicle structure is supported in one region in the vertical dz-direction, and additionally symmetry conditions were assumed in the Y –Z and X –Z planes (shown in Figure 7 as dx = 0 and dy = 0). Forces were assumed according to the requirements in the CEN standard (CEN, 2002). The main forces acting on the structure of the railcar body which were assumed for the topology optimization are: 2000 kN at the height of the buffers (F1 ), 1500 kN below the buffers (F2 ), 1500 kN on the axis of the railcar body (F3 ). The loads act in an asynchronous way (six load cases). The pressures acting on the walls and floor of the


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Figure 8.

Obtained layout of the railcar – isometric projection.

Figure 9.

Obtained layout of the railcar – left view.

railcar are a sample load resulting from the cargo. In Figures 8 and 9, the result of the optimization is presented. The solution under manufacturing constraints shown there was achieved after 31 iterations. By optimization, an initial design outline of a new railcar was acheived. The brighter areas indicate the most sensitive areas of the structure and in these areas the railcar main structure beams should be located. Based on the optimization results, a preliminary CAD design (see Figures 10 and 11) of a new railcar was prepared. In Figure 10, the CAD model of the structure resulting from topology optimization (Figure 8) is shown. In the CAD model (Figures 10 and 11), a standard dimensional profile was used for the structural beams. For the CAD modelling of a railcar concept body, catiaTM v.5 software was applied. The capacity of the preliminary railcar CAD model is 36 m3 . Figure 4 shows a hopper railcar, but the railcar obtained by topology optimization will require the use of additional unloading facilities.



Figure 10.

Preliminary CAD design of new rail railcar – isometric projection.

Figure 11.

Preliminary CAD design of new rail railcar – side view.

4.

9

Conclusions

A new design procedure for thin-walled structures with manufacturing constraints has been presented in this article. This unique capability has been created with the commercial software Altair Optistruct . As shown in an example of freight railcar optimization, the manufacturing constraints require a lot of attention at the conceptual design stage. In the presented methodology, a manufacturing constraints procedure with preprocessing of the design space was introduced. The manufacturing constraints have an important influence on shortening the distance between design proposal and reality. As shown in the example, the methodology turned out to be a very effective tool for the optimization of rail vehicles. It allows a suitable structure to be obtained in a short time for use as a CAD model base, built from parts widely available commercially. As a result of topology optimization, a new design proposition for the thin-walled structure was obtained. The proposed methodology allows an easily manufactured structure to be found. The methodology presented in the article can be easily extended to other types of thin-walled vehicle structure. The presented methodology was for the first time combined with standard topology optimization software.

Disclosure statement No potential conflict of interest was reported by the author.

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