Seminar Thesis. Topology Optimization using the SIMP method. Submitted by Daniel Löwen

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1 Seminar Thesis Topology Optimization using the SIMP method Submitted by Daniel Löwen Supervisor Dipl.-Ing. Anna-Lena Beger RWTH Aachen University Aachen, 17. November 2016 This paper was presented at the Center for Computational Engineering Science - mathematics division - RWTH Aachen Prof. Dr. Manuel Torrilhon

2 Abstract Topology optimization has experienced considerable publicity and growth in the past few decades with many successful implementations especially in the aviation and automotive sector. This thesis presents an introduction to topology optimization as a structural optimization tool, showing the underlying theory and considerations. In a case study, an ipad stand, a common consumer product, is developed using topology optimization and a prototype is produced with additive manufacturing. The major benefits and constraints of additive manufacturing are analyzed with a focus on topology optimization, yielding a new set of opportunities and drawbacks for the integration of additive manufacturing and topology optimization as a current field of study, and revealing significant potential for numerous future applications. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 2

3 Table of contents 1 Introduction Theory... 8 Structural Optimization Methods Sizing Optimization Shape Optimization... 8 Topology Optimization Topology Discrete Formulation SIMP Process Regularization Physical Significance of the SIMP Method Case Study Used Tools Case Modeling Topology Optimization Results Analysis Volume Fraction Study Feature Thickness Study Conclusion Additive Manufacturing Introduction Material Properties Support Structures Additive Manufacturing and Topology Optimization Case Study Prototype Conclusion and Outlook RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 3

4 Table of figures Figure 1: Use case example for a topology optimized part (Swerea 2009)... 7 Figure 2: Different methods of structural optimization (Bendsoe 2004)... 8 Figure 3: Set of equal forms... 9 Figure 4: Circle-point topologies in different spaces... 9 Figure 5: Removal of one point in a topology Figure 6: Penalization factor graphs (Günther 2014) Figure 7: Iterative process of a topology optimization (cf. Bendsoe 2004) Figure 8: MBB-Beam with singular load, two supports and a symmetry condition (Bendsoe 2004) Figure 9: Setting an Iso-value in post-processing (Bendsoe 2004) Figure 10: Problem modeling: Load case (top left), ipad Pro properties (top right), model right view (bottom left), model front view (bottom middle), model iso view (bottom right) Figure 11: Discretized design domain and parts (right view) Figure 12: Result of the topology optimization with different Iso-values Figure 13: Results of the topology optimization with a target volume fraction of 5% (top left), 20% (top right), 50% (bottom left) and 95% (bottom right) Figure 14: Results of the topology optimization with different feature thicknesses Figure 15: Areas of qualitative difference Figure 16: ipad stand comparison to the case study result (Left: Mirco 2017, Right: Bluelounge 2017) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 4

5 List of tables Table 1: Parameter study of topology optimizations for the MBB-Beam setup with varying filter sizes and mesh refinement, results of the Matlab implementation (cf. Sigmund 2001) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 5

6 Summary The paper begins with a motivation for using topology optimization and the introduction of a basic definition of topology. The theory of topology optimization and the difference to other structural optimization techniques are shown with a focus on the SIMP method and solutions to arising issues. The physical significance of the SIMP material model is explained. A case study is presented with an overview of used tools, the modeling of the problem with load case and conditions for the entailing topology optimization, followed by a results analysis and a conclusion. The case study is carried out with the hypothesis of the optimization yielding an unconventional design that is shown to be the optimum for the given loadcase. Additive manufacturing methods, benefits and constraints are outlined and analyzed with respect to producing topology optimized parts. The subsequent opportunities but also the drawbacks of the combination of additive manufacturing and topology optimization are presented. Promising directions for research and potential fields of application are highlighted in the conclusion and outlook. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 6

7 1 Introduction Topology optimization is used for many applications including product development, concept design or part optimization. In the automotive and aviation industry, a vehicle s or plane s performance and fuel consumption is affected for instance by rolling resistance and acceleration, both of which are dependent on mass. As topology optimization results in weight reduction of parts, the effects are fuel savings, emission reduction and better product performance as the material is used in a more efficient way (Figure 1). The conventional development cycle of a product includes the modeling of an idea, mostly computer aided, manufacturing, and validation by simulation or in a testing facility. Depending on the validation results the part often must be redesigned and subsequently manufactured and validated again. These iterations in the product development process lead to long development times and high cost. Topology optimization can be utilized as a starting point for product development or concept design. Iterations after validation by simulation or testing of a resulting part are reduced or prevented because the part is optimized for the specific load case. In concept design, product forms can be found by detaching the topology of the part from the influence of the designers or engineers ideas. Furthermore, results of topology optimizations often look very unconventional. Figure 1: Use case example for a topology optimized part (Swerea 2009) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 7

8 2 Theory In this section the topology optimization is differentiated from other structural optimization methods. The underlying theory is introduced and solutions to emerging issues are presented. The physical relevance of the material model is demonstrated at the end of the section. Structural Optimization Methods There are several methods available that can be used to optimize the form or the elements of a structure while specific constraints are met (Figure 2). Figure 2: Different methods of structural optimization (Bendsoe 2004) Sizing Optimization In sizing optimization, the domain of the design model and the state variables are known a priori and stay fixed throughout the optimization process. The sizing optimization of a truss structure as seen in Figure 2, a) takes elements of the structure as design variables, for instance the diameter of a rod or the thickness of a beam, with constraints such as equilibrium constraints and optimizes for an objective that can be, amongst others, mean compliance, peak stress, or deflection (cf. Bendsoe 2004) Shape Optimization The goal of shape optimization is finding the optimal shape of a domain with a known topology and the domain itself as design variable. This domain is defined by boundary curves or boundary surfaces and the optimal form of these boundaries is found by shape optimization (Figure 2, b)) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 8

9 Topology Optimization Ever since Bendsoe and Kikuchi presented the topology optimization method in 1988 (Bendsoe 1988) it has become a popular method for material distribution with several open source and commercial software packages implementing it as a design tool. It is not yet seen as a self-sufficient process with results being taken as the final product but rather as a starting point for any structural design process (cf. Bendsoe 2004). The goal of topology optimization is to find the optimal lay-out of a structure within a specified region (Bendsoe 2004). Known quantities are applied loads, support conditions, structure volume and additional design restrictions such as location and size of holes or solid areas that are exempt from the design domain. However, physical size, shape, and connectivity of the structure are unknown (Figure 2, c)) Topology Topology is the mathematical study of properties that are preserved through invertible deformations, twisting, and stretching of objects. However, tearing or gluing is not allowed. It can be used to abstract the inherent connectivity of objects, ignoring their detailed form (cf. Weisstein 2016). If two objects have the same topological properties, they are called homeomorphic. All forms in Figure 3 are homeomorphic with a possible physical representation of a rubber band, that can be stretched into each of these forms. Figure 3: Set of equal forms Topologies are dependent on the space, for instance the two objects in Figure 4 cannot be continuously transformed into another in 2D as tearing and subsequent gluing would be required, whereas in 3D in form B one could move the point out of the plane of the circle and lower it again inside the circle, meaning topological equivalence. Figure 4: Circle-point topologies in different spaces RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 9

10 On the other hand, a figure eight curve formed by two circles touching each other at a specific point is topologically distinct from a circle. Removal of a point in the circle preserves connectivity while removing the connection point of the figure eight results in two separate forms (Figure 5). Here, the number of connected forms is the non-preserved property of the topology. Figure 5: Removal of one point in a topology Discrete Formulation The given design domain is discretized for a finite element analysis. For each element in the discretized design domain the problem is whether there is material or not. The aim of topology optimization is to minimize an objective function such as the compliance of the structure: min : Φ(ρ, U(ρ)) ρ Compliance is defined as the work done by the external loads (Reiss 1976). In this problem, the compliance is dependent on the density vector ρ and the displacement vector U(ρ) out of the finite element analysis and is minimized under a volume constraint: N s. t. v e ρ e = v T ρ V e=1 where V* is the target volume, a fraction of the initial volume. Instead of a volume constraint, geometry, stress, or other constraints are possible. The density is set as constant in each element and can take the values 0 for void and 1 for material: This satisfies the equation ρ e = { 0 1, e = 1,, N K(ρ)U = F with K being the stiffness matrix, U the displacement vector and F the load vector. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 10

11 2.2.3 SIMP The discrete topology optimization problem is ill-posed and very computationally expensive (cf. Bruns 2005). For that reason, Bendsøe and Kikuchi moved from a discrete formulation to a gradient based approach, where continuous design variables are introduced. The density, the design function, is allowed to vary between a small number and one (cf. Bendsoe 1988): 0 < ρ min ρ 1 The assumption for the relation between the stiffness of an element and the density is that stiffness is linearly dependent on density. The power law approach, also called Solid Isotropic Material with Penalization (SIMP) method, uses this assumption: E(ρ e ) = ρ e p E 0, p > 1 Here, E 0 is the stiffness of solid material, E(ρ e ) the stiffness of element e and p is the penalization factor, penalizing the density ρ e in element e. In Figure 6, left, the relation between densities on the abscissa and the stiffness on the ordinate is shown for increasing penalization factors. x i is the design variable x i = ( ρ i ρ i 0). For the commonly used penalization factor p = 3 the right graph in Figure 6 shows the effect of the penalization. For an exemplary density of 0.5 a low stiffness value is obtained, thus having intermediate densities in the optimal design is uneconomical as it does not give much stiffness for the amount of available material. Subsequently the topology optimization algorithm will redistribute the material of the design domain such that the solution is penalized towards 0 and 1 as the stiffness values in elements. Figure 6: Penalization factor graphs (Günther 2014) For a penalization factor p = 1 the problem has a convex form. Therefore, independent from the starting point, the solution will converge to the global optimum. The approach to finding a good optimal design that is not only locally optimal is to find the global optimum with p = 1 and gradually increase the penalization factor. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 11

12 2.2.4 Process A topology optimization follows an iterative process that converges to an optimal design. This process is shown in Figure 7. Initialize FEM Finite Element Analysis Sensitivity Analysis Regularization (filtering) KU = F Optimization (material redistribution) Update Design variables no ρe converged? yes Plot results STOP Method of Moving Asymptotes Figure 7: Iterative process of a topology optimization (cf. Bendsoe 2004) The first step is defining the design domain, load case, boundary conditions and a objective function. The design domain is then discretized and a finite element analysis is carried out. The results of the FEA are used in a sensitivity analysis to compute the gradients of the objective function, for instance the compliance. To resolve the complications of mesh-dependency and checkerboarding, that are further discussed in section 2.2.5, a regularization filter is applied. After filtering, the material in the design domain is redistributed using the Method of Moving Asymptotes and the design variables in the elements are correspondingly updated. The topology optimization algorithm iterates until the densities in the elements converge Regularization The previously mentioned complications are demonstrated with the MBB-Beam setup in Figure 8. Figure 8: MBB-Beam with singular load, two supports and a symmetry condition (Bendsoe 2004) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 12

13 Table 1: Parameter study of topology optimizations for the MBB-Beam setup with varying filter sizes and mesh refinement, results of the Matlab implementation (cf. Sigmund 2001) Mesh-dependency The SIMP topology optimization problem [ ] lacks existence of solutions in its general continuum setting (Bendsoe 2004: 28-30). This is due to the fact that the introduction of more holes at a constant volume will generally increase efficiency of a structure. Subsequently, a structure does not converge to a global result with mesh refinement (cf. Bendsoe 2004). Qualitatively different optimal solutions are reached for different mesh-sizes and discretizations. This dependence of the solution on the refinement of the mesh is illustrated in Table 1, where a finer mesh results in a more detailed structure, smaller features and more holes. Bendsoe noted that mesh-refinement should ideally lead to a better finite element modelling of the same, globally optimal structure and a better boundary description, not a different structure. Checkerboarding Checkerboard patterns are areas where densities alternate between 0 and 1 between neighboring elements and occur in the results of the topology optimizations in the first row of Table 1. This problem mainly appears when optimizing a domain discretized with 4-node Q4 elements (cf. Bendsoe 2004). These areas have a high stiffness so the optimization problem includes the checkerboards in the optimal design. However, this stiffness is artificial as material only being connected on the edge has no physical stiffness. Thus, these patterns are not optimal and should be prevented in the optimal solution. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 13

14 Filtering Both presented issues are considered numerical instabilities. Filtering of sensitivities or densities is a filter-based method solving mesh-dependency as well as the checkerboard effect. The method introduces a neighborhood N e = {i x i x e R} with a defined radius R and updates the sensitivity (density) of an element as a weighted average of the sensitivities (densities) of the elements inside the neighborhood of that element: Sensitivity filtering: Density filtering: Φ ρ e = Φ i Ne H(x i )ρ i ρ i ρ e i N e H(x i ) E e (ρ) = ρ e p E 0, ρ e = H(x i Ne i)ρ i H(x i ) i N e Depending on the filter size the method yields a length scale for feature sizes in the optimal design by blurring edges and preventing the occurrence of very small features (cf. Bendsoe 2004). In Table 1 the effects of the filtering can be seen, as increasing the filter radius results in qualitatively similar optimized structures and the removal of checkerboard patterns Physical Significance of the SIMP Method A common point of criticism has been the physical significance of the stiffness to density relation, the intermediate densities in the SIMP method, as this can seem like an artificial material model. Following the transition from a discrete formulation to a gradient based approach there are no longer sharp boundaries between material and void areas. The greyscaled elements at the boundary are transformed to a sharp boundary by introducing an Iso-value as a border for density values (Figure 9). All densities from 0 to the border are set to 0 (void), all densities from the border to 1 are set to 1 (material). Through variation of the Iso-value, connectivity of the resulting design structure can be ensured. Setting the Iso-value to eliminate intermediate densities is done in post-processing. Figure 9: Setting an Iso-value in post-processing (Bendsoe 2004) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 14

15 3 Case Study The goal of the case study is optimizing the topology of an ipad Pro 12,9 stand that is later physically prototyped using additive manufacturing. The research question explores the benefits of using topology optimization as a concept design tool with the hypothesis that the resulting form of the ipad stand is very unconventional. For the case study two parameter studies are conducted, a variation of the volume fraction and a variation of the smallest feature thickness. Used Tools For the topology optimization, the commercial software SolidThinking Inspire (Version ) is used as it is straightforward to model the given case and sufficient options are available to carry out all aspects of the planned study. The implemented solver is OptiStruct, which uses the SIMP method for topology optimization. The printer used for additive manufacturing is a Ultimaker 2 with the software Cura (Version 2.1.2) to prepare the model for printing (Ultimaker 2016). Case Modeling The case is modeled with the ipad standing at a 20-degree angle with the weight force of 7N applied in the center of gravity that is in the middle of the ipad. A touch force, applied orthogonal on top of the ipad screen, 11 mm under the top edge, is modeled as a moment of 1,06 Nm around the bottom edge, with the weight force contributing to the moment (Figure 10, top left). The touch force amounts to 3,8 N as the Force Touch sensor included in the ipad s touchscreen does not weigh beyond a maximum weight of ~385 g (cf. Defauw 2015). The design domain is modeled large to not restrict the resulting topology in any way (Figure 10, bottom). The ipad is modeled with the sizes shown in Figure 10, top right, and a ground plane is modeled larger than the design domain. All contact areas are modeled as glued together. The design domain material is set to PLA (Formfutura 2015), for ipad and ground plane the existing material aluminum is chosen. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 15

16 Figure 10: Problem modeling: Load case (top left), ipad Pro properties (top right), model right view (bottom left), model front view (bottom middle), model iso view (bottom right) Topology Optimization Meshing setup and creation in Inspire has no user interaction possibilities. The domain is discretized using tetrahedral elements (Figure 11). The optimization target is maximizing stiffness, which is the same as minimizing compliance as compliance is the reciprocal of stiffness, under the constraint of a given volume fraction. Two parameter studies are carried out, the first study varying the target volume fraction between 5% and 95% in 5% steps with the steps 75% and 95% as control points for convergence with a fixed feature thickness of 0,03 m. In the second study the minimum feature thickness is varied between 0,015 m and 0,06 m at a fixed volume fraction of 20%. Figure 11: Discretized design domain and parts (right view) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 16

17 In the case study, all Iso-values are set to 0,5 to ensure comparability of results (Figure 12). Figure 12: Result of the topology optimization with different Iso-values Results Analysis Volume Fraction Study Figure 13 shows a selection of the results of the topology optimization for the volume fraction study. Structural connectivity with an Iso-value of 0,5 is lost under a volume fraction of 15%. Figure 13: Results of the topology optimization with a target volume fraction of 5% (top left), 20% (top right), 50% (bottom left) and 95% (bottom right) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 17

18 Compliance [m/n] In diagram 1 the compliance of the structure with varying volume fractions is shown. As expected, compliance increases as the volume fraction decreases. At the beginning the increase is very slowly as the lost material volume has no considerable influence on the compliance of the structure. After a volume fraction of 0,2 the increase rate begins to grow faster meaning a loss of bearing material. This can be interpreted as the algorithm lacking material to distribute to loaded areas resulting in high compliance values in the structure. 0,0012 Volume study - Compliance 0,001 0,0008 0,0006 0,0004 0, % 20% 40% 60% 80% 100% Volume Diagram 1: Compliance values for different volume fractions Feature Thickness Study In section the regularization for mesh dependencies was introduced, because a control over the minimum thickness of features in the resulting design was needed. To check the ipad-stand topology for dependency on feature thickness a study is conducted with the settings given in 3.3. The resulting topologies look qualitatively similar (Figure 14) with smaller feature size emphasizing certain areas, for instance the front part being more developed with a curve at the front base (Figure 15, A). The connecting volume (Figure 15, B) is also developed narrower in height and broader in width. No additional holes are introduced into the topology. Figure 14: Results of the topology optimization with different feature thicknesses RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 18

19 Figure 15: Areas of qualitative difference Topology optimizations for even smaller minimum feature thicknesses could not be carried out as the optimization time for the 0,015 m thickness already exceeded ten hours. Conclusion The goal of the case study, developing an ipad stand with an optimized topology, has been achieved. As a consumer product, a buyer usually chooses an ipad stand for the design or additional features. The resulting topology looks, in compliance with the hypothesis, very unconventional when compared to existing ipad stands, that tend to be slim, smooth, and minimalistic (Figure 16, left and middle). The overall form is bulky with an irregular, rough surface finish that depends on the Iso-value (Figure 16, right). Post-processing could smooth out the surface and provide a better surface quality. In both parameter studies results are qualitatively similar. For concept design this is an unsatisfying result as designers would rather choose from a selection of different design proposals considering that aesthetics is a deciding factor in the fields of customer products and product design. This could be achieved by modifying the load case, nevertheless resulting in a lot of additional work. Figure 16: ipad stand comparison to the case study result (Left: Mirco 2017, Right: Bluelounge 2017) RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 19

20 4 Additive Manufacturing In this section a general overview of methods, benefits, and constraints of additive manufacturing (AM) is given with the focus on producing topology optimized parts. Introduction A vast number of methods and associated materials are presently available ranging from Material Extrusion processing polymers, Binder Jetting processing ceramics or Powder Bed Fusion techniques processing metals. There are methods processing biological and bio-compatible materials or even producing edible items. Most methods produce a physical object from digital information in ways such as layer-bylayer, piece-by-piece, or line-by-line without the need of intermediate shaping tools. The method determines the objects geometry and material properties. Market factors such as shorter product development cycles and an increasing demand for customized or personalized products can be fulfilled using AM. These products, especially in the prototyping stage, can be produced in minutes or hours, not weeks. The resources needed for AM are often fewer, as is the need for special training (cf. Thompson 2016). When using AM, various constraints must be considered regarding build and surface quality, material property control and production cost, that also depend on the used AM method, material, and machine capabilities. A cost analysis for AM is not carried out in this thesis Material Properties Part quality and consistency depends on the chosen method. The boundaries between layers in AM parts are rarely seamless due to creating new material on already existing material. Often, material properties resulting out of AM processes are anisotropic. This issue can be addressed by modifying part orientation while printing to minimize the effect or post processing the part. The characterization of mechanical and optical material properties and the verification of internal features are not yet fully resolved issues with AM (cf. Thompson 2016) Support Structures During manufacturing, the part must be able to resist forces such as gravitational loads or internal forces from thermal and residual stresses. Sufficient resistance can be obtained by orienting the part or by adding support structures that support sections of the part with for instance an overhang structure, if the part is not self-supporting. Removal of support structures must be considered beforehand as it may be impossible in internal voids (cf. Thompson 2016). RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 20

21 Additive Manufacturing and Topology Optimization The typical workflow to produce a part by AM starts with a digital model that is prepared for printing, the resulting digital model of a topology optimization can be considered as this starting point. As opposed to conventional manufacturing methods, AM is able to produce most parts without the need of extensive post-processing, if the constraints of AM such as the critical angle of an overhang structure are met. Advantages Using topology optimization and AM, the product development cycle can be reduced significantly leading to cost savings and lower risk. The regular thinking of designers and engineers imposed by conventional manufacturing methods and past expertise is overcome by a design proposal optimized for a specific use case, which can lead to better product performance. As seen in section 2.2.6, micro-structures can be designed to give the SIMP method of topology optimization physical significance and manufactured considering that AM gives internal geometrical freedom in parts. Alternatively, sponge-like material can also be produced by AM leaving air filled pores in the material. It would be possible to reduce the penalization factor of the SIMP method to converge to the global optimum while assigning densities to micro-structures or sponge-densities, meaning that optimal designs are producible. Similarly, it can be achieved using different materials with disparate properties in the same part. Drawbacks Topology optimized parts often include small, overhanging features that prevent AM without support structures. It is possible that the support structures, especially with internal voids, cannot be removed. The manufacturing restriction of a critical angle must be included into the topology optimization algorithm. The properties of materials produced by AM processes must be considered as they can affect the final part. Optimization carried out with an isotropic material model will not result in the same material properties when manufactured with an AM process that produces anisotropic structures Case Study Prototype One of the hypothesis for the combination of methods is the manufacturing of topology optimized parts without the need for post-processing. To show this, the ipad stand prototype is exported as a STL file in the condition just after optimization with an Iso-value of 0.5. This file is then imported into Cura and scaled down to 20% due to the build volume of the Ultimaker 2 restricting a print of the model at full size. Support structures under the middle connection part must be added to account for the overhang. The model is printed using Formfutura EasyFil PLA (Formfutura 2015) and is shown in Figure 16, right. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 21

22 5 Conclusion and Outlook This thesis has demonstrated topology optimization to be a powerful tool for designers and engineers. The method can be used as a design tool in product development, resulting in unconventional designs that are detached from the influence of the user and his previous experiences. It is implemented in several open source and commercial software packages, making it available to designers and engineers for numerous applications. The integration of topology optimization and additive manufacturing provides new and promising opportunities for product development. The method could be used in highly demanding environments such as aviation or automotive, providing a method of weight reduction and producing better performing parts. A small overview of possible uses in other fields includes medical implants or prosthetics, home furnishing or product casings. There are numerous research opportunities in this field of study showing great potential to boost the growth of both methods. RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 22

23 Bibliography Altair Engineering: Altair OptiStruct. Concept Design with Topology and Topography Optimization, 2009 Bruns, T. E.: A reevaluation of the SIMP method with filtering and an alternative formulation for solid-void topology optimization, Structural and Multidisciplinary Optimization 30: , 2005 Bendsoe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, Volume 71, Issue 2: , November 1988 Bendsoe, M.P., Sigmund, O.: Topology Optimization. Theory, Methods and Applications, 2nd edition, 2004 Bluelounge, Mika, Defauw, M.: 3D Touch Introduction: Building a Digital Scale App and Quick Actions, 11/09/2015 Formfutura: EasyFil PLA Technical Data Sheet, 10/29/2015 Günther, M. und Tremmel, J.: Gewichtsoptimierung mit OptiStruct, 2014 Mirco, IPAD DOCK, Reiss, R.: Optimal Compliance Criterion for Axisymmetric Solid Plates, International Journal of Solids and Structures, 12: , 1976 Sigmund, O.: A 99 line topology optimization code written in matlab, Structural and Multidisciplinary Optimization, Volume Vol. 21, No. 2: , 2001 Swerea, Risberg, M.: Topology Optimization of Castings, Ludwigsburg, 2009 Thompson, M.K., et al.: Design for Additive Manufacturing: Trends, opportunities, considerations, and constraints, CIRP Annals Manufacturing Technology 65: , 2016 Timary, S.: Exploring Interaction as an Optimisation Tool, University College London, London, 2011 Ultimaker, 11/28/2016 Weisstein, Eric W. "Topology." From MathWorld--A Wolfram Web Resource. 11/19/2016 RWTH Aachen Univ.-Prof. Dr.-Ing. Georg Jacobs 23