1.4.2 TO described as a perturbation technique TO Practical use and impact on the CAD modelling workflow

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1 Part IV - Optimization methods for design 1.1 Introduction 1.2 Design for "X" 1.3 Local Approach 1.4 Topological Optimization (TO) Basic theory TO described as a perturbation technique TO Practical use and impact on the CAD modelling workflow TO: a case-study 1.5 References for Optimization and TO through Hyperworks/Optistruct 1.6 Exercises 1.1 Introduction After Preliminary lay-out, executive design occurs. It is devoted to define each details of the blueprint of both system and components, including manufacturing issues. In this step many optimization procedures may be adopted. The choice is left to the design according to the requirement list and her/his evaluation of the system critical aspects. A general overview about optimization in mechanical design can be summarized as follows: 1. Methods suitable to optimize specific design requirements --> "Design for X" techniques (and related Computer Aided "X") 2. Local or Perturbation Approach --> to analyse numerically the behaviour of a function that should be minimized according to some constraints. 3. Global Approach --> to analyse systematically by means of experiments (it does not matter if they are physical or virtual) one or more responses in a region of interest with the aim of: - evaluating predominant variables - evaluating response surface (=function able to model analytically the problem) - looking for optimal conditions These approaches are methodologies suitable to analyse different kinds of problems, according to responses specifically defined and adopted by the

2 engineer. On the contrary, other methods, such as Topological Optimization or Probabilistic Design, are specifically oriented to a design goal: defining topology and shape, the first one; evaluating the risk of failure, the second one. 1.2 Design for "X" Design for "X" stands for the group of methodologies suitable to optimize specific external properties, such as ergonomics, reliability, assembly, safety,... "X" stands for the property to be optimized. Each kind of Design for "X" methodology is set up according to these rules: 1. Define the "X" and its goal 2. Look for design rules necessary to implement it in the design and apply them 3. Apply specific methods suitable to measure and optimize the "X" Often Design for "X" is coupled with Computer Aided "X" techniques. They are CAD applications oriented to implement methods and design rules for the property to be optimized. Among Design for "x" methods, the most well known and applied are: Design for Assembly, Design for Safety, Design for Ergonomics. Design for Assembly aims to design systems with ease to assembly, so that part of the production costs are reduces as well as part of the maintenance efforts (that related to assembly/disassembly). This property is relevant to the costumer in terms of cost and maintenance efforts, and to the Enterprise according to the scheme provided in Fig. 1.

3 Fig. 1 - Simplified model of Enterprise (excerpted from "Product Design Methods and Practices", Henry W. Stoll, CRC Press, 1999) General rules for Design for Assembly (DfA) are: use standard components to apply interchangeability, reduce the number of components, optmize the assembly process (number of operations, type of operations, tools). Specific methods, that have been also implemented in CAx applications are: Hitachi method and Boothroyd and Dewhurst. For having a good overview of DfA and Design for X methods, please read 1 : Tsai-C. Kuo, Samuel H. Huang, Hong-C. Zhang, Design for manufacture and design for X : concepts, applications, and perspectives, Computers & Industrial Engineering, Volume 41, Issue 3, 2001, Pages , ISSN , Design for Ergonomics is another application of Design for "X". It aims to understand and optimize "interactions among humans and other elements of a system, applying theory, principles, data and methods to design in order to optimise human well-being and overall system performance., (International Ergonomics Association). In mechanical/industrial design it concerns human-machine interfaces for driving, checking, input-output, advising, and so on. A property extremely involved with ergonomics is safety of the persons who use of the system, other properties are aesthetic, manufacturing, standard. 1 Check it out in the class website - This reference is also useful for Design for Quality and Reliability

4 Among methods for Ergonomics, RULA (see for example: ) and push and pull analysis are two methods suitable to assess critical postures and loads during working. design for Ergonomics can be analysed by virtual manikins (e.g. Ergonomic workbench in Catia). 1.3 Local Approach During Design simulations have a relevant role to assess system/component physical behaviour. For this reason virtual prototyping is now a well-known practice to reduce the efforts for experimental prototyping, thus anticipating many optimizations that were made on the mock-up. Local approach, also known as perturbation approach is devoted to numerically optimize a design function that is estimated via simulation (e.g. deflection in a point as a function of section lengths). The design function is not analytically known, but it is evaluated as discrete by the simulation and then locally approximated to look for the minimum, by iterative algorithms like newton-raphson, simulated annealing or genetic algorithms 2. The method is called perturbation approach since it requires a first guess condition (= a simulation with a nominal value for the n input variables), then the input variables, vector x, are changed according to Δx (vector of pertubation): x i+1 =x i + Δx i i=1,n For each of the n perturbations, a simulation is run to quantify the objective function f, so that f(x) and f(x+ Δx) are evaluated and ready to be used for the gradient computation, necessary to locally estimate the function trend, e.g. via iterative algorithms. As shown in Fig. 2, this process is iteratively replicated until convergence is reached (= no more changes in f(x), or maximum number of allowable iterations is reached). Fig. 2 - Optimization via Local Approach: Workflow Fig. 2 shows the process in a schematic and rough way. In fact constraints and ranges for the input variables are also imposed during the optimization loop. Constraints are numerically evaluated starting from simulation results, ranges are defined according to the specific problem that is investigated. 2 See Matlab help for example of such optimization algorithms

5 The perturbation approach is also known as "local approach". This name is due to fact that starting from the first guess the domain of the input variables is investigated following the path imposed by the minimization algorithm, that means imposed by the shape of function f(x) nearby the first guess. It imposes the research of the optimal condition locally, around the first guess. Doing so, a local minimum of the objective function can be found, without any chance of understanding if other minima exist. A numerical technique useful to reduce the risk of finding a local minimum may be the multiple starting point selection, that requests multiple starting point (guess) points (see for example OptiStruct>User's Guide>Design Optimization: Global Search Option) Topological Optimization can be seen as a peculiar application of a perturbation approach. In the next sections basic notion about its goal and theory are given according to the paper Topological optimization in concept design: Starting approach and a validation case study, (2017) made by Bici et al Topological Optimization Additive technologies, generally, allow to add material where necessary, layer-by-layer, obtaining forms not otherwise producible. Obviously, also Additive Manufacturing (AM) has its own technological constraints so that it is impossible to produce everything with every AM technology. Nevertheless, it is important to understand that design dynamics change. In the AM field, the possibility of producing complex shapes and various features, regardless of process, reduces the effort of component modifications during the design phase. AM allows to move the designers focus from the executive design to the concept one, with a high reduction of the design for rules related to the process technology. This is why Topological Optimization (TO) procedures assume a substantial role, allowing, already in the concept phase, to obtain a component that has its own volume distributed in function of its conditions of load and use. It can be made through an extremely consistent mathematical formulation that is able to explore a very generic design space for a given set of loads and boundary conditions, so that the resulting layout satisfies a prescribed set of performance targets Basic theory TO is a structural optimization technique that started its evolution in the second half of the last century. According to Bensdoe 4, structural optimization pertains to topology, shape and size of a component. Topology extends the concepts of shape or geometry including the capability of adding and deleting volumes, that basically means changing the space connectivity via opening-closing operations (Figure 3). Shape and size optimization looks for mathematical conditions able to minimize (or maximize) a structural design objective through geometrical parameters like feature sizes (thickness, length, perimeter,...) or geometric shape, without changes in the topology (that means without opening/closing connections). 3 See on the course webpage for its extended version 4 Bendsoe, M. P., Sigmund, O., Topology Optimization Theory, Methods, and Applications, 2003, Springer

6 Fig. 3. Space connectivity examples (on the top); example of shape modification and topology change via opening operator (on the bottom - deformations are in blue and space reduction in the first object depicted in cyan). In other words: Topology defines geometrical variables that do not entail size variables. In other words it concerns solid-void connection sequences of the component. Shape is a parametric geometry defined through sizes and position in the space (= measures) From a mathematical point of view, topology studies geometrical properties that are invariant in the respect of measures. In this sense, a rectangular plate with a circular hole has different shape from a triangular one with an elliptical or hexagonal hole, but they are topologically equivalent, since they can be transformed one into the other through a transformation map (homeomorphism). Fig. 4 Example of topological and shape optimization At the beginning, the structural problem has been investigated looking for mathematical conditions able to delete or maintain material in each volume element of the assumed

7 design space. This leads to the microstructural or material approach that, together with the macrostructural approach, represent the most general and well-posed definition of the problem. SIMP, that stands for Solid Isotropic Material with Penalization, and the Level Set Method are two relevant examples of micro and macrostructural approaches. Other approaches with very intuitive formulations (e.g. hard-kill methods like Evolutionary Structural Optimization), can be seen as heuristic although many efforts made to extend or strictly define their field of applicability. Microstructure methods are also called density-based methods. Under the hypothesis of linear elastic behavior and assuming a FEM discretization, they are defined as: (1) where the objective function, f, can be the total design space mass, the natural frequencies, or cost function depending on stresses or compliance. u represents the nodal strain vector; K( ) the stiffness matrix in function of the density factor,. It is defined between 0 (void) and 1 (bulk). F represents the applied nodal loads and g i is the set of m constraints. The design variables able to look for the minimization of the material inside the design space, concern with the element stiffness matrices, generically named K el, which opportunely assembled define the stiffness matrix K. Each of them is function of by: where p stands for the penalty weight. p is able to transform the problem from discrete to continuous since p > 1 is able to penalize values far from 1. SIMP sets p=3. Each element of the mesh contributes to the component stiffness via (1) thus if the element does not result effective the p value will decreases its relevance, until to its deletion. As reported in literature, checkerboard patterns (Fig. 5) and mesh dependency are the major mathematical drawbacks of this formulation. Filtering techniques (Fig. 5) or mathematical relaxation of the optimization problem are two possible solutions to these problems. Both of them contribute to achieve a "well-posed" optimization problem, obtaining a reliable approach to face the TO via CAE. Filtering techniques reduce the set of possible solutions excluding, via filters, unphysical solutions. Many types of filter can be applied. (2)

8 Fig. 5 Checkerboard problem: examples and summary of filtering technques The mathematical relaxation of the minimization problem consists of adding new design variables. This is achieved putting aside the concept of solid isotropic distribution of material and defining an assigned microstructure of voids for each element. The new design variables can be represented by the sizes of the void areas (hole in cell approach) or by the configuration of a layer structure (layered structures of different ranks). The solution is then found by the so-called homogenization techniques. The macrostructure approach consists in boundary variation methods. They are based on implicit functions able to describe what happens on the edge of the design space. Doing so, changes of topology are linked to the distribution of the contours of the implicit function (x). In the conventional Level Set Method, this problem is described by the Hamilton-Jacobi-type equation that can be solved through sensitivity analysis on an assigned grid in the design-space domain. As already mentioned, the original formulation of these methods obviously assumes linear elastic behavior. Generalization to non-linear problems have been made. This reference has also an interesting overview of the possible fields of applications that range from MEMS to biomedical, to civil structure and multiphysics applications TO described as a perturbation technique. Under the hypothesis of linear elastic behavior and assuming a FEM discretization, they are defined as: (1) where the objective function, f, can be the total design space mass, the natural frequencies, or cost function depending on stresses or compliance. u represents the nodal strain vector; K( ) the stiffness matrix in function of the density factor,. It is defined between 0 (void) and 1 (bulk). F represents the applied nodal loads and g i is the set of m constraints.

9 The design variables able to look for the minimization of the material inside the design space, concern with the element stiffness matrices, generically named K el, which opportunely assembled define the stiffness matrix K. Each of them is function of by: where p stands for the penalty weight. p is able to transform the problem from discrete to continuous since p > 1 is able to penalize values far from 1. SIMP sets p=3. Each element of the mesh contributes to the component stiffness via (1) thus if the element does not result effective the p value will decreases its relevance, until to its deletion. So we have an objective function, f, an initial first guess (initial topology, Top.#0), n design variables (x= of each element) and constraints. Mathematical formula of f is unknown, so it has to be evaluated, and minimised, numerically. According to Fig. 2 the perturbation loop becomes the one of figure 6. (2) Fig.6 Specialization of the perturbation approach loop TO Practical use and impact on the CAD modeling workflow TO is available in many well-known FEM software (e.g. Ansys, Optistruct, Inspire, Nastran). Their practical use starts from a proper definition of the problem domain, in terms of preliminary envelope of the volume and lengths, functional surfaces (interface surface with specific contact conditions), geometrical conditions related to possible manufacturing constraints (e.g. symmetry, draft angle). In this sense, its applicability is possible for all kind of manufacturing, although in AM also TO solutions reached without specific manufacturing constraints can be obtained. Figure 7 gives some examples in term of envelop volume (Top.#0, constraints and final results).

10 Fig. 7 Examples of TO loops The preliminary envelope of the volume can be a geometrical entity or a dense FEM mesh, that has to be divided into design space and fixed volumes (e.g. functional surfaces or manufacturing constraints). Although this distinction is a quite natural concept, it can be implemented in different ways according to the adopted software. Starting from a mesh, it asks for a selection of two sets of element. Using CAD based software (e.g. Inspire), it may ask for splitting the volume in more than one set. As a consequence, contact conditions must be given not only among different parts of an assembly, according to FEM procedure, but also among different volumes of the same component. Contacts are part of the load conditions that may be split into loads and Degree Of Freedom (DOF) constraints. Also in this case, CAD-based software tend towards load applications not directly to mesh elements but on geometry entity, hiding the mesh loading results. It may reduce the knowledge of the effective load/constraint conditions that are applied, so that careful checks are required to evaluate the compliance of the model. Concerning loads, more than one operative condition can be analyzed and studied. The optimization can be applied on different loadcases, looking for a compromise solution. From the computational point of view, not so many input are required. The objective function and the constraints, usually taken from compliance, mass, natural frequency or a combination of them, Mesh size must be rather uniform and dense enough to reach the proper sensitivity to the element deletion, generally it must be equal or smaller than that used in a good structural analysis evaluation. In many cases a preliminary run is required to check the goodness of the model.

11 Optimal solution is provided in terms of final volume or density factor contour plot. It must be checked through safe factor or other design requirements. If it succeeds, other CAD activities are necessary: surface smoothing (since the computation mesh is rather rough after the optimization, due to its constant length), small area deletion or pocket closure if other manufacturing constraints are considered. In this scenario, new CAD technologies (e.g. curve and surface modeling, synchronous modeling) represent a relevant aid to reduce time, nevertheless the necessity of robust data exchange and a common user interface among CAD/CAE/CAM may represent one of the major drawbacks of the practical use of TO TO: a case study The goal of the case study is to give evidence of the workflow defined in the previous Section by using a commercial TO software (in the specific case solidthinking Inspire 2015) and to validate this design approach comparing the results to those previously achieved through a conventional trial and error design process. For this reason we develop two tests from a case study related to a suspension wishbone attachment of the Formula SAE car, named Gajarda, designed by the Sapienza Corse team. The attachment is used to connect the uniball joint to the monocoque chassis of the car at the end of each suspension s A-arm, as shown in the red circles of Figure 8. Fig. 8. Gajarda in two versions: the 2012 (on the left) and the 2013 (on the right). Red circles highlight some of the positions where the suspension wishbone attachments are located. The attachment is made of an aluminum alloy characterized by: E=70GPa, ρ=2700 Kg/mm 3, σ yield =260 MPa, ν=0.33. It has been developed and modified passing from car 2012 version to 2014, to reduce mass, saving functionalities and resistance. Figure 9 shows the design evolution made during these years, reducing the total mass from kg to kg. Fig. 9. Suspension wishbone attachment: interface surfaces (on the left) and shape evolution made by Sapienza Corse Formula SAE team from 2012 to 2014 According to the design intent, the interface surfaces are (Figure 9 on the left): counterbore holes that must be provided on the base, to allow the connection with the wall of the

12 chassis; a pocket that allows the insertion and assembly of the ball joint with various angular orientations of the arm; two aligned holes for the locking pin. The two tests are defined as follows: Test 1. Starting from the shape and the geometry of the 2012 version, we look for an optimal design reducing the original mass at the 50% (comparable with the reduction obtained in the 2014 version) and at the 20%, maximizing stiffness. Test 2. with the aim of decoupling the optimization process from the designer s choices, we give as input for the design space just the component s envelope volume. It has been defined as cylindrical since it should be made with almost axial-symmetric features, due to the fact that the actual component is manufactured by machining and the capability of fastening in different positions is required. Concerning the loads scheme, in the actual configuration they are applied through a locking pin that is inserted in the two aligned holes of the component. In the TO models, to take into account bending effects due to the pin deflection, they are applied on the middle point of the axis between the two holes and transferred to the corresponding cylindrical surfaces by a connector (thus defining the most severe condition of bending), as shown in figure 10. On the left, the three load conditions can be seen at the middle of the axis between the two holes for the locking pin. The central vertical plane has been defined as manufacturing constraint, since, as already said, the designer intent is also to maintain component symmetry, in both test cases. Fixed volumes are associated to the bolt interfaces necessary to link the component to the frame. The DOFs that are involved from these assembly constraints are clearly shown in red. Figure 10 shows also the design space for the two tests in brown and the fixed volumes, constrained as non-design space (in blue). Fig. 10 Test 1 (on the top) and Test 2 (on the bottom). Brown volumes are design spaces, blue volumes constrained areas, in red loads and constraints on DOF In both tests we have chosen to keep the number of fixing holes to three, differently to what developed by the design team, which has increased the number of connections reducing, at the same time, their diameter, always in a perspective of reduction of the masses.

13 Test 1 aims to investigate if, through TO, it is possible to include well-established design intents that are mainly imposed by the designer knowledge. For this reason, we setup the optimization to obtain comparable solutions starting from the virtual model of the 2012 attachment. Fig. 11. Test 1: mass reduction@50% (at the top); mass reduction@75% (at the bottom) The optimization problem has been defined as a max-stiffness research with the constraint of total weight at 50%. Figure 5 shows the final achievements for Test 1. Mass reduction@50% gives a total mass of kg in accordance with the imposed constraint. Starting from this solution an enhanced one has been investigated, moving the final mass constraint up to 75%, taking care that the hypothesis of linear elastic behavior is not missed. Figure 11, at the bottom, shows this result that set the mass to kg, basically sharpening and smoothing the volume around the pin hole. Test 2 aims to investigate the ability of the TO starting from the most general domain definition. Doing so a decoupling of the TO results and the designer knowledge is performed, to better assess the TO potentiality as concept design tool. Imposing the same load and constraint conditions (connected to the grey parts of non-design space) of Test 1, the optimization has been launched looking for the minimum achievable mass. Figure 12 shows the results. They refer to an 83% of mass reduction, starting from the CAD value of 0.14 kg. This leads to a final mass of kg. In both tests, the TO finds improved solutions. Nevertheless, Test 1 mass represents a shape refinement not a TO, since it does not change the geometric connectivity of the component.

14 Fig. 12. Test 2: Final solution Figure 13.a shows the von Mises equivalent stress of the original team s design, computed by FEM made by Ansys, Figure13.b the results of the 2 improved solutions investigated in Test 1 mass and Test 2, respectively (made by Inspire). Obviously, the most stressed areas are those with the greatest amount of material, around the pin hole and at the bolt localizations. The original design shows a stress below the yield stress (about 200 MPa). Similar stress values are found via TO, since red area must be considered outlier values due to stress concentration. Fig. 13. Von Mises equivalent stress: a) Sapienza Corse s model; b) Test 1 mass and Test 2 (with the connector for load transfer not hidden). Figure13.b shows the Von Mises equivalent stress, also for Test 2 (take care that in this case the locking pin seems to be present since it is graphically shown by the connector used in the load definition, as discussed in the previous section). This last result seems to exhibit a more uniform stress distribution since a smoother change of the model boundaries can be found when starting from a larger volume. Moreover, it must be pointed out that stress concentrations are always found near the non-design space at the bolts. This is another confirmation of the necessity of a subsequent smoothing, e.g. rounding, near abrupt changes of section or shapes. It can be seen as a shape and size refinement that is necessary for the next detailed design. 1.5 References for Optimization and TO through Hyperworks/Optistruct Inside the help guides of Optistruct go to: Optistruct/User sguide/design Optimization and check out the topics highlighted in figure 14

15 Figure 14: Topics on local optimization on the Optistruct User s guide These concepts find applications through the tutorials presented and explained in: OptiStruct > OptiStruct Tutorials and Examples: Topology Optimization 1.6 Exercises 1. Can you give example of a design for assembly application? How can be set-up a Boothroyd Dewhurst method? 2. In Design optimization, Iterative approach, local approach, perturbation approach are synonymous. Can you describe the meaning of these concepts? 3. An optimization problem has 3 input variables. In a perturbation approach, how many solver input files must be set-up if 10 iterations are necessary? 4. Using Optistruct make the tutorial related to TO, on the C Clip: OS-2000: Design Concept for a Structural C-clip