Shape Optimization

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, .... Furthermore the use and combination of optimization tools and innovation ...
Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

Shape Optimization Amir Moasvi, The University of British Columbia, 2329 West Mall, Vancouver, BC V6T 1Z4, Canada

Summary Design of profiles is important as by applying further surface design tools of CAD such as extrude, lofting and sweeping almost any shapes can be reached. Profile design is the foundation of shape design and has wide application in different disciplines of engineering. As long as the splines have found to be the best choice for modeling the fine, smooth and accurate profiles and furthermore can easily substitute the original profiles of the initial shape, the optimization the splines has got importance. In order to invent a general strategy for getting the optimal geometry of the profiles there have been many researches on optimization the spline, which is a multiobjetive and highly non-linear problem, but we haven’t reached the goals of an automatic and high performance design process yet. In this regard this paper aims to widen the awareness of the readers of the effective application of medeFRONTIER in optimization the splines. The combination of modeFRONTIER and splining is introduced for developing the profile design procedures which use CAD and CAE tools as an interface to the designer and splines for profile construction. After an introduction to multiobjective optimization (MOO) problem of the splining, the foundations of an automated engineering design environment are briefly presented. Then the former GA-based approach to splined profile optimization is reviewed. In following the selective approaches to profile design and surface deformation utilizing the general strategy of modeFRONTIER are reviewed and its benefits in developing mechanical design procedures are described.

Keywords Automated engineering design environment, profile design, modeFRONTIER

Introduction The discipline of computer aided geometric design (CAGD) deals with computational visual effects of geometry [1]. Several disciplines have interacted with CAGD e.g. computer graphics, computational geometry, solid modeling. Furthermore CAGD has also influenced fields such as medical imaging, geographic information systems, computer gaming, and scientific visualization.

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

The standard approach to surface design has been to design a network of curves and build a surface to cover the network utilizing computer tools. However currently many CAD software systems exist for this purpose, employing standard techniques of surface design on the basis of the profiles [1] [2] [3] [10]. As far as the 2D geometry of profiles is concerned, one of the major issues of CAGD applications is how to automatically reach to optimal curve shape using nonstandard data which is not ordered in a convenient way. Yet it depends critically on the designer, aesthetic stylists and manufacturing engineers [2] [3]. The major achievement in CAGD has been the theory of Bezier curves and surfaces which later was combined with spline methods. Utilizing the splining methods allows smooth shape changes via the control points. This characteristic cannot be achieved by arcs because they have very low degree of freedom. Instead splines easily substitute the original profiles of the initial shape. When a profile design cannot be based on features defined, optimization provides a tool for automatically achieving a desired geometry using limited design information. The essence of the method is to choose a single or multiple functions, called an objective function, whose value is determined by the control points of a spline. Then each objective function must attain a minimum or maximum value when the shape variables assume values that correspond to the desired shape. In order to find the optimal value for an objective function, one must solve simultaneous equations. Solving equations generally requires too much computer time, often hours of runtime, and sometimes no suitable solution is actually found [10] [11] [12]. In this regard GA has been seen as a solution for dealing with such complexity in CAGD. A survey by Gabor and Aniko [4] devoted to this topic. Goldenthal et al. [5]-[7] in 2003 utilized GA for dealing with MOO problems of curves and surfaces design. In their studies the geometry has been optimized respecting the design objectives i.e. approximation errors, elasticity energy, length and area. Although still there is not any straight solution for the MOO problem of curves. however because of the complexity of MOO problems; mainly nonlinearity, caused by multiple conflicting objectives, CAGD optimization has generally focused on simpler application problems with less objectives which can be solved by linear methods. Yet this has been an automatic technique for generating shapes [12].

The Automated Engineering Design Environment for Profiles An automated environment tries to push designs to reach the optimal solutions. Consequently, an efficient design can be achieved from an improvement process. In particular, the ability to apply automatic changes into our 2D design environment is linked to the concept of computer aided innovation (CAI), allowing for an exploration of a broader field for possible solutions to a design problem. Generally the conceptual framework of CAI includes evolutionary design and decision making [9]. An automated engineering design environment uses integrated CAD/CAE tools for providing support to the process in generating variants, simulations and decision making. This support, can improve the performance of the concepts by generating alternative solutions to optimization problems. Shape parameterization, evolutionary design process and optimization system can be considered as the foundations of creating an automated engineering design environment. An automated spline-based engineering design environment can guarantee the design efficiency of the different disciplines of engineering e.g. marine, appliance, multibody, crash, structural, vibro-acoustics, turbomachinery, civil engineering and aerospace.

Evolutionary design process The application of advanced computation methods in generating the optimal design is around since last three decades [12]. However a new area of development called evolutionary design has recently become a topic of intensive research. According to Bentley [11] evolutionary design process is capable of generating designs by optimizing the geometry. The ability of combining CAD and CAE which has been powered by managing the advanced computer science, geometric parameterization, design and evolutionary biology is well utilized in this application. Additionally the integrated CAD/CAE design method presents characteristics that add value to the product by creating the novel shapes which deliver higher performance.

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

Evolutionary design approach via experience and judgment leads to better profile design. In its most reviewed applications [5-19] judgment has been done by evolutionary algorithms -which are mostly GAs- when evaluating a fitness function and comparison against certain criteria.

Shape parameterization Optimization the existing designs by parameterization described and studied in study cases of Williams [15], Olhofer et al.[16], Obayashi et al.[17] and Sheriff [19] is the first type of evolutionary design and nowadays has widely application. In the mechanical field, geometric parameterization has been utilized to define the described changes by design variables. The methods used by the design developers of evolutionary design systems vary. However the spline parameterization approach [8-18] has the potential to be classified as creative where the shape optimization task is converted to a parameter value optimization task by using spline-based curves for profile representation. Furthermore it has been found that the spline method of parameterization is beneficial because its computational implementation is efficient and free of problems with numerical stability and parameterized splines allow smooth shape changes via the coordinates of their control points. Yet it is advantageous that the degree of the curve and the number of control points can be selected independently in order to satisfy curve smoothness and continuity for curve shape modifications.

Design optimization system A profile design optimization process is utilized as the basic of surface design for seeking optimal shapes of product geometry. The geometry of profile is defined in terms of spline parameters which define the external border of the product and allow more freedom to manipulate, see figure 1. Then optimization process is done by changing these external borders [9]. Typically, the process of optimization the splined profile is a MOO problem in a heavily constrained environment. In this regard the goal of MOO is to optimize a vector function whose elements are the objective functions often in conflicting each other [17] which is a difficult task to deal. The possible solutions of these problems are represented in a diagram called Pareto-solution diagram [12]. To solve this, a decision must be made by exploring the achievable limits of each particular attribute of parameter to find an ideal alternative. For this reason GAs are viewed as a mechanism for embedding the innovative principles in a CAD interface. Furthermore the use and combination of optimization tools and innovation capabilities is intended to provide a means for automatically varying the geometry from the evaluation made by CAE systems.

Automatic Optimization the Splined Profile by GA Engineering design community desperately needs an effective and automatic environment in surface deformation and design applications. In this regard there have been lots of researches so far and some methods, mostly GA-based, have been introduced. However there are also other methods for exploring the design space for shape optimization available [28, 35, 36, 37]. But still GA-based methods maintain their application because of benefits mentioned in [4, 7, 12 and 13]. Accordingly by using GA, it is possible to optimize the geometry and additionally automate the design process. In fact, the design automation is the main motivations for developing the GA-based approaches. GAs can be utilized to automatically produce alternative profile shapes for the simulator, and finally to evaluate the shapes on the basis of the simulator output data. The splining approach combined with GA-based optimization [5-19] is relatively new. The splined profile and its codified control points by genetic algorithms, form the basis of an evolutionary designed process. Furthermore a range of efforts [8- 10] have been applied so far for improving the strategy of splining approach. Lampinen [18] overviews such this approach for preliminary optimization of concept profiles for dynamic mechanisms. He creates a complete systematic approach for preliminary 2D designs with respect to the simulated computer models. A cam shape profile was given as an example to illustrate the proposed workflow process and its effectiveness. Albers et al. [8-10] also utilized similar strategy to develop an engine crankshaft. In these cases the 2D cross sections of the concept are represented by splines, which make it possible to freely express the shape in parametric forms.

Workflow The spline control points during optimization process are called floating-points which are actually variables for optimization. The splined profile is defined with a number of floating-points valued as parameters, see figure 1.

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

The idea is to convert a splined profile optimization task to a parameter value optimization task. Same as the presented general workflow in [5-19], parametric CAD software is automatically manipulated by the GAs via interface software. This interface allows the CAD software to run continually and get saved in the computer memory, therefore every time a solution is generated the geometry automatically adapts to the set of parameters. The process starts with an existing design, substitutes the current construction with splines and adds control points. The spline is modeled inside the tolerances of the original shape’s profile but changes during the development process. The floating points of the splines which are subjected to improvement are parameterized. A single coordinate of the spline floating points (for instance Y coordinate as in [18]) are encoded as genes. In other worlds each gene represents one floating-point of the spline curve. Three main genetic operators act on the genes of the geometry are selection, crossover, and mutation. Crossover allows the geometrical characteristics of selected splines to be merged in pairs and their properties to be extended to following generations. The crossover and mutation are responsible for generating new alternative shapes by altering the organization of floating-points.

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Figure 1: a: initial product (existing design) subjected to optimization b: digitizing process c: point cloud d: substitutes the current construction with splines, adds control points and parameterization the spline e: splined shape and parameterization the surfaces f: automatic optimization environment for profiles and covering surfaces g: optimal shape

Each individual of the population describes one complete concept shape with constant number of floating-point values in a parametric form. So, a chromosome, composed of many floating points valued genes, represents each individual shape. The GA-based optimization process attempts to find a series of shapes which satisfy the design objectives and meets all constraints. The objectives of the analysis are to develop the geometry in order to obtain the optimal results of emphasized CAE simulations. The objectives are introduced into the CAD and automatically provide the value of the fitness function. Individual shapes, represented by a vector of constant number of control points, will be evaluated with this fitness-function which is automatically updated every time the geometry is modified. When evaluating a fitness function, GA relies on judgment, based on evaluation and comparison against certain criteria. Yet it is supposed that experience and judgment the new shapes created by floating-point sets can lead to a good design.

Improving the workflow There have been lots of efforts by researchers for the reason of approaching the optimal geometry of splines via an automatic design environment in order to speed up the optimization process. As the result it was witnessed that the quality of optimized designs utilizing GA-based optimization was improved in comparison with the conventionally designed counterpart concepts. Obviously there are number of drawbacks to the process too. For instance in the applied workflow utilizing GA there is a weakness in handling numerous objectives which, makes the optimization process of more than two objectives very complicated. The other drawback is the complexity of workflow in setting a new project which might engineers couldn’t work with it properly. However it is expected to improve the process by satisfying some factors such as:       

Better defining the target value of CAE objectives  Maintaining the limitations concept design and manufacturability  Developing the interface for integration of the CAD, geometric evaluations, CAE and optimization  Complementing the innovation principles to apply to the geometry. 

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.



Obtaining new design concepts for the shapes 

In this regard lots of efforts have been done for the improving the whole process in terms of ease, automation, effectiveness, etc. As mentioned earlier the profile design is typically a multiobjective optimization problem in a heavily constrained environment. The objective functions are also highly multimodal and non-linear. For this reason it is expected to utilize more effective MOO algorithms, decision support tools, data assessment and sampling tools. In the next chapter modeFRONTIER is introduced as an improved design and optimization facility which can be completely responsible for an automatic design environment. This facility brings together techniques which have their origins in the fields of optimization, applied in [5-19], and new tools for innovation.

Splined Profile Optimization Utilizing modeFRONTIER modeFRONTIER is a multiobjective optimization and design environment, which can easily couple almost any CAE and CAD packages. One of the success reasons of it is the utilizing lots of optimization algorithms and tools including response surface modeling tool, MOGA, NSGA, NASH and B-BFGS in a hybrid form instead of a single algorithm. It acts as a robot controlling the design process [27]. Therefore the role of the users is limited to create a parameterized model, and to specify the objectives, which they wish to attain.. modeFRONTIER modifies the design variables till achieving the user-specified objectives. The use of it is put forward as a mechanism for mediating conflicts. The further modeFRONTIER extensive set of modules are listed as follow:         

Provides Design of Experiment (DOE) techniques  Includes standard applications such as Excel and Matlab  Robust design optimization  Multi Criteria Decision Making (MCDM)  Statistical analysis tools 

In order to be able to easily design and effectively optimize the complex 2D profiles the novel method of combined splining and modeFRONTIER is suggested. It is assumed that spline can deliver extraordinary results in an automated optimization environment such as modeFRONTIER. There have been some experiences in this regard too. Although according to author's knowledge there is not enough description available regarding the details of coupling splining and modeFRONTIER. However automatic shape optimization on the basis of the solid modeling tools is one of the well-known applications of modeFRONTIER. In most of the shape optimization cases utilizing modeFRONTIER, the shape has been modeled and parameterized by solid modeling tools. Lung design [29], MEMS design [38] and ball grid array design [39] are few examples in this regard. The applied strategy of modeFRONTIER in optimizing the spline has been around since 2003 in different applications including profile design. Pinto et al. [21] in optimization the profile of connective wavy channel of a heat exchanger utilize this strategy for modeling, parameterization and optimization. Applying splines for 2D periodic channel and further geometrical parameterizations presented better deformation to the shape though with more variables. With the aid of spline, lots of different possibilities were generated and the optimal geometry of profile applying modeFRONTIER was achieved. In the other project presented by Ciprian et al. [33] the geometry profile of a transonic airfoil with uncertainties has been optimized. For parameterization the upper and lower sides of the profile a Bezier curve, very similar to spline curve, has been utilized. The coordinate of their control points are the variables of optimization. The optimization goal was to find out an airfoil geometry which yields better results respecting to the performances and stability. Further applications which are reviewed below [20, 22 and 23] aren’t devoted directly on profile design but utilize same method and confirm the suitability of the applied strategy. For instance optimization and selecting advantageous ship routes on the basis of hydrodynamic simulations [22 and 23] which can also be considered as the early application of modeFRONTIER in optimizing the splines. The ship routes are actually in the form of open curves and easily are modeled by splines. In these studies high number of variables was manipulated and

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

possible solutions were introduced as a graph of Pareto optimal solutions. Furthermore Rousselon [20] aimed to increase the performance of sails by utilizing same tools. Splines have been applied for modeling via CAD, where each sail is parametrically defined as splines by control points. In order to have access to a wide range of high-performance optimization algorithms modeFRONTIER was applied. The application of utilized tools in the real proposed cases has been found to be robust, accurate and successful. The approach to MOO of splines and the benefits of the Pareto optimal solutions in identifying conflicts are presented in [25]. The role of splines in this context is found to be closely integrated with modeFRONTIER in enabling this development on a CAD&CAE software interface, and in enabling automation of the development. The optimization procedure utilizing modeFRONTIER freely explores a wide range of possible geometries. Dominique and G´abor [25] developed a flexible spline-based reconstruction technique utilizing mofeFRONTIER to reconstruct a distribution. The moment computation and the spline interpolation needed for a given set of control point positions are performed by Matlab. The modeFRONTIER easily was coupled with Matlab and the NSGA-II algorithm was applied to adjust the control points. It is concluded that optimization may support the development of an even more efficient procedures.

Surface Deformation Utilizing modefrontier In this part the application of modeFRONTIER in Surface design is reviewed. Surface design along with the confirmed profile design strategy would deliver a complete tool for 3D surface design. The potential of combination of modeFRONTIER and surface deformation methods for automating the process of optimization and design has been already proven. Surface deformation methods are divided in two different parts; mesh-based methods and CAD-based methods. modeFRONTIER is able to manage the optimization process with both methods. However the latter methods need less computation efforts. The combination of modeFRONTIER and meshing packages for surface deformation has found to be an ideal tool in CFD-based shape optimization problems. Paul and Mark [31] optimized an IP and console duct surface utilizing this tool. The focus of their study was to invent a procedure that accelerate the optimization process by utilizing arbitrary shape deformation (ASD) technology [32] coupled with CAE via modeFRONTIER. The ASD volume is defined by control points. It is the movement of these control points that changes the geometry of the ASD and the dependent mesh. Control points that are moved to deform the mesh are defined as design variables for the optimization process. The process of ASD has been mostly facilitated by mesh-deformation package of Sculptor. The combination of modeFRONTIER and Sculptor is in interest of researchers who look for flexible solution to optimize shapes with minimum parameters but great freedom, without involving CAD and mesh-generator software in the optimization loop. Additionally research group of Prof. Siniša Krajnović [34] utilized this coupled tool for shape optimization and active flow control for improved aerodynamic properties of car and fast train. However a drawback to the application of mesh-deformation in generating new geometry of shapes would be the low accuracy in modeling the complex shapes. The Study cases of CAD-based methods of surface deformation [26, 30, and 31] in optimization the complex 3D shapes utilize surface modeling tools integrated with modeFRONTIER. Cooling duct shape optimization [26] is one of the early documented examples which show the effectiveness of the method in 3D shape design. However in this study the subjected shape, because of the simplicity of its cross sections, has been modeled by surface sweeping tool based on the simple polygonal profiles. It was aimed to show the benefits of modeFRONTIER in design process, by using parametric models and distributed computational resources for flow analysis. The fast and easy integration of CAE packages in modeFRONTIER allows improvement in the geometry of the shape. The duct geometry was modified through surface parameters respecting the CFD-based objectives. Moreover robustness analysis was what modeFRONTIER offered on proposed solutions. In [30] in order to enable parameterized surface based shape modifications which can immediately be used in the CADbased surface deformation and design process, CatiaV5 as a powerful CAD package is deserved as a tool for managing the geometry via effective surface modeling tools.

Conclusions and Future Works CAI uses modeFRONTIER to provide support for the design process by letting an integrated computer tool of CAD and CAE takes part in ingenerating variants, simulations and decision making. This support, has improved the performance of the concepts by generating alternative solutions to optimization problems.

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Amir Mosavi, Shape Optimization, International modeFrontier Conference, 2010, Trieste, Italy.

The system of geometric shape optimization offer hints about the next generation of optimization tools. Yet in order to have a general and automated design procedure, the design environment must be effective in both profile design (2D) and then surface design (3D). It is concluded that modeFRONTIER can be highly effective in both fields. It was reviewed that surface modeling tools of powerful CAD packages or ASD facilitated by mesh-deformation packages can be utilized for 3D surface modeling and parameterization in modeFRONTIER environment for a standard and controlled optimization strategy. Furthermore the role of splines in this context is found to be closely integrated with modeFRONTIER in enabling concept’s profiles development on a CAD/CAE software interface, and in enabling automation of the development. The details of workflow and description of the combination of modeFRONTIER and splining for profile design would be the subject of future work which will be presented at the next modeFRONTIER user meeting.

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