Published on: **Mar 3, 2016**

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- 1. ME597 - Summary of Work IUPUI Zach Grey Application of Robust Topology Optimization for Space Exploration Vehicle (SEV) Mechanisms NASA Space Grant IUPUI Zach Grey Andres Tovar 06/27/2013
- 2. ME597 - Summary of Work IUPUI Zach Grey I. Generalized SEV Problem The generalized problem statement is to develop a method which accounts for the kinematic mechanism design considerations of Space Exploration Vehicles (SEV’s) which are subject to variation in environmental conditions and load conditions. The primary motivation of studying SEV’s is to relate the mechanism designs to topology designs of compliant mechanisms so the inherent system “stability” of transitioning various loading conditions is retained. The motivation in this research is to develop a method which captures the nature of the uncertainty in the mechanism designs. For example, in the design of Rover transmission mechanisms, the uncertainty in loading conditions (both magnitude and orientation) is of particular interest given the variations in the environment and obstacles the vehicle must maneuver. This particular research focuses on discussion a technique for using multi-objective solutions to educate the designer about the robustness of certain compliant mechanisms which can be related to the kinematic mechanisms. For simplicity, a typical academic problem (simple inverter) is used for demonstration purposes. II. Concept Topology optimization encompasses a variety of methods used to develop solutions for material distributions in defined design domains with associated boundary conditions and loading conditions. In particular, a popular method for the development of such solutions is the power-law or SIMP (Solid Isotropic Material with Penalization) approach. The development of the method is accredited to Bendsoe 1989; Zhou and Rozvany 1991; Mlejnek 1992. The basic technique employed in the SIMP approach is the penalization of relative material densities in the element mesh. Penalization is accomplished by means of scaling material density by a selected power that is considered physically permissible. The solution technique has been employed successfully in a multitude of designs involving multiple constraints, multiple physics and multiple materials (1). The robustness and simplicity of the SIMP approach make it ideal for educational and research purposes. The application of topology optimization for use in the design of mechanisms is a popular application of the study. Specifically, compliant mechanisms can be designed using a topology optimization approach by means of a sub set of input and output load cases. Such compliant mechanisms transfer input force or displacement to output force or displacement by means of elastic deformation. For simplicity, the deformations are considered simple linear elastic transformations with small magnitude. An example of a symmetric inverter compliant mechanism design is shown in Figure 1: Simple Inverter Design. Figure 1: Simple Inverter Design The plane of symmetry in the boundary conditions shown above for the design domain (Ω) is across the top boundary “y” direction constraints. The resulting compliant mechanism maximizes the Mutual Potential Energy (MPE) of the system which effectively maximizes the geometric advantage (maximum output displacement for a given input displacement). This solution is a replication of work described in the text Topology Optimization Theory, Methods and Applications by Bendsoe, M.P. and Sigmund, O.
- 3. ME597 - Summary of Work IUPUI Zach Grey The relevance of the described compliant mechanism and the objective of the research in this instance relates to the development of solutions that satisfy a greater degree of robustness in design. The inherent uncertainty in space craft missions demands the need for robust solutions. Thus far, the proposed solution above has been studied using a quasi-robust solution involving the development of a multi-objective tradeoff study. The multi-objective trade off study that has been proposed will investigate the feasibility of defining the sub set of solutions which satisfy minimum conditions between various weighted objectives for differing input/output load conditions. The intent is to study and understand the behavior and form of the solutions in an attempt to expand the results for replication with typical SEV boundary conditions and load conditions. III. Quasi-Robust Multi-Objective Solution The basis of formulating the fundamental multiple load case problems for use in the design of compliant mechanisms is shown below in Equation 1 (1). Equation 1 max n i ioutiuwc 1 10 0 :.. min 1 e N e ee Vv r ts In practice, this problem is solved by incorporating springs with stiffness to simulate the presence of linear strain based actuating systems. Graphically the solution to the simple linear elastic compliant mechanism known as the inverter can be formulated as shown below in Figure 2: Graphical Formulation of Simple Inverter. Figure 2: Graphical Formulation of Simple Inverter The solution for two load cases in the example shown is broken into two 2D linear elastic finite element solutions. Compliance is then summed over both solutions and the appropriate optimization algorithms
- 4. ME597 - Summary of Work IUPUI Zach Grey can be used in the methods discussed for solving minimum compliance (or maximum mutual potential energy). The formulation in Equation 1represents a simple weighted sum of output port displacements where iw represents the significance of a single displacement. It has been described (1) that the problem formulated above in Equation 1 is very similar to the minimization of end-compliance such that the problem can be reformulated for numerical analysis according to Equation 2 below (2). Equation 2 min iei T e k i N e eei k i ii T ii ukuEwKUUwc 0 1 11 10 :.. 0 1 e N e e vf V v ts In the formulation above, the “n” total indices have been replaced by “k” such that the new “k” total indices represent individual objectives of two separate compliant mechanism problems. The volume constraint has been reformulated to satisfy user constraints of the required volume fraction and the residual constraint has been nested in the problem formulation by means of assuming a linear system. A graphical representation of the formulation for the multi-objective problem is shown below (“A” has been replaced with the nomenclature “w”): Figure 3: Multi-Objective Problem Formulation The intent behind developing the formulation for a problem with varying sets of input and output load cases is the motivation to understand uncertainty in the loading conditions themselves. Even for this simple problem, the inverter topology will have a tendency to change significantly as the sets of input/output load cases change across the boundaries of the problem. In order to quantify the uncertainty behavior, the suggested formulation for obtaining a multi-objective solution by means of weighted sums is used to explore an uncertainty domain. One of the first fundamental steps in any robustness assessment is to quantify input values uncertainty in order to model the appropriate levels of input variation to the system. In the case of designing compliant mechanisms, the sets of input and output load cases are assumed to have variation in location along the design domain boundaries. For simplicity, the assumption is made that each set of load cases remains
- 5. ME597 - Summary of Work IUPUI Zach Grey symmetric across the design domain (paired symmetric load cases). Also, uncertainty in the magnitude and orientation of the load are not study in this instance. Graphically, the uncertainty in the sets of load cases can be represented by Figure 4: Uncertainty Assumptions for Sets of Load Cases below. Figure 4: Uncertainty Assumptions for Sets of Load Cases In order to combine the multi-objective compliant mechanism design with the input parameter uncertainty quantification a discrete uncertainty domain must be defined. The uncertainty domain is used to set discrete probabilistic limits on the input values such that the input domain represents an acceptable level of capability for the system. In the simple inverter example, the sets of load cases for the multi-objective problem are defined at +/- 3 sigma limits (~99.73% probability of observation) to represent extremum conditions. Alternative domains could be designed for use with methods of defining quadrature of input distributions or sets of known discrete input conditions. The uncertainty domain is then used to assign passive active density elements across the boundaries to represent an acceptable domain for unknown sets of input/output load cases. Once appropriate sets of input/output load cases are defined, the multi-objective formulation can be solved using a variety of optimization methods. For the purposes of replicating the work discussed in the text, the simple inverter problem with uncertain location of paired sets of load cases is solved using the SIMP methods described in the selected reference (1). After running 345 solutions of two weighted objectives for the simple inverter problem with two sets of input/output load cases at the input extremum values the Pareto behavior shown below is found.
- 6. ME597 - Summary of Work IUPUI Zach Grey Figure 5: Pareto Behavior of the Simple Inverter Problem The Pareto behavior gives particular insight into the robustness of the topologies. Developing a relationship between the penalized compliance provides insight into the convexity associated with the problem. In particular, this solution provides a description of how deflections at the output node are maximized with minimum compliance. The three topologies, shown in Figure 5: Pareto Behavior of the Simple Inverter Problem, detail solutions for three different locations on the Pareto front. Point “1” represents the optimal design for the first objective function. The first objective function is based on a topology solution for the typical inverter problem in which the applied set of load cases is at the top of the design domain boundary. The topology for the solution at point “1” resembles the expected topology for a typical simple inverter similar to the topology shown in Figure 1: Simple Inverter Design and hence provides a sense of validity to the model. The primary difference between the solution that was developed and the simple inverter design is the mesh size and aspect ratio which will produce slightly different solutions based on the input volume fractions. Point “3” shown in Figure 5: Pareto Behavior of the Simple Inverter Problem is the topology solution that minimizes the second objective function of compliance and maximizes deflections for the second set of load cases. In this case, the second set of load cases has been moved to the middle of the design space based on the uncertainty domain that was established by completing the uncertainty quantification of the input parameters and picking points to run at the extremum. Point “2” represents a single solution which falls on the Pareto front and is a good representation of the type of geometry that would satisfy both objective functions. In particular, point “2” represents the topology for the design point that is closest to utopia (the best compromise between both objective functions). This solution satisfies the intuitive understanding that a robust topology which is expected to support loads across the entire uncertainty domain (be insensitive to input variations) must retain a combination of topological features present in both of the extremum solutions. This is illustrated by the arms which clearly extend to both ends of the uncertainty domain. One particular refinement is related to the convergence criteria used for the topologies presented. Convergence is difficult to measure in the generation of topologies for compliant mechanisms. It would be advantageous to refine the convergence criteria for the specified problem to further improve some of the features of the topologies which may still be related to attributes of local minima.
- 7. ME597 - Summary of Work IUPUI Zach Grey The Pareto frontier and more generically the multi-objective solutions are advantageous for capturing robustness in topologies because they provide a more complete description of the convexity and the explicit relationships between outputs. This is of specific importance in the design when a single objective (or a set of transient objectives) has interactions with other sets of loading conditions and robustness is required across the system. Exploring the behavior of the Pareto frontier can then be used to make a true assessment of robustness by fitting an appropriate curve to develop a mathematical description of the tradeoff relationship. For example, a polynomial which minimizes the least square problem for the all or a subset of the 345 points representing the Pareto behavior could be used to describe the curvature in this instance. The Figure 6: Pareto Curvature Behavior below shows one approach to capturing the curvature behavior in the Pareto front. Figure 6: Pareto Curvature Behavior f2(f1) Figure 7: Pareto Curvature Behavior f1(f2) The 7th and 8th degree fits represent polynomials of equivalent order which describe the general trend of the Pareto curvature behavior. An assessment of Robustness can then be determine by solving an alternative optimization problem to find the minimum squared sum of slopes from the two complementary polynomial relationships either separately or jointly. The point which then minimizes the combined slope behavior of both objectives could be used to define the most robust solution. IV. Summary & Continued Work 1. Implement a method for interpolating solutions along the Pareto to develop a subset of continuous solutions that would define the morphology required in a kinematic mechanism a. Replicate the Pareto frontier behavior for comparison b. Supplement the design of a kinematic mechanism 2. Refine convergence criteria to further improve the bounds of the Pareto fronteir 3. Update boundary conditions to use single input load ports and distributed load output port to study the differences in topology using paired loading conditions 4. Run stochastic modeling assessment of the various designs and compare levels of capability (robustness) using a standard approach to hypothesis testing 5. Summarize results of robust Pareto locations using the suggested method of minimizing first order objective function curvature behavior
- 8. ME597 - Summary of Work IUPUI Zach Grey 6. Incorporate a method for transitioning topology solutions to kinematic mechanisms 7. Update boundary conditions to replicate real SEV designs 8. Update methodology to 3D
- 9. ME597 - Summary of Work IUPUI Zach Grey V. References 1. Bendsoe, M.P. and Sigmund, O, Topology Optimization Theory, Methods and Applications, Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2003 2. Andreassen, E Clausen, A Schevenels, M Lazarov, B. S. and Sigmund,O, Efficient topology optimization in MATLAB using 88 lines of code, privately published manuscript (Noname manuscript), http://www.topopt.dtu.dk/?q=node/751