Structural Shape Optimization Considering Both Performance and Manufacturing Cost Bill Nadir Advisors: Olivier de Weck and Il Yong Kim September 1, 2004

Bill Nadir, 9/1/2004Slide 2 Outline Problem motivation Manufacturing cost estimation Problem statement Optimization flow chart Example 1 description and results Example 2 description and results Manufactured example Post optimality Discussion and future work

Bill Nadir, 9/1/2004Slide 3 Problem Motivation There is a trade off between manufacturing cost and structural performance –Increased performance generally results from increased part complexity (Identical load and mass)

Bill Nadir, 9/1/2004Slide 4 Manufacturing Cost Estimation Module Manufacturing application: abrasive waterjet (AWJ) cutting Cutting time determined using cut length radius of curvature Manufacturing cost determined using cutting time and overhead cost

Bill Nadir, 9/1/2004Slide 5 Problem Statement Key Parameters Young’s modulus of structural material, E Material thickness for AWJ cutter Initial structural design Design variable scaling factor Definitions

Bill Nadir, 9/1/2004Slide 6 Optimization Flow Chart Gradient- based optimizer Finite Element Analysis Converged? no yes Mfg. Cost Estimation

Bill Nadir, 9/1/2004Slide 7 Example 1 Structural Design and Loading Asymmetric Material: A36 Steel Factor of safety = 1.5 Evenly distributed load 2-D model created using ANSYS elastic shell elements assigned a thickness of 1 cm

Bill Nadir, 9/1/2004Slide 8 Example 1 Initial Designs Four control points are used to determine the size and shape of each of the three holes Three different initial designs used to investigate a wide range of the design space and attempt to find a near- global optimal solution Proprietary ANSYS NURBS formulation used to create holes in ANSYS and MATLAB

Bill Nadir, 9/1/2004Slide 9 NURBS Non-uniform rational b-spline (NURBS) curves are used to define the cuts made by the abrasive waterjet cutter B-spline is a special case of NURBS Bezier curve is a special case of b- spline curves Example: order = 3, 6 control points

Bill Nadir, 9/1/2004Slide 10 Example 1 Side Constraints Side constraints defined to avoid hole collisions with each other and part boundary Restricted design space –Number of holes is fixed –Holes are forced to remain in distinct regions

Bill Nadir, 9/1/2004Slide 11 Example 1 Design Space Results Manufacturing cost and mass trade off evident Results not well distributed –Highly nonlinear objective functions Results not all in correct order –Too few initial designs investigated –Manufacturing cost is a function of radius of curvature as well as cutting length α = 0.2α = 0.8

Bill Nadir, 9/1/2004Slide 12 Example 1 Results Discussion A minimum cost radius of curvature exists This prevents the optimizer from fully exploiting reduction in hole size to minimize mass when mass is dominant in the weighted sum objective function Cutting speed reduction with reduced radius Radius of curvature manufacturing cost minimum

Bill Nadir, 9/1/2004Slide 13 Example 1 Convergence History Algorithm performs well for all considered weighted sum objective functions

Bill Nadir, 9/1/2004Slide 14 Example 2 Structural Design and Loading Simply supported bicycle frame-like structure Same material properties as example 1 Loads applied to simulate real bicycle riding conditions 2-D model created using ANSYS elastic shell elements assigned a thickness of 1 cm

Bill Nadir, 9/1/2004Slide 15 Example 2 Side Constraints Restrictive constraint boundaries Side constraints selected to avoid curve intersections with each other Design space limited to material between joints

Bill Nadir, 9/1/2004Slide 16 Example 2 Initial Designs Three initial designs from different regions of the design space were investigated as starting points for optimization

Bill Nadir, 9/1/2004Slide 17 Example 2 Results Minimize M Minimize C man

Bill Nadir, 9/1/2004Slide 18 Manufactured Examples Selected design solutions manufactured using abrasive waterjet Manufacturing cost model verified using actual manufacturing cost results Manufactured part (Omax) $2.91 Cost model (MATLAB) $2.96 Manufacturing Cost Model Validation* *Manufacturing cost results for part shown in figure

Bill Nadir, 9/1/2004Slide 19 Post Optimality Not confident that the global optimum has been found –KKT conditions have not been checked –Tightening objective function and constraint convergence tolerance settings result in improved solutions

Bill Nadir, 9/1/2004Slide 20 Discussion and Future Work Discussion –The consideration of manufacturing cost in the structural shape optimization process has been introduced –The trade off between structural performance and manufacturing cost is shown for two example metallic part structural optimization examples –Currently at a work-in-progress stage – additional work required Future Work –Implement the adaptive weighted sum (de Weck and Kim, 2004) method to help obtain well distributed Pareto frontiers –Include bicycle frame joints in optimization design space to allow for more interesting and larger variety of design solutions –Topology optimization: Include the number of holes as a design variable Include the ability to add or subtract holes from the structural design –Apply methodology to a different manufacturing process such as milling or stamping

Backup Slides

Bill Nadir, 9/1/2004Slide 22 Multidisciplinary Design Optimization The system model contains three main modules, each with it’s own discipline –Structures Finite element analysis (FEA) module using ANSYS software package –Industrial engineering disciplines Manufacturing cost estimation module Abrasive Waterjet Manufacturing FEA Visualization

Bill Nadir, 9/1/2004Slide 23 Optimization Algorithm Selection A gradient-based optimization algorithm was used to solve this problem –MATLAB function fmincom.m SQP algorithm Finds the constrained minimum of a function of several variables Why was this algorithm selected? –All design variables are continuous –Computation time is an issue –Relatively easy to integrate with MATLAB system model modules

Bill Nadir, 9/1/2004Slide 24 Manufacturing Cost Estimation Linear Cutting Speed Cutting speed, u, for a linear cut, is predicted using the following semi-empirical equation published by Zeng et al. in –u= the cutting speed (mm/min or inch/min) –f a = abrasive factor: value of 1 for garnet abrasive (known) –N m = machinability number: depends on material being used (known) –P w = water pressure: 40 kpsi (MPa or kpsi) (known) –d o = orifice diameter: 0.014” (mm or inch) (known) –M a = abrasive flow rate: 0.75 lb/min (g/min or lb/min) (known) –q= quality level index: input by user (known) –h = workpiece thickness: input by user (mm or inch) (known) –d m = mixing tube diameter (mm or inch): 0.030” (known) –C = system constant (788 for Metric units or 163 for English units) (known) © 2002 by OMAX Corporation ( 4 Zeng, J., and Kim, T., MECHANISMS OF BRITTLE MATERIAL EROSION ASSOCIATED WITH HIGH-PRESSURE ABRASIVE WATERJET PROCESSING: A MODELING AND APPLICATION STUDY (JET CUTTING), Ph.D. Thesis, The University of Rhode Island, 1992.

Bill Nadir, 9/1/2004Slide 25 Manufacturing Cost Estimation Cutting Speed Variation Equation for linear cutting speed prediction is modified to predict cutting speed for arc sections –The quality factor, q, is modified in the cutting speed prediction equation to account for the change in cutting quality based on the geometry of the cut being made –The modified q value is then plugged-into the linear cutting speed prediction equation to result in a cutting speed prediction for corner and arc cuts Arc section cut: A = Path angle change (sharp corner cut) E = Error limit R = Arc cut radius h = Thickness of material being cut © 2002 by OMAX Corporation ( Arc section cut speed

Bill Nadir, 9/1/2004Slide 26 MATLAB Cost Model Radius of Curvature Calculation The intersections of perpendicular lines to each pair of segments of the spline curve are assumed to be the center of a circle with a radius of the radius of curvature of the spline curve at a given point R1R1 u1 u2 (x1,y1) (x2,y2) R2R2 (x1c,y1c) (x2c,y2c) u3 (x3,y3)

Bill Nadir, 9/1/2004Slide 27 Manufacturing Cost Module Validation Module results were compared with Omax AWJ CAM software to verify the accuracy of the results Results agreed well Overhead cost for Aero/Astro machine shop AWJ cutter assumed