Structural Shape Optimization Considering Both Performance and

Manufacturing Cost

Bill Nadir

Advisors: Olivier de Weck and Il Yong Kim

September 1, 2004

Slide 2Bill Nadir, 9/1/2004

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

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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)

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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

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Problem Statement

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Key Parameters•Young’s modulus of structural material, E

•Material thickness for AWJ cutter

•Initial structural design

•Design variable scaling factor

Definitions

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Optimization Flow Chart

Gradient-based

optimizer

Finite Element Analysis

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Mfg. Cost Estimation

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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

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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

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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

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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

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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 = 0.2 α = 0.8

manij

ij CMyxJ 1,

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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

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Example 1 Convergence History

• Algorithm performs well for all considered weighted sum objective functions

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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

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Example 2 Side Constraints

• Restrictive constraint boundaries

• Side constraints selected to avoid curve intersections with each other

• Design space limited to material between joints

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Example 2 Initial Designs

• Three initial designs from different regions of the design space were investigated as starting points for optimization

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Example 2 Results

Minimize M

Minimize Cman

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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

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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

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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

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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

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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

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Manufacturing Cost EstimationLinear Cutting Speed

• Cutting speed, u, for a linear cut, is predicted using the following semi-empirical equation published by Zeng et al. in 19924

– u = the cutting speed (mm/min or inch/min)– fa = abrasive factor: value of 1 for garnet abrasive (known)– Nm = machinability number: depends on material being used (known)– Pw = water pressure: 40 kpsi (MPa or kpsi) (known)– do = orifice diameter: 0.014” (mm or inch) (known)– Ma = 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)– dm = 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 (www.omax.com)

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.

in/minormm/min15.1

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Manufacturing Cost EstimationCutting 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

22

182.0

RER

hq

Arc section cut:

A = Path angle change (sharp corner cut)E = Error limitR = Arc cut radiush = Thickness of material being cut

© 2002 by OMAX Corporation (www.omax.com)

Arc section cut speed

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MATLAB Cost ModelRadius 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

2)1(12

ciiicii xxxxR 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

X

Y

Waterjet Cutting Pattern Based on Control Points

R1

u1u2

(x1,y1)

(x2,y2)

R2

(x1c,y1c)

(x2c,y2c)

u3

(x3,y3)

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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