Institute of Computer-aided Product Development Systems

Optimizing Product Design using Response Surface Methodology

D. Roller , B. K. Ane

Variational geometry is often used in the application of computer-aided geometric design (CAGD) systems for applying dimensional constraints to free-form geometry in order to solve geometric constraint problems, to generate family of shapes, and to allow flexible modifica­tions of dimensions.
During design optimization, major engineering effort is directed towards parameter design in order to create a robust design. One of the well-known techniques for robust design is the so-called response surface methodology (RSM). This technique is used to identify critical design parameters, as well as to select optimal settings and to find compromise in the conflicting requirements.

Problem Definition

Amago (2001) developed First Order Analysis (FOA) on the basis of RSM as a new concept of Computer-Aided Engineering (CAE) which offered optimal calculation through the use of topology and size optimization.
Unfortunately, since the effort of Amago (2001) no research activity has been organized to improve the capability of RSM. The capability to depict the responses is still limited within E 3 Cartesian spaces which consist of two treatment factors. Analyzing response function consisting of more than two factors leads to a problem of excessive number of observations, as well as inability to visually analyze the performance response. Furthermore, as the number of factors understudy increases, there is an increasing probability that the fitted response does not attain equal precision at points that are at equal distances from the centre of the factor space. These problems create a possibility to fall into the trap of ‘local optima’.

Research objective

Methodology

Generally, the research project is organized through four main stages:

Stage 1: Experimental design

The objectives are to develop an orthogonal and rotatable design of an orthogonal array (OA) L 9 (3 4), and to construct a fundamental concept for a 5-axes response surface graph based on rotatable OA L 9 (3 4). An orthogonal array, or Taguchi Method, is a typical design of experiments where the columns for the independent variables are “orthogonal” to one another.

Stage 2: Mathematical modelling

The objective is to improve reliability in mapping the surface in the vicinity of the optimum through a set of numerical models development based on the method of steepest ascent.

Stage 3: Geometric modelling

This stage consists of three consecutive processes:

a) Points identification process

Points identification is done based on a transformation algorithm, which functions in translating the factor-levels and responses into a coded scale, as well as to identify spatial coordinates of the surface in a 5-axes spherical space.

unrotatable OA L 9 (3 4)

5-axes rotatable OA L 9 (3 4)

Figure 1. Basic Concept of 5-axes Response Surface Graph

b) Fitting Process

Surface fitting is done using deBoor Algorithm based on B-spline curve as a generalization of Bezier approach which uses blending functions to combine the influence of a series of parametric knots in an approximate curve that provides “local control” of the curve shape.

Figure 2. Application of de Boor Algorithm

A note: P_i¹ is calculated from P_i⁰ 0 or the control points, P_i² is calculated from P_i¹, and so on up to P₀ⁿ is obtained, then, the t P₀ⁿis the value of P(u).

c) Skinning Process Surface skinning is done on all of E 2 and E 3 arbitrary or regular profiles based on B-spline surface approach.

(a) Surface Skinning

(b) Fitted Surface

Figure 3. B-spline Response Surface

Note: The B-spline surface approximates a characteristic polyhedron and passes through the corner points of the polyhedron, where its edges are tangential to the edges of the polyhedron.

Stage 4: Simulation and Programming

The objective is to synthesize the whole points identification, fitting and skinning algorithms into an integral geometric modelling system for 5-axes response surface graph and to translate into a prototype computer simulation program using high-level programming language.