ME5542 Advanced Modelling and Design AME Mechanical Engineering School of Engineering and Design Brunel University Uxbridge UK Advanced Modelling and Design, Mechanical Engineering, March 2011 1 of 16 W.R. Shipway Design Optimisation – Use of golden section search method in the design of a new removable orthopedic device Conference on the Implementation of New Advances in Engineering Design 2011 W.R. Shipway Abstract The Golden Section Search Method is described, and employed for a number of variables in the design of a new removable orthopedic device. The pertaining variables involved in the method are elucidated, including the boundaries and tolerance, and derivation of the golden ratio. The boundaries and tolerance are discussed in detail and advice is formed and presented on the optimum selection of these variables. The qualities and limitations of the method are discussed, and the uncertainties in applying the method are deliberated. The efficiency is compared with a similar optimisation method (Fibonacci Method– not explained), and it is suggested when to use each method. The results of the design variables in the example application show convergence on the extremum of non-linear functions. Evidence is provided for the (already known) extremum‟s to show comparisons between the method and an exact solution obtained analytically. Engineering applications of the method are summarized, and more general applications are suggested, including economics and manufacturing, relating the use of the method to within an organisation. Excel is employed in the design application, but the use of a Matlab command is suggested as another way of utilizing GSS. Keywords: Golden Section Search Method, design factors, unimodal optimisation, orthopedic device 1. Introduction The (GSS) method is an iterative procedure for reducing the region where a solution to a unimodal function may lie. All iteration methods describe a procedure by which the root(s) of an equation (set of equations) can be found by repeatedly (iteratively) reducing the boundaries that the minimum (or maximum, named here as extremum) may lie. [Yakowitz, 1974, p. 234] The pertaining alterable quantities that define this procedure are the boundaries which the roots lie within, and the reduction method, irrespective of the equation itself. The two defining quantities are intrinsically related, with the boundary being thus reduced at each iteration. The procedure is to define the boundaries, inspect the solution at a (undefined thus far) point at a distance from the boundary, and adjust the boundaries so that the new point becomes a boundary itself. The new inspection point would be in-between the new boundary and the remaining original boundary. Which boundary is replaced is dependent on whether the solution is closer to the desired solution or not (i.e. the extremum).
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ME5542
Advanced Modelling and Design
AME Mechanical Engineering
School of Engineering and Design
Brunel University
Uxbridge
UK
Advanced Modelling and Design, Mechanical Engineering, March 2011 1 of 16 W.R. Shipway
Design Optimisation – Use of golden section
search method in the design of a new
removable orthopedic device
Conference on the Implementation of New Advances in Engineering Design 2011
W.R. Shipway
Abstract
The Golden Section Search Method is described, and employed for a number of variables in the
design of a new removable orthopedic device. The pertaining variables involved in the method
are elucidated, including the boundaries and tolerance, and derivation of the golden ratio. The
boundaries and tolerance are discussed in detail and advice is formed and presented on the
optimum selection of these variables. The qualities and limitations of the method are discussed,
and the uncertainties in applying the method are deliberated. The efficiency is compared with a
similar optimisation method (Fibonacci Method– not explained), and it is suggested when to
use each method. The results of the design variables in the example application show
convergence on the extremum of non-linear functions. Evidence is provided for the (already
known) extremum‟s to show comparisons between the method and an exact solution obtained
analytically. Engineering applications of the method are summarized, and more general
applications are suggested, including economics and manufacturing, relating the use of the
method to within an organisation. Excel is employed in the design application, but the use of a
Matlab command is suggested as another way of utilizing GSS.
Keywords: Golden Section Search Method, design factors, unimodal optimisation, orthopedic
device
1. Introduction
The (GSS) method is an iterative procedure for reducing the region where a solution to a
unimodal function may lie. All iteration methods describe a procedure by which the root(s) of
an equation (set of equations) can be found by repeatedly (iteratively) reducing the boundaries
that the minimum (or maximum, named here as extremum) may lie. [Yakowitz, 1974, p. 234]
The pertaining alterable quantities that define this procedure are the boundaries which the roots
lie within, and the reduction method, irrespective of the equation itself. The two defining
quantities are intrinsically related, with the boundary being thus reduced at each iteration. The
procedure is to define the boundaries, inspect the solution at a (undefined thus far) point at a
distance from the boundary, and adjust the boundaries so that the new point becomes a
boundary itself. The new inspection point would be in-between the new boundary and the
remaining original boundary. Which boundary is replaced is dependent on whether the solution
is closer to the desired solution or not (i.e. the extremum).
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Logically, the most obvious reduction method is to half the area in which the solution may lie
with each iteration, and is known as the bisection method [Press, 1992, p. 353]. It is considered
obvious as any method using a reduction of more than half may yield the solution in the other
region (i.e. 1 – (>0.5)) and thus be less than half. However, the GSS is known to be a more
efficient method of finding a solution iteratively, and was first deliberated in 1953 by Kiefer
[Kiefer, 1953, pp. 502-506]. Kiefer makes no mention of the golden ratio itself, but derives the
interval length similar to as shown in this paper.
The golden ratio is intuitively intriguing due to its abundance in natural phenomena. It is
defined in the context of this paper as (traced back to Phidias [Hemenway, 2005]);
Length of inspecting region
Length of larger interval=
Length of larger interval
Length of smaller interval
If the length of inspecting region is of unity and φ (phi – after Phidias) denotes the length of the
larger interval (so that the smaller interval is 1 – φ) then expressing mathematically,
1
φ=
φ
1 − φ (1)
The general form resolves to,
1
φ=
φ
1 − φ⟹ φ2 = 1 − φ ⟹ φ2 + φ − 1 = 0
With the solution to the quadratic equation found as,
ax2 + bx + c = 0 ⟹ x1, x2 =−b ± b2 − 4ac
2 × 1;
φ =−1 ± 12 − [4 × 1 × −1 ]
2 × 1=−1 ± 1 + 4
2
With the positive solution as the irrational number,
φ =−1 + 5
2= 0.61833988749895… (2)
- Procedure
Consider a function f(x) which follows the form shown in Fig. 1. The points x1 and x2 are at
lengths of the golden ratio from the boundaries (limits) B1 and B2.
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Figure 1. An arbitrary unimodal solution f(x), with upper and lower limits.
The lengths of the arrows can be calculated via different methods, all relating to φ. They are
shown in Fig. 2.
Figure 2. Various ways to calculate the inspection points x1 and x2 with the limits B1 and B2.
Once the initial inspection points have been determined from use of the equations stated in Fig.
2 they can be evaluated and the interval reduced. One of the inspection points will be less than
the second. The larger function root will then be replaced as the new limit. If, as is shown in
Fig. 2, x2 is smaller than x1 then x1 becomes B1 and a new inspection point will be created. This
point will be at a distance of the golden ratio from the new boundary. If x2 were larger than x1
then x2 would become the new boundary and a new inspection point would form at a distance
from x2.
Reiterating mathematically, if x2 < x1 then B1 = x1 with the new boundaries at B2 and x1, and the
new inspection point is x3 = (B2 – x1)(1 – φ). The two inspection points for the second iteration
are now (the same) x2 and (the new) x3, and will again be compared, with the appropriate
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(larger) inspection point being replaced to form the new boundary. Note in Fig. 2 the
symmetrical nature of the distances, so that the distance from B1 = (B2 – B1)(1 – φ) is the same
as the distance x1 from B1 (hence the use of the expression in the above iteration). The new
inspection points will then look as is shown in Fig. 3.
Figure 3. After one iteration the points of inspection will be (the same) x2 and (the new) x3.
Clearly, the reduction has „jumped‟ over the solution of minimising f(x), but as it still lies
within the boundaries the proceeding iterations will converge upon that point.
- Termination condition
At some point the iterations must be concluded, upon which the final value is found and the
decision is taken for that to be the solution. This must be decided upon before the method is
introduced to the problem, and can be as simple as a limit on the number of iterations.
- Applications
The main application of the GSS method may be used when the governing equations are not
differentiable, or when the solution is not easily obtainable, when differentiation proves too
complex practically. Examples include natural phenomenon: activation energy of a chemical
reaction [Cai, 2010], approximate language reasoning interpretations (fuzzy linguistics – a
pressing matter in quality of information) [Leephakpreeda, 2006]; partial differential equations
(PDE‟s): determining the shape of a statistical distribution in the use of mesh free methods for
numerical simulations of PDE‟s [Tsai, 2010]; and more generally in engineering it is applied to
solutions in complex fluid mechanics modelling (non-newtonian fluids [Ohen, 1990] and
thermal conductivity with unknown coefficients [Mierzwiczak, 2011]). But, it can be applied to
all extremum and optimisation methods of a unimodal form with no derivatives.
- A Note on the Golden Ratio The actual golden ratio is φR = 1.618… where φ = 1/φR and is the ratio of the larger to smaller
interval, not the length of the larger interval, and some sources derive the value at 1.618. But
when applied all cited sources should and will use 0.618 in application. This is because to have
a new inspection point at 1.618 times the interval would not reduce the inspection region at all!
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2. Method
As described in the previous section, applications of the GSS method are typically difficult to
solve with classical methods, and therefore the GSS is not utilized for common engineering
problems easily. Here the application of the method is straightforward for two reasons – 1. So
the application is understood to the reader, and 2. Verification is possible. If the application is
complex and “real world” then absolute verification would be complex, time consuming and/or
costly. Also, this application will highlight limitations in the method that may otherwise not be
apparent.
A method of tooth correction is envisaged by a new removable orthodontic design. The design
(shown in Fig. 4) enables the correction of a twisted tooth by applying a load on the tooth. By
applying the load (force) on the edge of the tooth only, it will inhibit the lateral movement of
the tooth whilst attaining maximum torisonal effect. Orthodontic practices or “physical human
factors” have not been considered in the design, although it is noted that currently an “S” type
shaped device has been patented [Andrews, 1981 – US patent No. 4249898] and shown in Fig.
5. The new device proposed here may provide an alternate method of application where the “S”
device is not appropriate.
a) b)
c)
Figure 4. Typical teeth view – a) inferior, and b) posterior, highlighting the twisted incisor with
c) inferior, highlighting the point of application of the orthodontic design. Drawings adjusted