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

4 of 16

(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!

5 of 16

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

from Isaacson, 2002.

Figure 5. Existing tooth adjustment design – Patent 4249898 [Andrews, 1981].

The new design would be attached to the gums with the usual method of a removable retainer,

with the force applied to the tooth end via a coiled spring load. In order to ensure the correct

6 of 16

load is achieved the deflection of the device is approximated if it were free. At this point the

load on the tooth is removed as the device has reached full extension, and the tooth has been

corrected. It is seen from Fig. 4b that the device can be modelled as a cranked cantilever with

the applied force in the perpendicular direction.

The use of the GSS method is applied to the cranked cantilever. An example design is shown in

Fig. 6. The free end is subjected to a load, W, and will deflect by Δ. This is dependent on a

number of factors. The contributing variables are:

Geometrically – cross sectional area (A)

second moment of area (I)

polar moment of inertia (J)

lengths of the bar sections (L1 and L2)

Materially – modulus of elasticity (E)

modulus of rigidity (G)

Figure 6. The cranked cantilever model of the orthopedic design.

The general governing equations that define the deflection according to strain energy methods

are (Rees, 1997),

Shear energy, U =1

2G

W2x

A

L

0

. dx (3)

Bending moment energy, U =1

2E

M2x

I

L

0

. dx (4)

Torsional moment energy, U =1

2G

T2x

J

L

0

. dx (5)

Where M is the moment and T is the applied torque at the end of the bar section.

7 of 16

The displacement is governed by Castigliano‟s second theorem,

Δ =∂U

∂W (6)

From Fig. 6 it is seen that the length L1 is subjected to shear and bending, and L2 is subjected to

shear, bending and torsion (where T = WL1). The explicit forms are thus,

UL1 =1

2G

W2x

A

L

0

. dx + 1

2E

M2x

I

L

0

. dx

=W2L1

2AG+

W2L13

6EI

= L1

2AG+

L13

6EI W2 (7)

UL2 =1

2G

W2x

A

L

0

. dx + 1

2E

M2x

I

L

0

. dx +1

2G

T2x

J

L

0

. dx

=W2L2

2AG+

W2L23

6EI+ WL1

2L2

2GJ

= L2

2AG+

L23

6EI+

L12L2

2GJ W2 (8)

Where,

A =πd2

4= πr2 (9)

I =πd4

64=πr4

4 (10)

J =πd4

32=πr4

2 (11)

The deflection is thus,

U =WΔ

2 ⟹ Δ =

2 U

W

Δ = 2 × L1

2AG+

L13

6EI+

L2

2AG+

L23

6EI+

L12L2

2GJ W 12

8 of 16

It is seen from equations 9 to 11 that the devices material and geometrical variables are only a

function of the diameter of the wire, but the form (graphical representation) will be different for

each function.

Considerations on the boundary conditions (constraints) include:

- The wire diameter should not be less than 0.5mm. As stated in Removable Orthodontic

Appliances - “the patient would readily distort thinner (than 0.5mm) wire and they are

not generally recommended” [Isaacson, 2002, p.18]

- The upper boundary of the wire diameter is 1.5mm. A larger size than this would

produce discomfort to the user.

- The lengths L1 and L2 are defined by a population evaluation of incisor size, and found

to be 8.47mm (SD 1.2) and width 6.84mm (SD 0.9) [Seper, 2006, pp. 59-60]. This

does not include children, as the use of an orthopedic device on deciduous teeth would

be fruitless. Also the variation does not include microdontia or macrodontia

(abnormally small/large teeth respectively). The limits are then for L1 = 5.94mm to

7.74mm, L2 = 7.27mm to 9.67mm.

The wire diameter minimum is defined due to the limits of elasticity. However, 0.5mm is not

necessarily the minimum as it is intrinsic to the material properties. Hence the boundaries

defined are a product of the elastic limit, which is in turn limited by the cost, manufacturability,

and legality regarding materials in-vivo the human body. Due to said legality reasons there is

only one spring material considered, that of stainless steel, with (approximate) properties of E =

203 GPa and G = 80 GPa. The wire diameter then determines the limits of A, I, and J as:

A = 0.196mm2 to 1.767mm

2

I = 0.00307mm4 to 0.249mm

4

J = 0.00614mm4 to 0.497mm

4

The deflection must also be considered. The maximum deflection necessary for the device

would be if the incisor were twisted by 90o. Beyond this point the location of the device would

be moved to the other side. Thus the maximum deflection would be a small amount less than

half the width of the incisor, because if the incisor were at 90o the device would no longer be

attributed to a twisting effect – it would be a lateral adjustment. However, the deflection is

considered a function of the lengths, as a smaller/larger length L1 and L2 will adjust the

deflection defined in equations 7 and 8.

The tolerance must now be defined, so that the iterations will be terminated whence the desired

accuracy is reached. Considering again manufacturability issues the wire diameter will only be

capable of certain finite changes in size. The lengths L1 and L2 are simultaneously products of

equation 12. Thus once the iterations reduce the variables by less than the defined tolerance any

further iterations would be ineffective as they will not be practical.

Tol. –

d = 0.05mm

Hence tol. –

A = 0.00196mm2

I = 3.068×10-7

mm4

J = 6.136×10-7

mm4

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The GSS method is then employed to use these limits and tolerances to define the minimum of

A, I, and J according to equations 7 through 11 by use of a table, with L1 = 7.74mm and L2 =

9.67mm. The deflection is then maximised by use of equations 7, 8 and 12, with the limits of

force defined as a function of the number of coils in the spring, from 0N to 4N.

3. Results

The results shown in Table 1 through 5 are that of the GSS method. The lengths are evaluated

from equation 12 once the minimum of A, I and J are defined. The results are compared with

that obtained analytically by use of equations 7 through 12.

Note: the extra columns are for procedural clarification, and in practice an algorithm would be

employed. The extra iterations are to show convergence is achieved, as is shown in Fig. 7.

Table 1. Iterations of GSS for the diameter as a function of the area.

Iteration

Boundaries Wire Diameter (mm) F(x) (mm2)

Tolerance

reached? Solution(mm

2)

Lower

(mm)

Upper

(mm) x1 x2 A1 A2

1 0.5 1.50000 0.88166 1.11834 0.61051 0.98228 NO N.A. 3

2 0.5 1.11834 0.73600 0.88234 0.42544 0.61146 NO N.A. 4

3 0.5 0.88234 0.64593 0.73642 0.32768 0.42593 NO N.A. 5

4 0.5 0.73642 0.59023 0.64619 0.27361 0.32795 NO N.A. 6

5 0.5 0.64619 0.55579 0.59039 0.24261 0.27376 NO N.A. 7

6 0.5 0.59039 0.53450 0.55589 0.22438 0.24270 NO N.A. 8

7 0.5 0.55589 0.52133 0.53456 0.21346 0.22443 NO N.A. 9

8 0.5 0.53456 0.51319 0.52137 0.20685 0.21349 NO N.A. 10

9 0.5 0.52137 0.50816 0.51321 0.20281 0.20687 NO N.A. 11

10 0.5 0.51321 0.50504 0.50817 0.20033 0.20282 NO N.A. 12

11 0.5 0.50817 0.50312 0.50505 0.19881 0.20034 YES 0.50312 13

12 0.5 0.50505 0.50193 0.50312 0.19787 0.19881 YES 0.50193 14

13 0.5 0.50312 0.50119 0.50193 0.19729 0.19787 YES 0.50119 15

14 0.5 0.50193 0.50074 0.50119 0.19693 0.19729 YES 0.50074 16

15 0.5 0.50119 0.50046 0.50074 0.19671 0.19693 YES 0.50046 17

16 0.5 0.50074 0.50028 0.50046 0.19657 0.19671 YES 0.50028 18

17 0.5 0.50046 0.50017 0.50028 0.19649 0.19657 YES 0.50017 19

18 0.5 0.50028 0.50011 0.50017 0.19643 0.19649 YES 0.50011 20

A B C D E F G H I

The formula employed in Excel is:

Lower Boundary = IF(F3<G3,B3,D3)

Upper Boundary = IF(F3<G3,E3,C3)

x1 = C4-(0.618339887498*(C4-B4)) % from Fig. 2

x2 = B4+(0.618339887498*(C4-B4)) % from Fig. 2

A1 = PI()*(D4/2)^2

A2 = PI()*(E4/2)^2

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Tolerance Reached? = IF((ABS(G4-F4)<0.00196),"YES","NO")

Solution = IF(H4="YES",IF(F4<G4,F4,G4),"N.A.")

Figure 7. A convergence check, ensuring that the diameter is converging upon the minimum

Table 2. Iterations of GSS for the diameter as a function of the second moment of area.

Iteration

Boundaries Wire Diameter (mm) F(x) (mm4)

Tolerance

reached?

Solution

(mm4)

Lower

(mm)

Upper

(mm) x1 x2 I1 I2

1 0.5 1.5 0.8816601 1.1183399 0.0296602 0.0767830 NO N.A.

2 0.5 1.1183399 0.7359957 0.8823442 0.0144036 0.0297524 NO N.A.

3 0.5 0.8823442 0.6459255 0.7364187 0.0085448 0.0144367 NO N.A.

4 0.5 0.7364187 0.5902316 0.6461871 0.0059574 0.0085586 NO N.A.

5 0.5 0.6461871 0.5557938 0.5903933 0.0046841 0.0059640 NO N.A.

6 0.5 0.5903933 0.5344995 0.5558938 0.0040064 0.0046875 NO N.A.

7 0.5 0.5558938 0.5213324 0.5345614 0.0036260 0.0040083 NO N.A.

8 0.5 0.5345614 0.5131907 0.5213707 0.0034047 0.0036271 NO N.A.

9 0.5 0.5213707 0.5081563 0.5132143 0.0032731 0.0034054 NO N.A.

10 0.5 0.5132143 0.5050434 0.5081710 0.0031936 0.0032735 NO N.A.

11 0.5 0.5081710 0.5031185 0.5050524 0.0031452 0.0031939 NO N.A.

12 0.5 0.5050524 0.5019283 0.5031241 0.0031156 0.0031454 NO N.A.

13 0.5 0.5031241 0.5011924 0.5019318 0.0030973 0.0031156 NO N.A.

14 0.5 0.5019318 0.5007373 0.5011945 0.0030861 0.0030974 NO N.A.

15 0.5 0.5011945 0.5004559 0.5007386 0.0030792 0.0030861 NO N.A.

16 0.5 0.5007386 0.5002819 0.5004567 0.0030749 0.0030792 NO N.A.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

Wir

e C

ross

sec

tio

nal

are

a (m

m2)

Iteration No.

Convergence on Minimum Wire X-secion Area

A1 A2

11 of 16

Table 2. Continued

17 0.5 0.5004567 0.5001743 0.5002824 0.0030722 0.0030749 NO N.A.

18 0.5 0.5002824 0.5001078 0.5001746 0.0030706 0.0030722 NO N.A.

19 0.5 0.5001746 0.5000666 0.5001080 0.0030696 0.0030706 NO N.A.

20 0.5 0.5001080 0.5000412 0.5000668 0.0030690 0.0030696 NO N.A.

21 0.5 0.5000668 0.5000255 0.5000413 0.0030686 0.0030690 NO N.A.

22 0.5 0.5000413 0.5000158 0.5000255 0.0030683 0.0030686 YES 0.5000158

Table 3. Iterations of GSS for the diameter as a function of the polar moment of inertia.

Iteration

Boundaries Wire Diameter (mm) F(x) (mm4)

Tolerance

reached?

Solution

(mm4)

Lower

(mm)

Upper

(mm) x1 x2 J1 J2

1 0.5 1.5000000 0.8816601 1.1183399 0.0593205 0.1535660 NO N.A.

2 0.5 1.1183399 0.7359957 0.8823442 0.0288072 0.0595048 NO N.A.

3 0.5 0.8823442 0.6459255 0.7364187 0.0170895 0.0288735 NO N.A.

4 0.5 0.7364187 0.5902316 0.6461871 0.0119149 0.0171172 NO N.A.

5 0.5 0.6461871 0.5557938 0.5903933 0.0093682 0.0119279 NO N.A.

6 0.5 0.5903933 0.5344995 0.5558938 0.0080129 0.0093749 NO N.A.

7 0.5 0.5558938 0.5213324 0.5345614 0.0072520 0.0080166 NO N.A.

8 0.5 0.5345614 0.5131907 0.5213707 0.0068095 0.0072541 NO N.A.

9 0.5 0.5213707 0.5081563 0.5132143 0.0065462 0.0068108 NO N.A.

10 0.5 0.5132143 0.5050434 0.5081710 0.0063873 0.0065470 NO N.A.

11 0.5 0.5081710 0.5031185 0.5050524 0.0062904 0.0063877 NO N.A.

12 0.5 0.5050524 0.5019283 0.5031241 0.0062311 0.0062907 NO N.A.

13 0.5 0.5031241 0.5011924 0.5019318 0.0061947 0.0062313 NO N.A.

14 0.5 0.5019318 0.5007373 0.5011945 0.0061722 0.0061948 NO N.A.

15 0.5 0.5011945 0.5004559 0.5007386 0.0061583 0.0061723 NO N.A.

16 0.5 0.5007386 0.5002819 0.5004567 0.0061498 0.0061584 NO N.A.

17 0.5 0.5004567 0.5001743 0.5002824 0.0061445 0.0061498 NO N.A.

18 0.5 0.5002824 0.5001078 0.5001746 0.0061412 0.0061445 NO N.A.

19 0.5 0.5001746 0.5000666 0.5001080 0.0061392 0.0061412 NO N.A.

20 0.5 0.5001080 0.5000412 0.5000668 0.0061379 0.0061392 NO N.A.

21 0.5 0.5000668 0.5000255 0.5000413 0.0061372 0.0061379 NO N.A.

22 0.5 0.5000413 0.5000158 0.5000255 0.0061367 0.0061372 YES 0.5000158

The diameter is thus the smallest of the values obtained in Tables 1 through 3. As I = J < A, so

d = 0.5000158. Note this is only due to the tolerance inaccuracies, with a solution of the area

found in 11 iterations.

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Table 4. Iterations of GSS for a variable load, showing the solution for deflection.

Iteration Boundaries Load (N) F(x) Tolerance

reached? Solution (N)

Lower (N) Upper (N) x1 x2 Δ1 Δ2

1 0 4 1.5266405 2.4733595 1.28426 2.08066 NO N.A.

2 1.52664 4 2.4706231 3.0560173 2.07836 2.57081 NO N.A.

3 2.47062 4 3.0543253 3.4162979 2.56939 2.87389 NO N.A.

4 3.05433 4 3.4152516 3.6390737 2.87301 3.06130 NO N.A.

5 3.41525 4 3.6384267 3.7768249 3.06075 3.17718 NO N.A.

6 3.63843 4 3.7764248 3.8620019 3.17684 3.24883 YES 3.7764248

The absolute limitations for d = 0.5mm and W = 4N are used to analytically solve equations 9

through 12 to expound the deflection, with the solution found as Δ = 3.3649mm (see appendix

Matlab algorithm for calculations).

Comparisons show the deflection to be GSSΔ = 3.776mm, and theoretical Δ = 3.365mm,

showing an inaccuracy of 12% from the absolute value.

4. Discussion

- Evaluation of GSS

From Tables 1 through 4 and Fig. 7 it is clear that the results do converge on the extremum. The

number of iterations is more dependent on the tolerance than on the interval length, and much

consideration should be spent on accurately defining the tolerance. Although this was attempted

it is seen from Fig.‟s 2 and 3 (for I and J respectively) this is not achieved. The tolerance should

have been a function of x rather than f(x), and hence determined by the wire diameter rather

than I or J. This is important as for complex functions much time would be spent on running the

applied algorithm. Indeed the tolerance can be expressed as [Kiusalaas, 2005, pp. 385-386],

|B2 – B1|φn = ε

∴ n =𝑙𝑛

ε |B2 − B1|

𝑙𝑛φ

Where B1 and B2 are the lower and upper limits respectively, n is the number of iterations

necessary, and ε is the error, or tolerance. The form taken for Table 1 was IF |f(x1) – f(x2)| <

manufacturing tolerance. This is comparable with above and user judgement must be taken in

each case.

It is seen that the lower limits are reached in all conditions for Section 2, which is of course true

as all variables have a positive gradient within the positive region. Hence the previous

derogatory descriptive term “uninspired”. But, by setting up the GSS method to solve an

equation of a non-linear positive form it highlights unequivocally convergence on the extremum

(see Fig. 7).

It is shown in Table 4 that the maximum is found by the GSS method, where all others locate

the minimum. This was achieved by setting the algorithm as,

B1 = IF(f(x1)> f(x2),B1, x1)

13 of 16

i.e. the limit is changed if the probing point is greater than the probing point x2. Nothing else is

necessary to change the algorithm to find the maximum.

The lengths L1 and L2 are a function of equation 12, and taken as constant. However the GSS

method should be capable of simultaneously minimising two or more variables at once. This

may be achieved by setting up the limits determined by the deflection, having two sets of x and

f(x) for L1 and L2, and having the boundaries reduced by which ever length is smaller (closer to

the minimum) at each iteration. But, in this instance it is not necessary as equation 12

determines a positive gradient throughout the limits. Also, more importantly, as the golden ratio

would not be used to reduce both variables simultaneously (only one of them at each iteration)

it proves inefficient.

The use of the GSS is more efficient than the bisection method as it only requires one iterative

procedure per reduction, where-as bisection requires two evaluations [Kiusalaas, 2005, p. 385].

This is because as shown in Fig. 3 the lower point is still at a golden ratio to the boundary, thus

only one new iteration evaluation is required when the larger of the two points becomes the

pseudo-limit. From further examination of equation 12 it is clear that minimising d (A, I, J) will

maximize the deflection as they are all denominators. If however a non-linear function were

obtained in which variables dependant of d but varied in their mathematical placement were

obtained, GSS would prove useful. The order of the polynomials of the dependant variables

would dictate which d would become the optimum value. Note this does not mean that the

extremum of d would be found, but the form of the equation would dictate its value.

Some text advise specifically to use a large approximation of the irrational number (a high

accuracy) [e.g. Chase, 2000, p. 3] whilst others use the ratio only to 3 d.p., but it is advised here

that user discretion is advocated. If a program is being used to implement the iterations then it

would do no harm to use a computed value, with the ratio described as equation 2 explicitly.

However if the user is, for example, using the method to suggest a new solution in an

experiment where the measured variable cannot be attainted to a high degree then there is no

point in striving for an accuracy beyond that of the experiment.

The GSS Method is an efficient method of reducing a unimodal function, and is thus commonly

described as a unimodal optimisation method. Importantly the method may be employed to find

the extremum of any unimodal function, as defined by Rao, which include a “nondifferentiable

or even a discontinuous function.” [Rao, 2009, p. 253], and may prove useful in its variety.

By re-assessing Fig. 3, it seems apparent that both points may become the boundary limits

assuming that f(x1) < f(x2) < f(B1) < f(B2). So that for the next iteration the boundaries become

f(x1) = f(B1) and f(x2) = f(B2). Indeed even if f(x1) > f(x2) then it holds true so long as f(x1) and

f(x2) < f(B1) and f(B2). This would provide full use of the initial data on the basis that the

solutions f(B1) and f(B2) are known. But, if B1 and B2 are not known this will produce the need

for two more inspection points and therefore would not necessarily increase the speed of

reduction. Also, it would be more costly if an algorithm were involved, at the price of memory

capacity. But it could furthermore be suggested that if the difference between f(x1) (or f(x2))

and f(B1) (or f(B2)) is great enough then it would provide a quicker method than the GSS

method.

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- Limitations

1. The use of “local” minimum brings some complexities to practical applications of the

method (and any other iteration extremum methods). How do you know you have

reached the minimum? Or most precisely how do you know the boundaries have

defined a local? Hence the term unimodal optimisation, that it only has one extremum.

However, in practical applications the general form of the solution (equation or

experimental variables) may not be known, thus it may not be a unimodal function,

both locally and globally.

2. The specific Matlab command (see „Implementation of GSS in a Co.‟ below) does not

inspect the solution at the boundaries, so in the example application the solution would

be limited by how close the inspection gets (known as the termination tolerance or

condition).

3. The general termination condition would suggest that the value produced by use of

GSS will never reach the theoretical extremum. This phenomenon is analogous to

many topics in physics and other sciences, for example attaining a zero absolute

temperature, or reaching the speed of light. Mathematically, the GSS unobtainable

extremum is (almost) an exact representation of an asymptote in analytical geometry.

An exception would be that if the evaluating point happens to be the minimum - by

luck.

- Implementation of GSS in a Co.

The GSS method could be applied within an engineering testing and experimentation

environment. The application of optimisation within experiments is well documented, including

the use of Taguchi Methods, for multivariable experiments. But finding the optimum solution to

an engineering problem with a single variable can apply this method to reduce the time and

assets. This is a practical application, where the solution is not defined analytically or

computationally. This was actually the original proposed application of GSS by Kiefer [Kiefer,

1953, p.505]. By having no previous experimental data the next potential root can be assessed

by this method. This may be a simple application (boundaries not withstanding), but said

applications are numerous. When the number of experiments is limited and defined, the

Fibonacci Search or other methods may be applied, but when the number of experiments are

unknown the Golden Section Search can be employed. Comparatively, GSS will produce a

length 1.17 times longer [Kiefer, 1953, p. 505] than from the use of the Fibonacci search, which

is a 17% longer interval [Beighter, C.S., 1979, p. 188]). This is one of the main advantages of

the GSS, that it requires the “least prior information” [Beighter, 1979, p.186].

The GSS method may be a particularly useful tool in the economics of a cost function. The cost

of manufacturing a product decreases as the quantity of produce increases. However, at some

point the form of the function will increase as the increase of cost out-weighs the decreasing

cost-per-item value. At this point there lies a minimum where the productivity/cost curve is at

an optimal. If this method were employed it could have massive effects on the profitability and

operation and management of the Co., but is completely dependent on the Co.‟s current

techniques used. Obviously the cost function is not employed as the example in the Method

Section as verification would prove impossible without access to a Co.‟s accounts. Another

potential use is in finding the robustness (performance criteria) of a design, in the form of a

Gaussian distribution.

The use of GSS in a company is easily employable to CAE with Matlab. There is an algorithm

“fminbnd” which employs the use of the GSS method (Mathworks, 2011). The algorithm is

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already written, with the single command above invoked to produce a result. Setup involves

creating a function handle of the function f(x), and entering the command fminbnd<name of

f(x)>(B1,B2) where, as before, Bn are the limits of the function. The tolerance is handled as

TolX. To see the description of the algorithm employed type doc fminbnd into Matlab.

5. Conclusions

The GSS method provides an iterative extremum solution. If a design problem is of a unimodal

form which is difficult or impossible to differentiate, or is discontinuous, and can be expressed

as a function then the implementation of said function with the Golden Section Search method

is of value. The method may be employed more usefully in an economic or productivity and

profitability environment. Matlab will readily invoke a rapid solution if setup is achieved. It

provides an efficient technique of locating the extremum, particularly if the number of

experiments (or iterations) is unknown. The tolerance is of great importance to efficiency of the

GSS method.

References

Andrews, L. F., 1981, Orthodontic rotation spring, La Jolla, CA, United States, PA.4249898

Beighter, C.S., Phillips, D. T., 1979, Foundations of Optimisation, 2nd

Ed., Prentice-Hall

Cai, j., et. al., 2010, Application of the golden section search algorithm in the nonlinear

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kinetic conversion data, Solid State Sciences, Volume 12, Issue 5, May 2010, pp. 829-833

Chase, T., 2000, Supplementary Notes on the Golden Section Search Technique, Computer

Aided Engineering, University of Minnesota, available online:

http://www.me.umn.edu/~trchase/me5241/goldSecImp.pdf, accessed 01.11.2010

Hemenway, P., 2005, Divine Proportion: Phi In Art, Nature, and Science, New York: Sterling,

pp. 20–21

Isaacson, K. G., et. al., 2002, Removable Orthodontic Appliances, Wright: Elsevier Science

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Appendix

Analytical calculations from Matlab are obtained from the following algorithm: -

%{ - Strain Energy of a Cranked Cantilever - W. Shipway - 03.12.2011 %} clear all clc format short %% - Original Formula W = 4; % N L1 = 5.94/1000; L2 = 7.27/1000; r = (0.5/2)/1000; % m E = 203E9; G = 80E9; % GPa A = pi*r^2; % mm^2 I = (pi*r^4)/4; J = (pi*r^4)/2; % mm^4 UAB = ((W^2*L1)/(2*A*G))+((W^2*L1^3)/(6*E*I)); % J UBC = ((W^2*L2)/(2*A*G))+((W^2*L2^3)/(6*E*I))+(((W*L1)^2*L2)/(2*G*J)); % J deflection = ((2*(UAB+UBC))/W)*1000 %mm %% - Strain Energy / W^2 UAB2 = ((L1)/(2*A*G))+((L1^3)/(6*E*I)); %J UBC2 = ((L2/(2*A*G))+(L2^3/(6*E*I))+((L1^2*L2))/(2*G*J)); % J %% - Deflection for re-arranged formula (same result) Deflection2 = 1000*2*(UAB2+UBC2)*W %mm %% - Plots plot (W,Deflection2)