the optimal design of drainage systems

Engineering Optimization

ISSN: 0305-215X (Print) 1029-0273 (Online) Journal homepage: http://www.tandfonline.com/loi/geno20

THE OPTIMAL DESIGN OF DRAINAGE SYSTEMS A. J. WILSON , A. L BRITCH & A. B. TEMPLEMAN To cite this article: A. J. WILSON , A. L BRITCH & A. B. TEMPLEMAN (1974) THE OPTIMAL DESIGN OF DRAINAGE SYSTEMS, Engineering Optimization, 1:2, 111-123, DOI: 10.1080/03052157408960581 To link to this article: http://dx.doi.org/10.1080/03052157408960581

Published online: 21 May 2007.

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Date: 26 December 2015, At: 09:56

© Gordonand Breach Science Publishers Ltd. Printed in Great Britain

Engineering Optimization 1974, Vol. I, pp. 111-123

THE OPTIMAL DESIGN OF DRAINAGE SYSTEMS A: J. WILSON, A. L. BRITCH Centre for Computer Aided Building Design and

A. B. TEMPLEMAN

Department of Civil Engineering, University ofLiverpool

Downloaded by [Andrew Templeman] at 09:56 26 December 2015

(Received July. 1973)

This paper derives from a research project investigating the systematic use of formal optimization techniques in computer aided building design. A comprehensive modelof drainage design suitablefor optimization is developed. After limited success with a very powerful general purpose optimizationprogram based upon the Geometric Programming technique, a problem dependent algorithm is developed based upon Dynamic Programming. The latter technique is shown in actual designs to produce optimal drainage designs simply and cheaply.

INTRODUCTION This paper describes a study made of computer techniques by which the optimal drainage design for residential building developments may be established. The study arose as part of wider research into the general use of formal optimization techniques in computer aided building design. 1 From this wider research it appeared that if optimization was to be of general use in building design, and economic trends suggested that its general use was desirable, then a more rigorous approach to the marriage of optimization techniques and design problems than had hitherto been the case was needed. It seemed rational to propose an intermediate "model" stage between design problem and optimization technique to provide a formal information interface and to regard the optimization technique as a "black-box." Clearly, such an approach may not be capable of exploiting the particular idiosyncrasies of certain design problems to full advantage, but since few practising architects and engineers have either the time or inclination to gain the familiarity with complex optimization techniques which such ad hoc exploitation presumes, this approach was deemed a suitable starting point since the designer need not understand the mechanism of the black-box but needs only to express his design problem in the model form. The optimization technique chosen as the core of the general purpose optimizing device was Geometric Programming." Accordingly, a comprehensive computer program was written to implement the technique as a general purpose optimizing aid." 111

Drainage system design has much to recommend it as a suitable design problem for optimization; like many problems of building design any improvement would result in significant benefits on a national scale, but unlike other aspects of building design it is a totally objective design problem. When this research began there were at least two quite well-known automatic drainage design programs'v" yet although each adopted the heuristic of minimum slope neither claimed to identify optimal solutions. The traditional drainage designer does not think of his design problem in terms of a formal quantitative model suitable for optimization and it will be useful to list in detail the transformation by which the economic and functional design considerations of drainage design may be adapted to such a model. The Geometric Programming technique is designed to operate upon a problem expressible as the minimization of a polynomial objective function subject to polynomial constraint functions as follows.

i= 1,2""J m

where x l s X1, , XN are the problem (primal) variables and the func tions fj are of the form

Ii

Tm

=

2:

t=1

N

em

I1 x~mtn n=1

i

= 0, ..., m

112

A. J. WILSON, A. L BRITeH AND A. B. TEMPLEMAN

And this is the unaccustomed form in which it is necessary to express a drainage design problem if it is to be optimized by Geometric Programming.

Downloaded by [Andrew Templeman] at 09:56 26 December 2015

MODEL FORMULATION 'Traditionally the drainage designer regards his problem largely in an elemental manner. Realizing he is unable to determine quantitatively the effect of decisions in one element upon decisions in a far removed element he artificially simplifies the problem by designing each 'element in isolation. However the decision to design the system by optimization necessitates a different approach; for the entire system must be defined in terms of explicit relationships before entry to the optimization process. Explicit relationships require the definition of system variables. In drainage, as in most engineering problems, there are many dependent variables and many independent variables and the choice of which to incorporate as system variables in the model formulation is not obvious. Yet the choice is important, since the form of the objective and constraint functions is very much dependent upon the variables chosen, and hence the theoretical integrity and computational effectiveness of the model formulation largely depends upon the system variables. In general two types of information are necessary to define a pipe run; these are the pipe size (only circular section pipes are considered in the present formulation) and the end conditions of the pipe. By end conditions, is implied the position of the pipe in both horizontal and vertical planes. However, such freedom is rare and in the majority of cases by the time the drainage system is to be designed the positions of the manholes, or nodes, on plan have been determined within often quite close limits. (The academically attractive problem of network optimization where the plan positions of the nodes are variable has been considered, I but after limited success the study was terminated when a survey revealed that the practical value of such techniques in drainage design was small.) The end conditions of pipes in a Fixed Pathway System must then be defined in the vertical plane only. There are two ways in which this may be achieved; either by regarding the height above datum HI of the inverts at each node as design variables, in which case the slope would be defined as (H I I - H1 1)I L or, by regarding the pipe slope S, itself defined as (HI I .- H 1 1)IL as the independent variable. The choice is resolved by reference to the Geometric Programming optimization process. The former alternative has two major disadvantages; it involves two polynomial terms and since

slope occurs in many of the functions to be derived it means an increased degree of difficulty, and since it involves a negative sign the complementary geometric programming form would be necessary. Hence, pipe slope S becomes a system variable. But it is still necessary to express the upstream invert elevation to be used with slope to define depth at any point in the system and whereas this is'constant in the case of primary pipes (minimum cover), for secondary pipes it is necessary to introduce a new variable q to denote the difference between ground elevation and the outgoing invert elevation at all internal nodes. Constraints may conveniently be divided into two types; hydraulic and geometric. There are many well known formulae describing flow in pipes, and after a detailed comparison of the relative merits of each it was decided to make use of the Crimp and Bruges' formula. The first constraint is one upon minimum velocity and derives from the need to prevent blockage.

49.W 2 / 3 • SI/2 ~ Vmin(fps units)

(1)

The value of V min advised by Code of Practice" is 2.5 ft/sec. Also, it is considered unwise to allow velocities above a certain limit in order to prevent scouring and stranding (although the latest code of practice gives no explicit recommendation). Hence

49.W 2! 3 • SI!2 ~ Vmax

(2)

Assuming that steady, Uniform, unsurcharged, full bored flow exists in the pipes at peak flow.

38.65D 8/3 SI!2 ~ Q

(3)

Where Q is the design discharge capacity required of the pipe. The Simplest geometric constraint is that pipe sizes should lie in the practical range. D~Dmin

(4)

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