# Investigation of Viability of Genetic Programming for Symbolic Regression of Engineering – Math Problem Example

The paper “ Investigation of Viability of Genetic Programming for Symbolic Regression of Engineering”   is an apposite version example of a math problem on logic & programming. Controlling the flow or discharge of rivers and canals is essential in flood control, irrigation, and navigation. This can be done by modeling using various mathematical software and equations. The common model is GPLAB genetic programming software created by Silva (2007). GPLAB genetic programming software is popular for approximating volumetric outflow rate of rivers channels and rivers in consideration with width and depth.

In this paper will try to model the volumetric outflow rate of the canal is computed from the following equation Q=bCd Where Q is the discharge (volumetric outflow rate) (m3/s), b is the width of the channel (m), Cd is the coefficient of discharge, yG is the height of the gate opening above the bed of the channel (m), g is the acceleration due to gravity (m/s2) y1 is the upstream depth of water above the bed of the channel (m) and y2 is the depth just downstream of the gate for free outflow. Represents velocity of flow when there is a restriction to the flow of water.

The continuity equation states that the reduction in diameter would cause an increase in the fluid flow speed. This is where when the water depth some distance downstream of the gate (y3) is raised by blocking the channel during a flood, high tide levels. If this depth is large enough such that y3 > y2 then submerged outflow occurs Under submerged flow conditions equation A1 no longer applies. The submerged discharge, Q, is now a function of the “ head difference” y1 – y3.

The approach that has been used in this paper to obtain an empirical equation for the submerged discharge under a radial gate is to carry out a symbolic regression using genetic programming of observed head differences and discharges for different gate opening heights. This leads to the use of the global expression for predicting the critical depth in channels with different cross-sections and flow regimes, two separate analyses were performed to verify the selected “ best” expression and also investigate the underlying state of the expression’ s coefficients. A dimensional analysis will be carried out to determine the relationship between the upstream depth of water of the channel, discharge, and width of the river.

Any increase in the flow speed to the point that viscous dissipation can no longer stabilize the flow, the macroscopic balance between mean flow inertia and viscous effects breaks down. At this point, there is a transition from purely laminar flow to turbulence. Errors in the analysis can be minimized by quality control of the survey and are generally within acceptable tolerances. However, gross errors do sometimes occur and these need to be eliminated either by quality control procedures at the data input stage (for example by plotting all the cross-sections on the computer screen) or during the model calibration procedure.

Errors in the hydrological data are not so easy to eliminate and may, therefore, need to be catered for by a sensitivity analysis. For example, if say a 100-year return period flood hydrograph has been generated, it may be subject to quite wide confidence limits. Thus it would be advisable to run the computational model with several flood hydrographs to establish the sensitivity of flood stage to flood discharge.

References

Abril, J.B. & Knight, D.W., 2004. Stage-discharge prediction for rivers in flood applying a depth-averaged model. Journal of Hydraulic Research, IAHR, 42 (6): 616-629.

Aytek, A. & Kisi, O., 2008. A genetic programming approach to suspended sediment modeling. Journal of Hydrology, Elsevier, 351 (3-4): 288-298.

Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge: Cambridge University Press.

Chadwick, A., Morfett, J. & Borthwick, M. 2004. Hydraulics in Civil and Environmental Engineering. London: Spon Press.

Chaudhry, M.H., 2008. Open-Channel Flow. New York: Springer.

Dey, S., 2001. Flow measurement by the end-depth method in inverted semicircular channels. Flow Measurement and Instrumentation, Elsevier, 12 (4): 253–258.

Dodge, J. C. I. & O'Kane, J. P. 2003.Deterministic Methods in Systems Hydrology. Balkema, Lisse.

Finnemore, E.J. & Franzini, J.B., 2002. Fluid mechanics with engineering applications. 10th edition, McGraw-Hill, Boston.

Giustolisi, O. & Savic, D. A., 2006. A symbolic data-driven technique based on evolutionary polynomial regression, J. ydroinform., 8(3), 207–222, DOI:10.2166/hydro.

Halberstein, J.H., 1966. Recursive, complex Fourier analysis for real-time applications, Proc. IEEE. 54(6) 903-4

McGahey, C.and Knight, D. & Samuels, P.G. 2009. Advice, methods, and tools for estimating channel roughness. Proc. Instn. Civ. Engrs., Water Management, 162(6), 353-362.

Novak, P. Guinot, V. Jeffrey, A. & Reeve, D. 2010. Hydraulic Modelling – An Introduction: Principles, Methods and Applications. Spon Press, London.

Parasuraman, K., Elshorbagy, A., & Si, B. C., 2007. Estimating saturated hydraulic conductivity using genetic programming, Soil Sci. Soc. Am. J., 71, 1676–1684.

Slavs, L. & Brilly, M., 2007. Development of a low-flow forecasting model using the M5 machine learning method, Hydrology. Sci.J0, 52(3), 466–477.

Zhao, M., Cheng, L., & Teng, B., 2007, Numerical Modeling of Flow and Hydrodynamic Forces around a Piggyback Pipeline near the Seabed, Journal of Waterway, Port, Coastal, and Ocean Engineering, pp.286-294.