Groundwater Flow Algorithm (GWFA): Inspired by the Movement of Groundwater Flow Applied to Several Standard Engineering Problems

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1. Introduction

Groundwater is one of the most valuable natural resources especially in arid regions due to negligible rainfall and the scarcity of surface water resources [1]. Groundwater models are computer models of groundwater flow systems, and are used by various types of numerical solutions like the finite difference method and the finite element method.

2. Optimization Approach Technique

Optimization deals with the process of searching for the best solution in a given scenario [2]. An algorithm employed to find such a feasible solution is called an optimization algorithm. The focus is on the recent development to tackle the challenges that an algorithm might face when solving real-world problems [3].

Fig 1: Optimization Approach Technique

3. What is Ground Water?

Groundwater is the water found underground in the cracks and spaces in soil, sand and rock. It is stored in and moves slowly through geologic formations of soil, sand and rocks called aquifers [4]. Groundwater flows underground in response to elevation difference and pressure difference (from areas of high pressure to areas of low pressure).

Fig 2: What is Ground water?

4. Inspiration of GWFA

The algorithm named as Ground Water Flow Algorithm (GWFA), is inspired from the movement of groundwater from recharge areas to discharge areas following Darcy’s law. Most of this water can be seen with bare eyes when people go to beaches or stand at the side of rivers or ponds [5]. But still there is water which cannot be seen unless a hole is dug up in the ground deep enough to reach the water tables. This water is known as groundwater [6]. The area where the precipitated water flows past the unsaturated zone to the water table is called the Recharge area (RA) while the areas where the groundwater flows to (streams, lakes etc.) are called Discharge Areas (DAs). Groundwater flow is known as the flow of groundwater from RAs to DAs.

Fig 3: Inspiration of GWFA

5. Darcy’s Experimental set up for groundwater flows

The most important aspect of the GWFA optimization approach is the ground water flows direction which is guided through Darcy’s law [7]. During experimentation, Darcy had discovered that the velocity with which groundwater flows is dependent on two major factors height difference and gap in positions. Darcy’s experimental setup has been presented in Figure below;

Fig 4: Darcy’s law for groundwater flow

6. Groundwater Flow Algorithm

Initialized the function under consideration describing the optimization objective and has dimensions (dim) [8]. Then the candidate solution will be represented as:

where  represents the value of m the candidate solution in n the dimension, (i) and (i)  are the upper bound and lower bound of function i respectively and  (1) is a random number restricted in [0,1]. Each candidate solution is guided by a velocity term. The velocity of every individual GWS is also initialized with 0, and it is represented as

Here  (0) represents the initial velocity of the candidate solution.

After getting the initial GWSs, their positions are updated based on groundwater flow rules and abiding by Darcy’s law. Groundwater is mainly guided by two important terms: height difference (∆h) and length of gap (L).The mathematical representations for the current candidate solution in terms of current functional values are shown below

Dis   represent the selected Discharged area. R and t represent the discharge area water flow and ground water flow respectively. Darcy’s law states that the discharge velocity is directly proportional to the Hydraulic gradient (hg).

Therefore, by multiplying the hg with a, discharge velocity (vel) in respect to the discharge area (DA) and ground water (GW) for every individual is computed. P is also known as the coefficient of permeability. Vel is the discharge velocity. Therefore, overall time complexity (TC) all the model can be calculated as:

Where GW is the number of ground water, Iter denotes the number of iterations, dim denotes the number of dimension in the objective function [9].

7. Advantages of GWFA

Fig 5: Advantages of GWFA

8. Flowchart of GWFA

Fig 6: Flowchart of GWFA

9. Applications of GWFA

Engineering problems are some real world problems that involve designing and building of systems and/or products [10]. It is basically a decision making process that contains complex objective functions and large number of decision variables. Thus the applications implemented in the GWFA are as following below;

  • Welded beam design problem
  • Gear train design problem
  • Tension/ compression spring design problem
  • Pressure design vessel problem
  • Pressure design vessel problem
Fig 7: Applications of GWFA

Reference

[1]U. Zinko, J. Seibert, M. Dynesius and C. Nilsson, “Plant Species Numbers Predicted by a Topography-based Groundwater Flow Index”, Ecosystems, vol. 8, no. 4, pp. 430-441, 2005. Available: 10.1007/s10021-003-0125-0.

[2]L. Townley and J. Wilson, “Computationally Efficient Algorithms for Parameter Estimation and Uncertainty Propagation in Numerical Models of Groundwater Flow”, Water Resources Research, vol. 21, no. 12, pp. 1851-1860, 1985. Available: 10.1029/wr021i012p01851.

[3]F. Sajedi-Hosseini et al., “A novel machine learning-based approach for the risk assessment of nitrate groundwater contamination”, Science of The Total Environment, vol. 644, pp. 954-962, 2018. Available: 10.1016/j.scitotenv.2018.07.054.

[4]H. Namdar, “Developing an improved approach to solving a new gas lift optimization problem”, Journal of Petroleum Exploration and Production Technology, vol. 9, no. 4, pp. 2965-2978, 2019. Available: 10.1007/s13202-019-0697-7.

[5]P. Huyakorn, B. Jones and P. Andersen, “Finite Element Algorithms for Simulating Three-Dimensional Groundwater Flow and Solute Transport in Multilayer Systems”, Water Resources Research, vol. 22, no. 3, pp. 361-374, 1986. Available: 10.1029/wr022i003p00361.

[6]J. Kukačka, V. Altová, J. Bruthans and O. Zeman, “Groundwater Flow in Crystalline Carbonates (Jeseniky Mts., Chech Rep.): Using Stream Thermometry and Groundwater Balance for Catchment Delineation”, Acta Carsologica, vol. 37, no. 1, 2008. Available: 10.3986/ac.v37i1.164.

[7]K. Katsifarakis, D. Karpouzos and N. Theodossiou, “Combined use of BEM and genetic algorithms in groundwater flow and mass transport problems”, Engineering Analysis with Boundary Elements, vol. 23, no. 7, pp. 555-565, 1999. Available: 10.1016/s0955-7997(99)00011-9.

[8]D. Bertsekas and P. Tseng, “Partial Proximal Minimization Algorithms for Convex Pprogramming”, SIAM Journal on Optimization, vol. 4, no. 3, pp. 551-572, 1994. Available: 10.1137/0804031.

[9]M. Lin, J. Tsai and C. Yu, “A Review of Deterministic Optimization Methods in Engineering and Management”, Mathematical Problems in Engineering, vol. 2012, pp. 1-15, 2012. Available: 10.1155/2012/756023.

[10]V. Lo, “Heuristic algorithms for task assignment in distributed systems”, IEEE Transactions on Computers, vol. 37, no. 11, pp. 1384-1397, 1988. Available: 10.1109/12.8704.

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