Case Study Finite Element Method And Artificial Neural-Books Pdf

Case Study Finite Element Method and Artificial Neural
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been applied In addition it is not clear how they would compare The hydraulic conductivity k h is related to the soil water. with FEM These issues are addressed in this case study where pressure h as Van Genuchten 1979. both the developed models of FEM and ANN were applied to a. predict seepage through the body of Jeziorsko earthfill dam in k h ks 1 4. Poland Piezometers placed on the section of Jeziorsko dam for 1 h n 1 h n. monitoring seepage have been used since 1995 The model results where p parameter which can take on a value of 1 Burdine. were compared using the data obtained from these piezometers 1953 or 2 Mualem 1976. The performance of the two models was also quantitatively ana The solution of Eq 1 yields the spatial and temporal distri. lyzed and compared bution of the soil water pressure field in the domain of interest. Consequently it is possible to determine the position of the water. Downloaded from ascelibrary org by IZMIR YUKSEK TEKNOLOJI ENSTITUSU on 07 20 16 Copyright ASCE For personal use only all rights reserved. level corresponding to the zero pressure i e isoline h 0 and to. Seepage Flow Models find other quantities describing the soil infiltration and seepage. characteristics in the dam such as the spatial and temporal distri. Unsaturated Flow bution of the water content and hydraulic head. A two dimensional unsteady flow through an earthfill dam can be. described by the Richards equation Neuman 1975 as Finite Element Model. Eq 1 was solved using the finite element method FEM Ac. h h cordingly Eq 1 was reduced to the following system of first. kr h kxx kxz kxz, x x z order nonlinear differential equations Neuman 1975. kr h kzx kzz kzz, S Aijh j Fij, Q i B i D i i j 1 2 m 5. C h SwSs 1, where h soil water pressure h 0 in the saturated zone h 0 in Aij e 4 k r kxxbib j kxz bic j b jci kzzcic j. the unsaturated zone h 0 at the water table kr h relative hy. draulic conductivity expressed as kr h k h ks where k h. hydraulic conductivity and ks saturated hydraulic conductivity i j 1 2 m 6. kr h 1 in the saturated zone C h differential water capacity. characterizing the change in the water retention due to the change. in the water content i e C h d dh where water content Fij e 12 2Ci C j1 C j2 Ss 2Swi Swj1 Swj2. and C h is equal to zero in the saturated zone Sw water satura. tion ratio which is equal to s where s saturated water con. tent and s in the saturated zone and Sw is equal to 1 in the for i j otherwise Fij 0 7. fully saturated zone and it is equal to 0 in the fully unsaturated. zone Ss specific water retention S source water discharge Lq i. such as seepage from a ditch per unit volume per unit time and Qi e 2. K kzx kzz two dimensional tensor of hydraulic conductivity. Eq 1 can be employed to simulate two dimensional unsteady Le. state water flow through a nonhomogeneous anisotropic. saturated unsaturated porous media receiving lateral flow S It is. Bi e k r kszci 9, usually assumed that in the unsaturated zone the impact of the. consolidation on the water retention is negligible compared to the Le. effect of changes in retention resulting from the change in the. water content Hence it is assumed that Ss 0 in the unsaturated. Di e 3 Si 10, The relation between the water content and the soil water 1.
pressure h can be described using the empirical formula of Van k r kri krj1 krj2 11. Genuchten 1979, where Lq i depicts the flow at the boundary of the length L at. element e where the Neumann boundary condition is posed j1. j2 number of the remaining nodes in the element m number of. nodes Le number of elements area of the triangular element. where r residual water content n and parameters and i j k ai x jzk xkz j where i 1 2 3 j 2 3 1 k 3 1 2 bi. is expressed as 1 a n where a parameter which can take z j zk and ci xk x j Note that Di 0 for nodes where the. on a value of 1 Mualem 1976 or 2 Burdine 1953 source function S 0 is determined. Following Eq 2 one can find the following expression for The time derivative in Eq 5 was approximated by the back. C h ward difference method The predictor corrector and Picard s it. eration methods were employed for the solution of the resulting. d n 1 n s r system of algebraic nonlinear equations The predictor corrector. C h h n 1 3, dh 1 h n 1 method linearizes the system of equations at each time step and. 432 JOURNAL OF HYDRAULIC ENGINEERING ASCE JUNE 2005. J Hydraul Eng 2005 131 6 431 440, xi are fed into the input layer neurons which in turn pass them. on to the hidden layer neurons zi after multiplication by connec. tion weights vij Fig 1, net j xivij b j 16, A hidden layer neuron adds up the weighted input received from. each input neuron xivij associates it with a bias b j and then. passes the result net j on through a generally employed nonlinear. sigmoid transfer function, Downloaded from ascelibrary org by IZMIR YUKSEK TEKNOLOJI ENSTITUSU on 07 20 16 Copyright ASCE For personal use only all rights reserved.
f net j 17, The learning of ANNs is generally accomplished by the most. commonly used supervised training algorithm of the back. propagation algorithm The objective of the back propagation al. gorithm is to find the optimal weights that would generate an. Fig 1 Representation of three layer feed forward artificial neural. output vector Y y 1 y 2 y p as close to the target values of the. output vector T t1 t2 t p as possible with the selected accu. racy The optimal weights are found by minimizing a predeter. mined error function E of the following form ASCE Task Com. Picard s method iteratively solves the resulting system which has mittee 2000. a large and sparse coefficient matrix utilizing the method of suc. cessive overrelaxation SOR E P p yi ti 2 18, The right choice of the time step t is essential in order to. have a stable numerical scheme The right time step taking into where y i component of an ANN output vector Y ti. account the change in the water content in the flow region was component of a target output vector T p number of output. selected following Belmans et al 1983 as neurons and P number of training patterns. In the back propagation algorithm the effect of the input is. max first passed forward through the network to reach the output layer. After the error is computed it is then propagated back towards. t max the input layer with the weights being modified The gradient. descent method along with the chain rule of differentiation was. where max maximum incremental increase in the water con employed to modify the network weights as ASCE Task Com. tent The value selected from the range of 0 001 max mittee 2000. 0 002 resulted in stable numerical solutions, The value of C h determined from Eq 3 leads to large er E. vij n m vij n 1 19, rors in the numerical model Celia et al 1990 Ross 1990 Pani vij. coni et al 1991 Li 1993 Rathfelder and Abriola 1994 and Tocci. where vij n and vij n 1 weight increments between node i. et al 1997 Therefore C h was evaluated effectively by follow. and j during the nth and n 1 th pass or epoch learning rate. ing Cooley 1983 and Abriola and Rathfelder 1993 as. and m momentum factor, An equation similar to Eq 19 was also used to correct the.
Dt mi mi i t, Ci hmi hi t 13 bias values The learning rate was used to increase the likeli. Dthmi hmi hi t hood of avoiding the training process being trapped in a local. where minimum instead of a global minimum However it is possible. that the training process can still be trapped in a local minimum. despite the use of a learning rate The solution often follows a. Dt mi 14 zigzag path while trying to reach a minimum error and this may. t slow down the training process The momentum factor m can. be employed to speed up training in very flat regions of the error. hmi hi t surface and help prevent oscillations in the weights ASCE Task. Dthmi 15 Committee 2000, The network learns by adjusting biases and weights that link. Note that when hmi hi t then Ci is evaluated from Eq 3 its neurons Before training weights and biases of the network. must be set to small random values Also due to the nature of the. sigmoid function used in the back propagation algorithm all ex. Artificial Neural Networks, ternal input and output values before passing them into a network. ANNs have an ability to identify relationships from given patterns should be standardized Without standardization large values of. and hence they have an ability to solve large scale complex prob input into an ANN would require extremely small weighting fac. lems such as pattern recognition nonlinear modeling classifica tors to be applied and this could cause a number of problems. tion association and control Their hydraulic applications gener Dawson and Wilby 1998 Since sigmoid function extends to. ally consider a three layer feedforward artificial neural network minus infinity and plus infinity asymptotically it never reaches. as shown in Fig 1 In a feedforward ANN the input quantities zero or one Therefore in most cases it is better to compress the. JOURNAL OF HYDRAULIC ENGINEERING ASCE JUNE 2005 433. J Hydraul Eng 2005 131 6 431 440, Downloaded from ascelibrary org by IZMIR YUKSEK TEKNOLOJI ENSTITUSU on 07 20 16 Copyright ASCE For personal use only all rights reserved. Fig 2 Detailed cross section sketch of the Jeziorsko earth fill dam with depicted soil layers. data into the 0 1 0 9 range Eq 20 which compresses all the the cross section of the dam The slope of the upstream side is 1 3. data into the range of 0 1 0 9 was employed in this study for while the inclination of the downstream side is 1 2 5 The cross. standardization section has two different layers of the geological formation The. lower layer 35 m thick represents an alluvial deposit that over. 0 8 xi xmini, xi 0 1 20 lies a chalk formation and the upper layer represents a quaternary.
xmaxi xmini formation medium grained sand Fig 2 The upper part of the. chalk layer is impermeable and therefore the bottom part of the. where xmaxi and xmini are the maximum and minimum values of. alluvial deposit forms the model boundary The infiltration model. the ith neuron in the input layer for all the feed data vectors. parameters n r s and ks for the two layers are given in. respectively, Table 1 The geological material at the dam toe involves rocky. sediments of chalk clay dust glacier formations and sand gravel. alluvial deposits, Application A Case Study of Jeziorsko Dam. On the downstream side of the toe of the dam at a height of. The Jeziorsko earthfill dam located in the central part of Poland 112 7 m and at a distance of about 77 m from the upstream side. was employed in this study to calibrate and verify the FEM and of the dam a stoneware drainage of 30 cm diameter pipe is in. ANN models The dam partitions the Warta River valley near stalled Fig 2 At about every 80 m there are openings carrying. Uniejow City and forms with other lateral dams a reservoir area away the water from the drainage pipe down to the drainage ditch. of 42 3 km2 The maximum water rise is 121 5 m above the mean Ditch A in Fig 2 situated at about a height of 112 3 m and a. sea level and its reservoir capacity is 202 million m3 The dam distance of 5 m from the drainage pipe Fig 2 The second drain. body is homogeneous constructed with medium grained sand age ditch Ditch B in Fig 2 runs parallel to the first Ditch A at. The length of the dam is 2 720 m and its height is 12 m The a height of 112 m and a distance of about 35 m from Ditch A. upstream slope is secured with a tight ferroconcrete screen joined The bottom and the slopes of the ditches are secured with ferro. with a clay cutoff wall of 0 5 m thickness and 50 m width A seal concrete panels separated by openwork panels. made up of a film and extending down to 800 m inside the reser The infiltrated water flows in the direction from the upstream. voir forms an extension of the clay cutoff wall The cross section side towards the downstream side What affects the infiltration. 1 900 of the Jeziorsko dam was considered for determining in and seepage is the pressure gradient due to the difference in the. filtration and seepage Fig 2 shows a schematic representation of water levels in the upstream and downstream sides of the dam. Table 1 Hydraulic Parameters of the Soil Layers, Layer type n cm3 cm3 cm day cm3 cm3. Upper 0 02307 1 46826 0 0012 172 8 0 364, medium grain sand. Lower 0 17327 1 82043 0 003 1728 0 395, alluvial deposit.
434 JOURNAL OF HYDRAULIC ENGINEERING ASCE JUNE 2005. J Hydraul Eng 2005 131 6 431 440, Downloaded from ascelibrary org by IZMIR YUKSEK TEKNOLOJI ENSTITUSU on 07 20 16 Copyright ASCE For personal use only all rights reserved. Fig 3 Temporal variations of water level in piezometers and in the upper and lower reservoirs. tubular drainage and two drainage ditches Four piezometers for this segment of the boundary At the untight screen on the. were placed in the dam in order to monitor the flow of infiltrated upstream side it was assumed that the leakage was uniformly. water through the dam body Fig 2 Three piezometers labeled distributed and the Cauchy boundary condition of nonzero water. as P37 P38 and P39 were placed on the dam body whereas one flux was employed for this segment of the boundary The Cauchy. piezometer labeled as P148 was placed in the alluvial deposit boundary condition assumes that the difference between the. layer see Fig 2 The water levels in the piezometers have been know. This study developed a numerical model using the FEM for two dimensional unsteady state seepage through the saturated unsaturated zone in an earth ll dam The FEM model can be more effective when data on the spatial variation of the actual model parameters at every element of the numerical mesh is available

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