Scholarly article on topic 'Penalty Function Element Free Method to Solve Complex Seepage Field of Earth Fill Dam'

Penalty Function Element Free Method to Solve Complex Seepage Field of Earth Fill Dam Academic research paper on "Economics and business"

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Abstract of research paper on Economics and business, author of scientific article — Tang Jing, Liu Yongbiao

Abstract Element Free Method only needs to solve node information in domain and calculate domain boundary condition without need of any element information and it is easy to handle, with high computing efficiency. Dynamic radius of influence selection scheme is adopted and nodes are arranged flexibly, which can effectively reduce leveling effect. Penalty Function Element-free Method is used in this paper to study the steady seepage field in main dam that is with protective concrete slab and structure of hydrofracture grouting cutoff wall, of Dashi Bu reservoir located in Donghai County, when it is normal water level for the reservoir, and compare the errors between measured value and computed value from piezometer tube. The analysis shows that computed value is fully in line with measured value; distribution of seepage and free surface conforms to universal law, so Element Free Method can solve seepage issue of fill dam under complex condition.

Academic research paper on topic "Penalty Function Element Free Method to Solve Complex Seepage Field of Earth Fill Dam"

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IERI Procedía 1 (2012) 117 - 123

2012 2nd International Conference on Mechanical, Industrial, and Manufacturing

Engineering

Penalty Function Element Free Method to Solve Complex Seepage Field of Earth Fill Dam

Tang Jinga,Liu Yongbiaoa*

aSichuan College of Architecture Technology,Deyang 618000, P.R. China

Abstract

Element Free Method only needs to solve node information in domain and calculate domain boundary condition without need of any element information and it is easy to handle, with high computing efficiency. Dynamic radius of influence selection scheme is adopted and nodes are arranged flexibly, which can effectively reduce leveling effect. Penalty Function Element-free Method is used in this paper to study the steady seepage field in main dam that is with protective concrete slab and structure of hydrofracture grouting cutoff wall, of Dashi Bu reservoir located in Donghai County, when it is normal water level for the reservoir, and compare the errors between measured value and computed value from piezometer tube. The analysis shows that computed value is fully in line with measured value; distribution of seepage and free surface conforms to universal law, so Element Free Method can solve seepage issue of fill dam under complex condition.

© 2012 Published bb Elsevier 13 .V.

Selection and peer review under responsibility of Information Engineering Research Institute

Keywords: element free method; penalty function; steady seepage; free surface;

Element Free Method is a newly developed engineering numerical calculation method , started late in study of simulating seepage field and it is needed to be further improved for such method to solve complex seepage filed. Guangxin Li and Jinhong Ge used Element Free Method to solve steady seepage filed of homogenous fill dam ; Xiaochun Li applied Element Free Method to solving seepage in crack and considered to use Penalty Function Method to deal with boundary condition, but it is a pity that he did not present us derivative process; while, Xiaohu Chen, Zhenzhong Shen and others adopted Penalty Function to handle seepage boundary condition and put forward exact expressions. However, leveling effect shall be generated when there

* Corresponding author. Tel.: +086-159-8291-0782; fax:+086-838-265-1997. E-mail address: liuyb@scac.edu.cn.

2212-6678© 2012 Published by Elsevier B.V. Selection and peer review under responsibility of Information Engineering Research Institute doi:10.1016/j.ieri.2012.06.019

is great difference in arrangement density of nodes in neighboring areas or in osmotic coefficient of different materials, which will have greater influence on computed results. According to actual situation of main dam of the reservoir in Dashi Bu, Donghai County, aiming at the situation of protective concrete slab and hydrofracture grouting of the dam body, Penalty Function Method is used in this paper to deal with boundary condition, and selection scheme of dynamic radius of influence is used and nodes are arranged flexibly, which effectively reduces leveling effect. Steady seepage in the main dam of Dashi Bu Reservoir is calculated and analyzed when it is normal water level, and compare the error between measured value and computed value of piezometer tube. Analyzed results shows that computed value is fully in line with the measured value and this method can be used to solve the seepage issue in dam under complex condition.

1. Penalty Function Element Free Method for Solving Seepage Issue

Moving Least Square Method is mathematics foundation of Element Free Galerkin Method and such factors as selection of basic function, determination of influence radius and weight function, solutions for discontinuity, introduction of essential boundary condition, selection of integration scheme, node arrangement scheme and more, are the key factors that affect computed accuracy of Element Free Method.

As for two-dimensional homogeneous incompressible fluid, its basic differential equation for saturated steady seepage, without taking endogenesis into consideration,

, d'H , S'H „

+ = 0 k k m

Where, H denotes water head field function; x and z are osmotic coefficient in x and z direction accordingly.

According to Theory of Moving Least Square Method, approximate representation of head field function can be

H = NT H = W.H, +••• + n. H.

1 1 J J (2)

Total energy of system can be expressed as

7T = — HTKH-HTF+C

Global Balancing Equation can be obtained by variational principle KH+F=0

n ; (4)

Where, H denotes array of water head; j denotes influenced node numbers of evaluation point; . denotes shape function of i; K is seepage tensor; F is right hand side.

1.1. Differential Equation

Energy functional in Expression (3) can be expressed as

-if 4kS^

According to variation principle, solve extreme value for above expression and we can find

K«= JJn B(x)TKB(x)dQ (6)

F(1)= (0,0,—,0) (7)

Where: B(x) = [Bi(x), B2(x), -, Bn(x)]; B,.(x)=

d.n dn T K = ~kx 0"

_dx dz_ 0

1.2. How to deal with Boundary Condition

1.2.1 Given Water Head Boundary

Fixed solution problem of steady seepage with free surface, generally, contains four kinds of boundary conditions: given water head boundary, impervious boundary, exit face boundary and free surface boundary. Penalty function is used to solve boundary condition of given water head. Assume that there exists powerful pervious bed in the boundary, thickness is AS, osmotic coefficient is ks, following can be found by deriving:

K(2) = f Ndr 1 Jri AS 1

F(2) = f -^NHdr,

Where, N denotes matrix of shape function, Tj denotes given water head boundary, HB denotes the

h_s_ ks = c-k

given water head function, ££ denotes penalty factor. In this paper, ££ Ab is selected, where

c=10~100 (10 is chosen), Ab = min(dn), dn denotes interval between nodes, k denotes normal osmotic

coefficient.

1.2.2 Impervious Boundary

Penalty function method is adopted to solve impervious boundary condition, consider that there is one layer of aquitard on the impervious boundary and outer boundary of aquitard is denoted as T2 , inner boundary as r2, normal flow q on T2 as 0. Assume that normal osmotic coefficients of weakly-permeable layer and seepage area are ks and k accordingly, and thickness of aquitard is Ad. Following can be found by deriving,

F3) =(0,0—o)

Where, -is penalty factor, R is rotation matrix that is

K(3)=Jr — B( x)TKTRKB(x)^ 2

(10) (11)

-sin a cos a

sin a cos a cos a a is the included angle of impervious boundary and X-axis.

Ad c•Ab

Selection of penalty factor is similar with penalty factor of given water head, assume-=-,

where c =10~100 (10 is used here), Ab = Rmax , Rmax is max. radius of influence, k is normal osmotic

x ' max y max y

coefficient.

To sum up, following can be found by K = K(1) + K(2^ K(3), F = F(1) + F(2^ F(3)

- kxky sin a cos a

1.3. Boundary of Exit Face

Exit point is the particular point in seepage free surface, and it can not only be used as node of given water head, whose position can be modified according to the position of the nodes in neighboring free surface, but also be used as node of unknown water head in seepage free surface and be iterated in the same way with other nodes in the free surface. It is tested and verified from number of computed examples, that such exit point elevation which is obtained by selecting exit point as node of given water head to consider is much approximate, compared to result by finite element. Therefore, exit point is used in this paper as node of given water head to consider.

1.4. Determination of Free Surface and Exit Point

Position of seepage free surface is unknown, so it is necessary for the position to be determined by iterative computation. Element free method only arranges nodes, and it is mutual independence between computation of nodes and integration grid, so distortion for finite element grid cannot take place, moreover, difficulty of reconstructing grid does not exist and arrangement of nodes can be adjusted from time to time during computing. Therefore, moving grid method in traditional finite element computation in seepage is borrowed, in this paper, to undertake iteration computation to free surface.

First, nodes are generated in free surface, especially, more nodes shall be generated in boundary of different material, so as to guarantee its computation accuracy. After initial free surface is determined, head water field in initial seepage field is calculated to obtain the head water value of nodes in free surface and judge if the value is equal to the coordinate value in Z direction; if the value is not equal to the coordinate value, node position in free surface shall be adjusted to generate new seepage field and then compute water head value in each node again, and if the values are still not equal, above mentioned shall be adjusted and computed again and again, till difference between node water head in free surface and its coordinate value in Z direction is within control accuracy. Exact steps are shown as below: water head value of node i in free surface obtained by No. n iteration, is noted as

Я( n) (n) I j

i — z{ \< € (sis control accuracy of iteration ), stop computing; if such condition cannot be satisfied, coordinate of Z direction in free surface shall be adjusted in accordance with following expression:

zt(n+1)1 = z(n ^ mt (Ht (n z(n))

Where, mi is correction coefficient, and it can be 0.5 < < 1.

2. Dynamic Radius of Influence and Selection of Node Arrangement Scheme

2.1. Dynamic Radius of Influence

Selection of radius of influence is closely related to weight function. Proper radius of influence should make the node in domain of influence as many as possible but as few as possible, as a result, computation accuracy can not only be guaranteed, but Gauss point cannot be influenced by other neighboring nodes that do not need considering. Therefore, when such thin-layer structure as protective concrete slab, cutoff wall is dealt with, smaller radius of influence of Gauss Point in this area shall be selected to make all the nodes surrounded by it be the nodes in the area; meanwhile, encrypted degree for the nodes shall be properly controlled to make sure that there are approximate 20 nodes in domain of influence. As for Gauss points in other areas, large radius of influence shall be used and their node density shall be also controlled well, so as to make sure that there are also about 20 nodes in domain of influence. It is tested and verified from a large number of computed examples, that in this way leveling effect can be reduced to get much more ideal results.

2.2. Scheme of Node Arrangement

Scheme of node arrangement has much greater impact on final computed results of element free method, so proper arrangement of nodes is the key factor to guarantee computed accuracy. Following suggestions are presented according to experience of the Author: coincidence of background grid and integration grid; homogeneous arrangement of nodes, that is, intervals between nodes in horizontal and vertical directions shall be approximately equal; number of influence nodes in each Gauss Point shall be as the same as possible, and 20 is chosen in this paper; partial area of thin-layer structure such as protective concrete slab shall be properly encrypted, and according to computing experience, generally, if there exists more than one order of magnitude between osmotic coefficient in neighboring areas, node density shall be increased by a factor of two, requirement of computation accuracy can be met.

3. Engineering Application

Regarding Dashi Bu Reservoir in Donghai County, elevation of its main dam is 55.32m, max. Height of dam is 10.32m, width of dam crest is 5.0m, gradient of upstream dam slope is 1:2.5, gradient of downstream dam slope is 1:3.0. Normal water level in upstream of the reservoir is 52.00m, and it is 45.00m in the downstream. Hydrofracture clay grouting is done to dam body by dam crest, with thickness of 100cm and concrete poured in the upstream can reach 15cm thick. Computed parameter adopts design value, shown in Table I.

When computing, because such thin-layer structure as protective concrete slab and cutoff wall exists, node shall be encrypted based on above mentioned requirement and radius of influence in the two areas shall be reduced, too. The form of time square-base shall be adopted for calculation of free surface, and initial seepage height 45.50m shall be selected, tolerance shall be controlled within 0.01m, free surface and equipotential line obtained after calculation are shown as in Figure 1, comparison between calculated value and measured value of piezometer tube is shown in Table 2.

We can see, from Figure I, that distribution of free surface in seepage filed and equipotential line confirms to the distribution laws of general seepage field. From Table I, we can see that calculated value and observed value fit much better, so penalty function element free method is used to compute analyzed results, which can basically reflect the actual situation of seepage field of the dam.

Table I. Osmotic Coefficient of Material in Each Section

Soil Layer

Osmotic Coefficient / (cm/s)

Fillings for Dam Body 5.5x10 7 m / s

Differentiation Rock Zone of Dam Base 7.5x10 "7 m /s

Concrete Slab 1.0x10" 9 / m/s

Hydrofracture Grouting Cutoff Wall 9.8x10 ■8 / m/s

Table 2. Comparison of Observed Value and Computed Value for Drill Water Level

Drill Water Level /m Relative Error /(%)

Observed Value Computed Value

ZK02 50.70 50.58 "1.71

ZK05 48.80 49.01 3.86

ZK07 46.90 46.67 "3.29

Where, Relative Erroi=(Computed Value-Observed Value)/ difference of water head in upstream and downstream.

Fig.1. Distribution Diagram for Free Surface and Equipotential Line

4. Conclusions

According to actual situation of main dam of Dashi Bu reservoir in Donghai County, complex fill dam with protective concrete slab for dam body and hydrofracture grouting in dam body is studied in this paper, methods of dynamic radius of influence and flexible arrangement of nodes are put forward, penalty function Element Free Method is used to analyze the steady seepage field in main dam of the reservoir under normal water level and steady seepage field & free surface of the dam body are obtained. The result shows that distribution of seepage field of dam body & dam base and free surface confirms to general laws, so element free method based on penalty function is adopted to calculate and analyze the results, which can basically reflect the actual situation of the dam seepage field, showing that leveling effect can be reduced effectively by selection of dynamic radius of influence and flexible arrangement of nodes, and there is greater application prospect for Element Free Method in actual engineering.

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