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Physics Procedía 24 (2012) 1171 - 1177

Physics

Procedía

2012 International Conference on Applied Physics and Industrial Engineering

Fast Algorithms of the Simulation Analysis of the Thermal Stresses on Concrete Dams during Construction Periods

Li Yangboab, Huang Dahaib, Ouyang Jianshub

"College of Water Resources and Hydropower; Wuhan University; Wuhan; China; bCollege of Civil &Hydroelectric Engineering; China Three Gorges University; _Yichang; China_

Abstract

During the simulation analysis solution on temperature control and thermal stresses fields of concrete dams' construction, it is so huge finite element meshes, complex environmental conditions of construction, difficulties in boundary features description that its computational process is time-consuming and consumes far too much storage space. To solve the problem above and reduce calculations, two fast algorithms are introduced. One is the Overall Planning algorithm which makes the simulation process eliminate repetitive costs; the other is the Incomplete Cholesky Conjugate Gradient algorithm including element by element which accelerates the solution of systems of linear equations of the finite element method and requires less storage space. Based on two algorithms above, the calculation program is developed to process the simulation analysis of the finite element model of a million DOF on personal computers and applied to the temperature forecasts and optimized design of temperature control measures in several high concrete dams. It is shown that the calculation results are reliable and accurate, the acceleration is very efficient.

© 2011 Published bd Elsevies B. V Selection and/or peer-review under responsibility of ICAPIE Organization Committee.

Keywords:Simulation Analysis; Concrete Dam; Temperature Field; Stress Field; ICCG.

1. Introduction

Simulation analysis has been widely applied to the optimization design of temperature control of concrete dams and accepted by engineers in wide range. Currently, hydroelectric development in china reaches its peak season, temperature control and crack control of concrete dam is an important task during construction period. Particularly, a number of high arc dams that started sequentially in southwest china have met with crack problems. The trend of development needs simulation analysis to serve for project as a constructive tool. Simulation analysis can forecast cracking risk; provide rapid warning of concrete and

1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of ICAPIE Organization Committee. doi:10.1016/j.phpro.2012.02.175

reference for construction. Accordingly, the software of simulation analysis need meet the requirements that correlation parameters of construction, climate, and temperature control and so on be described truly, calculation accuracy be improved and the scale of calculation be expanded for the fast calculation speed.

When calculating temperature and stress field, traditional software of simulation analysis is based on classic algorithm flow chart. The coefficient matrix of linear equations is stored in one-dimensional array with variable semi-bandwidth and linear equations are solved in the LU decomposition. Array storage requirements depend directly on how the nodes are numbered. And consequently nodal numbering is very important with banded solutions. However, the concrete dam during construction period is rising constantly, and new nodes and new elements join in calculation continuously. Even though optimally re-ordering the nodal numbers, lots of zero coefficients are included in the banded envelope in the majority of time steps, and it spends plenty of time in decomposing zeros in the matrix with the LU decomposition method. Consequently, with the number of the nodes increased, the occupied time and storage space get more, and the occupied memory even exceeds computer memory. The band-with minimization is difficult so that the experienced analysts adopt other solution schemes.

In order to achieve the significant economy, the text introduces two fast algorithms. One is the Overall Planning algorithm and the other is the Incomplete Cholesky Conjugate Gradient method accompanied with element by element. Based on calculation procedures of temperature and stress fields, the algorithm one summarizes the characteristics of the simulation analysis and reorganizes calculation procedures. The other algorithm is the Incomplete Cholesky Conjugate Gradient algorithm, a method of linear finite element equations solving. Directed by the two algorithms above, the author developed the simulation analysis program of temperature control and accomplished the optimization design on such concrete dams' temperature control as Xiluodu, Xiangjiaba, Zangmu, Shenxigou, Tiechuanqiao projects and so on, the achievements are credible.

2. The characteristics of simulation analysis about concrete dam construction

Temperature and stress fields of concrete dams during both construction and service periods have the characteristics below.

(1) Transient behaviors. Simulation analysis includes the whole process of construction and service periods, and is connected with the thermal stress of a time series, so it is time-dependant and has transient behavior.

(2) The finite element calculation. A finite element calculation is performed in every discrete time step.

(3) "Loop". Concrete blocks of dams are accumulated from bottom to top, so the sub-regions of dams tend to be involved in calculation repeatedly. A calculation is done in every discrete time step; it is called "a loop".

(4) Transient and steady. In time series, some variables are time-invariant, and the others are time-dependant. So in the course of accelerating the calculation, firstly, the calculation of invariant should be put up the outside of the loop and the transient is inside, and then the crucial part of time-consuming would be found out. After examine the process of system, it is not difficult for us to find that the linear finite element equations solving are the most time-consuming process of simulation analysis.

3. Overall planning algorithm

The aim of the overall planning algorithm is to divide invariants and transients in simulation analysis. Relatively to the loop, the invariants calculations are lain outside and the transients calculations are lain inside. The invariants are excluded from the "loop" and stored in memory. Once the "loop" needs these invariants, the relevant data can be read directly from memory and only take opearionts on additions and

substractions. So, these data used to be avoided calculating repeatedly. Those measurements are taken target of saving the computation time, because it spends more time multiplying and dividing than adding and substracting. The overall planning algorithm would omit the time which is consumed in repeated calculations and thus accelerate the calculation.

In the process of simulation analysis, the first step is to generate the finite element meshes, and to assign the property of casting time to every element. The clock variable is defined to stand for the time of a step in the loop(the whole process). In every time step, the elements whose values of casting time is less than the clock variables constitute the dam shape in the step. In this way, once a concrete block is casted, its elements' location in space is unsteady according to the small deformation hypothesis of elastic finite element methods, whereas the shape of arc dam is variable. Therefore, the location of element is independent of time, yet the shape of arc dam is time-dependent.

With the support of theory in ref. [3] , overall solving equation of unstable temperature field in finite element method is denoted as

2 2 ([H ] +——[R]){T„+1} + ([H ]-—[R]){T} + F} + {F„+i} = 0

Ar„ Ar„ (1)

[H] = 2 [H]e ,[R] = 2 [R]e ,{F} = ^{Fy . [R] is the heat capacity matrix, [H] is the heat

conduction coefficient matrix, all of these are the symmetric positive definite matrices. {F} is the temperature load vector, {Tn} is the node temperature vector in the time step tn.

In every element, [H]e and [R]e are independent of time t , and [F] is time-dependent, because adiabatic temperature rise, temperature of cooling water in pipes and air temperature are functions of time t . Therefore, in order to achieve the goal of out-lain the invariants and in-lain the transients, program flow chart can be changed to the form as figure 1.

Similarly, in the process of thermal stress fields solving, overall solving equation in finite element method is expressed as

[K]{A^} = {APn}L + {APn}C + {APn}T + {AP/ + {APn}S (2)

where [K] is global stiffness matrix of a time step tn, {asn } is displacement increment vector of a time step tn., {APn }L, {APn }C , {APn }T, {APn }0 and {APn }S are node load increment vectors of external load, creep, temperature, self-grown volume deformation and shrinkage. [Dn ] is related to time and can be factorized as

[Dn ] = En [D], D = [0]-1 (3)

where [D] is elasticity matrix and is independent of the time. So the stiffness matrix of every element [kr]e can be given as:

[k]e = JJJ[ B]T [ D ][ B]dxdydz

[k ]e = Ejk'r (5)

[k']e is independent of the time, so it is lain out in the calculation loop. Moreover, equivalent load of

element self-weight is time-invariant and can be lain out, program flow chart of stress field can be changed to figure 2.

In the course of programming, two different meshes are defined, one is the whole mesh and the other is the mesh of current time step. Then the array is introduced to make writing of the relations between the whole mesh and the current time step tn. When calculated in current time step, the invariables of the whole mesh are read by means of the relation array. The [H], [R]e and [k']e are calculated outside of "loop" and stored in memory. When an element is in its casting time, they directly join the calculation in the "loop".

4. Preconditioned conjugate gradient algorithm

The overall planning algorithm aims at reducing the computational burden of simulation analysis, then the goal of the preconditioned conjugate gradient algorithm is up-shifting the simulation analysis. Simulation analysis repeats finite element calculation many times, yet in finite element calculation, solving linear equations (6) is most time-consuming.

For isotropic non-fractural structure issue , A in (6) is sparse symmetric positive definite matrix.

The preconditioned conjugate gradient algorithm is a fast way to solve the positive sparse linear equations, especially for strictly diagonally dominant symmetric positive definite matrices, the preconditioned conjugate gradient algorithm have good numerical stability. The preconditioned conjugate gradient algorithm falls into two categories: the Incomplete Cholesky Conjugate Gradient algorithm and Symmetric Successive Over Relaxation algorithm. Lin Shaozhong firstly introduced the revised SSOR-PCG to simulation analysis, the acceleration effect is good, and then Chen Guorong improved it. This text introduces the ICCG to simulation analysis, two equations, (7) and (8), are used in this text of preconditioned conjugate gradient algorithm. Compared to ref [6-9], equations (7) and (8) reduced the calculation amount of decomposing upper and lower triangular matrices, and directly decomposed the principal diagonal elements of matrix A, don't need to assembly overall heat conduct matrix or global stiffness matrix. The operation is easy and the storage is low, meanwhile, equations (7) and (8) may add the condition number of matrix A and accelerate convergence. Algorithm flow chart is shown in figure 3.

Ax = b

J5»l . tf't*'

II-{l™. *"»)/(/»». ii») jM.^t+^Jt*

Figure 3 Flow chart of the ICCG method

Processing mode of the ICCG method is as follows:

M = diag (A) (7)

M = LLT (8)

where diag (A) is the principal diagonal elements array of matrix A, {aii} , L is column vector {li} , h •

5. Numerical examples

In Visual Fortran 6.6 and Visual C++ net programming environment, the author developed the software of simulation analysis, and took the 3m embedded slab of ref [3] as an example. Compared with the results in ref [3], the calculation result of software is proved correct and valid. The basic parameters of concrete and the rocks are as below.

Table 1 basic parameters of concrete and the rocks

Material Item Rocks Concrete

coefficient of heat conductivity X ( kJ/m • d • C ) 360 220

thermal diffusivity a ( m2/d ) 0.1754 0.1

volume weight y (kg/m3) 2700 2450

specific heat c (kJ/ (kg .C)) 0.76 0.94

coefficient of inertia expansion a ( 10"6/C) 6 8

Poisson's Ratio 0.20 0.17

heat emission coefficient p (kJ/ (m2 . d . C)) 2000 2000

The adiabatic temperature, rise of concrete is 0(t) = &0t/(1 .0 + t) .The elastic modulus of concrete is E(t) = E0 (1 - e~0 4t ) .The coefficient of stress relaxation is

K(t,T) = 1 - (0.4 + 0.6e0 62T°17 )[1 - e-0 2-0 27 ^3 (t -t)036]The temperature and stress process lines of embedded 3 m thick concrete slab are shown as below.

To verify the efficiency of the calculation acceleration, take the simulation analysis of a RCC arch dam as an example, which would be constructed with non-joint continuously. The finite element model of the arc dam is shown as figure 6. Hexahedral iso-parametric element with 8 nodes is used in meshing, the total number of nodes is 443962, and the total number of elements is 411264. The time span of construction and operating period is 1140 days, the date of commence is January 1st, 2009, and the completion date would be August 30th, 2009. The time step is 0.5 days. No transverse joint is built in the arc dam. The dam height is 94.5m, bottom thickness is 20m, top thickness is 6m, and the arc length of the top is 230.68m. The concreting temperature is 18 °C . Elastic modulus formula of RCC is 0(t) = 31.9t/(8.0 + t) GPa, coefficient of inertia expansion is 9 X10-6/C, adiabatic temperature rise formula is $(t) = 25.1t/(1.56 + t) C , air temperature formula is Ta = 19.5 + 7.0cos(2n(T-180)/365)°C, heat emission coefficient is 1000kJ/ (m2 • d -°C) , creep is left out.

When running in a PC with 2.50GHZ CPU and 2G memory, the calculation process of temperature field takes 3 hours, and the process of stress field takes 8 hours. After 5 to 20 iterations, the calculation results can obtain the design accuracy. Because of occupying huge memory storage space, traditional

programs that are based on LU decomposition are unable to run.

6. Conclusion

The software of simulation analysis which is developed on the basis of overall planning and ICCG algorithm, is applied to the optimization design of temperature control in Xiluodu, Xiangjiaba, Zangmu, Shenxigou, Tiechuanqiao and so on, the achievements are reliable and accuracte. The numerical examples that are analyzed by using the method mentioned in this paper, testify that simulation calculation with millions of degrees of freedom can be completed in ordinary PC, and Accelerating effect is very good. Currently, interrelated simulation software is being applied in project of temperature forecast in Xiluodu high arc dam during construction period, and well received by all concerned.

Acknowledgement

The authors gratefully acknowledge Professor Zheng Hong for his guidance to algorithms, and appreciate Liu Xinting, Han Yan, Huang Wei, Yang Xianyuan, Xu Qing, Wang Xiangfeng and other members in this organization for their support and help in the program debugging.

Reference

[1] Huang Dahai, Song Yupu, Zhao Guofan. Advancement of Thermal Creep Stress Analysis for RCC Dam[J] China Civil Engineering Journal. 2000: 33(4), 97-100;

[2] Liu Defu, Huang Dahai, Tian Bin. The Optimization of Joint Closure Temperature Field and Temperature Control of Arc Dam[M]. Beijing, China Waterpower Press. 2008. 141-144;

[3] Zhu Bofang. Thermal Stresses and Temperature Control of Mass Concrete[M]. Beijing, China Waterpower Press, 1998;

[4] Meijerink J. A. and Van Der Vorst H. A. An iterative solution method for linear system of which the coefficient matrix is a symmetric M-Matrix, Math. Comp, 1977, 31: 148-162;

[5] Manteuffel T. A. An incomplete factorization technique for positive definite linear systems, Math.Comp, 1980, 34: 473-497;

[6] Lin Shaozhong, Su Donghai. Fast Algorithms for Stress Analysis Simulating Construction Process of Massive Concrete Structures and Application[J]. Journal of Yangtze River Scientific Research Institute. 2003: 20(6),19-22;

[7] Lin Shaozhong. Application of Preconditioned Conjugated Gradient Method to Finite Element Equations and Programme Design[J]. Hohai University Journals(Science edition). 1998:26(3), 112-115;

[8] Chen Guorong, Li Huangsheng, Li Hongjian. Improvement of fast solution method for mass concrete temperature fields[J]. Hohai University Journals(Science edition). 2009:37(4), 396-399;

[9] Zhang Yongjie, Sun Qin. A New ICCG Method of Large Scale Sparse Linear Equations[J]. Journal on Numerical Methods and Computer Applications. 2007: 28 (6), 133-137;

[10] Li Yangbo. Acceleration Algorithm of Structural Numerical Simulation Analysis of Mass Concrete[D]. China Three Gorges University. 2005, 36-38.

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