2　Temperature field analysis

Based on finite element method，stress analysis of the structure was carried out to reproduce the thermal cracking in massive RC structures.It is well known that the temperature field is required for the subsequent thermal stress calculation.Therefore，further reinforcement design must give priority to reproduction of the temperature field of the entire structure.

In the temperature field analysis，the measured temperature can be obtained by optical fibre in concrete.The thermal parameters of concrete（i.e.thermal conductivity，convection heat exchange coefficient，heat capacity and thermal expansion coefficient）are fundamental importance for the temperature field.However，it is usually difficult to obtain the thermal parameters of concrete in actual projects.In the study，based on the temperature data measured by distributed temperature system（DTS）using optical fibre（as shown in Figure 1），the values of the thermal parameters were obtained by the GA.Moreover，GA plays critical roles in the inversion of thermal parameters.These steps are shown in detail in the part one of Figure 2.

2.1　Calculation principles

Concrete thermal properties are affected by various factors，which make concrete thermal field analysis complicated（Neville，1997）.Since the thermal conductivity coefficient of concrete is one of the thermal parameters to predict the temperature variation during hydration，the reliability of the proposed numerical approach depends on the thermal conductivity coefficient of concrete，especially at the very early age.

In this paper，the boundary condition is assumed that the concrete surface heat flow is proportional to the difference between concrete surface temperature and atmospheric temperature，which can be expressed as follows，

where λis thermal conductivity；nis the surface normal direction；Tdenotes the concrete surface temperature，Tadenotes atmospheric temperature and βrepresented surface heat transfer coefficient.

The instantaneous temperature value of one point in the homogeneous solid can be represented as Ti=f（x，y，z，τ）.Based on the Fourier heat conduction theory，three dimensional（3D）unstable heat conduction equation can be obtained as Equation 2（Zhu，2003a）.

Owing to the hydration heat of cement，the heat conduction equation of 3D unsteady temperature field of concrete can be expressed as Equation 3.

where T，t，α，θ，and τmean temperature，time，thermal diffusivity，adiabatic temperature rise and concrete age，respectively.

According to the variation principle，the methods of spatial discretisation and time domain difference are used to obtain the temperature field governing equations with the consideration of initial and boundary conditions（Zhu，2003a）.

where[H]means heat conduction coefficient matrix；[R]express heat conduction added matrix；{Fk+1} is node temperature load array；{Tk+1} and {Tk} are thermal load arrays in the time k+1 and k，respectively.

2.2　Calculation method of cooling water pipes

Considering the cooling effect of pipe in concrete，temperature simulation in the code is based on the equivalent heat conduction equation，in which the cooling water is regarded as a negative heat resource and its cooling effect is equivalent to the temperature reduction.The equivalent heat conduction is quoted as follows（Zhu，2003b）.

where Tw，denoting the temperature of cooling water；ϕis the function related to pipe cooling.The function ϕin Equation（5）is the function related to pipe cooling.In detail，it can be expressed as，

where ameans thermal diffusivity；λis the thermal conductivity；Ddenotes the diameter of the cooling pipe；Lis the length of the cooling pipe and cw，ρw，qwrepresent the specific heat，density and flow of the cooling water，respectively.

2.3　Thermal parameters inversion analysis

The adiabatic temperature rise of concrete can be expressed as

where θdenotes the transient adiabatic temperature rise in time（t）；θ0 represents the ultimate adiabatic temperature rise；both aand bare the temperature variation corresponding parameters.

It is necessary to obtain the thermal parameters of the concrete insituto predict the temperature field.The quadratic sum of the temperature error between the actual measurement and the finite element analysis result is defined as objective function，and thermal parameters α，β，θ0，a，bare the design variables.Therefore，the objective function Er，Tcan be described as

where α，βmean thermal diffusivity and surface heat transfer coefficients.

In practice，the design parameters are constrained in ranges，which can be written as

2.4　Working mechanism of GA

There are various methods to solve the constrained nonlinear programming problem（Equation 8），such as direct，（Gao，2009）indirect，（Mantia & Casalino，2006）hybrids，（Mengali & Quarta，2007）etc.However，good initial guesses are the requirements of the previous method，which is their main drawback.In contrast，GA can obtain satisfactory results without any prior knowledge（Whitley，1994），which exists in three operations to generate additional control parameters：reproduction，crossover and mutation.Therefore，in this paper，GA is used to solve the programming problem with the corresponding objective function（Equation 8）and design parameters（Equations 9 and 10）.

Remark Figure 3：The flow diagram of GA is shown in Figure 3，which clearly demonstrated the working mechanism of GA.In addition，the main processes include：（i）give the first generation to initialisation；（ii）evaluate the current generation using the fitness function Er；（iii）decide whether or not the GA should be end；（iv）create the（i）generation by using“reproduction”，“crossover”and“mutation”operations until the new generation satisfied the accuracy requirement.Moreover，the detailed GA tutorial and theory is presented in literature（Fogel，1994；Srinivas & Patnaik，1994）.