# Optimization of Mass Concrete Construction Using a Twofold Parallel Genetic Algorithm
## 2 Principles of the Thermo-Chemo-Mechanical Numerical Model  
[PERSON]'s model has been described in detail elsewhere [9; 10; 18; 19; 20]. Therefore, in the present paper only the equations necessary to understand the optimization process are presented.  
The equation for the transient thermal and hydration fields that expresses the thermo-chemical coupling is:  
\[c\gamma\dot{T}=Q_{\xi}\dot{\xi}+k\
abla^{2}T \tag{1}\]  
where: \(c\) is the specific heat; \(\gamma\) is the density; \(T\) is the temperature; \(Q_{\xi}\dot{\xi}\) represents the heat generated by the exothermic reaction, being \(Q_{\xi}\) a material constant. \(0\leq\xi\leq 1\) is the degree of hydration, a normalized variable that accounts for the evolution of the hydration reaction, and \(k\) is the thermal conductivity.  
The chemo-mechanical coupling is represented, in a simplified form, by the following equation:  
\[\dot{\sigma}=\mathbf{C}(\xi)\mathbf{:}(\dot{\varepsilon}-\alpha\mathbf{1} \dot{T}-\beta\mathbf{1}\dot{\xi}) \tag{2}\]  
where: \(\sigma\) is the stress tensor; \(\varepsilon\) is the strain tensor; \(\alpha\) is the thermal dilatation coefficient; \(\beta\mathbf{1}\dot{\xi}\) is the variation of autogenous shrinkage strain, considered, for the sake of simplicity, linear with the variation of the hydration degree; \(\beta\) is a material constant; and \(\mathbf{C}(\xi)\) is the elastic compliance tensor, dependent on the degree of hydration. As Poisson's ratio \(\
u\) is considered to be constant with the degree of hydration, one can write, formally, \(\mathbf{C}(\xi)=\mathbf{C}(E(\xi))\).  
Therefore, for a given point of the structure, and for a given time, the cracking index, that represents the cracking tendency at this point, can be written as:  
\[I_{c}(x,y,z,t)=\frac{<\sigma_{1}(x,y,z,t)>^{+}}{f_{t}(\xi(x,y,z,t))} \tag{3}\]  
where \(<\sigma_{1}>^{+}\) is the maximal positive principal stress and \(f_{t}\) is the tensile strength of the concrete.  
It should be stressed that the cracking index is always computed in the elastic range even if it is greater than 1, i.e., even if the tensile stresses exceed the tensile strengths. This strategy was used because, within the framework of the optimization process, there is no need to precisely compute the post-cracking behavior (cracking evolution) of the structures at hand. The cracking index is just a parameter that is used to penalize sets of variables that tend to crack the structure at the early ages for the first generations of the GA. For the last generations and for the best case, the structure does not present any element where the strength is reached. In this way cracking indexes greater than 1 do not have a precise physical meaning and there is no need to use a sophisticated microcracking model that simulates the very complex damage evolution of cracked concrete.  
This numerical model was implemented in a massively parallelized 3D FEM code using linear tetrahedral elements. It is composed by two modules: thermo-chemical and mechanical. The code allows for the simulation of a layered construction, i.e., the geometry of the structure may change to simulate the sequential pouring of the several concrete lifts. The parallel implementation is designed for a distributed memory architecture and the data structure known as compressed storage row/columns (CSRC) is used to store the coefficient matrix and obtain iterative solutions, in a subdomain-by-subdomain approach (SBS) using the method of preconditioned conjugate gradients (PCG) [16; 21].