## 2 Basic Transport Formulation of Unsaturated Concrete
### Overlapping Domain Decomposition Method  
In this study, a parallel program was developed by dividing the program into multiple fragments, each of which could execute simultaneously on its own processor. This was achieved using the domain decomposition (DD) method, which is well suited for parallel finite element computations. The DD method is commonly employed to solve large  
Figure 1: Framework of parallel FEM with hygro-chemo-thermo transport model.  
scale system equations, enabling sub-domain operations to be executed concurrently on separate processors [7].  
There are two main types of domain decomposition methods: overlapping and non-overlapping. In this study, the overlapping method was preferred for solving the linear sparse matrix. This method offers a simpler implementation for algebraic approaches and results in faster convergence compared to the non-overlapping DD method. Additionally, the extended subdomains in the overlapping approach have smoother boundaries than those of non-overlapping subdomains, leading to a better overall performance.  
To create the algorithm structure, the overlapping DD method consists of four parts: subdomain preparation, subdomain solver, data exchange, and global management. The process starts with a set of overlapping subdomains, each with its own solver. During each iteration, the boundary conditions need to be updated, requiring data to be exchanged between neighboring subdomains. To coordinate the solvers and data exchange, the global management part controls the progress of the DD iterations, monitors overall convergence, starts the subdomain solvers, and manages the data exchange [46].  
To solve the system of equations required by the DD method, an iterative solver is necessary. In this study, the GMRES (Generalized Minimal Residual) method was primarily used, as it is particularly effective for solving non-symmetric linear systems that involve both global and local matrices. Furthermore, to improve the convergence in solving the multi-species transport problem, the additive Schwarz preconditioner was applied.  
The mathematical formulations of the overlapping domain decomposition (DD) method, along with details of the preconditioned Schwarz method, can be found in [46; 47]. To achieve overlapping decomposition of the domain, each subdomain \(\Omega_{i}\) is extended into a larger domain \(\Omega_{i}^{\prime}\), where, \(\Omega_{i}\subset\Omega_{i}^{\prime}\). The overlap is uniform and \(V_{i}\subset V^{h}\) is the discretized space over \(\Omega_{i}^{\prime}\). Therefore, Equation (19) can be rewritten as below:  
\[\Omega\ =\ \bigcup_{e=1}^{net}\Omega_{e}^{\prime} \tag{35}\]  
\[V^{h}=V_{1}+\cdots+V_{N} \tag{36}\]  
where \(V^{h}\) denotes a finite dimensional subspace of the Sobolev space \(V\), and subscript \(N\) represents the number of subspaces, and superscript \(h\) is the element size.  
Equation (32) is simplified as follows:  
\[Au=f \tag{37}\]  
Since Equation (37) is typically ill-conditioned, a preconditioner such as the additive Schwarz preconditioner is necessary to solve it using any iterative method.  
The additive Schwarz preconditioned system can be written as  
\[M^{-1}Au\equiv(T_{1}+\cdots+T_{N})u=g \tag{38}\]  
\(T_{i}\) denotes the operators and can be written in matrix form as \(T_{i}=M^{-1}A=R_{i}^{T}A_{i}^{-1}R_{i}A_{i}\), where \(R_{i}\) is the restriction matrix and \(A_{i}^{-1}\) is the solver for the subdomain problem. In the PFE program, for the additive scheme, the number of subdomains, \(N\), is typically set as equal to the number of available processors.