# Synthetic Microstructure Generation and Multiscale Analysis of Asphalt Concrete
## 2 Materials and Methods
### Multiscale Modeling  
In the 2D linear elastic computations with relatively small analyzed domain (which is the case of the numerical test presented in this paper), no multiscale approach is necessary. However, addressing the proposed framework to the 3D nonlinear analysis of the whole pavement structure, it is unavoidable.  
In our research [15; 19], we develop the multiscale finite element method (MsFEM) that is a versatile tool, e.g., for the AC modeling. In the aforementioned papers, the method (its extension to the higher order approximation) is introduced and thoroughly described. In this paper, we restrict ourselves to a brief description of its workflow. We proceed with two meshes: a coarse mesh used for the main problem solution and a set of fine meshes within coarse elements. The fine meshes comply with the microstructure but the solution obtained on a global fine mesh (with gluing all fine meshes) is typically very expensive or infeasible. Thus, we take advantage of the microresolution information to compute the effective coarse element stiffness matrices and load vectors. Finally, we solve the main problem using the coarse mesh with the effective quantities and reasonable number of DOFs (degrees of freedom) reduction.  
The idea of the MsFEM is based on the concept of the special shape functions (c.f. [20; 21]). Starting with the standard coarse element shape functions, we modify them to account for the oscillations of the material parameters. The price of the MsFEM is the solution of the N local boundary value problems (BVP) recapitulated below, where N is the number of the coarse element shape functions. The strong form of the BVP to be solved within the subdomain \(\Omega_{s}\) (a coarse element) is recalled.  
Given \(\boldsymbol{\Psi}\), which is a coarse mesh vector-valued shape function, we look for its vector-valued counterpart \(\boldsymbol{\Phi}\) that is a discrete solution of the following Dirichlet boundary value problem with \(\Phi_{k}\)\(\in\)\(C^{0}(\Omega)\)  
\[\begin{cases}\frac{\partial}{\partial x_{j}}\left(\zeta_{ijkl}\frac{\partial \Phi_{k}}{\partial x_{l}}\right)\!\!=\!\!f(\Psi_{k})&\forall i,j,k,l=\!1,\!2,\! 3,\;\boldsymbol{x}\in\Omega_{s}\subset\Omega\\ \boldsymbol{\Phi}=\boldsymbol{\hat{\Phi}}&\text{on}&\partial\Omega_{s}\end{cases} \tag{1}\]  
The exemplary function \(f(\Psi_{k})\) and essential boundary conditions (\(\boldsymbol{\hat{\Phi}}\)) can be found, e.g., in [15; 19]. Also, we refer the reader to the aforementioned papers for a thorough description on how the Dirichlet boundary conditions are defined on \(\partial\Omega_{s}\). In [15], we presented also a number of examples of the modified shape functions and further numerical tests. Solution to N Problems (1) constitutes the prolongation operator \(\boldsymbol{P}\) and allows one for the computation of the effective coarse element stiffness matrix and load vector, respectively,  
\[\boldsymbol{K}_{H} =\boldsymbol{P}^{T}\boldsymbol{K}_{hl}\boldsymbol{P} \tag{2}\] \[\boldsymbol{f}_{H} =\boldsymbol{P}^{T}\boldsymbol{f}_{h} \tag{3}\]  
in terms of \(\boldsymbol{K}_{h}\) and \(\boldsymbol{f}_{h}\), which denote the fine mesh stiffness matrix and load vector assembled within the domain occupied by the analyzed coarse mesh element in such a setting.  
The generation of the (only) locally non periodic microstructure, we advocated on in the previous chapter, is justified by the computation speed-up; thus, the effective coarse mesh element quantities have to be computed only once and used for each of these elements.