The alkali-silica reaction (ASR), also known as concrete cancer, is one of the most prevalent causes of concrete degradation. In this chemical reaction, amorphous silica in the aggregates reacts with alkalis in the pore solution. By absorbing water, hydrophillic ASR products generate highly localized internal pressure that causes expansion and cracking. The detrimental effects of ASR on concrete pose a major threat to the safety and operability of concrete infrastructure in all parts of the world. The long lifespan of concrete structures and their high economic significance make it crucial to evaluate the effect of ASR-induced degradation. ASR has therefore been the subject of extensive research over the past few decades. Modeling and experimental studies have provided fundamental insight into the physics of ASR at the meso-scale of concrete. However, the impact of the mesoscopic ASR damage evolution on the macro-scale, or structural scale, on concrete is not well understood yet. Investigating the structural outcome of the ASR damage necessitates robust meso-scale solvers faster than the existing meso-scale models that conventionally use finite element method (FEM) as the solution scheme. Over the past 30 years, fast Fourier transform (FFT)-based methods have gained much attention as fast and reliable alternatives to conventional FEM solvers because they can exploit regular grid structures, use lightweight iterative solvers, and speed up meso-scale simulations by orders of magnitude. However, it has not been feasible to effectively utilize them in damage mechanics problems due to two shortcomings: ringing artifacts and incapability to solve non-elliptic problems, in their recent efficient implementations, where conjugate gradient (CG) is used as the linear solver. In the present thesis, we have resolved the shortcomings of the FFT-based solution scheme for being effectively used in damage mechanics problems. All of the developed methods are implemented in an open-source FFT-accelerated software package capable of solving generic homogenization problems for other use cases as well. The developed library is capable of solving non-convex problems (non-elliptic partial differential equation (PDE)) containing phases with extremely high contrast with a ringing-free scalable FFT-accelerated solver. The developed fast and robust numerical framework is employed to conduct ASR meso-scale simulations. The obtained results show good agreement with the results obtained using the conventional FEM solver. The developed solution scheme is 200 times faster than the solution of the same problem with conventional FEM solvers. Therefore, it enables comprehensive multi-scale structural ASR damage modeling with reasonable computational costs.