Cementitious materials such as concrete are intrinsically heterogeneous and include internal structures and constituents across length scales ranging from nanometers to millimeters. These materials are widely used as protective materials for military applications. They need not only to withstand conventional quasi-static loadings but also to defeat extreme loadings such as high-rate blast, impact, and penetration. To fully explore the design and application of these materials in war-fighting efforts, it is essential to understand the deformation and failure mechanisms of multiscale internal structures and constituents under different loading conditions. Mesoscale structures and constituents of cementitious materials include mesoscale particles such as aggregates, sand, fibers, mesoscale porosities, and cracks. Several numerical methods have been developed to investigate the deformation and failure mechanisms of mesoscale structures and constituents under different loading conditions. In this report, we explored the Lattice Discrete Particle Method (LDPM) and the Finite Element Method (FEM). The work provides some basic knowledge on these methods and aids in formulating a path forward in the next phase of the research. DISCLAIMER: The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. All product names and trademarks cited are the property of their respective owners. The findings of this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.