Due to the emergence of technologies enabling the fabrication of complex cellular materials, new materials with higher mechanical efficiency than the constituent material have been introduced. Combination of optimized cellular architectures with high-performance metals and composites can result in lightweight materials with mechanical properties previously unattainable at low densities. Consequently, new materials can be designed to maximally fit the target application. There is a wide range of important applications including energy absorption, metamaterial, thermal management, and bioscaffold for ultralight architected cellular metals as well as airframes and shape morphing for ultralight architected cellular composites. Therefore, it is of importance to propose a nearly exact and highly efficient methodology to study low-mass metal and composite systems with architected cellular structures. We present the simplest initial computational framework for the analysis, design, and topology optimization of such cellular materials. In the present methodology, the repetitive Representative Volume Element (RVE) approach is employed to model the actual cellular metallic/composite micro-lattices. Each member of the cellular material is modeled using only one finite beam element with 12 degrees of freedom (DOF), and the nonlinear coupling of axial, bidirectional-bending, and torsional deformations is studied for each spatial three-dimensional (3D) beam element. The large deformation analysis of the cellular strut members is performed utilizing mixed variational principle in the updated Lagrangian co-rotational reference frame. The explicit form of the stiffness matrix is calculated under the effect of plasticity for the case of the cellular metals and under the effect of nonlinear flexible connections for the case of the cellular composites. Then, we use newly proposed homotopy methods to solve the algebraic equations.