Mechanical metamaterials are used in various applications to control bulk wave propagation through elastic materials. Controlling both the longitudinal and transverse waves at the same time by designing a single phononic device is challenging through theoretical, computational, and experimental perspectives. Suppression of system generated second harmonics due to interaction of monochromatic longitudinal wave and during one-way two-wave mixing of two longitudinal waves by designing effective phononic material is demonstrated in previous work by the author [Ghodake, J. Acoust. Soc. Am. 150, A149 (2021)]. In this talk, the design of linear periodic metamaterials which can control both the longitudinal and transverse waves, their second harmonics during propagation of monochromatic waves, and one-way two-wave mixing between longitudinal and transverse waves, is discussed by solving inverse problems using finite element analysis, effective boundary conditions, and optimization techniques. Along with the widths of periodic elastic materials, the number of repeated periodic cells (N) in a phononic lattice is defined as a design parameter. Different strong and weak contains are implemented so that in every iteration either during or after the iteration N will remain integer. A gradient-free optimization algorithm is used during constrained optimization. Strong constraint solves the inverse problem in a few minutes.