This paper explores the use of optimization to design multifunctional metamaterials, and proposes a methodology for constructing a design envelope of potential properties. A thermal-mechanical metamaterial, proposed by Ai and Gao (2017), is used as the subject of the study. The properties of the metamaterial are computed using finite element-based periodic homogenization, which is implemented in Abaqus utilizing an open-source plugin (EasyPBC). Several optimization problems are solved using a particle swarm-based optimization method from the pyOpt package. A series of constrained optimization problems are used to construct a design envelop of potential properties. The design envelope more fully captures the potential of the metamaterial, compared with the current practice of using parametric studies. This is because the optimizer can change all parameters simultaneously to find the optimal design. This demonstrates the potential of using an optimization-based approach for designing and exploring multifunctional metamaterial properties. This proposed approach is general and can be applied to any metamaterial design, assuming an accurate numerical model exists to evaluate its properties.