This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices