While it might appear almost hopeless to look for theoretical (closed form) solutions, there are some available that are both powerful and practical. Most finite difference and finite element elastic solutions can incorporate temperature effects and are used for a wide range of opto-mechanical structures. In addition to the temperature, one must also know the corresponding material parameter (for example, the coefficient of thermal expansion) These parameters are ofter themselves temperature dependent and are not constant either throughout the structure or even within a single (non-homogeneous) component.
Since temperature distributions can be irregular, variable and difficult to predict exactly, orthogonal functions can often be analyzed. Then the thermal distribution can be approximated by a sum of these functions thereby predicting the whole response.