There are several types of analyses that interest engineers, all of which involve mathematical modeling. *Structural analysis* deals with linear and nonlinear models and stresses on a material. The linear models assume the material does not plastically deform (the remaining deformation after the load causing it is removed); nonlinear models stress the material past its elastic capabilities. The stresses in the material then vary with the amount of deformation. *Vibrational analysis* deals with possible resonance and subsequent failure. It is used to test a material that may experience random vibrations, impacts, or shocks. *Fatigue analysis* is used to determine the life of a material or structure. It shows the effects of occasional (periodic) or cyclic loading on a structure or object, pointing out where cracks or fractures are most likely to occur. Engineers measure *heat transfer* to determine a material or structure’s conductivity or thermal fluid dynamics. In this way, researchers understand how a material will respond to various hot and cold conditions—or even how it diffuses heat and cold—over time.