## 1 Introduction  
A disk brake unit stops a moving car by dissipating the kinetic energy of the car into frictional heat energy, which is generated by sliding friction between the disc and pads, when a caliper moves the brake pads into contact with the frictional surface of the disc. The contact positions and areas change during braking due to the thermo-elastic behavior of the disc and pads, the vibration of the brake unit, variation in the friction coefficient, uneven wear of the contacting surfaces, etc. Contact pressure between the two contacting surfaces also varies at every moment of the braking process. Estimating these contact positions, and pressure distribution during a braking application is a prerequisite for calculating thermal stress and designing a brake unit. In the late fifties, to obtain temperature distributions, a brake disc was modeled as a solid bound by two parallel infinite planes. A heat flux linearly decreasing with time was input to one face, while at the other face an adiabatic boundary condition was applied. To solve the one-dimensional transient heat equation, a Fourier series, the methods of images, or Laplace transform [1] was applied. Solutions showed that the disc's temperature rise was linearly proportional to the heat flux input. In the 2000s 2-dimensional transient heat problems for ideal brake discs were solved analytically by using [PERSON]'s function [2] or [PERSON]'s integral [3]. Though the discs used in their models were similar to commercial brake discs, the mounting seat and bolts were not included. More realistic 3-dimensional transient heat problems have been conducted using finite element analysis. One of the main merits of finite element analysis is that it can easily reflect non-linear physical or/and mechanical properties in simulation. [PERSON] et al. [4; 5; 6; 7] investigate the influence of temperature dependent properties such as friction coefficient, thermal conduction coefficient, and specific heat on the temperatures and wear of brake discs. Their numerical results showed that fluctuating properties had an effect on the temperature fields and wear in the disc. Computer enhancement has allowed researchers to deal with coupled thermo-mechanical problems. [PERSON] and [PERSON] [8] first obtained heat convection coefficients on every surface of the disc by air flow analysis, then performed a thermo-mechanical coupled analysis for full and ventilated brake discs. The maximum temperature on the ventilated disc surface was higher by 60 \({}^{\circ}\)C at 100 km/h. [PERSON] and [PERSON] [9] applied a multi-body technique and a thermo-mechanical coupled model. According to their results, there were fluctuations in temperatures and stresses in the disc. [PERSON] and [PERSON] [10] also performed a thermo-mechanical coupled simulation for a disc/pad unit and compared their numerical and experimental results. They showed that the two results were in good agreement, and temperatures on the disc surface fluctuated during braking. [PERSON] et al. [11] simulated turbulent flow on a ventilated disc with cross-drilled holes using finite element analysis. The cross-drilled holes provided 15-17% higher cooling capacity. [PERSON] et al. [12] showed that X-type cooling vanes were more efficient than conventional radial vanes, increasing cooling capacity by 18-21%. Some researchers have studied thermal fatigue in brake discs. [PERSON] et al. [13] predicted the fatigue crack propagation of a high speed railway brake disc using an extended finite element. [PERSON] et al. [14] showed that maximum thermal stress of the disc during a single brake application was linearly proportional to the temperature rise of the disc. They measured the temperature variation of a commercial train disc, and predicted the crack initiation life of the disc using the relation between the temperature rise and thermal stress. According to [PERSON] et al. [15], thermal stress in the disc was very sensitive to the profile of the contact pressure distribution in the radial direction.