Microprocessor Silicon Die Thermal Distribution (Mechanical & Materials Engineering)

A mechanical thermal packaging engineer at the UNLV Howard R. Hughes College of Engineering is designing a vapor chamber cooling solution for a high-performance computer graphics processor (GPU). The silicon processor die occupies a closed, square planar domain:

D = \{ (x, y) \in \mathbb{R}^2 \mid 0 \le x \le 4, \, 0 \le y \le 4 \}

where

x

and

y

are spatial coordinates measured in centimeters (

\text{cm}

Non-uniform core clocking generates a continuous temperature distribution across the silicon surface modeled by:

T(x, y) = 2x^2 + 2y^2 - 4x - 8y + 65 \quad (\text{in }^\circ\text{C})

Thermal reliability standards require the cooling package to maintain peak temperatures below silicon breakdown limits while identifying the lowest-temperature location for thermal reference diode placement.

Verify that the domain $D$ satisfies the conditions of the Extreme Value Theorem.
Locate any critical points within the interior of the silicon die and evaluate the local temperature.
Analyze the temperature along each of the four boundary edges ( $y=0$ , $x=4$ , $y=4$ , and $x=0$ ) to identify perimeter thermal candidates.
Evaluate the temperature at the four mechanical corner mounting posts.
Identify the absolute maximum junction temperature $T_{\max}$ and absolute minimum temperature $T_{\min}$ , stating their precise coordinate locations.