Microprocessor Silicon Die Thermal Distribution (Mechanical & Materials Engineering) Solution

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:

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

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.

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

Step-by-Step Solution:

1. Verify Extreme Value Theorem Conditions: State the Extreme Value Theorem for Functions of Two Variables (defined in Extrema on Closed and Bounded Regions):

   • A continuous function on a closed and bounded domain achieves absolute extrema.
   • $T(x, y)$ is a polynomial and continuous on $\mathbb{R}^2$. The die domain $D = [0, 4] \times [0, 4]$ includes all four bounding edges (closed) and has finite diameter $4\sqrt{2}\text{ cm} \approx 5.66\text{ cm}$ (bounded). Thus, absolute extrema exist.

2. Tier 1: Interior Critical Points: Compute the partial derivatives of $T(x, y)$:

   $$T_x = \frac{\partial}{\partial x}(2x^2 + 2y^2 - 4x - 8y + 65) = 4x - 4$$
   $$T_y = \frac{\partial}{\partial y}(2x^2 + 2y^2 - 4x - 8y + 65) = 4y - 8$$
   Set both partial derivatives to zero:
   $$4x - 4 = 0 \implies x = 1\text{ cm}$$
   $$4y - 8 = 0 \implies y = 2\text{ cm}$$
   The point $(1, 2)$ lies strictly within the interior $0 < 1 < 4$ and $0 < 2 < 4$. Evaluate temperature at $(1, 2)$:
   $$T(1, 2) = 2(1)^2 + 2(2)^2 - 4(1) - 8(2) + 65 = 2 + 8 - 4 - 16 + 65 = 55^\circ\text{C}$$

3. Tier 2: Boundary Edge Search:

   • Bottom Edge ($y = 0, 0 \le x \le 4$):
     $$T(x, 0) = 2x^2 - 4x + 65 \implies T'(x, 0) = 4x - 4 = 0 \implies x = 1$$
     Candidate: $(1, 0)$. Value: $T(1, 0) = 2(1)^2 - 4(1) + 65 = 63^\circ\text{C}$.
   • Right Edge ($x = 4, 0 \le y \le 4$):
     $$T(4, y) = 2(4)^2 + 2y^2 - 4(4) - 8y + 65 = 2y^2 - 8y + 81$$
     $$T'(4, y) = 4y - 8 = 0 \implies y = 2$$
     Candidate: $(4, 2)$. Value: $T(4, 2) = 2(2)^2 - 8(2) + 81 = 8 - 16 + 81 = 73^\circ\text{C}$.
   • Top Edge ($y = 4, 0 \le x \le 4$):
     $$T(x, 4) = 2x^2 + 2(4)^2 - 4x - 8(4) + 65 = 2x^2 - 4x + 65$$
     $$T'(x, 4) = 4x - 4 = 0 \implies x = 1$$
     Candidate: $(1, 4)$. Value: $T(1, 4) = 2(1)^2 - 4(1) + 65 = 63^\circ\text{C}$.
   • Left Edge ($x = 0, 0 \le y \le 4$):
     $$T(0, y) = 2y^2 - 8y + 65 \implies T'(0, y) = 4y - 8 = 0 \implies y = 2$$
     Candidate: $(0, 2)$. Value: $T(0, 2) = 2(2)^2 - 8(2) + 65 = 8 - 16 + 65 = 57^\circ\text{C}$.

4. Tier 3: Corner Mounting Posts Check: Evaluate temperature at each vertex:

   • $(0, 0)$: $T(0, 0) = 65^\circ\text{C}$
   • $(4, 0)$: $T(4, 0) = 2(16) - 4(4) + 65 = 32 - 16 + 65 = 81^\circ\text{C}$
   • $(4, 4)$: $T(4, 4) = 2(16) + 2(16) - 4(4) - 8(4) + 65 = 32 + 32 - 16 - 32 + 65 = 81^\circ\text{C}$
   • $(0, 4)$: $T(0, 4) = 2(16) - 8(4) + 65 = 32 - 32 + 65 = 65^\circ\text{C}$

5. Master Comparison & Thermal Management Conclusion:

   Candidate Point $(x, y)$	Domain Location	Temperature $T(x, y)$
   $(1, 2)$	Interior Critical Point	$55^\circ\text{C}$ (Absolute Minimum)
   $(0, 2)$	Left Boundary Edge	$57^\circ\text{C}$
   $(1, 0)$	Bottom Boundary Edge	$63^\circ\text{C}$
   $(1, 4)$	Top Boundary Edge	$63^\circ\text{C}$
   $(0, 0)$	Bottom-Left Corner	$65^\circ\text{C}$
   $(0, 4)$	Top-Left Corner	$65^\circ\text{C}$
   $(4, 2)$	Right Boundary Edge	$73^\circ\text{C}$
   $(4, 0)$	Bottom-Right Corner	$81^\circ\text{C}$ (Absolute Maximum)
   $(4, 4)$	Top-Right Corner	$81^\circ\text{C}$ (Absolute Maximum)

   • Absolute Minimum Temperature: $T_{\min} = 55^\circ\text{C}$ at location $(1\text{ cm}, 2\text{ cm})$, an optimal placement site for baseline reference thermal sensors.
   • Absolute Maximum Temperature: $T_{\max} = 81^\circ\text{C}$ at corner standoff locations $(4\text{ cm}, 0\text{ cm})$ and $(4\text{ cm}, 4\text{ cm})$, requiring focused heat pipe placement to prevent edge hotspots.