Partial Differential Equations Final Project

Final Project Topic:

Optimal Control of Systems Governed by Partial Differential Equations, Heat Sink Shape Optimization using the Adjoint Method

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The purpose of the MATLAB files are to a) solve 2D Laplace Equation with Dirichlet and convective boundary conditions (Math_112C_Project_Main.m) (constant heat emitted from the bottom of the heat sink in ambient temperature), b) calculate the defined objective function (objectiveFunction.m), $$J = \lambda_{1}\frac{area_{k}}{area_{ref}} + \lambda_{2}\frac{temp_{k}}{temp_{ref}}$$ where $\lambda_{1}$ and $\lambda_{2}$ are weights placed on the normalized area and average temperature (to be minimized), and c) generate the updated geometry using the adjoint method with a gradient descent algorithm (yGradient.m).

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Background:

Final project for UC Irvine's Math 112C—Intro to Partial Differential Equations III course.

Material covered in the course:

1. Numerical Methods for ODEs and PDEs
   • Euler's Method
   • Error and Stability
   • Finite Differences for Heat Equation and 2D Poisson Equation
2. Reaction-Diffusion Equations
   • Fisher-KPP Equation
   • Wave Speed
   • Turing Pattern Formation
3. Random Walks
   • Brownian Motion
   • Stochastic Differential Equations
   • Fokker-Planck Equation
   • Ornstein-Uhlenbeck Process
4. Black-Scholes Equation
   • Ito's Lemma
5. Fourier Transforms for PDEs