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Applied Math Seminar
The Mathematics of Microchip Failure
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Microchips often fail when the (tiny!) wires connecting the transistors and diodes on the chip degrade due to extremely high current densities. The physics of this process is quite interesting; it is a non-local moving interface problem involving elastic deformation and diffusion. Stress singularities can develop which make boundary conditions difficult to understand and numerical simulation difficult to implement reliably. I will describe a mathematical framework which handles these difficulties cleanly and leads to a nice qualitative understanding of the behavior of this system. Along the way we will encounter a Dirichlet to Neumann map involving the equations of elasticity, a bit of semigroup theory (needed to prove well-posedness), and a rich dynamics which can include flux reversal and pseudo-steady states resembling continental drift in plate tectonics. I will also discuss some of the numerical challenges which arise due to stress singularities and show how to solve them. |