Optimization of the finite-element solution of the semiconductor-device Poisson equation

A mesh-optimization technique is applied to the numerical simulation of semiconductor devices. The technique consists of moving the mesh-nodes while keeping their number constant, and is based upon the maximization of a functional related to the RHS of Poisson's equation. The result is equivalent to the minimization of the seminorm of
u - u_{t}
, where
u
is the normalized electric potential and u
T
its discretization over mesh
T
. The nodal-coordinate variations
\Delta\bar{x}
induce variations
\Delta\bar{u}
onto the electric potential, yielding a system of algebraic equations where both unknown vectors
\Delta\bar{x}
,
\Delta\bar{u}
appear. A suitable technique avoids any matrix inversion and allows application of the gradient method for the maximization procedure. The method has been tested on a one-dimensional Poisson solver for bipolar transistors.