A stochastic Lagrangian approach for geometrical uncertainties in electrostatics

This work proposes a general framework to quantify uncertainty arising from geometrical variations in the electrostatic analysis. The uncertainty associated with geometry is modeled as a random field which is first expanded using either polynomial chaos or Karhunen-Loève expansion in terms of independent random variables. The random field is then treated as a random displacement applied to the conductors defined by the mean geometry, to derive the stochastic Lagrangian boundary integral equation. The surface charge density is modeled as a random field, and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method, respectively. Various numerical examples are presented to study the effect of uncertain geometry on relevant parameters such as capacitance and net electrostatic force. The results obtained using the proposed method are verified using rigorous Monte Carlo simulations. It has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters.