Petri Nets for Detecting a 3D Deadlock Problem in Hp-adaptive Finite Element Simulations

The papers describes a Petri net model for mesh adaptation in 3D self-adaptive hp Finite Element Method (hp-FEM) simulations. The method is applied to a model 3D problem consisting of the simulation of direct current (DC) measurements in a borehole environment with a deviated steel-cased well This model problem requires a high accuracy numerical solution, that can is delivered by the self-adaptive hp-FEM algorithm. We show that during the execution of the self-adaptive hp-FEM algorithm a deadlock problem occurs, that is, due to incompatibility of refinements, a given mesh cannot be further refined. We employ the Petri net model to identify the reason of the deadlock and introduce additional mesh transformations to overcome the problem and continue the simulations without occurrence of the deadlock