A particle tracking technique for the Lagrangian-Eulerian finite element method in multi-dimensions

This paper presents a multi-dimensional particle tracking technique for applying the Lagrangian-Eulerian finite element method to solve transport equations in transient-state simulations. In the Lagrangian-Eulerian approach, the advection term is handled in the Lagrangian step so that the associated numerical errors can be considerably reduced. It is imponani to have an adequate particle tracking technique for computing advection accurately in the Lagrangian step. The parade tracking technique presented here it designed to trace fictitious particles in the real-world flow field where the flow velocity is either measured or computed at a limited number of locations. The technique, named 'in-element' particle tracking, traces fictitious particles on an element-by-element basis. Given a velocity field, a fictitious particle is traced one element by one element until either a boundary is encountered or the available time is completely consumed. For the tracking within an element, the element is divided into a desired number of subelements with the interpolated velocity computed at all nodes of the subelements. A fictitious particle, thus, is traced one subelement by one subelement within the element. The desired number of subelements can be determined based on the complexity of the flow field being considered. The more complicated the flow field is, the more subelements are needed to achieve accurate particle tracking results. A single-velocity approach can be used to efficiently perform particle tracking in a smooth flow field, while an average-velocity approach can be employed to increase the tracking accuracy for more complex flow fields.