Abstract
A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach.
Highlights
Dynamic, adaptive meshing is often used during numerical simulations of transient fluid flows to improve accuracy [1,2]
This is achieved by first mapping the control-volume field into a finite element representation on the donor mesh by Galerkin projection, interpolating onto the target mesh by constructing a finite element supermesh from the intersection of the donor and target meshes and projecting back to a control-volume representation on the target mesh
A series of test cases have demonstrated that the method is superior to both node-wise and Grandy interpolation for control-volume fields with a discontinuous dual finite element representation
Summary
Adaptive meshing is often used during numerical simulations of transient fluid flows to improve accuracy [1,2]. The mesh may change every time-step depending on the error metrics used to control refinement and coarsening This inevitably means that data must be mapped from one mesh to another using an interpolation algorithm [3]. Interfaces between fluids as well as material property boundaries in both inertial and porous media flow [4,5,6,7,8] In the latter example, discontinuous representations of control-volume fields (that is to say when the dual finite element representation is discontinuous) may further help to capture abrupt changes in material properties (e.g. rock permeability) [9,10].
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