On a reduced sparsity stabilization of grad-div type for incompressible flow problems

We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar

Finite element methods for multicomponent convection-diffusion

We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons

Alternative Solution Algorithms for Primal and Adjoint Incompressible Navier-Stokes

Regardless of the specific discretisation framework, the discrete incompressible Navier-Stokes equations present themselves in the form of a non-linear, saddle-point Oseentype system. Traditional CFD codes typically solve the system via the well-known SIMPLE-like algorithms, which are essentially block preconditioners based on Schur complement theory. Due to their “segregated” nature, which reduces to iteratively solving a sequence of linear systems smaller than the full Oseen and better conditioned, traditional SIMPLE-like algorithms have long been considered as the only viable strategy. However, recent progress in computational power and linear solver capabilities has led researchers to develop, for Oseen-type systems (and discrete Navier-Stokes in particular), a number of alternative preconditioners and solution schemes, found to be more efficient than SIMPLE-like strategies but previously deemed practically unfeasible in industrial contexts. The improved efficiency of novel preconditioners entails a) faster, more stable convergence and b) the possibility of driving residuals below more strict tolerances, which is sometimes difficult with SIMPLE due to stagnating behaviour. The second aspect in particular is extremely relevant in the context of adjoint-based optimisation, as evidence suggests that an adjoint system may be affected by convergence issues when the primal flow solution is not well converged. In this work, we present some solution schemes (both traditional and novel) implemented for the Mixed Hybrid Finite Volumes Navier-Stokes solver we introduced in our previous work. Performance, in terms of robustness and convergence properties, is assessed on a series of benchmark test cases. We also turn our attention to the discrete adjoint Navier-Stokes problem itself, which in essence requires solving a linear system similar to the original Oseen and therefore may benefit from the same preconditioning techniques. We show how the primal algorithms are adapted to the adjoint system, and we run a series of adjoint test cases to compare performance of various solution scheme