Advances in the simulation of viscoplastic fluid flows using interior-point methods

Abstract

We present a primal-dual interior point algorithm for the resolution of steady-state viscoplastic fluid flows formulated as a conic optimization problem. We give a complete description of the algorithm including some advanced aspects such as a predictor-corrector and scaling scheme to improve its efficiency. Our interior-point approach is shown to be largely more efficient than Augmented Lagrangian (AL) approaches which are traditionally used to solve such problems. In particular, the interior-point approach is roughly 5 times faster than the modern accelerated version of AL algorithms. The yield surfaces are shown to be accurately predicted and various examples ranging from channel flows to three-dimensional flows through a porous medium demonstrate its efficiency.

Hybrid Discretization Methods with Adaptive Yield Surface Detection for Bingham Pipe Flows

“…The above results on the velocity field and the localization of the yield surface also match well with the predictions reported in [7]. To provide a more quantitative comparison, we present in Table 4 the value of the flux Q = (u h , 1) Ω across the cross-section as predicted using the methodology from [7] (recall that it is based on a conforming finite element discretization, and the numerical solver uses either the ADMM or second-order cone programming) and the present one.…”

“…We consider flows in pipes with a circular cross-section and an eccentric annular cross-section. The first setting leads to one of the few Bingham pipe flow problems with known exact solution, whereas the second setting is well-documented in the literature (see, among others, [7,44,47]). In all cases, the external force f , which represents the transverse pressure gradient forcing the flow, is constant over the cross-section.…”

“…First, the computational effectiveness of the actual numerical procedure can be enhanced by using hp-refinement techniques, a posteriori error estimation in the flowing (yielded) regions, and some of the acceleration techniques mentioned in the introduction for the ADMM [2,45] or second-order cone programming [7,8]. Second, future work can include solving Stokes-like viscoplastic flows and studying their interaction with gas bubbles.…”

“…To provide a more quantitative comparison, we present in Table 4 the value of the flux Q = (u h , 1) Ω across the cross-section as predicted using the methodology from [7] (recall that it is based on a conforming finite element discretization, and the numerical solver uses either the ADMM or second-order cone programming) and the present one. We consider uniformly refined meshes as the conforming finite element solver does not support locally refined meshes with hanging nodes.…”

“…Recent advances leading to faster convergence rates include the accelerated ADMM from [45] and the ADMM with variable step sizes from [2]. We also mention the recent interior-point methods combined with second-order cone programming [7,8].…”