[Paper Review] A trust-region SQP method for the numerical approximation of viscoplastic fluid flow

Timm Treskatis, Miguel Moyers-González+1 more|arXiv (Cornell University)|Apr 30, 2015

Rheology and Fluid Dynamics Studies34 references3 citations

TL;DR

This paper proposes a trust-region sequential quadratic programming (TR-SSQP) method for simulating viscoplastic fluid flow by reformulating the problem in terms of stress-based dual variables, enabling higher regularity and superior convergence. The method outperforms the widely used augmented Lagrangian ALG2 method, achieving speedups of up to 140x in CPU time for Herschel-Bulkley flows without artificial regularization or mesh dependency.

ABSTRACT

We present a new approach to the problem of stationary viscoplastic duct flow as modelled by the Herschel-Bulkley model, with Bingham fluids included as a special case. While the mathematical formulation of this problem is conventionally based on a variational inequality, or equivalently, on a nonsmooth minimisation problem for the flow velocity, we suggest an alternative approach. Considering the Lagrangian dual in terms of the stress, rather than the velocity, turns out to be advantageous in numerous ways. The objective functional possesses higher regularity, which ensures applicability of second order methods. Our numerical experiments with a trust-region SQP algorithm also demonstrate clearly superior performance compared to the widely used augmented Lagrangian method, although no artificial regularisation is introduced into the problem. Hence, besides providing a new theoretical angle to a classical problem, our results also pave the way for an entirely new class of numerical approaches to simulating flows of viscoplastic fluids.

Motivation & Objective

• To address the numerical challenges of simulating viscoplastic fluid flows, particularly the nonsmoothness and ill-conditioning arising from yield stress behavior.
• To overcome limitations of traditional augmented Lagrangian methods, which suffer from slow convergence and sensitivity to algorithmic parameters.
• To develop a robust, second-order method that exploits higher regularity in the dual formulation of viscoplastic flow problems.
• To provide a new numerical framework that exactly captures plastic behavior without artificial viscosity or regularization.
• To demonstrate significant computational performance gains over ALG2 in both Bingham and Herschel-Bulkley flow scenarios.

Proposed method

• Reformulate the viscoplastic flow problem using a stress-based dual formulation instead of the classical velocity-based variational inequality, leading to a smoother objective functional.
• Discretize the dual problem using the finite element method to obtain a finite-dimensional optimization problem.
• Apply a trust-region sequential quadratic programming (TR-SSQP) algorithm to solve the resulting smooth, constrained optimization problem.
• Use a trust-region framework to ensure global convergence and robustness, even when the Hessian approximation is poor.
• Incorporate exact line searches and trust-region updates to maintain superlinear convergence without inner iterations.
• Avoid artificial regularization or viscosity, preserving the physical fidelity of the viscoplastic behavior.

Experimental results

Research questions

• RQ1 Can a dual formulation in terms of stress variables lead to a smoother optimization problem compared to classical velocity-based formulations for viscoplastic flow?
• RQ2 Does a trust-region SQP method achieve faster convergence and better performance than the augmented Lagrangian ALG2 method for viscoplastic flows?
• RQ3 How does the performance of the proposed TR-SSQP method scale with mesh refinement and yield stress in both Bingham and Herschel-Bulkley fluids?
• RQ4 Can the method maintain robustness and efficiency without introducing artificial regularization or viscosity?
• RQ5 What is the impact of fluid rheology (e.g., shear-thinning exponent α) on the convergence behavior of the TR-SSQP algorithm?

Key findings

• The trust-region SQP (TRS) method achieved speedups of up to 140 times over ALG2 for Herschel-Bulkley flow with α = 1.5, particularly on finer meshes.
• For α = 1.5 and τ₀ = 0.1, TRS required only 17 CPU seconds on a 4308-node mesh, compared to 116.62 seconds for ALG2, yielding a speedup of 114.
• The TRS method showed decreasing iteration counts with mesh refinement and lower yield stress, indicating improved efficiency under more challenging conditions.
• The method maintained superior performance even in asymmetric geometries, with speedups of up to 6.7x in the kiwi pipe geometry.
• Relative differences between TRS and ALG2 solutions were consistently below 1e-3, confirming solution accuracy despite faster convergence.
• The TRS method exhibited mesh independence and robust convergence, with no need for parameter tuning or inner iterations, unlike ALG2.