An efficient split-step framework for non-Newtonian incompressible flow problems with consistent pressure boundary conditions

Pacheco, Douglas R. Q.; Schussnig, Richard; Fries, Thomas‐Peter

doi:10.1016/j.cma.2021.113888

Cited by 17 publications

(17 citation statements)

References 32 publications

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“…gives the Neumann condition for the PPE when restricted to Γ D . For a detailed derivation, the interested reader is referred to our recent work [23], while we herein focus directly on a weak formulation of the split-step scheme. Let us denote the L 2 (Ω) and L 2 (Γ D ) scalar products by ⟨•, •⟩ and ⟨•, •⟩ Γ D , respectively, and start off by multiplying the PPE (9) with a test function q ∈ H 1 (Ω), q| Γ N = 0, integrating by parts and inserting the Neumann boundary condition (11), thereby yielding…”

Section: Time-splitting Schemementioning

confidence: 99%

“…omitting some of the details from [23] for brevity. Other vital ingredients of the split-step scheme are (i) the full decoupling of momentum balance and PPE through explicit treatment of the pressure gradient term in ( 1), (ii) projection of the PPE Dirichlet condition on Γ N (10), (iii) recovering the viscosity µ via an L 2 projection and (iv) improving conservation of mass using divergence damping [20,22].…”

Section: Time-splitting Schemementioning

confidence: 99%

“…Note here, that the Poisson problems ( 15)-( 16) can be combined, solving only one Poisson problem per time step (cf. [18,23]) and for Newtonian fluids, the viscosity projection step ( 14) is skipped. Also, the rheological law is easily replaced by swapping the right-hand side of ( 14) and was actually lumped in our experiments.…”

Section: Ppe Dirichlet Conditionmentioning

confidence: 99%

“…CFL e > 1 is admissible in the split-step scheme (cf. Pacheco et al [23]), but only at the cost of increasing iteration counts in the momentum balance solve.…”

Section: Abdominal Aortic Aneurysmmentioning

confidence: 99%

“…By contrast, we focus herein on the recent extension of the PPE scheme [18] to generalised Newtonian fluids [23,24]. This new framework allows for continuous equal-order finite elements, is higher-order accurate, iteration-free, and consists of an advection-diffusion equation, a single PPE, and two mass matrix solves to recover pressure Dirichlet data and viscosity.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

Schussnig

Pacheco

Fries

2021

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

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In various practically relevant incompressible flow problems, such as polymer flow or biomedicalengineering applications, the dependence of fluid viscosity on the local shear rate plays an impor-tant role. Standard techniques using inf-sup stable finite elements lead to saddle-point systemsposing a challenge even for state-of-the-art solvers and preconditioners.For efficiency, projection schemes or time-splitting methods decouple the governing equations forvelocity and pressure, resulting in more, but easier to solve linear systems. Doing so, boundaryconditions and correction terms at intermediate steps have to be carefully considered in order toprohibit spoiling accuracy. In the case of Newtonian incompressible fluids, pressure and velocitycorrection schemes of high-order accuracy have been devised (see, e.g. [1, 2]). However, the exten-sion to generalised Newtonian fluids is a non-trivial task and considered an open question. Deteixet al. [3] successfully adapted the popular rotational correction scheme to consider for shear-ratedependent viscosity, but this resulted in substantial numerical overhead caused by necessarily pro-jecting viscous stress components.In this contribution we address this shortcoming and present a split-step scheme, extending pre-vious work by Liu [4]. The new method is based on an explicit-implicit treatment of pressure,convection and viscous terms combined with a Pressure-Poisson equation equipped with fully con-sistent Neumann and Dirichlet boundary conditions. Through proper reformulation, the use ofstandard continuous finite element spaces is enabled due to low regularity requirements. Addition-ally, equal-order velocity-pressure pairs are applicable as in the original scheme.The stability, accuracy and efficiency of the higher-order splitting scheme is showcased in challeng-ing numerical examples of practical interest.[1] Karniadakis, G. E., Israeli, M. and Orszag, S. A. High-order splitting methods for the incom-pressible Navier-Stokes equations. J. Comput. Phys., (1991).[2] Timmermans, L.J.P., Minev, P.D. and Van de Vosse, F. N. An approximate projection schemefor incompressible flow using spectral elements. Internat. J. Numer. Methods Fluids, Vol.22(7), pp. 673–688, (1996).[3] Deteix, J. and Yakoubi, D. Shear rate projection schemes for non-Newtonian fluids, Comput.Methods Appl. Mech. Engrg., Vol. 354, pp. 620–636, (2019).[4] Liu, J. Open and traction boundary conditions for the incompressible NavierStokesequations.J. Comput. Phys., Vol. 228(19), pp. 7250..7267, (2009).

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Section: Time-splitting Schemementioning

confidence: 99%

Section: Time-splitting Schemementioning

confidence: 99%

Section: Ppe Dirichlet Conditionmentioning

confidence: 99%

“…CFL e > 1 is admissible in the split-step scheme (cf. Pacheco et al [23]), but only at the cost of increasing iteration counts in the momentum balance solve.…”

Section: Abdominal Aortic Aneurysmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

Schussnig

Pacheco

Fries

2021

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

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Consistent splitting schemes for incompressible viscoelastic flow problems

Pacheco

Castillo

2023

Numerical Meth Engineering

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Viscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining the Navier–Stokes system with a dynamic tensorial equation, increasing mathematical and computational demands. Hence, fractional‐step methods decoupling the calculation of the flow quantities are an attractive option. In consistent fractional‐step schemes, the splitting of the equations is derived from the continuous level, so that neither mass nor momentum balance is sacrificed. Thus, no corrections or velocity projections are needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees the absence of both numerical boundary layers and splitting errors, enabling high‐order accuracy in space and time. This article introduces the first consistent splitting methods for incompressible flows of viscoelastic fluids, for which arbitrary constitutive laws are allowed. We present the formulation and the algorithm, along with various numerical examples testing their accuracy. First‐, second‐ and third‐order backward‐differentiation schemes are numerically tested, and optimal convergence is confirmed for several spatial and temporal discretizations. Furthermore, good numerical stability is verified in challenging benchmark problems, including steady and time‐dependent solutions.

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On the role of tissue mechanics in fluid–structure interaction simulations of patient‐specific aortic dissection

Schussnig,

Rolf‐Pissarczyk,

Bäumler

et al. 2024

Numerical Meth Engineering

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Modeling an aortic dissection represents a particular challenge from a numerical perspective, especially when it comes to the interaction between solid (aortic wall) and liquid (blood flow). The complexity of patient‐specific simulations requires a variety of parameters, modeling assumptions and simplifications that currently hinder their routine use in clinical settings. We present a numerical framework that captures, among other things, the layer‐specific anisotropic properties of the aortic wall, the non‐Newtonian behavior of blood, patient‐specific geometry, and patient‐specific flow conditions. We compare hemodynamic indicators and stress measurements in simulations with increasingly complex material models for the vessel tissue ranging from rigid walls to anisotropic hyperelastic materials. We find that for the present geometry and boundary conditions, rigid wall simulations produce different results than fluid–structure interaction simulations. Considering anisotropic fiber contributions in the tissue model, stress measurements in the aortic wall differ, but shear stress‐based biomarkers are less affected. In summary, the increasing complexity of the tissue model enables capturing more details. However, an extensive parameter set is also required. Since the simulation results depend on these modeling choices, variations can lead to different recommendations in clinical applications.

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An efficient split-step framework for non-Newtonian incompressible flow problems with consistent pressure boundary conditions

Cited by 17 publications

References 32 publications

Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

Efficient and Higher-Order Accurate Split-Step Methods for Generalised Newtonian Fluid Flow

Consistent splitting schemes for incompressible viscoelastic flow problems

On the role of tissue mechanics in fluid–structure interaction simulations of patient‐specific aortic dissection

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