A global residual‐based stabilization for equal‐order finite element approximations of incompressible flows

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

doi:10.1002/nme.6615

Cited by 8 publications

(9 citation statements)

References 54 publications

(184 reference statements)

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“…We will next present a generalisation of the boundary vorticity stabilisation (BVS) by Pacheco et al [45] to quasi-Newtonian problems. The BVS is a residual-based formulation containing a first-order boundary term proportional to the vorticity ∇ × u, which guarantees consistency even for linear elements.…”

Section: The Generalised Boundary Vorticity Stabilisation Methodsmentioning

confidence: 99%

“…The BVS is a residual-based formulation containing a first-order boundary term proportional to the vorticity ∇ × u, which guarantees consistency even for linear elements. Details on the method for the Newtonian case can be found in our previous article [45].…”

Section: The Generalised Boundary Vorticity Stabilisation Methodsmentioning

confidence: 99%

“…Finally, we have attained a stabilised formulation retaining full consistency even for linear elements, since all second-order derivatives have been eliminated and no part of the residual is lost. This, along with the fact that our stabilisation term is constructed from a Poisson equation with consistent pressure boundary conditions, results in a formulation that is free from spurious pressure boundary layers [45]. This will be illustrated with a numerical example.…”

Section: Weak Formulationmentioning

confidence: 96%

“…with X h being once again a continuous finite element space. If we have a homogeneous Newtonian fluid, then ∇µ ≡ 0 and we recover the original boundary vorticity stabilisation [45]. We therefore denote the present method as generalised boundary vorticity stabilisation (GBVS).…”

Section: Weak Formulationmentioning

confidence: 99%

“…Of course, for finer meshes and smaller stabilisation parameters the difference tends to become less visible, but the spurious boundary behaviour will always be there (the thickness of the boundary layer in PSPG is actually proportional to √ αh 2 [46]). A quantitative assessment of the boundary accuracy can be found in the first study [45] where the Newtonian BVS was originally proposed. Q being the inflow rate.…”

Section: Numerical Examplesmentioning

confidence: 99%

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An Equal-Order Finite Element Framework for Incompressible Non-Newtonian Flow Problems

Pacheco¹,

Steinbach²

2021

14th WCCM-ECCOMAS Congress

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Various materials of engineering and biomedical interest can be modelled as generalised Newtonian fluids, i.e., via an apparent viscosity depending locally on the flow field. In spite of the particular features of those models, they are often handled in practice by classical numerical techniques originally conceived for Newtonian fluids. Methods designed specifically for the generalised case are rather scarce in the literature, as well as their use in practical applications. As it turns out, tackling non-Newtonian problems with standard finite element formulations can have undesired consequences such as the induction of spurious pressure boundary layers and the emergence of natural boundary conditions not suitable for realistic flow scenarios. In this context, we introduce a novel framework that deals with those issues while maintaining simplicity and low computational cost. The new stabilised formulation is based on a modified system combining the continuity equation with a Poisson equation for the pressure and consistent pressure boundary conditions. A weak enforcement of the rheological law is employed to enable full consistency even for first-order finite element pairs. Simple numerical examples are provided to demonstrate the potential of our method in yielding accurate solutions for relevant problems.

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Section: The Generalised Boundary Vorticity Stabilisation Methodsmentioning

confidence: 99%

Section: The Generalised Boundary Vorticity Stabilisation Methodsmentioning

confidence: 99%

Section: Weak Formulationmentioning

confidence: 96%

Section: Weak Formulationmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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An Equal-Order Finite Element Framework for Incompressible Non-Newtonian Flow Problems

Pacheco¹,

Steinbach²

2021

14th WCCM-ECCOMAS Congress

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On the numerical treatment of viscous and convective effects in relative pressure reconstruction methods

Pacheco

2021

Numer Methods Biomed Eng

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The mechanism of many cardiovascular diseases can be understood by studying the pressure distribution in blood vessels. Direct pressure measurements, however, require invasive probing and provide only single-point data. Alternatively, relative pressure fields can be reconstructed from imaging-based velocity measurements by considering viscous and inertial forces. Both contributions can be potential troublemakers in pressure reconstruction: the former due to its higher-order derivatives, and the latter because of the quadratic nonlinearity in the convective acceleration. Viscous and convective terms can be treated in various forms, which, although equivalent for ideal measurements, can perform differently in practice. In fact, multiple versions are often used in literature, with no apparent consensus on the more suitable variants. In this context, the present work investigates the performance of different versions of relative pressure estimators. For viscous effects, in particular, two new modified estimators are presented to circumvent second-order differentiation without requiring surface integrals. In-silico and in-vitro data in the typical regime of cerebrovascular flows are considered, allowing a systematic noise sensitivity study. Convective terms are shown to be the main source of error, even for flows with pronounced viscous component. Moreover, the conservation (often integrated) form of convection exhibits higher noise sensitivity than the standard convective description, in all three families of estimators considered here. For the classical pressure Poisson estimator, the present modified version of the viscous term tends to yield better accuracy than the (recently introduced) integrated form, although this effect is in most cases negligible when compared to convection-related errors.

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Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

Pacheco

Schussnig

2022

Comput Mech

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For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.

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A global residual‐based stabilization for equal‐order finite element approximations of incompressible flows

Cited by 8 publications

References 54 publications

An Equal-Order Finite Element Framework for Incompressible Non-Newtonian Flow Problems

An Equal-Order Finite Element Framework for Incompressible Non-Newtonian Flow Problems

On the numerical treatment of viscous and convective effects in relative pressure reconstruction methods

Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

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