An efficient semi‐implicit method for three‐dimensional non‐hydrostatic flows in compliant arterial vessels

Fambri, Francesco; Dumbser, Michael; Casulli, Vincenzo

doi:10.1002/cnm.2651

Cited by 18 publications

(22 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…the pressure is a nonnegative function of bounded variation. 13,14,[81][82][83][84][85] Due to the properties of T r , the linear subproblems within the Newton-type algorithm can be solved at the aid of a matrix-free conjugate gradient method or with the Thomas algorithm for tridiagonal systems in the one-dimensional case. It is therefore possible to employ the same (nested) Newton-type techniques for the solution of (23) as those proposed and analyzed by Casulli et al [37][38][39][40] For all implementation details and a rigorous convergence proof of the (nested) Newton method, the reader is referred to the above references.…”

Section: Pressure Subsystemmentioning

confidence: 99%

“…The iterative Newton-type techniques of Casulli et al have already been used with great success as building block of semi-implicit finite volume schemes in different application contexts, see other works. 13,14,[81][82][83][84][85] Due to the properties of T r , the linear subproblems within the Newton-type algorithm can be solved at the aid of a matrix-free conjugate gradient method or with the Thomas algorithm for tridiagonal systems in the one-dimensional case. Note that, for the ideal gas EOS, the resulting system (23) becomes linear in the pressure; hence, one single Newton iteration is sufficient to solve (23).…”

Section: Pressure Subsystemmentioning

confidence: 99%

See 1 more Smart Citation

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

Summary In this paper, we present a novel pressure‐based semi‐implicit finite volume solver for the equations of compressible ideal, viscous, and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum, and total energy, and in multiple space dimensions, it is constructed in such a way as to respect the divergence‐free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multidimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi‐implicit method for the MHD equations proposed here discretizes the nonlinear convective terms as well as the time evolution of the magnetic field explicitly, whereas all terms related to the pressure in the momentum equation and the total energy equation are discretized implicitly, making again use of a properly staggered grid for pressure and velocity. Inserting the discrete momentum equation into the discrete energy equation then yields a mildly nonlinear symmetric and positive definite algebraic system for the pressure as the only unknown, which can be efficiently solved with the (nested) Newton method of Casulli et al. The pressure system becomes linear when the specific internal energy is a linear function of the pressure. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfvén wave speed and is not based on the speed of the magnetosonic waves. Being a semi‐implicit pressure‐based scheme, our new method is therefore particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well known that explicit density‐based Godunov‐type finite volume solvers become increasingly inefficient and inaccurate because of the more and more stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi‐implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense, our new scheme is a true all Mach number flow solver.

show abstract

Section: Pressure Subsystemmentioning

confidence: 99%

Section: Pressure Subsystemmentioning

confidence: 99%

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

Dumbser

Balsara

Tavelli

et al. 2018

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

show abstract

“…Numerical simulations of blood flow in veins or arteries have been carried out by several authors, see e.g. [17,15,1,2,16]. In contrast our mathematical model can be solved with a simple and very efficient numerical method.…”

Section: Numerical Implementationmentioning

confidence: 99%

A one-dimensional model of viscous blood flow in an elastic vessel

Berntsson

Karlsson

Kozlov

et al. 2016

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessel's wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel.

show abstract

“…The θ -method is widely used in the semi-implicit schemes (see [8,12,14,17,30,32]) because is it quite a simple and cheap strategy to achieve second order of accuracy in time.…”

Section: Semi-implicit Finite Volume Schemesmentioning

confidence: 99%

“…To improve the information that regards the radial velocity profile, one can use a semi-implicit scheme for a twodimensional model such as the one suggested in [14]. This method is an extension to weakly compressible flows of the family of semi-implicit methods for blood flow presented in a series of recent papers [8,17,32]. Under the hypothesis of hydrostatic radial pressure equilibrium and when the longitudinal scale is much larger than the radial one, the motion of a compressible barotropic fluid in a circular elastic duct is described by the following system of equations, which is a simplification of the Navier-Stokes equation in cylindrical coordinates:…”

Section: Semi-implicit Finite Volume Schemesmentioning

confidence: 99%

A comparison of explicit and semi‐implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime

2017

Self Cite

View full text Add to dashboard Cite

Key words Finite volume schemes, explicit path conservative schemes, semi-implicit staggered schemes, viscous compressible laminar flow in elastic tubes, unsteady friction models, highly oscillatory flow, water hammer. This paper compares the accuracy and computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric compliant tubes. There are essentially two main classes of mathematical models that can be used to predict the pressure and velocity distribution along the tube: one class is based on the full compressible Navier-Stokes equations in an axially symmetric geometry, leading to a two-dimensional governing PDE system with moving boundaries, and the other class uses a simpler, cross-sectionally averaged version of the Navier-Stokes equations, which leads to a non-conservative PDE system in only one space dimension along the axial direction of the tube. Within the first class of models, the influence of the wall friction on the flow field is directly obtained from first principles, without any further modelling assumptions and is thus valid even for highly unsteady flows. In the second case, only averaged flow quantities are available, and it is well known from previous studies published in the literature that the correct representation of the wall friction needs to be frequency dependent, since the use of a simple steady friction model, like the classical Darcy-Weisbach law, is not sufficient to reproduce the wall friction effects in highly transient flows. For the cross-sectionally averaged Navier-Stokes equations, there are again two main classes of frequency-dependent wall friction models: convolution integral (CI) models and instantaneous acceleration (IA) models.In this paper we provide a very thorough and critical comparison of all the above-mentioned methods for the simulation of highly oscillatory flows in rigid and compliant tubes concerning accuracy and computational efficiency. From our numerical results we can conclude that the convolution integral models are significantly superior to instantaneous acceleration models concerning accuracy. Furthermore, the CI models require only a slight computational overhead if they are properly implemented by solving a set of additional ODEs for appropriate auxiliary variables, instead of directly computing the convolution integrals. We also find that semi-implicit finite volume methods are clearly superior to conventional explicit finite volume schemes concerning computational efficiency, however, providing the same level of accuracy.

show abstract

An efficient semi‐implicit method for three‐dimensional non‐hydrostatic flows in compliant arterial vessels

Cited by 18 publications

References 47 publications

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

A divergence‐free semi‐implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics

A one-dimensional model of viscous blood flow in an elastic vessel

A comparison of explicit and semi‐implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime

Contact Info

Product

Resources

About