Element Spatial Discretization Research Articles

A note on the accuracy of the generalized‐α scheme for the incompressible Navier‐Stokes equations

AbstractWe investigate the temporal accuracy of two generalized‐ schemes for the incompressible Navier‐Stokes equations. In a widely‐adopted approach, the pressure is collocated at the time step tn + 1 while the remainder of the Navier‐Stokes equations is discretized following the generalized‐ scheme. That scheme has been claimed to be second‐order accurate in time. We developed a suite of numerical code using inf‐sup stable higher‐order non‐uniform rational B‐spline (NURBS) elements for spatial discretization. In doing so, we are able to achieve high spatial accuracy and to investigate asymptotic temporal convergence behavior. Numerical evidence suggests that only first‐order accuracy is achieved, at least for the pressure, in this aforesaid temporal discretization approach. On the other hand, evaluating the pressure at the intermediate time step recovers second‐order accuracy, and the numerical implementation is simplified. We recommend this second approach as the generalized‐ scheme of choice when integrating the incompressible Navier‐Stokes equations.

Wake transition in the flow around two circular cylinders in staggered arrangements

The wake transition of the flow around two circular cylinders placed in staggered arrangements with fixed streamwise separation of 5Dand cross-stream separation varying from 0 to 3Dhas been studied. The wake transition is compared to that of a single isolated cylinder. Linear stability analysis utilizing Floquet theory and direct numerical simulations using a spectral/hp element spatial discretization were carried out. The unstable modes that first appear in the wake transition of the flow around a single cylinder, which are the long-spanwise-wavelength mode A and the short-spanwise-wavelength mode B, are also found in the flow around the staggered arrangements. However, a third mode, referred to as mode C, is also present in the wake transition of the flow around staggered arrangements, depending on the relative positioning of the cylinders. This mode has an intermediate spanwise wavelength and period-doubling character. The structure and onset characteristics of mode C are analysed and the nonlinear character of the bifurcation for this mode is investigated.

One-dimensional modelling of a vascular network in space-time variables

In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled. In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A,Q), area-velocity (A,u), pressure-velocity (p,u) and pressure-mass flux(p,Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A,u) variables the extension of the single-vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves. The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second-order Adams-Bashforth time-integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)). Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.

Computational haemodynamics: geometry and non-newtonian modelling using spectral/hp element methods

The increased availability of large scale computational facilities is enabling the use of computational modelling for the treatment of complex problems of industrial interest. Computational haemodynamics is an area where improved computational power and algorithmic development have permitted the application in biomechanics of techniques originally devised for direct numerical simulations in others areas of scientific interest. In this paper we present a highly accurate algorithm for the numerical solution of the incompressible Navier–Stokes equations which employs a spectral/hp element spatial discretisation and a high-order time splitting. Two important developments discussed in this paper are the coupling of these techniques with unstructured mesh generation, used in many everyday engineering applications, and the introduction of a non-Newtonian viscosity model to account for the shear thinning properties of blood. The proposed method will be applied to the analysis of shear stress distributions in flows of interest in haemodynamics.

Numerical algorithm for local failure mechanism of pre-stressed concrete containment vessel wall with penetration

The pre-stressed concrete containment vessel is one of the most important structures in nuclear power plants, which is required to prevent release of fission products to the environment even in the case of a severe accident. This paper deals with existing problems concerning numerical analysis for reinforced concrete composite structures. Finite element spatial and consistent material discretizations are summarized to solve these types of problem. Simple and real complex problems are illustrated in numerical experiments.