922 research outputs found
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
A "well-balanced" finite volume scheme for blood flow simulation
We are interested in simulating blood flow in arteries with a one dimensional
model. Thanks to recent developments in the analysis of hyperbolic system of
conservation laws (in the Saint-Venant/ shallow water equations context) we
will perform a simple finite volume scheme. We focus on conservation properties
of this scheme which were not previously considered. To emphasize the necessity
of this scheme, we present how a too simple numerical scheme may induce
spurious flows when the basic static shape of the radius changes. On contrary,
the proposed scheme is "well-balanced": it preserves equilibria of Q = 0. Then
examples of analytical or linearized solutions with and without viscous damping
are presented to validate the calculations. The influence of abrupt change of
basic radius is emphasized in the case of an aneurism.Comment: 36 page
Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid
This is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordA numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible 1D shallow-water equations in a moving vessel with a rigid-lid with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale et al. (2002). The method splits the jump in the fluxes and source terms including the pressure gradient at the rigid-lid into waves propagating away from each grid cell interface. For the influx-efflux boundary conditions the time dependent source terms are handled via a fractional step approach. In the linear case the numerical solutions are validated by comparison with the exact analytical solutions. Numerical solutions presented for the nonlinear case include shallow-water sloshing waves due to prescribed surge motion of the vessel.The research reported in this paper is supported by the Engineering and Physical Sciences Research Council Grant EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]
Modelling of nonlinear wave-buoy dynamics using constrained variational methods
We consider a comprehensive mathematical and numerical strategy to couple water-wave motion with rigid ship dynamics using variational principles. We present a methodology that applies to three-dimensional potential flow water waves and ship dynamics. For simplicity, in this paper we demonstrate the method for shallow-water waves coupled to buoy motion in two dimensions, the latter being the symmetric motion of a crosssection of a ship. The novelty in the presented model is that it employs a Lagrange multiplier to impose a physical restriction on the water height under the buoy in the form of an inequality constraint. A system of evolution equations can be obtained from the model and consists of the classical shallow-water equations for shallow, incompressible and irrotational waves, and relevant equations for the dynamics of the wave-energy buoy. One of the advantages of the variational approach followed is that, when combined with symplectic integrators, it eliminates any numerical damping and preserves the discrete energy; this is confirmed in our numerical results
Resonant effects in weakly nonlinear geophysical fluid dynamics
Many observed phenomena in geophysical systems, such as quasigeostrophy, and turbulence effects in rotating fluids, can be attributed to the resonances that emerge from multiple scale analysis. In this thesis the multiple scale method of asymptotic expansion is used to study resonant wave interactions in the context of quasigeostrophic geophysical systems, including and extending triad interactions. The one and two layer rotating shallow water equations and the equations for uniformly stratified fluid under the Boussinesq assumption are studied in detail, we evaluate their asymptotic expansions, and analyse their behaviour. Of particular interest is the expansion for the two layer equations, where we investigate a resonance not previously considered in the literature. We formulate general theory concerning the behaviour of the splitting of the dynamics into the fast and slow parts of the systems. We find that all layered shallow water type equations cannot have any interaction between a set of fast waves that produces a slow wave, regardless of whether they are resonant or non-resonant. In the stratified case we find that this is not true, although these interactions are constrained to a slow timescale. Building on the resonant expansion, we then reformulate the expansions to allow the inclusion of near resonant interactions. We detail a new formulation of the near resonances as the representation of higher order interactions that are sufficiently fast acting to be included at the triad order of interaction. We then demonstrate the effectiveness of this near resonant expansion by direct numerical simulation and evaluation of the rotating shallow water equations. We derive qualitatively different behaviour, found analytically in the near resonant expansion of the stratified equations, showing that many higher order interactions not accessible in the layered equations are possible in the stratified case. Finally we consider the expansions in the wavepacket framework, with the introduction of multiple spatial scales. We find that consideration of the magnitude of the difference between the group velocities of component wavepackets in a quartet interaction is sufficient to derive the higher order behaviour previously found by other methods in the literature. It then follows from this that the near resonant expansion can contain many types of interaction that are not possible between wavepackets if only exact resonances are considered
Higher-order fem for nonlinear hydroelastic analysis of a floating elastic strip in shallow-water conditions
The hydroelastic response of a thin, nonlinear, elastic strip floating in shalow-water
environment is studied by means of a special higher order finite element scheme. Considering
non-negligible stress variation in lateral direction, the nonlinear beam model, developed by
Gao, is used for the simulation of large flexural displacement. Full hydroelastic coupling
between the floating strip and incident waves is assumed. The derived set of equations is
intended to serve as a simplified model for tsunami impact on Very Large Floating Structures
(VLFS) or ice floes. The proposed finite element method incorporates Hermite polynomials of
fifth degree for the approximation of the beam deflection/upper surface elevation in the
hydroelastic coupling region and 5-node Lagrange finite elements for the simulation of the
velocity potential in the water region. The resulting second order ordinary differential
equation system is converted into a first order one and integrated with respect to time with the
Crank-Nicolson method. Two distinct cases of long wave forcing, namely an elevation pulse
and an N-wave pulse, are considered. Comparisons against the respective results of the
standard, linear Euler-Bernoulli floating beam model are performed and the effect of large
displacement in the beam response is studied
The numerical simulation of nonlinear waves in a hydrodynamic model test basin
This thesis describes the development of a numerical algorithm for the fully nonlinear simulation of free-surface waves. The aim of the research is to develop, implement and investigate an algorithm for the deterministic and accurate simulation of twodimensional nonlinear water waves in a model test basin. The simulated wave field may have a broad-banded spectrum and the simulations should be carried out by an efficient algorithm in order to be applicable in practical situations. The algorithm is based on a combination of Runge-Kutta (for time integration), Finite Element (boundary value problem) and Finite Difference (velocity recovery) methods. The scheme is further refined and investigated using different models for wave generation, propagation and absorption of waves
A compatible finite element discretisation for the nonhydrostatic vertical slice equations
We present a compatible finite element discretisation for the vertical slice
compressible Euler equations, at next-to-lowest order (i.e., the pressure space
is bilinear discontinuous functions). The equations are numerically integrated
in time using a fully implicit timestepping scheme which is solved using
monolithic GMRES preconditioned by a linesmoother. The linesmoother only
involves local operations and is thus suitable for domain decomposition in
parallel. It allows for arbitrarily large timesteps but with iteration counts
scaling linearly with Courant number in the limit of large Courant number. This
solver approach is implemented using Firedrake, and the additive Schwarz
preconditioner framework of PETSc. We demonstrate the robustness of the scheme
using a standard set of testcases that may be compared with other approaches.Comment: Response to reviewers. Thanks to Golo Wimmer for pointing out the
wrong factor of h in the interior penalty for diffusion - this was also wrong
in the codes and we reran the dense bubble testcase
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