19,566 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
Multi-Adaptive Time-Integration
Time integration of ODEs or time-dependent PDEs with required resolution of
the fastest time scales of the system, can be very costly if the system
exhibits multiple time scales of different magnitudes. If the different time
scales are localised to different components, corresponding to localisation in
space for a PDE, efficient time integration thus requires that we use different
time steps for different components.
We present an overview of the multi-adaptive Galerkin methods mcG(q) and
mdG(q) recently introduced in a series of papers by the author. In these
methods, the time step sequence is selected individually and adaptively for
each component, based on an a posteriori error estimate of the global error.
The multi-adaptive methods require the solution of large systems of nonlinear
algebraic equations which are solved using explicit-type iterative solvers
(fixed point iteration). If the system is stiff, these iterations may fail to
converge, corresponding to the well-known fact that standard explicit methods
are inefficient for stiff systems. To resolve this problem, we present an
adaptive strategy for explicit time integration of stiff ODEs, in which the
explicit method is adaptively stabilised by a small number of small,
stabilising time steps
High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of and (div) methods
The accurate numerical simulation of turbulent incompressible flows is a
challenging topic in computational fluid dynamics. For discretisation methods
to be robust in the under-resolved regime, mass conservation as well as energy
stability are key ingredients to obtain robust and accurate discretisations.
Recently, two approaches have been proposed in the context of high-order
discontinuous Galerkin (DG) discretisations that address these aspects
differently. On the one hand, standard -based DG discretisations enforce
mass conservation and energy stability weakly by the use of additional
stabilisation terms. On the other hand, pointwise divergence-free
-conforming approaches ensure exact mass conservation
and energy stability by the use of tailored finite element function spaces. The
present work raises the question whether and to which extent these two
approaches are equivalent when applied to under-resolved turbulent flows. This
comparative study highlights similarities and differences of these two
approaches. The numerical results emphasise that both discretisation strategies
are promising for under-resolved simulations of turbulent flows due to their
inherent dissipation mechanisms.Comment: 24 pages, 13 figure
Multiphase modelling of vascular tumour growth in two spatial dimensions
In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters are investigated
Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media
This paper proposes to address the issue of complexity reduction for the
numerical simulation of multiscale media in a quasi-periodic setting. We
consider a stationary elliptic diffusion equation defined on a domain such
that is the union of cells and we
introduce a two-scale representation by identifying any function defined
on with a bi-variate function , where relates to the
index of the cell containing the point and relates to a local
coordinate in a reference cell . We introduce a weak formulation of the
problem in a broken Sobolev space using a discontinuous Galerkin
framework. The problem is then interpreted as a tensor-structured equation by
identifying with a tensor product space of
functions defined over the product set . Tensor numerical methods
are then used in order to exploit approximability properties of quasi-periodic
solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to
regularity) and the argumentation thereof. Corrected proof of proposition 3.
Corrected wrong multiplicative factor in proposition 4 and its proof (was 2
instead of 1). Added remark 6 at the end of section 2. Extended remark 7.
Added references. Some minor improvements (typos, typesetting
Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
We describe a compatible finite element discretisation for the shallow water
equations on the rotating sphere, concentrating on integrating consistent
upwind stabilisation into the framework. Although the prognostic variables are
velocity and layer depth, the discretisation has a diagnostic potential
vorticity that satisfies a stable upwinded advection equation through a
Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at
the gridscale whilst retaining optimal order consistency. We also use upwind
discontinuous Galerkin schemes for the transport of layer depth. These
transport schemes are incorporated into a semi-implicit formulation that is
facilitated by a hybridisation method for solving the resulting mixed Helmholtz
equation. We illustrate our discretisation with some standard rotating sphere
test problems.Comment: accepted versio
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