194 research outputs found
Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation
This paper presents an a-posteriori goal-oriented error analysis for a
numerical approximation of the steady Boltzmann equation based on a
moment-system approximation in velocity dependence and a discontinuous Galerkin
finite-element (DGFE) approximation in position dependence. We derive
computable error estimates and bounds for general target functionals of
solutions of the steady Boltzmann equation based on the DGFE moment
approximation. The a-posteriori error estimates and bounds are used to guide a
model adaptive algorithm for optimal approximations of the goal functional in
question. We present results for one-dimensional heat transfer and shock
structure problems where the moment model order is refined locally in space for
optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131
An Entropy Stable Discontinuous Galerkin Finite-Element Moment Method for the Boltzmann Equation
This paper presents a numerical approximation technique for the Boltzmann
equation based on a moment system approximation in velocity dependence and a
discontinuous Galerkin finite-element approximation in position dependence. The
closure relation for the moment systems derives from minimization of a suitable
{\phi}-divergence. This divergence-based closure yields a hierarchy of
tractable symmetric hyperbolic moment systems that retain the fundamental
structural properties of the Boltzmann equation. The resulting combined
discontinuous Galerkin moment method corresponds to a Galerkin approximation of
the Boltzmann equation in renormalized form. We present a new class of
numerical flux functions, based on the underlying renormalized Boltzmann
equation, that ensure entropy dissipation of the approximation scheme.
Numerical results are presented for a one-dimensional test case.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0518
A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media
In this paper we consider a multiscale phase-field model for
capillarity-driven flows in porous media. The presented model constitutes a
reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model,
valid in situations where interest is restricted to dynamical and equilibrium
behavior in an aggregated sense, rather than a precise description of
microscale flow phenomena. The model is based on averaging of the equation of
motion, thereby yielding a significant reduction in the complexity of the
underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its
macroscopic dynamical and equilibrium properties. Numerical results are
presented for the representative 2-dimensional capillary-rise problem
pertaining to two closely spaced vertical plates with both identical and
disparate wetting properties. Comparison with analytical solutions for these
test cases corroborates the accuracy of the presented multiscale model. In
addition, we present results for a capillary-rise problem with a non-trivial
geometry corresponding to a porous medium
Condition number analysis and preconditioning of the finite cell method
The (Isogeometric) Finite Cell Method - in which a domain is immersed in a
structured background mesh - suffers from conditioning problems when cells with
small volume fractions occur. In this contribution, we establish a rigorous
scaling relation between the condition number of (I)FCM system matrices and the
smallest cell volume fraction. Ill-conditioning stems either from basis
functions being small on cells with small volume fractions, or from basis
functions being nearly linearly dependent on such cells. Based on these two
sources of ill-conditioning, an algebraic preconditioning technique is
developed, which is referred to as Symmetric Incomplete Permuted Inverse
Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the
SIPIC preconditioner in improving (I)FCM condition numbers and in improving the
convergence speed and accuracy of iterative solvers is presented for the
Poisson problem and for two- and three-dimensional problems in linear
elasticity, in which Nitche's method is applied in either the normal or
tangential direction. The accuracy of the preconditioned iterative solver
enables mesh convergence studies of the finite cell method
Extension of a fast method for 2D steady free surface flow to stretched surface grids
Steady free surface flow is often encountered in marine engineering, e.g. for calculating ship hull resistance. When these flows are solved with CFD, the water-air interface can be represented using a surface fitting approach. The resulting free boundary problem requires an iterative technique to solve the flow and at the same time determine the free surface position. Most such methods use a time-stepping scheme, which is inefficient for solving steady flows. There is one steady technique which uses a special boundary condition at the free surface, but that method needs a dedicated coupled flow solver. To overcome these disadvantages an efficient free surface method was developed recently, in which the flow solver can be a black-box. It is based on quasi-Newton iterations which use a surrogate model in combination with flow solver inputs and outputs from previous iterations to approximate the Jacobian. As the original method was limited to uniform free surface grids, it is extended in this paper to stretched free surface grids. For this purpose, a different surrogate model is constructed by transforming a relation between perturbations of the free surface height and pressure from the wavenumber domain to the spatial domain using the convolution theorem. The method is tested on the 2D flow over an object. The quasi-Newton iterations converge exponentially and in a low number of iterations
Discontinuities without discontinuity: The Weakly-enforced Slip Method
Tectonic faults are commonly modelled as Volterra or Somigliana dislocations
in an elastic medium. Various solution methods exist for this problem. However,
the methods used in practice are often limiting, motivated by reasons of
computational efficiency rather than geophysical accuracy. A typical
geophysical application involves inverse problems for which many different
fault configurations need to be examined, each adding to the computational
load. In practice, this precludes conventional finite-element methods, which
suffer a large computational overhead on account of geometric changes. This
paper presents a new non-conforming finite-element method based on weak
imposition of the displacement discontinuity. The weak imposition of the
discontinuity enables the application of approximation spaces that are
independent of the dislocation geometry, thus enabling optimal reuse of
computational components. Such reuse of computational components renders
finite-element modeling a viable option for inverse problems in geophysical
applications. A detailed analysis of the approximation properties of the new
formulation is provided. The analysis is supported by numerical experiments in
2D and 3D.Comment: Submitted for publication in CMAM
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