17 research outputs found
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation
Stable, accurate, divergence-free simulation of magnetized supersonic
turbulence is a severe test of numerical MHD schemes and has been surprisingly
difficult to achieve due to the range of flow conditions present. Here we
present a new, higher order-accurate, low dissipation numerical method which
requires no additional dissipation or local "fixes" for stable execution. We
describe PPML, a local stencil variant of the popular PPM algorithm for solving
the equations of compressible ideal magnetohydrodynamics. The principal
difference between PPML and PPM is that cell interface states are evolved
rather that reconstructed at every timestep, resulting in a compact stencil.
Interface states are evolved using Riemann invariants containing all transverse
derivative information. The conservation laws are updated in an unsplit
fashion, making the scheme fully multidimensional. Divergence-free evolution of
the magnetic field is maintained using the higher order-accurate constrained
transport technique of Gardiner and Stone. The accuracy and stability of the
scheme is documented against a bank of standard test problems drawn from the
literature. The method is applied to numerical simulation of supersonic MHD
turbulence, which is important for many problems in astrophysics, including
star formation in dark molecular clouds. PPML accurately reproduces in
three-dimensions a transition to turbulence in highly compressible isothermal
gas in a molecular cloud model. The low dissipation and wide spectral bandwidth
of this method make it an ideal candidate for direct turbulence simulations.Comment: 28 pages, 18 figure
On the Proper Setup of the Double Mach Reflection as a Test Case for the Resolution of Gas Dynamics Codes
This note discusses the initial and boundary conditions as well as the size
of the computational domain for the double Mach reflection problem when set up
as a test for the resolution of an Euler scheme for gas dynamics
Hybrid finite difference/finite element immersed boundary method
The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian
description of the structural deformations, stresses, and forces along with an Eulerian description of the
momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary
methods described immersed elastic structures using systems of flexible fibers, and even now, most
immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This
work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian
variables that facilitates independent spatial discretizations for the structure and background grid. This
approach employs a finite element discretization of the structure while retaining a finite difference scheme
for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively
contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases
in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors
that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes.
The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse
structural meshes with the immersed boundary method. This work also contrasts two different weak forms
of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations
facilitated by our coupling approach
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
Quasi-static imaged-based immersed boundary-finite element model of human left ventricle in diastole
SUMMARY:
Finite stress and strain analyses of the heart provide insight into the biomechanics of myocardial function and dysfunction. Herein, we describe progress toward dynamic patient-specific models of the left ventricle using an immersed boundary (IB) method with a finite element (FE) structural mechanics model. We use a structure-based hyperelastic strain-energy function to describe the passive mechanics of the ventricular myocardium, a realistic anatomical geometry reconstructed from clinical magnetic resonance images of a healthy human heart, and a rule-based fiber architecture. Numerical predictions of this IB/FE model are compared with results obtained by a commercial FE solver. We demonstrate that the IB/FE model yields results that are in good agreement with those of the conventional FE model under diastolic loading conditions, and the predictions of the LV model using either numerical method are shown to be consistent with previous computational and experimental data. These results are among the first to analyze the stress and strain predictions of IB models of ventricular mechanics, and they serve both to verify the IB/FE simulation framework and to validate the IB/FE model. Moreover, this work represents an important step toward using such models for fully dynamic fluid鈥搒tructure interaction simulations of the heart
Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library
In this paper we employ two implementations of the fictitious domain (FD)
method to simulate water-entry and water-exit problems and demonstrate their
ability to simulate practical marine engineering problems. In FD methods, the
fluid momentum equation is extended within the solid domain using an additional
body force that constrains the structure velocity to be that of a rigid body.
Using this formulation, a single set of equations is solved over the entire
computational domain. The constraint force is calculated in two distinct ways:
one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method
and another using a fully-Eulerian approach of the Brinkman penalization (BP)
method. Both FSI strategies use the same multiphase flow algorithm that solves
the discrete incompressible Navier-Stokes system in conservative form. A
consistent transport scheme is employed to advect mass and momentum in the
domain, which ensures numerical stability of high density ratio multiphase
flows involved in practical marine engineering applications. Example cases of a
free falling wedge (straight and inclined) and cylinder are simulated, and the
numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some
parts of it for the reader's convenienc
A Model of Fluid-Structure and Biochemical Interactions for Applications to Subclinical Leaflet Thrombosis
Subclinical leaflet thrombosis (SLT) is a potentially serious complication of
aortic valve replacement with a bioprosthetic valve in which blood clots form
on the replacement valve. SLT is associated with increased risk of transient
ischemic attacks and strokes and can progress to clinical leaflet thrombosis.
SLT following aortic valve replacement also may be related to subsequent
structural valve deterioration, which can impair the durability of the valve
replacement. Because of the difficulty in clinical imaging of SLT, models are
needed to determine the mechanisms of SLT and could eventually predict which
patients will develop SLT. To this end, we develop methods to simulate leaflet
thrombosis that combine fluid-structure interaction and a simplified thrombosis
model that allows for deposition along the moving leaflets. Additionally, this
model can be adapted to model deposition or absorption along other moving
boundaries. We present convergence results and quantify the model's ability to
realize changes in valve opening and pressures. These new approaches are an
important advancement in our tools for modeling thrombosis in which they
incorporate both adhesion to the surface of the moving leaflets and feedback to
the fluid-structure interaction.Comment: 29 pages, 11 figure