27,021 research outputs found
Adaptive computation of gravitational waves from black hole interactions
We construct a class of linear partial differential equations describing
general perturbations of non-rotating black holes in 3D Cartesian coordinates.
In contrast to the usual approach, a single equation treats all radiative modes simultaneously, allowing the study of wave perturbations of black
holes with arbitrary 3D structure, as would be present when studying the full
set of nonlinear Einstein equations describing a perturbed black hole. This
class of equations forms an excellent testbed to explore the computational
issues of simulating black spacetimes using a three dimensional adaptive mesh
refinement code. Using this code, we present results from the first fully
resolved 3D solution of the equations describing perturbed black holes. We
discuss both fixed and adaptive mesh refinement, refinement criteria, and the
computational savings provided by adaptive techniques in 3D for such model
problems of distorted black holes.Comment: 16 Pages, RevTeX, 13 figure
An object-oriented approach for parallel self adaptive mesh refinement on block structured grids
Self-adaptive mesh refinement dynamically matches the computational demands of a solver for partial differential equations to the activity in the application's domain. In this paper we present two C++ class libraries, P++ and AMR++, which significantly simplify the development of sophisticated adaptive mesh refinement codes on (massively) parallel distributed memory architectures. The development is based on our previous research in this area. The C++ class libraries provide abstractions to separate the issues of developing parallel adaptive mesh refinement applications into those of parallelism, abstracted by P++, and adaptive mesh refinement, abstracted by AMR++. P++ is a parallel array class library to permit efficient development of architecture independent codes for structured grid applications, and AMR++ provides support for self-adaptive mesh refinement on block-structured grids of rectangular non-overlapping blocks. Using these libraries, the application programmers' work is greatly simplified to primarily specifying the serial single grid application and obtaining the parallel and self-adaptive mesh refinement code with minimal effort. Initial results for simple singular perturbation problems solved by self-adaptive multilevel techniques (FAC, AFAC), being implemented on the basis of prototypes of the P++/AMR++ environment, are presented. Singular perturbation problems frequently arise in large applications, e.g. in the area of computational fluid dynamics. They usually have solutions with layers which require adaptive mesh refinement and fast basic solvers in order to be resolved efficiently
Space-time adaptive solution of inverse problems with the discrete adjoint method
Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method.
This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.
In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, -refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem
Adaptive mesh refinement computation of acoustic radiation from an engine intake
A block-structured adaptive mesh refinement (AMR) method was applied to the computational problem of acoustic radiation from an aeroengine intake. The aim is to improve the computational and storage efficiency in aeroengine noise prediction through reduction of computational cells. A parallel implementation of the adaptive mesh refinement algorithm was achieved using message passing interface. It combined a range of 2nd- and 4th-order spatial stencils, a 4th-order low-dissipation and low-dispersion Runge–Kutta scheme for time integration and several different interpolation methods. Both the parallel AMR algorithms and numerical issues were introduced briefly in this work. To solve the problem of acoustic radiation from an aeroengine intake, the code was extended to support body-fitted grid structures. The problem of acoustic radiation was solved with linearised Euler equations. The AMR results were compared with the previous results computed on a uniformly fine mesh to demonstrate the accuracy and the efficiency of the current AMR strategy. As the computational load of the whole adaptively refined mesh has to be balanced between nodes on-line, the parallel performance of the existing code deteriorates along with the increase of processors due to the expensive inter-nodes memory communication costs. The potential solution was suggested in the end
Refinement strategies for polygonal meshes applied to adaptive VEM discretization
In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well and states new issues, here tackled, concerning good quality mesh elements and reliability of the simulations. In this paper we propose several new polygonal refinement strategies and numerically investigate the quality of the meshes generated by an adaptive mesh refinement process, as well as optimal rates of convergence with respect to the number of degrees of freedom. Among the several possible problems in which these strategies can be applied, here we have considered a geometrically complex geophysical problem as test problem that naturally yields to a polygonal mesh and tackled it by the Virtual Element Method. All the adaptive strategies here proposed, but the “Trace Direction strategy”, can be applied to any problem for which a polygonal element method can be useful and any numerical method based on polygonal elements and can generate good quality isotropic mesh elements
Massive black hole and gas dynamics in galaxy nuclei mergers. I. Numerical implementation
Numerical effects are known to plague adaptive mesh refinement (AMR) codes
when treating massive particles, e.g. representing massive black holes (MBHs).
In an evolving background, they can experience strong, spurious perturbations
and then follow unphysical orbits. We study by means of numerical simulations
the dynamical evolution of a pair MBHs in the rapidly and violently evolving
gaseous and stellar background that follows a galaxy major merger. We confirm
that spurious numerical effects alter the MBH orbits in AMR simulations, and
show that numerical issues are ultimately due to a drop in the spatial
resolution during the simulation, drastically reducing the accuracy in the
gravitational force computation. We therefore propose a new refinement
criterion suited for massive particles, able to solve in a fast and precise way
for their orbits in highly dynamical backgrounds. The new refinement criterion
we designed enforces the region around each massive particle to remain at the
maximum resolution allowed, independently upon the local gas density. Such
maximally-resolved regions then follow the MBHs along their orbits, and
effectively avoids all spurious effects caused by resolution changes. Our suite
of high resolution, adaptive mesh-refinement hydrodynamic simulations,
including different prescriptions for the sub-grid gas physics, shows that the
new refinement implementation has the advantage of not altering the physical
evolution of the MBHs, accounting for all the non trivial physical processes
taking place in violent dynamical scenarios, such as the final stages of a
galaxy major merger.Comment: 11 pages, 11 figures, 1 table, it matches the published versio
Planet-disc interaction on a freely moving mesh
General-purpose, moving-mesh schemes for hydrodynamics have opened the
possibility of combining the accuracy of grid-based numerical methods with the
flexibility and automatic resolution adaptivity of particle-based methods. Due
to their supersonic nature, Keplerian accretion discs are in principle a very
attractive system for applying such freely moving mesh techniques. However, the
high degree of symmetry of simple accretion disc models can be difficult to
capture accurately by these methods, due to the generation of geometric grid
noise and associated numerical diffusion, which is absent in polar grids. To
explore these and other issues, in this work we study the idealized problem of
two-dimensional planet-disc interaction with the moving-mesh code AREPO. We
explore the hydrodynamic evolution of discs with planets through a series of
numerical experiments that vary the planet mass, the disc viscosity and the
mesh resolution, and compare the resulting surface density, vortensity field
and tidal torque with results from the literature. We find that the performance
of the moving-mesh code in this problem is in accordance with published
results, showing good consistency with grid codes written in polar coordinates.
We also conclude that grid noise and mesh distortions do not introduce
excessive numerical diffusion. Finally, we show how the moving-mesh approach
can naturally increase resolution in regions of high densityaround planets and
planetary wakes, while retaining the background flow at low resolution. This
provides an alternative to the difficult task of implementing adaptive mesh
refinement in conventional polar-coordinate codes.Comment: 21 pages, 15 figures, 2 tables. Updated to match version published by
MNRA
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