Riemann-Cartan Geometry of nonlinear dislocation mechanics

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold – where the body is stress free – is a Weitzenbock manifold, i.e. a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance

Hydrodynamic simulations with the Godunov SPH

We present results based on an implementation of the Godunov Smoothed Particle Hydrodynamics (GSPH), originally developed by Inutsuka (2002), in the GADGET-3 hydrodynamic code. We first review the derivation of the GSPH discretization of the equations of moment and energy conservation, starting from the convolution of these equations with the interpolating kernel. The two most important aspects of the numerical implementation of these equations are (a) the appearance of fluid velocity and pressure obtained from the solution of the Riemann problem between each pair of particles, and (b the absence of an artificial viscosity term. We carry out three different controlled hydrodynamical three-dimensional tests, namely the Sod shock tube, the development of Kelvin-Helmholtz instabilities in a shear flow test, and the "blob" test describing the evolution of a cold cloud moving against a hot wind. The results of our tests confirm and extend in a number of aspects those recently obtained by Cha (2010): (i) GSPH provides a much improved description of contact discontinuities, with respect to SPH, thus avoiding the appearance of spurious pressure forces; (ii) GSPH is able to follow the development of gas-dynamical instabilities, such as the Kevin--Helmholtz and the Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl structures in the shear-flow test and the dissolution of the cold cloud in the "blob" test. We also discuss in detail the effect on the performances of GSPH of changing different aspects of its implementation. The results of our tests demonstrate that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled to an N-body solver, for astrophysical and cosmological applications. [abridged]Comment: 19 pages, 13 figures, MNRAS accepted, high resolution version can be obtained at http://adlibitum.oats.inaf.it/borgani/html/papers/gsph_hydrosim.pd

Progress in multi-dimensional upwind differencing

Multi-dimensional upwind-differencing schemes for the Euler equations are reviewed. On the basis of the first-order upwind scheme for a one-dimensional convection equation, the two approaches to upwind differencing are discussed: the fluctuation approach and the finite-volume approach. The usual extension of the finite-volume method to the multi-dimensional Euler equations is not entirely satisfactory, because the direction of wave propagation is always assumed to be normal to the cell faces. This leads to smearing of shock and shear waves when these are not grid-aligned. Multi-directional methods, in which upwind-biased fluxes are computed in a frame aligned with a dominant wave, overcome this problem, but at the expense of robustness. The same is true for the schemes incorporating a multi-dimensional wave model not based on multi-dimensional data but on an 'educated guess' of what they could be. The fluctuation approach offers the best possibilities for the development of genuinely multi-dimensional upwind schemes. Three building blocks are needed for such schemes: a wave model, a way to achieve conservation, and a compact convection scheme. Recent advances in each of these components are discussed; putting them all together is the present focus of a worldwide research effort. Some numerical results are presented, illustrating the potential of the new multi-dimensional schemes

TESS: A Relativistic Hydrodynamics Code on a Moving Voronoi Mesh

We have generalized a method for the numerical solution of hyperbolic systems of equations using a dynamic Voronoi tessellation of the computational domain. The Voronoi tessellation is used to generate moving computational meshes for the solution of multi-dimensional systems of conservation laws in finite-volume form. The mesh generating points are free to move with arbitrary velocity, with the choice of zero velocity resulting in an Eulerian formulation. Moving the points at the local fluid velocity makes the formulation effectively Lagrangian. We have written the TESS code to solve the equations of compressible hydrodynamics and magnetohydrodynamics for both relativistic and non-relativistic fluids on a dynamic Voronoi mesh. When run in Lagrangian mode, TESS is significantly less diffusive than fixed mesh codes and thus preserves contact discontinuities to high precision while also accurately capturing strong shock waves. TESS is written for Cartesian, spherical and cylindrical coordinates and is modular so that auxilliary physics solvers are readily integrated into the TESS framework and so that the TESS framework can be readily adapted to solve general systems of equations. We present results from a series of test problems to demonstrate the performance of TESS and to highlight some of the advantages of the dynamic tessellation method for solving challenging problems in astrophysical fluid dynamics.Comment: ApJS, 197, 1

Geometric Algebra Techniques for General Relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.Comment: 34 pages, 0 figures; submitted to Annals of Physic

PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection-diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier-Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org)