The method known as Smoothed Particle Hydrodynamics (SPH) is an important tool in
modern numerical Astrophysics. It has been extensively used to simulate a large number
of systems ranging from planets to clusters of galaxies. Nevertheless current applications of the method are restricted to dynamical situations because of the limitations in the time-step imposed by the Courant condition. Here we describe the main features of a new implicit SPH code which is able to handle with several thousand particles and, therefore, it can be used to simulate slowly evolving systems.Peer ReviewedPostprint (published version

Bravo Guil, Eduardo

Escartín Vigo, José Antonio

García Senz, Domingo

English

The method known as Smoothed Particle Hydrodynamics (SPH) is an important tool in

modern numerical Astrophysics. It has been extensively used to simulate a large number

of systems ranging from planets to clusters of galaxies. Nevertheless current applications of the method are restricted to dynamical situations because of the limitations in the time-step imposed by the Courant condition. Here we describe the main features of a new implicit SPH code which is able to handle with several thousand particles and, therefore, it can be used to simulate slowly evolving systems.Peer Reviewe

UPCommons

Computer simulations using implicit Lagrangian hydrodynamics in 3D

UPCommons. Portal del coneixement obert de la UPC

	  	  Highlights	  of	  Spanish	  Astrophysics	  VI,	  Proceedings	  of	  the	  IX	  Scientific	  Meeting	  of	  the	  Spanish	  Astronomical	  Society	  held	  on	  September	  13	  -­	  17,	  2010,	  in	  Madrid,	  Spain.	  M.	  R.	  Zapatero	  Osorio	  et	  al.	  (eds.)	  Computer simulations using implicit Lagrangianhydrodynamics in 3DJ. A. Escart´ın1, D. Garc´ıa-Senz1,2, and E. Bravo11 Departament de F´ısica i Enginyeria Nuclear. Universitat Polite`cnica de Catalunya. C/.Compte d’Urgell 187, 08036 Barcelona, Spain.2 Institut d’Estudis Espacials de Catalunya. C/. Gran Capita` 2-4, 08034 Barcelona, Spain.AbstractThe method known as Smoothed Particle Hydrodynamics (SPH) is an important tool inmodern numerical Astrophysics. It has been extensively used to simulate a large numberof systems ranging from planets to clusters of galaxies. Nevertheless current applications ofthe method are restricted to dynamical situations because of the limitations in the time-stepimposed by the Courant condition. Here we describe the main features of a new implicitSPH code which is able to handle with several thousand particles and, therefore, it can beused to simulate slowly evolving systems.1 IntroductionThe SPH is a Lagrangian method of calculation used in the simulation of motion flows. Thesystem is represented by a grid of points where the coordinates move with the fluid. If we havea good resolution the grid will reproduce the behavior of a continuous fluid more accuracy.Each point is associated with a number of local properties, including the mass, so that fromnow it will identify as a particle. The fluid properties are calculated as the interpolated sumof the properties of particles in their environment.The advantages of this method are that it allows an implementation in three dimensionsmore easily than Eulerian methods and, by construction, it exactly conserves mass, linearmomentum and angular momentum of the system.The simplest implementation is through an explicit method and its main disadvantageis that to preserve the model stability, the step of time between iterations can not exceed theCourant condition. This restriction forces to use very small time steps, limiting the methodto the modeling of dynamic processes of very short duration (such as collisions or mergers).704 Implicit SPH in 3D2 ObjectivesThe main objective of this work is double. On the one hand, build and implement a mul-tidimensional implicit SPH code incorporating the needed physics to simulate quasi-static 1pre-explosive phases in compact stellar objects: novae, supernovae and X-ray burst. Moreoverapplying the SPH code to the study of the ignition conditions of thermonuclear supernovae[3, 4], starting from massive white dwarf, initially in hydrostatic equilibrium and unbalancedby accretion of material.This pre-explosive phase is barely known and the simulation will be studied based onhighly simplified initial conditions. Phase space to explore is very complex, so this type ofstudy will continue calling the attention of the scientific community for many years. Ourintention is that the code developed in this work will endure and will be applied in suchstudies.Given the huge effort involved in constructing an implicit SPH code, we have intentionthat it could be used in the study of many astrophysical scenarios. To this end we have chosena modular design that separates the treatment of basic evolutionary equations (Section 5)from the material properties (equation of state, nuclear reactions, etc.).The code should be operational with the resources of the GAA (Section 4.2) and avail-able in supercomputers (such as MareNostrum).3 The state of the artThere have been several attempts to get a functional implicit SPH code. So far the problemhas always been the high computational cost of such methods. This is because the propertiesof the next time step depend on the values of the unknown variables in this time and thereforeall must be calculated simultaneously. The consequence is that the whole system of equationsmust be solved together by inverting a huge sparse matrix (in three dimensions the array sizeis (n ∗ 8)2, where n is the number of particles).In his thesis, [5] proposes an implicit method based on iterative solvers of Krylov andon Newton-Rapshon corrections. His code has been developed from a previous explicit SPHone: SPHINX of Los Alamos National Laboratory. The Knapp code was developed in 2000and used a maximum of 6 000 particles, an amount entirely inadequate to describe complexastrophysical scenarios.Our goal is to work with an amount between 10 000 and 100 000 particles in threedimensions, depending on the complexity of the physics of the problem.Moreover we include in our implicit SPH code basic physics of the explicit SPH codesavailable from our group. This has not been done yet, because this basic physics influenceson the calculation time, turning it very hard to compute.1Characterized by an evolution time greater than the time that a sound signal takes across the system.J. A. Escart´ın et al. 7054 Code ISFAABased on this previous work, we developed an implicit SPH code where, using the experiencein the explicit SPH code of GAA and providing new ideas we have improved the capacity ofcalculation of the code.The progress made until today has resulted in a SPH code which meets non-functionalrequirements proposed and is able to pass a series of basic tests. These tests control thecorrectness of the code in key functional areas (Section 6).The current code contains the basic equations of SPH and improvements of the codedescribed by Knapp:• Multidimensional Code: we have done a general solution to the code so it is easy tochange the number of dimensions to be addressed in the physical model.• Artificial Viscosity [6, 1]: astrophysical processes that are required to be simulated needthis improvement to avoid usual problems such as particle interpenetration.• Gravity : as we work with models where gravity is an essential force our equations needto contemplate this phenomenon.• Equation of state (EOS): in contrast to Knapp, which uses an ideal gas EOS, we haveimplemented an EOS that takes into account contributions from the electrons, ions andradiation fitting to the conditions studied.• Treatment of the temperature: needed for EOS and heat diffusion.• Centered scheme: we have modified and implemented the scheme of calculation so thatthe equations are calculated by leaps centered steps. In terms of numerical accuracy,values of the particle properties are correct until second order.• Management of the matrix according to Morton method (Z-order): the main problemof solving the system of equations is the dispersion of the matrix to invert. This orderensures the proximity of the particles reducing the dispersion and the computationtime.• Sparse matrix storage according to the CSR format (compressed by rows): you canreduce the amount of memory reserved because only the nonzero cells in the array arestored. The memory savings produced by this property is very important because therate of filling of the matrix in astrophysical models that treat is about 2%.• Resolution of sparse matrices using parallel computing libraries: the resolution of thesystem of equations is computationally the most expensive part of the calculation (morethan 95% of the time of calculation time). After making a detailed study of parallelcomputation packages available, we decided to use the PARDISO 4.0 library of theUniversity of Basel [7, 8].706 Implicit SPH in 3D4.1 Functional requirementsThe planned physical model is very developed. The requirements not implemented yet andwhich will improve the behavior of the model approaching it to a more realistic scenario arethe following: 1) heat diffusion, and 2) network of nuclear reactions.4.2 Non-functional requirementsThe developed code implements the following requirements: modularity, maintainability,extensibility, scalability, robustness, documentation, efficiency, performance and code correc-tion.Regarding the software, we can say that the code has been developed in Fortran95 andruns on systems GNU/Linux.To make the calculations we are using a shared memory and parallel computer machine,purchased by the GAA in November 2009, to support this work. To make this acquisitionwe made a study of professional processors looking for the best ratio: power versus costcalculation. This study concluded that the best option would be to assemble a workstation“by pieces”: it consists of 16 cores (2 Intel Xeon processors at 3.2 Ghz W5580 8-core (4 realcores + 4 virtual cores)) and 48GB of RAM DDR3-1333Mhz.5 Physical modelEach particle of the model is associated with a group of basic evolutionary equations, derivedfrom the Lagrangian fluid equations [6]:Continuity equation:dρidt=N∑j=1mj (vi − vj)∇iW˜ij . (1)Moment equation:dvidt= −N∑j=1mj(Piρ2i+Pjρ2j+ Πij)∇iW˜ij − g (ri) . (2)Energy equation:duidt=N∑j=1mj(Piρ2i+Πij2)(vi − vj)∇iW˜ij , (3)being ∇iW˜ij the gradient of the interpolating function; mi mass; vi speed; ρi density; Pipressure; ui internal energy; Πij artificial viscosity; g (ri) gravity.6 Tests casesTo check the validity of the code we have done some tests whose solution is known:J. A. Escart´ın et al. 707Figure 1: Left: Sedov test, density profile with across time t = 2.16×10−2 s. Right: Collapseof a cylinder of infinite height. It represents the simulated evolution with ISFAA and thetheoretical trajectory, beginning with different initial radii.6.1 Sedov testThe test [9] examines the ability of the artificial viscosity to represent shock waves. Wehave a two-dimensional rectangular grid, balanced and equally spaced particles. There is acentral disturbance, increasing the density a factor 104 by a Gaussian function. The systempropagates the density wave to the outside. An analytically description of the change indensity between the pre-shock and the post-shock zone could be done and the result is thatis a factor of four, as befits a strong shock wave in an ideal gas (see Fig. 1, left).6.2 Freefall testThe test examines the correction of gravity force in the code. We have a circular grid in twodimensions, balanced and equally spaced particles. (It would be equivalent to a cylindricalthree-dimensional distribution of infinite height). The system is allowed to evolve withoutpressure or artificial viscosity, so it will be a free fall toward the center of mass, which shouldmatch with the analytical result [2] (see Fig. 1, right).6.3 Stability of a white dwarfWe have a white dwarf star in three dimensions, built with a realistic EOS and a mass of 1.2solar masses, simulated and relaxed with an explicit SPH. The system is allowed to evolve,from a very small step of time and zero speed until the dynamics of the system is relaxed,using steps of time larger than the Courant time and getting the stability (see Fig. 2, left).708 Implicit SPH in 3DFigure 2: Left: Evolution of internal, gravitational and kinetics energy of a 1.15 M whitedwarf. The total energy (E) remains constant. Right: The time step vs. the Courant time.7 Future workBelow we show the temporary planning to completation of the work: 1) add the remainderof the planned physics and calculations for the study of the previous stages to the ignitionof supernovas, 2) documentation and reading thesis, and 3) applying the method to otherscenarios.AcknowledgmentsWith the support of Comisionado para las Universidades e Investigacio´n del Departament d’Innovacio´,Universitats i Empresa de la Generalitat de Catalunya y del Fondo social europeo”.References[1] Darelid, J. 2007, UPC, Barcelona[2] Garc´ıa-Senz, D., Relan˜o, A. Cabezo´n, R. M., & Bravo, E. 2008, Royal Astron. Society, 392[3] Hillebrandt, W., & Niemeyer, J. C. 2000, ARAA, 38[4] Ho¨flich, P., & Stein, J. 2001, ApJ, 568[5] Knapp, C. E. 2000, Los Alamos National Laboratory[6] Monaghan, J. J. 1992, ARAA, 365, 199[7] Schenk, O., & Ga¨rtner, K. 2004, Journal of Future Generation Computer Systems, 20(3)[8] Schenk, O., & Ga¨rtner, K. 2006, Elec. Trans. Numer. Anal. 23[9] Sedov, L. I., 1959, Similarity and Dimesional Methods in Mechanics, Academic Press Inc.

https://upcommons.upc.edu/bitstream/2117/15372/1/F-escartinja.pdf

Computer simulations using implicit Lagrangian hydrodynamics in 3D

Abstract

Similar works

Full text

Available Versions

UPCommons

UPCommons. Portal del coneixement obert de la UPC