Smoothed Particle Hydrodynamics Code Research Articles

Moving-least-squares-particle hydrodynamics?I. Consistency and stability

The Smooth-Particle-Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite-element method. This derivation is modified to replace the SPH interpolant with the Moving-Least-Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable-rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable-h SPH. The new MLSPH method is conservative if an anti-symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one-point collocation rule is used to retain similarity with SPH, leading to a non-conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non-linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd.

Accuracy of Mesh‐based Cosmological Hydrocodes: Tests and Corrections

We perform a variety of tests to determine the numerical resolution of the cosmological TVD Eulerian code. Tests include 5123 and 2563 simulations of a Pk∝k−1 spectrum to check for self-similarity and a comparison of results with those from higher resolution smooth-particle hydrodynamics (SPH) codes and grid-based calculations. We conclude that in regions where density gradients are not produced by shocks, the code degrades resolution with a Gaussian smoothing (radius) length of 1.7 cells. At shock-caused gradients (for which the code was designed) the smoothing length is 1.1 cells. Finally, for β-model-fitted clusters, we can approximately correct the numerical resolution by the transformation R2core→R2core-(CΔl)2, where Δl is the cell size and C=1.1-1.7. When we use these corrections on our previously published computations for the standard cold dark matter (SCDM) and ΛCDM models, we find luminosity-weighted zero-redshift X-ray cluster core radii of 210 ± 86 h−1 and 280 ± 67 h−1 kpc, respectively, which are marginally consistent with observed values of 50-200 h−1 kpc. Using the corrected core radii, the COBE-normalized SCDM model predicts a number of bright (LX>1043 ergs s-1) clusters that is too high by a factor of ~20, and the ΛCDM model is consistent with observations.

The formation of disc galaxies

We investigate the influence of the cooling epoch on the formation of galaxies in a cold dark matter dominated universe. Isolated haloes, with circular speeds typical of spiral galaxies, have been selected from a low resolution numerical simulation for re-simulation at higher resolution with dark matter and gas components. The initial conditions are evolved with two smoothed particle hydrodynamics codes, TREESPH and GRAPESPH. In previous SPH simulations, strong outward transport of angular momentum has led to the formation of disc-like systems with much smaller angular momenta than observed in real disc galaxies. Here we investigate whether this problem can be circumvented if feedback processes prevent disc formation until late epochs. The results of varying the cooling epoch for each of five different haloes are analysed. When cooling and star formation occur at early times, stellar discs are destroyed during merger events and we observe similar catastrophic transport of angular momentum as seen in previous work. With cooling suppressed until z=1, discs can form by the present day with angular momenta comparable to those of observed disc galaxies. We conclude that feedback processes, which prevent gas from collapsing until late epochs, are an essential ingredient in disc galaxy formation.

Chemo -- Dynamical evolution of disk galaxies, smoothed particles hydrodynamics approach

A new Chemo -- Dynamical Smoothed Particle Hydrodynamic (CD -- SPH) code is presented. The disk galaxy is described as a multi -- fragmented gas and star system, embedded into the cold dark matter halo. The star formation (SF) process, SNII, SNIa and PN events as well as chemical enrichment of gas have been considered within the framework of standard SPH model. Using this model we try to describe the dynamical and chemical evolution of triaxial disk -- like galaxies. It is found that such approach provides a realistic description of the process of formation, chemical and dynamical evolution of disk galaxies over the cosmological timescale.

Cosmological Simulations with Scale‐Free Initial Conditions. I. Adiabatic Hydrodynamics

We analyze hierarchical structure formation based on scale-free initial conditions in an Einstein-de Sitter universe, including a baryonic component with Ωbary = 0.05. We present three independent, smoothed particle hydrodynamics (SPH) simulations, performed at two resolutions (323 and 643 dark matter and baryonic particles) and with two different SPH codes (TreeSPH and P3MSPH). Each simulation is based on identical initial conditions, which consist of Gaussian-distributed initial density fluctuations that have a power spectrum P(k) ∝ k-1. The baryonic material is modeled as an ideal gas subject only to shock heating and adiabatic heating and cooling; radiative cooling and photoionization heating are not included. The evolution is expected to be self-similar in time, and under certain restrictions we identify the expected scalings for many properties of the distribution of collapsed objects in all three realizations. The distributions of dark matter masses, baryon masses, and mass- and emission-weighted temperatures scale quite reliably. However, the density estimates in the central regions of these structures are determined by the degree of numerical resolution. As a result, mean gas densities and Bremsstrahlung luminosities obey the expected scalings only when calculated within a limited dynamic range in density contrast. The temperatures and luminosities of the groups show tight correlations with the baryon masses, which we find can be well represented by power laws. The Press-Schechter (PS) approximation predicts the distribution of group dark matter and baryon masses fairly well, though it tends to overestimate the baryon masses. Combining the PS mass distribution with the measured relations for T(M) and L(M) predicts the temperature and luminosity distributions fairly accurately, though there are some discrepancies at high temperatures/luminosities. In general the three simulations agree well for the properties of resolved groups, where a group is considered resolved if it contains more than 32 particles.

The formation of disc galaxies

We investigate the influence of the cooling epoch on the formation of galaxies in a cold dark matter dominated universe. Isolated haloes, with circular speeds typical of spiral galaxies, have been selected from a low resolution numerical simulation for re-simulation at higher resolution with dark matter and gas components. The initial conditions are evolved with two smoothed particle hydrodynamics codes, TREESPH and GRAPESPH. In previous SPH simulations, strong outward transport of angular momentum has led to the formation of disc-like systems with much smaller angular momenta than observed in real disc galaxies. Here we investigate whether this problem can be circumvented if feedback processes prevent disc formation until late epochs. The results of varying the cooling epoch for each of five different haloes are analysed. When cooling and star formation occur at early times, stellar discs are destroyed during merger events and we observe similar catastrophic transport of angular momentum as seen in previous work. With cooling suppressed until z=1, discs can form by the present day with angular momenta comparable to those of observed disc galaxies. We conclude that feedback processes, which prevent gas from collapsing until late epochs, are an essential ingredient in disc galaxy formation.

The Opacity of the Lyα Forest and Implications for Ωband the Ionizing Background

We have measured the distribution function of the flux decrement D = 1 - e-τ caused by Lyα forest absorption from intervening gas in the lines of sight to high-redshift QSOs from a sample of seven high-resolution QSO spectra obtained with the Keck telescope. The observed flux decrement distribution function (FDDF) is compared with the FDDF from two simulations of the Lyα forest: a ΛCDM model (with Ω = 0.4, Λ = 0.6), computed with the Eulerian code of Cen & Ostriker, and a standard cold dark matter (SCDM) model (with Ω = 1), computed with the smoothed particle hydrodynamics code of Hernquist et al. Good agreement is obtained between the shapes of the simulated and observed FDDFs for both simulations after fitting only one free parameter, which controls the mean flux decrement. The difference between the predicted FDDFs from the two simulations is small, and we show that it arises mostly from a different temperature in the low-density gas (caused by different assumptions that were made about the reionization history in the two simulations), rather than differences between the two cosmological models or numerical effects in the two codes, which use very different computational methods. A measurement of the parameter μΩ -->2b h -->3/Γ (where Γ is the H I ionization rate due to the ionizing background) is obtained by requiring the mean flux decrement in the simulations to agree with the observed one. Estimating the lower limit Γ > 7 × 10-13 s-1 from the abundance of known QSOs, we derive a lower limit on the baryonic matter density, Ωbh2 > 0.021 (0.017) for the ΛCDM (SCDM) model. The difference between the lower limits inferred from the two models is again due to different temperatures in the low-density gas. We give general analytical arguments for why this lower limit is unlikely to be reduced for any other models of structure formation by gravitational collapse that can explain the observed Lyα forest. When combined with constraints from big bang nucleosynthesis, the large Ωb we infer is inconsistent with some recent D/H determinations (Rugers & Hogan), favoring a low deuterium abundance as reported by Tytler, Fan & Burles. Adopting a fixed Ωb, the measurement of μ(z) allows a determination of the evolution of the ionizing radiation field with redshift. Our models predict an intensity that is approximately constant with redshift, which is in agreement with the assumption that the ionizing background is produced by known quasars for z < 3, but requires additional sources of ionizing photons at higher redshift given the observed rapid decline of the quasar abundance.

Evolution of Stellar Collision Products in Globular Clusters. I. Head‐on Collisions

We explore the evolution of collisionally merged stars in the blue straggler region of the H-R diagram. The starting models for our stellar evolution calculations are the results of the smoothed particle hydrodynamics (SPH) simulations of parabolic collisions between main-sequence stars performed by Lombardi, Rasio, & Shapiro. Since SPH and stellar evolution codes employ different and often contradictory approximations, it is necessary to treat the evolution of these products carefully. The mixture and disparity of the relevant timescales (hydrodynamic, thermal relaxation, and nuclear burning) and of the important physical assumptions between the codes makes the combined analysis of the problem challenging, especially during the initial thermal relaxation of the star. In particular, the treatment of convection is important and semiconvection must be modeled in some detail. The products of seven head-on collisions are evolved through their initial thermal relaxation and then through the main-sequence phase to the base of the giant branch. Their evolutionary tracks are presented. In contrast to what was assumed in previous work, these collision products do not develop substantial convective regions during their thermal relaxation and therefore are not mixed significantly after the collision.

Resolution requirements for smoothed particle hydrodynamics calculations with self-gravity

We determine a new resolution requirement for the smoothed particle hydrodynamics (SPH) numerical method when self-gravity is included. Comparison calculations between an SPH code and an Eulerian grid code are performed. The calculations are of a computationally demanding molecular cloud collapse and fragmentation problem. We demonstrate that the results given by the two different hydrodynamic methods are in good agreement, so long as the minimum resolvable mass in the SPH calculations is always less than the Jeans mass. If this criterion is not satisfied, SPH may give incorrect results, with the stability of near-Jeans-mass clumps depending on the details of how the SPH code is implemented rather than on physical processes. We give examples, from the literature, of problems that have been encountered in simulations of star, galaxy and cosmological-structure formation where this resolution requirement is ignored.

The Origin of the Moon and the Single Impact Hypothesis V

Previous papers in this series have described the smooth particle hydrodynamics (SPH) method, which has been employed to explore the possibility that a major planetary collision may have been responsible for the formation of the Moon. In those simulations the SPH code used particles of equal mass and fixed smoothing lengths; I have found that the results obtained were reliable regarding what happens to the interiors of the colliding planets. Because the particles placed into the surrounding space were isolated rather than overlapping, however, that part of the calculation was unreliable. Ten additional cases have been run with 5000 particles in the Protoearth and 5000 in the Impactor, with variable smoothing lengths. Three of the cases had Protoearth/Impactor mass ratios of 5:5, 6:4, and 7:3. The other cases had a mass ratio of 8:2 and a variety of angular momenta. All cases had zero velocity at infinity. In every case the product of the collision became surrounded by evaporated particles of rock vapor, forming an extended atmosphere; however, relatively little mass extended beyond the Roche lobe. If the Moon formed from a rock disk in orbit around the Earth, then some other mechanism would be needed to transport angular momentum and mass outward in the equatorial plane, so that rock condensates from the hot atmosphere would be precipitated beyond the Roche limit, thus providing material for collection into the Moon. Most of this atmospheric material was originally in the Impactor and was mixed with terrestrial rock before evaporation. Recent calculations by R. M. Canup and L. W. Esposito (1996, Icarus119, 427–446) have shown that it is very difficult to form the Moon from a gaseous disk largely confined to within the Roche lobe. On the other hand, higher-angular-momentum collisions can leave a quite massive body in orbit about the Earth, and this could form much or essentially all of the Moon. These questions remain challenging and require further investigations.

Adaptive Smooth Particle Hydrodynamics and Particle-Particle Coupled Codes: Energy and Entropy Conservation

We present and test a general purpose code, called PPASPR, for evolving self-gravitating fluids in astrophysics, both with and without a collisionless component. In PPASPH, hydrodynamical properties are computed by using the SPH (smoothed particle hydrodynamics) method while, unlike most previous implementations of SPH, gravitational forces are computed by a PP (particle-particle) approach. Other important features of this code are as follows: (1) PPASPR takes into account the contributions of all particles to the gravitational and hydrodynamical forces on any other particle. This results in a better energy conservation. (2) Smoothing lengths are updated by an iterative procedure that ensures an exactly constant number of neighbors around each gas particle. (3) Cooling processes have been implemented in an integrated form that includes a special treatment to avoid a nonphysical catastrophic cooling phenomenon. Such a procedure ensures that cooling does not limit the time step. (4) Hydrodynamics equations optionally include the correction terms (hereafter ∇h terms) appearing when h(t, r) is not constant. <P />Our code has been implemented by using the data parallel programming model on the Connection Machine (CM), which allows for an efficient unification of the SPH and PP methods with costs per time step growing as ∼N. <P />PPASPR has been applied to study the importance of adaptive smoothing correction terms on the entropy conservation. We confirm Hernquist's interpretation of the entropy violation observed in previous SPR simulations as a result of having neglected these terms. An improvement on the entropy conservation is not found by merely considering larger numbers of particles or different N<SUB>s</SUB> choices. The correct continuum description is only obtained if the ∇h correction terms are included. Otherwise, the entropy conservation is always rather poor as compared to that found for the total energy.

Simulations of Nonaxisymmetric Instability in a Rotating Star: A Comparison between Eulerian and Smooth Particle Hydrodynamics

We have carried out 3-D numerical simulations of the dynamical bar instability in a rotating star and the resulting gravitational radiation using both an Eulerian code written in cylindrical coordinates and a smooth particle hydrodynamics (SPH) code. The star is modeled initially as a polytrope with index $n = 3/2$ and $T_{\rm rot}/|W| \approx 0.30$, where $T_{\rm rot}$ is the rotational kinetic energy and $|W|$ is the gravitational potential energy. In both codes the gravitational field is purely Newtonian, and the gravitational radiation is calculated in the quadrupole approximation. We have run 3 simulations with the Eulerian code, varying the number of angular zones and the treatment of the boundary between the star and the vacuum. Using the SPH code we did 7 runs, varying the number of particles, the artificial viscosity, and the type of initial model. We compare the growth rate and rotation speed of the bar, the mass and angular momentum distributions, and the gravitational radiation quantities. We highlight the successes and difficulties of both methods, and make suggestions for future improvements.

Smooth Particle Hydrodynamics Simulations of Deflagrations in Supernovae

Some results of a set of tridimensional calculations concerning the explosion of a white dwarf by means of a smooth particle hydrodynamics (SPH) code are presented. The burning front advance is solved along with the hydrodynamics of the whole star, and a characterization of the front surface in terms of fractal concepts is done. Our main result indicates that the surface of the deflagration front sought at the largest scales of the Rayleigh-Taylor instability can be characterized as a fractal with a time-dependent dimension, D(t), whose value is 2 ≤ D(t) ≤ 2.4.

Numerical aspects of the smoothed particle hydrodynamics method for simulating accretion disks

The derivation of the Smoothed Particle Hydrodynamics (SPH) method is reviewed. In particular, the problem of second-order derivative terms is investigated. Applying these considerations to the Navier-Stokes equations, a physical viscosity is constructed which can be used to perform simulations of viscous fluids within the framework of SPH. With such a viscous stress tensor the energy balance and the angular momentum conservation for the particle and the continuum representations are compared. An SPH code based on these results is tested on different problems, especially on an analytically solvable problem, namely the spreading of a ring of gas moving with Keplerian speed around a point mass. Additionally, some examples for the dynamics of accretion disks in close binary systems are presented. Finally, the efficient implementation of this SPH code is discussed in some detail, in particular by a comparison between scalar and vector computers.

A technique for the investigation of organic reactions driven by shock waves in liquids

A method has been developed for investigating the chemical reactions of organic compounds under the dynamic shock loading conditions of detonation. Sealed capsules are placed along the axis of a cylindrical charge which is initiated at one end. As the detonation wave progresses along the charge, a major part of the sample inside each capsule is subjected to a well-balanced annular compression with peak pressure approximately equal to the Chapman–Jouguet pressure of the explosive used. The geometry of the system ensures both minimal structural deformation and low residual velocity of the capsules; consequently high survival rates have been obtained. The temperature and pressure distribution within the sample space has been calculated as a function of time by a computer simulation using MAGI, a three-dimensional Smooth Particle Hydrodynamics code. The temperature has also been measured experimentally with a variety of organic reactions known to be insensitive to pressure, whose Arrhenius parameters are obtained from the literature and measurements in our laboratory.

Softened Lagrangian hydrodynamics for cosmology

view Abstract Citations (95) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Softened Lagrangian Hydrodynamics for Cosmology Gnedin, Nickolay Y. Abstract A new approach to cosmological hydrodynamics is discussed that is based on a moving, quasi-Lagrangian mesh. The softened Lagrangian hydrodynamics (SLH) method utilizes a high-resolution Lagrangian hydrodynamic code combined with a low-resolution Eulerian solver to deal with severe mesh distortions. Most of the volume of a simulation is treated with the Lagrangian code and only in sites where the Lagrangian approach fails, due to mesh distortions, does the Eulerian part of the code step in. This approach utilizes a high-resolution gravity solver without use of TREE or P3M methods; Poisson's equation is solved on the moving baryonic mesh using a simple relaxation technique. The dark matter is included by means of the cloud-in-cell method on the Lagrangian mesh. All three components of the cosmological code--gravity, dark matter, and baryons--are thus treated selft-consistently with exactly the same resolution. The computer code based on the SLH approach is described in detail, and comparison with existing Eulerian and smooth particle hydrodynamics (SPH) codes is presented. For most purposes the SLH approach turns out to be the intermediate between Eulerian and SPH codes, but it outperforms both of these approaches in resolving caustics. Thus, it may turn out to be a valuable tool to study galaxy formation. Publication: The Astrophysical Journal Supplement Series Pub Date: April 1995 DOI: 10.1086/192141 Bibcode: 1995ApJS...97..231G Keywords: Astronomical Models; Computational Grids; Computerized Simulation; Cosmology; Euler-Lagrange Equation; Hydrodynamics; Lagrangian Function; Mathematical Models; Baryons; Dark Matter; Galactic Evolution; Gravitation; Missing Mass (Astrophysics); Poisson Equation; Astrophysics; COSMOLOGY: THEORY; HYDRODYNAMICS; METHODS: NUMERICAL full text sources ADS |

A method of smoothed particle hydrodynamics using spheroidal kernels

We present a new method of three-dimensional smoothed particle hydrodynamics (SPH) designed to model systems dominated by deformation along a preferential axis. These systems cause severe problems for SPH codes using spherical kernels, which are best suited for modeling systems which retain rough spherical symmetry. Our method allows the smoothing length in the direction of the deformation to evolve independently of the smoothing length in the perpendicular plane, resulting in a kernel with a spheroidal shape. As a result the spatial resolution in the direction of deformation is significantly improved. As a test case we present the one-dimensional homologous collapse of a zero-temperature, uniform-density cloud, which serves to demonstrate the advantages of spheroidal kernels. We also present new results on the problem of the tidal disruption of a star by a massive black hole.

Calculation of oblique impact and fracture of tungsten cubes using smoothed particle hydrodynamics

The oblique impact and fracture of tungsten cubes is simulated using the Smoothed Particle Hydrodynamics code MAGI . A series of experiments with well defined and repeatable results provide the basis for validation of an evolving scalar damage model within SPH.

Postexplosion hydrodynamics of supernovae in red supergiants

Shock propagation, mixing, and clumping are studied in the explosion of red supergiants as Type II supernovae using a two-dimensional smooth particle hydrodynamic (SPH) code. We show that extensive Rayleigh-Talor instabilities develop in the ejecta in the wake of the reverse shock wave. In all cases, the shell structure of the progenitor is obliterated to leave a clumpy, well-mixed supernova remnant. However, the occurrence of mass loss during the lifetime of the progenitor can significantly reduce the amount of mixing. These results are independent of the Type II supernova explosion mechanism.

SELF-CONSISTENT MODELS OF BARRED SPIRAL GALAXIES

ABSTRACT Self-consistent models of barred spiral galaxies based on the observed properties of NGC3992, NGC1073, and NGC1398 are constructed and analyzed. The method of model construction is a slight modification of the technique developed by Contopoulos and Grosbol for the case of unbarred spirals. The main factors which influence self-consistency are the amplitude, pitch angle, scale length and z-thickness of the spirals, the mass of the bar, the angular velocity of the bar/spiral pattern, the central surface density and scale length of the disk, and the central value and slope of the velocity dispersion. Stochastic orbits whose Jacobi constants lie between the values at the Lagrange points L1 and L4 are found to play a significant role in supporting self-consistent spiral structure, especially in the regions just beyond the ends of the bar. Stochastic orbits whose Jacobi constants lie below this interval tend to fill more or less uniformly either rings in the outer disk or ovals in the bar region, depending on the regions to which they are confined. Stochastic orbits whose Jacobi constants lie above that of L4 also tend not to support any imposed structure. The model bars are predominantly comprised of elongated orbits trapped around the x1 family and terminate close to corotation. The response of gas to the forces of the most successful models is calculated using a two-dimensional smoothed particle hydrodynamics code. The results confirm that a bar alone is not sufficient to drive a strong spiral response in the gas of the outer disk. An underlying spiral structure in the more massive stellar component appears to be required. If stellar spirals are present, strong gas spirals may persist for long times.