Individual Time Steps Research Articles

An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems

SummaryWe propose the study of a posteriori error estimates for time‐dependent generalized finite element simulations of heat transfer problems. A residual estimate is shown to provide reliable and practically useful upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. Two sets of numerical experiments are presented. First, the error estimate is shown to capture the decrease in the error as the number of enrichment functions is increased or the time discretization refined. Second, the estimate is used to predict the behaviour of the error where no exact solution is available. It also reflects the errors incurred in the poorly conditioned systems typically encountered in generalized finite element methods. Finally, we study local error indicators in individual time steps and elements of the mesh. This creates a basis towards the adaptive selection and refinement of the enrichment functions. Copyright © 2016 John Wiley & Sons, Ltd.

Pervasive, yet idiosyncratic, epistatic pleiotropy during adaptation in a behaviourally complex microbe.

Understanding how multiple mutations interact to jointly impact multiple ecologically important traits is critical for creating a robust picture of organismal fitness and the process of adaptation. However, this is complicated by both environmental heterogeneity and the complexity of genotype-to-phenotype relationships generated by pleiotropy and epistasis. Moreover, little is known about how pleiotropic and epistatic relationships themselves change over evolutionary time. The soil bacterium Myxococcus xanthus employs several distinct social traits across a range of environments. Here, we use an experimental lineage of M.xanthus that evolved a novel form of social motility to address how interactions between epistasis and pleiotropy evolve. Specifically, we test how mutations accumulated during selection on soft agar pleiotropically affect several other social traits (hard agar motility, predation and spore production). Relationships between changes in swarming rate in the selective environment and the four other traits varied greatly over time in both direction and magnitude, both across timescales of the entire evolutionary lineage and individual evolutionary time steps. We also tested how a previously defined epistatic interaction is pleiotropically expressed across these traits. We found that phenotypic effects of this epistatic interaction were highly correlated between soft and hard agar motility, but were uncorrelated between soft agar motility and predation, and inversely correlated between soft agar motility and spore production. Our results show that 'epistatic pleiotropy' varied greatly in magnitude, and often even in sign, across traits and over time, highlighting the necessity of simultaneously considering the interacting complexities of pleiotropy and epistasis when studying the process of adaptation.

Enhancing weak transient signals in SEVIRI false color imagery: Application to dust source detection in southern Africa

A method is described to significantly enhance the signature of dust events using observations from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI). The approach involves the derivation of a composite clear-sky signal for selected channels on an individual time step and pixel basis. These composite signals are subtracted from each observation in the relevant channels to enhance weak transient signals associated with either (a) low levels of dust emission or (b) dust emissions with high salt or low quartz content. Different channel combinations, of the differenced data from the steps above, are then rendered in false color imagery for the purpose of improved identification of dust source locations and activity. We have applied this clear-sky difference (CSD) algorithm over three (globally significant) source regions in southern Africa: the Makgadikgadi Basin, Etosha Pan, and the Namibian and western South African coast. Case study analyses indicate three notable advantages associated with the CSD approach over established image rendering methods: (i) an improved ability to detect dust plumes, (ii) the observation of source activation earlier in the diurnal cycle, and (iii) an improved ability to resolve and pinpoint dust plume source locations.

Semi-implicit anisotropic cosmic ray transport on an unstructured moving mesh

In the interstellar medium of galaxies and the intracluster gas of galaxy clusters, the charged particles making up cosmic rays are moving almost exclusively along (but not across) magnetic field lines. The resulting anisotropic transport of cosmic rays in the form of diffusion or streaming not only affects the gas dynamics but also rearranges the magnetic fields themselves. The coupled dynamics of magnetic fields and cosmic rays can thus impact the formation and evolution of galaxies and the thermal evolution of galaxy clusters in critical ways. Numerically studying these effects requires solvers for anisotropic diffusion that are accurate, efficient, and robust, requirements that have proven difficult to satisfy in practice. Here, we present an anisotropic diffusion solver on an unstructured moving mesh that is conservative, does not violate the entropy condition, allows for semi-implicit time integration with individual timesteps, and only requires solving a single linear system of equations per timestep. We apply our new scheme to a large number of test problems and show that it works as well or better than previous implementations. Finally, we demonstrate for a numerically demanding simulation of the formation of an isolated disk galaxy that our local time-stepping scheme reproduces the results obtained with global time-stepping at a fraction of the computational cost.

Incorporating structural uncertainty of hydrological models in likelihood functions via an ensemble range approach

ABSTRACTModel ensembles are possibly the most powerful tool to assess uncertainties in runoff predictions stemming from inadequacies in model structure. But in many applications little knowledge is gained about the specific weaknesses of the individual models. Here we introduce the ensemble range approach (ERA). Compared to other ensemble techniques, ERA is primarily intended to facilitate hydrological reasoning about model structural uncertainty. This is attempted by separate modelling of data uncertainty and structural uncertainty with two different error density functions that are combined in one likelihood function. The width of the structural error density is in accordance with the range of runoff predictions calculated by a small model ensemble at each individual time step. Albeit not the only choice, this study is restricted on the use of a modified beta density to represent structural uncertainty. The performance of ERA is assessed in some synthetic and real data case studies. Ensembles of two structurally identical models are applied, made possible by estimating the parameters of ERA and both models simultaneously.Editor D. Koutsoyiannis; GUEST editor S. Weijs

Adaptively stepped SPH for fluid animation based on asynchronous time integration

We present a novel adaptive stepping scheme for SPH fluids, in which particles have their own time steps determined from local conditions, e.g. courant condition. These individual time steps are constrained for global convergence and stability. Fluid particles are then updated asynchronously. The approach naturally allocates computing resources to visually complex regions, e.g. regions with intense collisions, thereby reducing the overall computational time. The experiments show that our approach is more efficient than the standard method and the method with globally adaptive time steps, especially in highly dynamic scenes.

Hierarchical tree algorithm for collisional N-body simulations on GRAPE

Abstract We present an implementation of the hierarchical tree algorithm on the individual timestep algorithm (the Hermite scheme) for collisional N-body simulations, running on the GRAPE-9 system, a special-purpose hardware accelerator for gravitational many-body simulations. Such a combination of the tree algorithm and the individual timestep algorithm was not easy on the previous GRAPE system mainly because its memory addressing scheme was limited only to sequential access to a full set of particle data. The present GRAPE-9 system has an indirect memory addressing unit and a particle memory large enough to store all the particle data and also the tree node data. The indirect memory addressing unit stores interaction lists for the tree algorithm, which is constructed on the host computer, and, according to the interaction lists, force pipelines calculate only the interactions necessary. In our implementation, the interaction calculations are significantly reduced compared to direct N2 summation in the original Hermite scheme. For example, we can achieve about a factor 30 of speedup (equivalent to about 17 teraflops) against the Hermite scheme for a simulation of an N = 106 system, using hardware of a peak speed of 0.6 teraflops for the Hermite scheme.

Ambient Seismic Imaging Throughout the Unconventional Field’s Life Cycle

Technology Update Newly developed ambient seismic imaging methods provide valuable information throughout the life cycle of an unconventional field. Methods developed by Global Geophysical Services enable new insights for exploration, development planning, fracture and refracture design, reservoir management, production forecasting, and reserves estimation. Ambient seismic imaging has also been used to monitor conventional reservoirs, waterfloods, CO2 sequestration, mine stability, and water influx into mines. Additional applications include geothermal systems and induced seismicity monitoring. Many assume that the term “ambient seismic” (imaging artificial or natural seismic emissions) is synonymous with “microseismic” (imaging earthquakes with a magnitude less than 0, i.e., microearthquakes or MEQs). However, advances in the understanding of natural and induced seismic emissions and new imaging and analytical methods have differentiated these terms. Microseismic methods typically locate MEQs by use of classical seismological methods: polarization angles and arrival times of P and S waves. Thus, the term “microseismic” implies methods relying only on MEQs, the “dots-in-a-box” long familiar to fracturing engineers and development geologists. In recent years, research has shown that the majority of the seismic energy radiated during a fracture treatment is in seismic phenomena other than MEQs. Other emission sources include longperiod, long-duration (LPLD) activity (Das and Zoback 2013a and 2013b), long-duration signals (LDS) (Sicking et al. 2014), and fluid resonance in the fracture system (Tary and van der Baan 2012). These types of emissions cannot be analyzed with classical methods because they do not have distinct, noticeable first arrivals or recognizable P/S wave separation. Such emissions can persist or fade in and out for minutes to hours, rather than milliseconds, and therefore cannot be classified as distinct events such as MEQs. Furthermore, there may be other seismic phenomena that have yet to be identified and categorized. Seismic emission tomography (SET) is an alternative to locating earthquakes with P and S wave arrivals and polarization angles. It is in wide use for surface monitoring and most recently for downhole monitoring. SET focuses total trace energy by means of ray tracing through the velocity model in short time steps (on the order of 100 ms), thus imaging energy from many types of seismic emissions, including MEQs, LPLD, LDS, fluid resonance, and possibly unknown phenomena. Each computed SET time step produces a voxel-by-voxel depth image of seismic emissions in the same way that a 3D surface reflection seismic survey produces a 3D time image. MEQs located in individual time steps are termed “proxy hypocenters” because they represent brief, high-energy events. However, they are not identified uniquely as MEQs by classical methods.

Time-averaged droplet size distribution in steady-state dropwise condensation

This work focused on the time-averaged droplet size distribution in steady-state dropwise condensation. Through theoretical analysis, we derived an expression to describe the droplet size distribution, which is a function of the surface fraction, fractal dimension, and the maximum radius of droplets. Compared with the well-known semi-empirical model by Le Fevre and Rose (1966), the present model has no empirical constant and all parameters in the model have clearly physical meaning. Computer simulations were further conducted for different initial configuration of primary droplets, growth rate of an individual droplet and time step. It was found that the simulation results corresponded well with the available experimental data. The predictions from the semi-empirical model by Le Fevre and Rose and from the present model were also compared with the simulation results and the measured data. A good agreement between them implies that both the two models are valid to be used to describe the time-averaged droplet size distribution under various conditions used in experiments and simulations.

EUNHA: A NEW COSMOLOGICAL HYDRODYNAMIC SIMULATION CODE

We develop a parallel cosmological hydrodynamic simulation code designed for the study of formation and evolution of cosmological structures. The gravitational force is calculated using the TreePM method and the hydrodynamics is implemented based on the smoothed particle hydrodynamics. The initial displacement and velocity of simulation particles are calculated according to second-order Lagrangian perturbation theory using the power spectra of dark matter and baryonic matter. The initial background temperature is given by Recfast and the temperature fluctuations at the initial particle position are assigned according to the adiabatic model. We use a time-limiter scheme over the individual time steps to capture shock-fronts and to ease the time-step tension between the shock and preshock particles. We also include the astrophysical gas processes of radiative heating/cooling, star formation, metal enrichment, and supernova feedback. We test the code in several standard cases such as one-dimensional Riemann problems, Kelvin-Helmholtz, and Sedov blast wave instability. Star formation on the galactic disk is investigated to check whether the Schmidt-Kennicutt relation is properly recovered. We also study global star formation history at different simulation resolutions and compare them with observations.

Using dynamic Brownian bridge movement modelling to measure temporal patterns of habitat selection

Accurately describing animal space use is vital to understanding how wildlife use habitat. Improvements in GPS technology continue to facilitate collection of telemetry data at high spatial and temporal resolutions. Application of the recently introduced dynamic Brownian bridge movement model (dBBMM) to such data is promising as the method explicitly incorporates the behavioural heterogeneity of a movement path into the estimated utilization distribution (UD). Utilization distributions defining space use are normally estimated for time-scales ranging from weeks to months, obscuring much of the fine-scale information available from high-volume GPS data sets. By accounting for movement heterogeneity, the dBBMM provides a rigorous, behaviourally based estimate of space use between each set of relocations. Focusing on UDs generated between individual sets of locations allows us to quantify fine-scale circadian variation in habitat use. We used the dBBMM to estimate UDs bounding individual time steps for three terrestrial species with different life histories to illustrate how the method can be used to identify fine-scale variations in habitat use. We also demonstrate how dBBMMs can be used to characterize circadian patterns of habitat selection and link fine-scale patterns of habitat use to behaviour. We observed circadian patterns of habitat use that varied seasonally for a white-tailed deer (Odocoileus virginianus) and coyote (Canis latrans). We found seasonal patterns in selection by the white-tailed deer and were able to link use of conifer forests and agricultural fields to behavioural state of the coyote. Additionally, we were able to quantify the date in which a Rio Grande wild turkey (Meleagris gallopavo intermedia) initiated laying as well as when during the day, she was most likely to visit the nest site to deposit eggs. The ability to quantify circadian patterns of habitat use may have important implications for research and management of wildlife. Additionally, the ability to link such patterns to behaviour may aid in the development of mechanistic models of habitat selection.

Long-term socio-economic and spatial pattern drivers of land cover change in a Caribbean tropical moist forest, the Cockpit Country, Jamaica

Very little research has considered the underlying drivers of land cover change in Caribbean islands, particularly in those islands that are still experiencing a net loss of forest cover. We investigated the underlying driving forces (socio-economic drivers) and spatial pattern drivers (biophysical features) of both deforestation and reforestation in the Cockpit Country, Jamaica. This area is one of the most globally important sites for plant diversity, but is threatened by clearance for small-scale agriculture. Drivers of change were assessed for both the individual time steps within the study period (1942–2010) and for the entire 68 years using multivariate, spatially explicit, statistical models. The primary drivers of deforestation over the study period were accessibility (gentler slopes, closer to forest edges, more fragmented forests) and greater relative wealth/socio-economic status (increased access to piped water). Reforestation generally increased closer to forest edges and in areas with lower market access (greater distances to roads and towns) and lower wealth/status (increased reliance on pit latrines). We found considerable temporal variation among the most important drivers for each time step, including climate, employment status, population density, population age structure and relative wealth. Forest reserve status was not a key determinant of deforestation but did increase the probability of reforestation between 1961 and 1980. During the final time step (2001–2010) access was less important as a deterrent to deforestation, which increased within the most contiguous forest blocks. If the deforestation drivers of the last decade do not change, deforestation is predicted to occur within the forest reserves, and in the largest, least fragmented forest blocks. Thus, conservation and management strategies for our study site must seek to address issues related to both enforcement and the socio-economic factors that influence deforestation and habitat fragmentation.

Comment on ‘ShangS. 2012. Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step. Hydrological Processes26: 3338–3343. DOI: 10.1002/hyp.8405’

AbstractA previous study analyzed errors in the numerical calculation of actual crop evapotranspiration (ETa) under soil water stress. Assuming no irrigation or precipitation, it constructed equations for ETa over limited soil‐water ranges in a root zone drying out due to evapotranspiration. It then used a single crop‐soil composite to provide recommendations about the appropriate usage of numerical methods under different values of the time step and the maximum crop evapotranspiration (ETc). This comment reformulates those ETa equations for applicability over the full range of soil water values, revealing a dependence of the relative error in numerical ETa on the initial soil water that was not seen in the previous study. It is shown that the recommendations based on a single crop‐soil composite can be invalid for other crop‐soil composites. Finally, a consideration of the numerical error in the time‐cumulative value of ETa is discussed besides the existing consideration of that error over individual time steps as done in the previous study. This cumulative ETa is more relevant to the final crop yield. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.

Employing Per-Component Time Step in DSMC Simulations of Disparate Mass and Cross-Section Gas Mixtures

AbstractA new approach to simulation of stationary flows by Direct Simulation Monte Carlo method is proposed. The idea is to specify an individual time step for each component of a gas mixture. The approach consists of modifications mainly to collision phase simulation and recommendations on choosing time step ratios. It allows lowering the demands on the computational resources for cases of disparate collision diameters of molecules and/or disparate molecular masses. These are cases important e.g., in vacuum deposition technologies. Few tests of the new approach are made. Finally, the usage of new approach is demonstrated on a problem of silver nanocluster diffusion in argon carrier gas under conditions of silver deposition experiments.

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double-sided node-to-surface contact or self-contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time-stepping scheme. The time step may be fixed or time-adaptive. New demands on global collision detection are discussed exemplified by position codes and node-to-segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.

Asynchronous variational integration using continuous assumed gradient elements

Asynchronous variational integration (AVI) is a tool which improves the numerical efficiency of explicit time stepping schemes when applied to finite element meshes with local spatial refinement. This is achieved by associating an individual time step length to each spatial domain. Furthermore, long-term stability is ensured by its variational structure. This article presents AVI in the context of finite elements based on a weakened weak form (W2) Liu (2009) [1], exemplified by continuous assumed gradient elements Wolff and Bucher (2011) [2]. The article presents the main ideas of the modified AVI, gives implementation notes and a recipe for estimating the critical time step.

Phantom-GRAPE: Numerical software library to accelerate collisionless N-body simulation with SIMD instruction set on x86 architecture

We have developed a numerical software library for collisionless N-body simulations named “Phantom-GRAPE” which highly accelerates force calculations among particles by use of a new SIMD instruction set extension to the x86 architecture, Advanced Vector eXtensions (AVX), an enhanced version of the Streaming SIMD Extensions (SSE). In our library, not only the Newton’s forces, but also central forces with an arbitrary shape f(r), which has a finite cutoff radius rcut (i.e. f(r)=0 at r>rcut), can be quickly computed. In computing such central forces with an arbitrary force shape f(r), we refer to a pre-calculated look-up table. We also present a new scheme to create the look-up table whose binning is optimal to keep good accuracy in computing forces and whose size is small enough to avoid cache misses. Using an Intel Core i7–2600 processor, we measure the performance of our library for both of the Newton’s forces and the arbitrarily shaped central forces. In the case of Newton’s forces, we achieve 2×109 interactions per second with one processor core (or 75 GFLOPS if we count 38 operations per interaction), which is 20 times higher than the performance of an implementation without any explicit use of SIMD instructions, and 2 times than that with the SSE instructions. With four processor cores, we obtain the performance of 8×109 interactions per second (or 300 GFLOPS). In the case of the arbitrarily shaped central forces, we can calculate 1×109 and 4×109 interactions per second with one and four processor cores, respectively. The performance with one processor core is 6 times and 2 times higher than those of the implementations without any use of SIMD instructions and with the SSE instructions. These performances depend only weakly on the number of particles, irrespective of the force shape. It is good contrast with the fact that the performance of force calculations accelerated by graphics processing units (GPUs) depends strongly on the number of particles. Substantially weak dependence of the performance on the number of particles is suitable to collisionless N-body simulations, since these simulations are usually performed with sophisticated N-body solvers such as Tree- and TreePM-methods combined with an individual timestep scheme. We conclude that collisionless N-body simulations accelerated with our library have significant advantage over those accelerated by GPUs, especially on massively parallel environments.

Accelerating nbody6 with graphics processing units

We describe the use of Graphics Processing Units (GPUs) for speeding up the code NBODY6 which is widely used for direct $N$-body simulations. Over the years, the $N^2$ nature of the direct force calculation has proved a barrier for extending the particle number. Following an early introduction of force polynomials and individual time-steps, the calculation cost was first reduced by the introduction of a neighbour scheme. After a decade of GRAPE computers which speeded up the force calculation further, we are now in the era of GPUs where relatively small hardware systems are highly cost-effective. A significant gain in efficiency is achieved by employing the GPU to obtain the so-called regular force which typically involves some 99 percent of the particles, while the remaining local forces are evaluated on the host. However, the latter operation is performed up to 20 times more frequently and may still account for a significant cost. This effort is reduced by parallel SSE/AVX procedures where each interaction term is calculated using mainly single precision. We also discuss further strategies connected with coordinate and velocity prediction required by the integration scheme. This leaves hard binaries and multiple close encounters which are treated by several regularization methods. The present nbody6-GPU code is well balanced for simulations in the particle range $10^4-2 \times 10^5$ for a dual GPU system attached to a standard PC.

N-body integrators with individual time steps from Hierarchical splitting

We review the implementation of individual particle time-stepping for N-body dynamics. We present a class of integrators derived from second order Hamiltonian splitting. In contrast to the usual implementation of individual time-stepping, these integrators are momentum conserving and show excellent energy conservation in conjunction with a symmetrized time step criterion. We use an explicit but approximate formula for the time symmetrization that is compatible with the use of individual time steps. No iterative scheme is necessary. We implement these ideas in the HUAYNO1Available online at www.amusecode.org.1 code and present tests of the integrators and show that the presented integration schemes shows good energy conservation, with little or no systematic drift, while conserving momentum and angular momentum to machine precision for long term integrations.

PSDF: Particle Stream Data Format for N-body simulations

We present a data format for the output of general N-body simulations, allowing the presence of individual time steps. By specifying a standard, different N-body integrators and different visualization and analysis programs can all share the simulation data, independent of the type of programs used to produce the data. Our Particle Stream Data Format, PSDF, is specified in YAML, based on the same approach as XML but with a simpler syntax. Together with a specification of PSDF, we provide background and motivation, as well as specific examples in a variety of computer languages. We also offer a web site from which these examples can be retrieved, in order to make it easy to augment existing codes in order to give them the option to produce PSDF output.