Tuning the Coarse Space Construction in a Spectral AMG Solver
Abstract not provided
97 MATHEMATICS AND COMPUTING↗
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TA_solver -- Traffic Assignment solver package
TA_solver is a modular software framework for solving user equilibrium traffic assignment problems, both static and dynamic, utilizing the OTM (Open Traffic Models) API to encode traffic scenarios. The design is based on the formulation of the problem as a variational inequality. Unlike most existing traffic assignment software which focus on specific traffic models, TA_solver accommodates various traffic models, but also enables using parallel computation in high performance computing environments to speed up large-scale equilibrium calculations.
Ugirumurera, Juliette↗
Scalability of high-performance PDE solvers
Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.
97 MATHEMATICS AND COMPUTING↗
A Geometric Volume of Fluid-Based Multiphase Flow Solver Extension to the Reacting Flow Solver, PeleLM
A new algorithm is presented to simulate multiphase flows with surface tension in a pathway for spray combustion simulation. The algorithm combines capabilities from two open-source packages, including the interface reconstruction library (IRL), a library of computational geometry routines to enable the volume of fluid (VOF) method, and PeleLM, a solver for the reacting Navier-Stokes equations. Additionally, surface tension is implemented using the continuum surface force (CSF) model with an improved height function technique in the volume fraction field. Spurious errors in volume fraction arising from our combined strategy are corrected through a topology-based method that improves both numerical stability and accuracy. Multiple validation simulations are conducted, including (i) translations and rotations of Zalesak's disk, (ii) a stationary circular droplet with surface tension, (iii) an oscillating elliptical droplet, and (iv) three-dimensional deformation of a spherical droplet. Results indicate that the combined scheme retains the favorable properties of each of the component algorithms.
42 ENGINEERING↗
Stable large-scale solver for Ginzburg–Landau equations for superconductors
Understanding the interaction of vortices with inclusions in type-II superconductors is a major outstanding challenge both for fundamental science and energy applications. At application-relevant scales, the long-range interactions between a dense configuration of vortices and the dependence of their behavior on external parameters, such as temperature and an applied magnetic field, are all important to the net response of the superconductor. Capturing these features, in general, precludes analytical description of vortex dynamics and has also made numerical simulation prohibitively expensive. Here we report on a highly optimized iterative implicit solver for the time-dependent Ginzburg-Landau equations suitable for investigations of type-II superconductors on massively parallel architectures. Its main purpose is to study vortex dynamics in disordered or geometrically confined mesoscopic systems. In this work, we present the discretization and time integration scheme in detail for two types of boundary conditions. We describe the necessary conditions for a stable and physically accurate integration of the equations of motion. Using an inclusion pattern generator, we can simulate complex pinning landscapes and the effect of geometric confinement. We show that our algorithm, implemented on a GPU, can provide static and dynamic solutions of the Ginzburg-Landau equations for mesoscopically large systems over thousands of time steps in a matter of hours. Using our formulation, studying scientifically-relevant problems is a computationally reasonable task.
Sadovskyy, I. A.↗
Accurate derivative evaluation for any Grad–Shafranov solver
We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad–Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A scalable geometric multigrid solver for nonsymmetric elliptic systems with application to variable-density flows
A geometric multigrid algorithm is introduced for solving nonsymmetric linear systems resulting from the discretization of the variable density Navier–Stokes equations on nonuniform structured rectilinear grids and high-Reynolds number flows. The restriction operation is defined such that the resulting system on the coarser grids is symmetric, thereby allowing for the use of efficient smoother algorithms. To achieve an optimal rate of convergence, the sequence of interpolation and restriction operations are determined through a dynamic procedure. A parallel partitioning strategy is introduced to minimize communication while maintaining the load balance between all processors. To test the proposed algorithm, we consider two cases: 1) homogeneous isotropic turbulence discretized on uniform grids and 2) turbulent duct flow discretized on stretched grids. Testing the algorithm on systems with up to a billion unknowns shows that the cost varies linearly with the number of unknowns. This O (N) behavior confirms the robustness of the proposed multigrid method regarding ill-conditioning of large systems characteristic of multiscale high-Reynolds number turbulent flows. The robustness of our method to density variations is established by considering cases where density varies sharply in space by a factor of up to 104, showing its applicability to two-phase flow problems. Strong and weak scalability studies are carried out, employing up to 30,000 processors, to examine the parallel performance of our implementation. Excellent scalability of our solver is shown for a granularity as low as 104 to 105 unknowns per processor. At its tested peak throughput, it solves approximately 4 billion unknowns per second employing over 16,000 processors with a parallel efficiency higher than 50%.
Computer Science↗
Balancing conflicting requirements for grid and particle decomposition in continuum-Lagrangian solvers
Not Available
Sitaraman, Hariswaran↗
An AMR capable finite element diffusion solver for ALE hydrocodes [An AMR capable diffusion solver for ALE-AMR]
Here, we present a novel method for the solution of the diffusion equation on a composite AMR mesh. This approach is suitable for including diffusion based physics modules to hydrocodes that support ALE and AMR capabilities. To illustrate, we proffer our implementations of diffusion based radiation transport and heat conduction in a hydrocode called ALE-AMR. Numerical experiments conducted with the diffusion solver and associated physics packages yield 2nd order convergence in the L 2 norm.
42 ENGINEERING↗
Thermal stitching: Combining the advantages of different quantum fermion solvers
For quantum fermion problems, many accurate solvers are limited by the temperature regime in which they can be usefully applied. The Mermin theorem implies the uniqueness of an effective potential from which both the exact density and free energy at a target temperature can be found, via a calculation at a different, reference temperature. We derive exact expressions for both the potential and the free energy in such a calculation and introduce three controllable approximations that reduce the cost of such calculations. In conclusion, we illustrate the effective potential and its free energy, and test the approximations, on the asymmetric two-site Hubbard model at finite temperature.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
A Performance Model for Allocating the Parallelism in a Multigrid-in-Time Solver
The traditional way to numerically solve time-dependent problems is to sequentially march through time, solving for one time step and then the next. The parallelism in this approach is limited to the spatial dimension, which is quickly exhausted, causing the gain from using more processors to solve a problem to diminish. One approach to overcome this barrier is to use methods that are parallel in time. These methods have the potential to achieve dramatically better performance compared to time-stepping approaches, but achieving this performance requires carefully choosing the amount of parallelism devoted to space versus the amount devoted to time. Here, we present a performance model that, for a multigrid-in-time solver, makes the decision on when to switch to parallel-in-time and on how much parallelism to devote to space vs. time. In our experiments, the model selects the best parallel configuration in most of our test cases and a configuration close to the best one in all other cases.
97 MATHEMATICS AND COMPUTING↗
Steady-State Calibration of a Diesel Engine in CFD Using a GPU-Based Chemistry Solver
The prospect of analysis-driven pre-calibration of a modern diesel engine is extremely valuable in order to significantly reduce hardware investments and accelerate engine designs compliant with stricter EPA fuel economy regulations. Advanced modeling tools, such as CFD, are often used with the goal of streamlining significant portions of the calibration process. The success of the methodology largely relies on the accuracy of analytical predictions, especially engine-out emissions. However, the effectiveness of CFD simulation tools for in-cylinder engine combustion is often compromised by the complexity, accuracy, and computational overhead of detailed chemical kinetics necessary for combustion calculations. The standard approach has been to use skeletal kinetic mechanisms (~50 species) which consume acceptable computational time but with degraded accuracy.In this work, a comprehensive demonstration and validation of the analytical pre-calibration process is presented for a passenger car diesel engine using CFD simulations with CONVERGE™ and a GPU-based chemical kinetics solver (Zero-RK, developed at Lawrence Livermore National Laboratory) on high performance computing resources to enable the use of detailed kinetic mechanisms. Diesel engine combustion computations have been conducted over 600 operating points spanning in-vehicle speed-load map, using massively parallel ensemble simulation sets on the Titan supercomputer located at the Oak Ridge Leadership Computing Facility. The results with different mesh resolutions have been analyzed to compare differences in combustion and emissions (NO x , Carbon Monoxide CO, Unburned Hydrocarbons UHC, and Smoke) with actual engine measurements. The results show improved agreement in combustion and NO x predictions with a large n-heptane mechanism consisting of 144 species and 900 reactions with refined mesh resolution; however; agreement in CO, UHC and Smoke remain a challenge.
Gao, Jian↗
BeamDyn: A High-Fidelity Wind Turbine Blade Solver in the FAST Modular Framework
Explore the source record for details and available documents.
17 WIND ENERGY↗
Fourier Neural Networks as Function Approximators and Differential Equation Solvers
We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a straightforward architecture with a single hidden layer. The simplicity of this network architecture facilitates the integration with any other higher-complexity networks, at a data pre- or postprocessing stage. We validate this FNN on naturally periodic smooth functions and on piecewise continuous periodic functions. We showcase the use of this FNN for modeling or solving partial differential equations with periodic boundary conditions. The main advantages of the current approach are the validity of the solution outside the training region, interpretability of the trained model, and simplicity of use.
Fourier decomposition↗