Results for “Iterative solvers” — Page 30

DOE OSTI2015

BiGlobal Secondary Instabilities of Roughness-Induced Transient Growth

Roughness-induced transient disturbances distort laminar boundary layers and can lead to premature transition if the roughness is of sufficient height and shape. If transition does occur, it does so through a high-frequency "secondary" instability of the roughness wake. A BiGlobal secondaryinstability code has been developed to solve the stability problem downstream of roughness. The code implements multiple solvers for incompressible/compressible, inviscid/viscous, and temporal/spatial stability. High-order finite differences, sparse matrix linear algebra, LU decomposition, and Arnoldi iterations are used to solve the resulting generalized eigenvalue problem. The code has been used to evaluate the stability properties of experimental roughness-induced transient growth on a flat-plate at low-speed and a slightly blunted cone at M∞= 6. The resulting mode shapes, growth rates, and spatial N-factors show excellent agreement with experimental data. The viscous mode shapes show the best agreement with the experimental disturbance profiles. Yet, execution time can be reduced by an order-of-magnitude by using the inviscid solvers with only modest differences in growth rates.

BiGlobal↗

DOE OSTI2019

Sparse, iterative simulation methods for one-dimensional laminar flames

Sparse, iterative simulation methods for one-dimensional laminar flames are proposed. The resulting steady and unsteady flame solvers exploit approximate Jacobians to greatly reduce the computational cost associated with matrix operations. The constant, non-unity Lewis number assumption is introduced to further reduce the computational cost. The solvers are also parallelized to reduce time-to-solution on distributed memory computer systems. Computed laminar flame speeds and species profiles for a range of chemical mechanisms (from 10 to 2878 species) are compared against a well-validated commercial code and are found to be consistent within solver tolerances. The computation times of both the unsteady and steady solutions increase only linearly with the number of species, which is a significant improvement over the quadratic or cubic scaling of existing steady-state flame solvers. For the largest mechanism tested, the steady-state flame solver is two orders of magnitude faster than commonly-used codes. The use of an approximate Jacobian is shown to reduce the rate of convergence for the steady-state solver, but does not significantly affect the domain of convergence. The steady-state solver with approximate Jacobian is thus well suited for computationally efficient laminar flame speed sweeps with large kinetic mechanisms.

42 ENGINEERING↗

DOE OSTI2013

Hierarchical Krylov and nested Krylov methods for extreme-scale computing

The solution of large, sparse linear systems is typically a dominant phase of computation for simulations based on partial differential equations, which are ubiquitous in scientific and engineering applications. While preconditioned Krylov methods are widely used and provide many advantages for solving sparse linear systems that do not have highly convergent, geometric multigrid solvers or specialized fast solvers, Krylov methods encounter well-known scaling difficulties for over 10,000 processor cores because each iteration requires at least one vector inner product, which in turn requires a global synchronization that scales poorly because of internode latency. To aid in overcoming these difficulties, we have developed hierarchical Krylov methods and nested Krylov methods in the PETSc library that reduce the number of global inner products required across the entire system (where they are expensive), though freely allow vector inner products across smaller subsets of the entire system (where they are inexpensive) or use inner iterations that do not invoke vector inner products at all. Nested Krylov methods are a generalization of inner-outer iterative methods with two or more layers. Hierarchical Krylov methods are a generalization of block Jacobi and overlapping additive Schwarz methods, where each block itself is solved by Krylov methods on smaller blocks. Conceptually, the hierarchy can continue recursively to an arbitrary number of levels of smaller and smaller blocks. As a specific case, we introduce the hierarchical FGMRES method, or h-FGMRES, and we demonstrate the impact of two-level h-FGMRES with a variable preconditioner on the PFLOTRAN subsurface flow application. We also demonstrate the impact of nested FGMRES, BiCGStab and Chebyshev methods. These hierarchical Krylov methods and nested Krylov methods significantly reduced overall PFLOTRAN simulation time on the Cray XK6 when using 10,000 through 224,000 cores through the combined effects of reduced global synchronization due to fewer global inner products and stronger inner hierarchical or nested preconditioners.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nosetip, and the mechanisms underlying the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nosetip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the perturbation form of the 2D, harmonic Navier-Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton-Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the GMRES iterative method with a preconditioner corresponding to a simplified Jacobian without the cross derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation (NPSE) results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over the blunt cone. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher frequency Mack’s second-mode instabilities further downstream.

boundary layer transition↗

NASA NTRSDate not supplied

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nosetip, and the mechanisms underlying the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification for moderate to large nosetip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the perturbation form of the 2D, harmonic Navier-Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton-Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the GMRES iterative method with a preconditioner corresponding to a simplified Jacobian without the cross derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation (NPSE) results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over the blunt cone. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher frequency Mack’s second-mode instabilities further downstream.

boundary layer transition↗

DOE OSTI2022

A low-rank power iteration scheme for neutron transport criticality problems

Computing effective eigenvalues for neutron transport often requires a fine numerical resolution. Here, the main challenge of such computations is the high memory effort of classical solvers, which limits the accuracy of chosen discretizations. In this work, we derive a method for the computation of effective eigenvalues when the underlying solution has a low-rank structure. This is accomplished by utilizing dynamical low-rank approximation (DLRA), which is an efficient strategy to derive time evolution equations for low-rank solution representations. The main idea is to interpret the iterates of the classical inverse power iteration as pseudo-time steps and apply the DLRA concepts in this framework. In our numerical experiment, we demonstrate that our method significantly reduces memory requirements while achieving the desired accuracy. Analytic investigations show that the proposed iteration scheme inherits the convergence speed of the inverse power iteration, at least for a simplified setting.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Full time-dependent SOLPS-ITER simulation of the SPARC tokamak: actuator design for particle and divertor condition control *

Abstract This paper presents the application of full time-dependent SOLPS-ITER simulations for actuator design in the SPARC tokamak. This study employs both the EIRENE module, a neutral solver, and the B2.5 plasma module in a time-dependent mode. This is in contrast to most SOLPS simulations, which focus on steady-state solutions, where the neutral distribution is evolved without any time limit or for a time step of 1 ⋅ 10 − 3 second, which is several orders of magnitude larger than the fluid plasma time step. The time-dependent EIRENE was tested with a fixed B2.5 background and compared with a simple conductance based model in a simplified pump chamber geometry. This comparison aimed to verify the reliability of the neutral relaxation timescale derived from the time-dependent EIRENE. Subsequently, a full time-dependent simulation was performed in a realistic geometry, with the Monte-Carlo neutral time step synchronized with the plasma fluid time step. The numerical setup of the code, including relative time steps and the size of the census data used to store Monte-Carlo particle information is considered. The full-time dependent simulations are then applied to inform the design of the SPARC louver structure, which affects divertor plasma parameters by regulating the neutral conductance from the divertor to the pump. The response of the plasma and neutral parameters was captured on a timescale that enables the design of the actuator to consider time-dependent control capability. It was found that changing the louver opacity has an equivalent effect as varying the gas throughput via puff actuation. Therefore, equivalent divertor plasma conditions can be obtained from both actuators, while the neutral pressure distribution in the pump and divertor differs for each actuator.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2016

Pin-by-Pin Core Calculation with an NEM-Based Two-Level Hybrid CMFD Algorithm

In the near future, systematic pin-by-pin core calculations are expected to provide a high level of accuracy. These are being developed in pursuit of high-fidelity tools for analyzing the reactor configurations. One such tool that has recently gained significant attention is the pin-by-pin core calculation via local-global iteration using nonlinear acceleration scheme, such as the One-Node CMFD (Coarse-Mesh Finite Difference) nonlinear algorithm. The One-Node CMFD method uses two correction factors to preserve the interface current, in contrast to the one correction term employed by conventional nonlinear iterative methods such as that proposed by Kord. S. Smith. These correction factors are updated with surface flux and net current which are obtained from a high order local problem. With two correction factors, One-Node CMFD can reduce computing time of local FMFD (Fine-Mesh Finite Difference) as it nonlinearly couples the CMFD and FMFD methods in global-local iterations. Nonetheless, very small mesh size is still required and it consequently requires long computing time for high accuracy. As such, a more flexible approach is needed to address this challenge. This paper presents one such possible approach named the hCMFD (Hybrid CMFD) method. In this method, One-Node CMFD is used to solve the global problem, while Two-Node NEM (Nodal Expansion Method) CMFD replaces FMFD as the solver for the local problem. In this paper, two kinds of CMFD methods has been simultaneously implemented for a global-local nonlinear iteration in order to do pin-by-pin heterogeneous reactor calculations. The global problem is solved by a one-node CMFD which uses two correction factors, while the local fixed-source problem is solved by the conventional two-node NEM CMFD which use one correction factor on a surface. From solution of the global eigenvalue problem, incoming partial current is obtained and it is used as boundary condition of the local fixed-source problems. It has been shown that over 99% of the whole calculations can be done in parallel with the new CMFD method for 2-D problems and the algorithm can be easily parallelized by using the OpenMP parallel scheme. The hCMFD method will enable an efficient pin-by-pin reactor analysis since a very efficient parallel calculation is possible with the hCMFD method. In future works, the hCMFD method will be extended to three-dimensional reactor problem. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1986

A new approach for solving the three-dimensional steady Euler equations. I - General theory

The present iterative procedure combines the Clebsch potentials and the Munk-Prim (1947) substitution principle with an extension of a semidirect Cauchy-Riemann solver to three dimensions, in order to solve steady, inviscid three-dimensional rotational flow problems in either subsonic or incompressible flow regimes. This solution procedure can be used, upon discretization, to obtain inviscid subsonic flow solutions in a 180-deg turning channel. In addition to accurately predicting the behavior of weak secondary flows, the algorithm can generate solutions for strong secondary flows and will yield acceptable flow solutions after only 10-20 outer loop iterations.

Chang, S.-C.↗

DOE OSTI2014

A Flexible CUDA LU-based Solver for Small, Batched Linear Systems

This chapter presents the implementation of a batched CUDA solver based on LU factorization for small linear systems. This solver may be used in applications such as reactive flow transport models, which apply the Newton-Raphson technique to linearize and iteratively solve the sets of non linear equations that represent the reactions for ten of thousands to millions of physical locations. The implementation exploits somewhat counterintuitive GPGPU programming techniques: it assigns the solution of a matrix (representing a system) to a single CUDA thread, does not exploit shared memory and employs dynamic memory allocation on the GPUs. These techniques enable our implementation to simultaneously solve sets of systems with over 100 equations and to employ LU decomposition with complete pivoting, providing the higher numerical accuracy required by certain applications. Other currently available solutions for batched linear solvers are limited by size and only support partial pivoting, although they may result faster in certain conditions. We discuss the code of our implementation and present a comparison with the other implementations, discussing the various tradeoffs in terms of performance and flexibility. This work will enable developers that need batched linear solvers to choose whichever implementation is more appropriate to the features and the requirements of their applications, and even to implement dynamic switching approaches that can choose the best implementation depending on the input data.

bached linear solvers↗

DOE OSTI2021

Beyond Taylor’s hypothesis: a novel volumetric reconstruction of velocity and density fields for variable‑density and shear flows

Abstract This work presents a novel numerical procedure for reconstructing volumetric density and velocity fields from planar laser-induced fluorescence (PLIF) and stereoscopic particle image velocimetry (SPIV) data. This new method is theoretically and practically demonstrated to provide more accurate 3D vortical structures and density fields in high shear flows than reconstruction methods based on the mean convective velocity. While Taylor’s hypothesis of frozen turbulence is commonly applied by using the local mean streamwise velocity, the proposed algorithm uses the measured local instantaneous velocity for data convection. It consists of a step-by-step reconstruction based on a mixed Lagrangian–Eulerian solver that includes the 3D interpolation of scattered flow data and that relaxes the Taylor’s hypothesis by iterative enforcement of the incompressibility constraint on the velocity field. This methodology provides 3D fields with temporal resolution, spatial resolution, and accuracy comparable to that of real 3D snapshots, thus providing a practical alternative to tomographic measurements. The procedure is validated using numerical data of the constant-density channel flow available on the Johns Hopkins University Turbulence Database (JHTDB), showing the accurate reconstruction of the 3D velocity field. The algorithm is applied to an experimental dataset of PLIF and SPIV measurements of a variable-density jet flow, demonstrating its capability to provide 3D velocity and density fields that are more consistent with the Navier–Stokes equations compared to the mean flow convective method. Graphic abstract

42 ENGINEERING↗

DOE OSTI2017

HFEM3D

Software solves the three-dimensional Poisson equation div(k(grad(u)) = f, by the finite element method for the case when material properties, k, are distributed over hierarchy of edges, facets and tetrahedra in the finite element mesh. Method is described in Weiss, CJ, Finite element analysis for model parameters distributed on a hierarchy of geometric simplices, Geophysics, v82, E155-167, doi:10.1190/GEO2017-0058.1 (2017). A standard finite element method for solving Poisson’s equation is augmented by including in the 3D stiffness matrix additional 2D and 1D stiffness matrices representing the contributions from material properties associated with mesh faces and edges, respectively. The resulting linear system is solved iteratively using the conjugate gradient method with Jacobi preconditioning. To minimize computer storage for program execution, the linear solver computes matrix-vector contractions element-by-element over the mesh, without explicit storage of the global stiffness matrix. Program output vtk compliant for visualization and rendering by 3rd party software. Program uses dynamic memory allocation and as such there are no hard limits on problem size outside of those imposed by the operating system and configuration on which the software is run. Dimension, N, of the finite element solution vector is constrained by the the addressable space in 32-vs-64 bit operating systems. Total storage requirements for the problem. Total working space required for the program is approximately 13*N double precision words.

Weiss, Chester J↗

NASA NTRSDate not supplied

Edge-Based Viscous Method for Mixed-Element Node-Centered Finite-Volume Solvers

A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turbulence-model and chemistry-model diffusion terms, and the corresponding Jacobian contributions. Initially, the EBV method had been implemented for tetrahedral grids and demonstrated multifold acceleration of all viscous-kernel computations. This paper presents an extension of the EBV method for mixed-element grids. In addition to the primal edges of a given mixed-element grid, virtual edges are introduced to connect cell nodes that are not connected by a primal edge. The EBV method uses an efficient loop over all (primal and virtual) edges and features a compact discretization stencil based on the nearest neighbors. This study verifies the EBV method and assesses its efficiency on mixed-element grids by comparing the EBV solution accuracy and iterative convergence with those of well-established solutions obtained using a cell-based viscous (CBV) discretization method. The EBV solver’s memory footprint is optimized and often smaller than the memory footprint of the CBV solver. A multifold speedup is demonstrated for all viscous-kernel computations resulting in significant reduction of the time to solutions for several benchmark mixed-element-grid computations, including simulations of a flow around NASA’s juncture-flow model and a hypersonic, chemically reacting flow around a blunt body.

CFD↗

NASA NTRSDate not supplied

Edge-Based Viscous Method for Mixed-Element Node-Centered Finite-Volume Solvers

A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turbulence-model and chemistry-model diffusion terms, and the corresponding Jacobian contributions. Initially, the EBV method had been implemented for tetrahedral grids and demonstrated multifold acceleration of all viscous-kernel computations. This paper presents an extension of the EBV method for mixed-element grids. In addition to the primal edges of a given mixed-element grid, virtual edges are introduced to connect cell nodes that are not connected by a primal edge. The EBV method uses an efficient loop over all (primal and virtual) edges and features a compact discretization stencil based on the nearest neighbors. This study verifies the EBV method and assesses its efficiency on mixed-element grids by comparing the EBV solution accuracy and iterative convergence with those of well-established solutions obtained using a cell-based viscous (CBV) discretization method. The EBV solver’s memory footprint is optimized and often smaller than the memory footprint of the CBV solver. A multifold speedup is demonstrated for all viscous-kernel computations resulting in significant reduction of the time to solutions for several benchmark mixed-element-grid computations, including simulations of a flow around NASA’s juncture-flow model and a hypersonic, chemically reacting flow around a blunt body.

Edge-based viscous method↗

DOE OSTI2018

A non-intrusive parallel-in-time approach for simultaneous optimization with unsteady PDEs

This paper presents a non-intrusive framework for integrating existing unsteady partial differential equation (PDE) solvers into a parallel-in-time simultaneous optimization algorithm. The time-parallelization is provided by the non-intrusive software library XBraid [Parallel multigrid in time, software available at http://llnl.gov/casc/xbraid], which applies an iterative multigrid reduction technique to the time domain of existing time-marching schemes for solving unsteady PDEs. Its general user-interface has been extended in [S. Günther, N.R. Gauger, and J.B. Schroder, A non-intrusive parallel-in-time adjoint solver with the XBraid library, Comput. Vis. Sci. 19 (2018), pp. 85–95. Available at arXiv, math.OC/1705.00663] for computing adjoint sensitivities such that gradients of output quantities with respect to design changes can be computed parallel-in-time alongside with the primal PDE solution. In this paper, the primal and adjoint XBraid iterations are embedded into a simultaneous optimization framework, namely the One-shot method. In this method, design updates towards optimality are employed after each state and adjoint update such that optimality and feasibility of the design and the PDE solution are reached simultaneously. Finally, the time-parallel optimization method is validated on an advection-dominated flow control problem which shows significant speedup over a classical time-serial optimization algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Coupled Modeling of Hydrodynamics and Sound in Coastal Ocean for Renewable Ocean Energy Development

An underwater sound model was developed to simulate sound propagation from marine and hydrokinetic energy (MHK) devices or offshore wind (OSW) energy platforms. Finite difference methods were developed to solve the 3D Helmholtz equation for sound propagation in the coastal environment. A 3D sparse matrix solver with complex coefficients was formed for solving the resulting acoustic pressure field. The Complex Shifted Laplacian Preconditioner (CSLP) method was applied to solve the matrix system iteratively with MPI parallelization using a high performance cluster. The sound model was then coupled with the Finite Volume Community Ocean Model (FVCOM) for simulating sound propagation generated by human activities, such as construction of OSW turbines or tidal stream turbine operations, in a range-dependent setting. As a proof of concept, initial validation of the solver is presented for two coastal wedge problems. This sound model can be useful for evaluating impacts on marine mammals due to deployment of MHK devices and OSW energy platforms.

Long, Wen↗

DOE OSTI2015

The Development of a Finite Volume Method for Modeling Sound in Coastal Ocean Environment

: As the rapid growth of marine renewable energy and off-shore wind energy, there have been concerns that the noises generated from construction and operation of the devices may interfere marine animals’ communication. In this research, a underwater sound model is developed to simulate sound prorogation generated by marine-hydrokinetic energy (MHK) devices or offshore wind (OSW) energy platforms. Finite volume and finite difference methods are developed to solve the 3D Helmholtz equation of sound propagation in the coastal environment. For finite volume method, the grid system consists of triangular grids in horizontal plane and sigma-layers in vertical dimension. A 3D sparse matrix solver with complex coefficients is formed for solving the resulting acoustic pressure field. The Complex Shifted Laplacian Preconditioner (CSLP) method is applied to efficiently solve the matrix system iteratively with MPI parallelization using a high performance cluster. The sound model is then coupled with the Finite Volume Community Ocean Model (FVCOM) for simulating sound propagation generated by human activities in a range-dependent setting, such as offshore wind energy platform constructions and tidal stream turbines. As a proof of concept, initial validation of the finite difference solver is presented for two coastal wedge problems. Validation of finite volume method will be reported separately.

Coastal↗

DOE OSTI2019

Low-order multiphysics coupling techniques for nuclear reactor applications

The accurate modeling and simulation of nuclear reactor designs depends greatly on the ability to couple differing sets of physics together. Current coupling techniques most often use a fixed-point, or Picard, iteration scheme in which each set of physics is solved separately, and the resulting solutions are passed between each solver. In the work presented here, two different coupling techniques are investigated: a Jacobian-Free Newton-Krylov (JFNK) approach and a new methodology called Coarse Mesh Finite Difference Coupling (CMFD-Coupling). In this work, both techniques were applied to the low-order CMFD system of equations. This allows for the multiphysics feedback effects to be captured on the low-order system without having to perform a neutron transport solve. The JFNK and CMFD-Coupling approaches were implemented in the MPACT neutron transport code, which is being developed for the Consortium for Advanced Simulation of Light Water Reactors (CASL). These methods were tested on a wide range of practical reactor physics problems, from a single 3D fuel pin to a massively parallel 3D full core problem. When coupled neutronics-thermal hydraulics problems were investigated with both JFNK and CMFD-Coupling, it was concluded that CMFD-Coupling outperformed JFNK in terms of both accuracy and runtime for every problem. When applied to large full core problems with multiple sources of strong feedback enabled, CMFD-Coupling reduced the overall number of transport sweeps required for convergence.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗