SEARCH · Search NASA

Results for “nested iteration”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

15 records

All sources NASA NTRS DOE OSTI

NASA NTRS2009

Perl Modules for Constructing Iterators

The Iterator Perl Module provides a general-purpose framework for constructing iterator objects within Perl, and a standard API for interacting with those objects. Iterators are an object-oriented design pattern where a description of a series of values is used in a constructor. Subsequent queries can request values in that series. These Perl modules build on the standard Iterator framework and provide iterators for some other types of values. Iterator::DateTime constructs iterators from DateTime objects or Date::Parse descriptions and ICal/RFC 2445 style re-currence descriptions. It supports a variety of input parameters, including a start to the sequence, an end to the sequence, an Ical/RFC 2445 recurrence describing the frequency of the values in the series, and a format description that can refine the presentation manner of the DateTime. Iterator::String constructs iterators from string representations. This module is useful in contexts where the API consists of supplying a string and getting back an iterator where the specific iteration desired is opaque to the caller. It is of particular value to the Iterator::Hash module which provides nested iterations. Iterator::Hash constructs iterators from Perl hashes that can include multiple iterators. The constructed iterators will return all the permutations of the iterations of the hash by nested iteration of embedded iterators. A hash simply includes a set of keys mapped to values. It is a very common data structure used throughout Perl programming. The Iterator:: Hash module allows a hash to include strings defining iterators (parsed and dispatched with Iterator::String) that are used to construct an overall series of hash values.

Tilmes, Curt↗

NASA NTRS1995

Development of Fast Algorithms Using Recursion, Nesting and Iterations for Computational Electromagnetics

In the first phase of our work, we have concentrated on laying the foundation to develop fast algorithms, including the use of recursive structure like the recursive aggregate interaction matrix algorithm (RAIMA), the nested equivalence principle algorithm (NEPAL), the ray-propagation fast multipole algorithm (RPFMA), and the multi-level fast multipole algorithm (MLFMA). We have also investigated the use of curvilinear patches to build a basic method of moments code where these acceleration techniques can be used later. In the second phase, which is mainly reported on here, we have concentrated on implementing three-dimensional NEPAL on a massively parallel machine, the Connection Machine CM-5, and have been able to obtain some 3D scattering results. In order to understand the parallelization of codes on the Connection Machine, we have also studied the parallelization of 3D finite-difference time-domain (FDTD) code with PML material absorbing boundary condition (ABC). We found that simple algorithms like the FDTD with material ABC can be parallelized very well allowing us to solve within a minute a problem of over a million nodes. In addition, we have studied the use of the fast multipole method and the ray-propagation fast multipole algorithm to expedite matrix-vector multiplication in a conjugate-gradient solution to integral equations of scattering. We find that these methods are faster than LU decomposition for one incident angle, but are slower than LU decomposition when many incident angles are needed as in the monostatic RCS calculations.

Chew, W. C.↗

NASA NTRS1996

A Note on Multigrid Theory for Non-nested Grids and/or Quadrature

We provide a unified theory for multilevel and multigrid methods when the usual assumptions are not present. For example, we do not assume that the solution spaces or the grids are nested. Further, we do not assume that there is an algebraic relationship between the linear algebra problems on different levels. What we provide is a computationally useful theory for adaptively changing levels. Theory is provided for multilevel correction schemes, nested iteration schemes, and one way (i.e., coarse to fine grid with no correction iterations) schemes. We include examples showing the applicability of this theory: finite element examples using quadrature in the matrix assembly and finite volume examples with non-nested grids. Our theory applies directly to other discretizations as well.

Douglas, C. C.↗

NASA NTRS1986

Multigrid method for a vortex breakdown simulation

An inviscid model for a steady axisymmetric flow with swirl was studied. The governing equation is a nonlinear elliptic equation which has more than one solution for a certain range of the swirl parameter. The physically interesting solutions have closed streamlines that look like vortex breakdown (bubble-like solutions). A multigrid method is used to find these solutions. Using an FMG algorithm (nested iteration), the problem is solved in just a few multigrid cycles.

Taasan, S.↗

NASA NTRS2000

Interference Lattice-based Loop Nest Tilings for Stencil Computations

A common method for improving performance of stencil operations on structured multi-dimensional discretization grids is loop tiling. Tile shapes and sizes are usually determined heuristically, based on the size of the primary data cache. We provide a lower bound on the numbers of cache misses that must be incurred by any tiling, and a close achievable bound using a particular tiling based on the grid interference lattice. The latter tiling is used to derive highly efficient loop orderings. The total number of cache misses of a code is the sum of (necessary) cold misses and misses caused by elements being dropped from the cache between successive loads (replacement misses). Maximizing temporal locality is equivalent to minimizing replacement misses. Temporal locality of loop nests implementing stencil operations is optimized by tilings that avoid data conflicts. We divide the loop nest iteration space into conflict-free tiles, derived from the cache miss equation. The tiling involves the definition of the grid interference lattice an equivalence class of grid points whose images in main memory map to the same location in the cache-and the construction of a special basis for the lattice. Conflicts only occur on the boundaries of the tiles, unless the tiles are too thin. We show that the surface area of the tiles is bounded for grids of any dimensionality, and for caches of any associativity, provided the eccentricity of the fundamental parallelepiped (the tile spanned by the basis) of the lattice is bounded. Eccentricity is determined by two factors, aspect ratio and skewness. The aspect ratio of the parallelepiped can be bounded by appropriate array padding. The skewness can be bounded by the choice of a proper basis. Combining these two strategies ensures that pathologically thin tiles are avoided. They do not, however, minimize replacement misses per se. The reason is that tile visitation order influences the number of data conflicts on the tile boundaries. If two adjacent tiles are visited successively, there will be no replacement misses on the shared boundary. The iteration space may be covered with pencils larger than the size of the cache while avoiding data conflicts if the pencils are traversed by a scanning-face method. Replacement misses are incurred only on the boundaries of the pencils, and the number of misses is minimized by maximizing the volume of the scanning face, not the volume of the tile. We present an algorithm for constructing the most efficient scanning face for a given grid and stencil operator. In two dimensions it is based on a continued fraction algorithm. In three dimensions it follows Voronoi's successive minima algorithm. We show experimental results of using the scanning face, and compare with canonical loop orderings.

VanderWijngaart, Rob F.↗

NASA NTRS1999

Power System Optimization Codes Modified

A major modification of and addition to existing Closed Brayton Cycle (CBC) space power system optimization codes was completed. These modifications relate to the global minimum mass search driver programs containing three nested iteration loops comprising iterations on cycle temperature ratio, and three separate pressure ratio iteration loops--one loop for maximizing thermodynamic efficiency, one for minimizing radiator area, and a final loop for minimizing overall power system mass. Using the method of steepest ascent, the code sweeps through the pressure ratio space repeatedly, each time with smaller iteration step sizes, so that the three optimum pressure ratios can be obtained to any desired accuracy for each of the objective functions referred to above (i.e., maximum thermodynamic efficiency, minimum radiator area, and minimum system mass). Two separate options for the power system heat source are available: 1. A nuclear fission reactor can be used. It is provided with a radiation shield 1. (composed of a lithium hydride (LiH) neutron shield and tungsten (W) gamma shield). Suboptions can be used to select the type of reactor (i.e., fast spectrum liquid metal cooled or epithermal high-temperature gas reactor (HTGR)). 2. A solar heat source can be used. This option includes a parabolic concentrator and heat receiver for raising the temperature of the recirculating working fluid. A useful feature of the code modifications is that key cycle parameters are displayed, including the overall system specific mass in kilograms per kilowatt and the system specific power in watts per kilogram, as the results for each temperature ratio are computed. As the minimum mass temperature ratio is encountered, a message is printed out. Several levels of detailed information on cycle state points, subsystem mass results, and radiator temperature profiles are stored for this temperature ratio condition and can be displayed or printed by users.

Juhasz, Albert J.↗

NASA NTRS2002

Automating CPM-GOMS

CPM-GOMS is a modeling method that combines the task decomposition of a GOMS analysis with a model of human resource usage at the level of cognitive, perceptual, and motor operations. CPM-GOMS models have made accurate predictions about skilled user behavior in routine tasks, but developing such models is tedious and error-prone. We describe a process for automatically generating CPM-GOMS models from a hierarchical task decomposition expressed in a cognitive modeling tool called Apex. Resource scheduling in Apex automates the difficult task of interleaving the cognitive, perceptual, and motor resources underlying common task operators (e.g. mouse move-and-click). Apex's UI automatically generates PERT charts, which allow modelers to visualize a model's complex parallel behavior. Because interleaving and visualization is now automated, it is feasible to construct arbitrarily long sequences of behavior. To demonstrate the process, we present a model of automated teller interactions in Apex and discuss implications for user modeling. available to model human users, the Goals, Operators, Methods, and Selection (GOMS) method [6, 21] has been the most widely used, providing accurate, often zero-parameter, predictions of the routine performance of skilled users in a wide range of procedural tasks [6, 13, 15, 27, 28]. GOMS is meant to model routine behavior. The user is assumed to have methods that apply sequences of operators and to achieve a goal. Selection rules are applied when there is more than one method to achieve a goal. Many routine tasks lend themselves well to such decomposition. Decomposition produces a representation of the task as a set of nested goal states that include an initial state and a final state. The iterative decomposition into goals and nested subgoals can terminate in primitives of any desired granularity, the choice of level of detail dependent on the predictions required. Although GOMS has proven useful in HCI, tools to support the construction of GOMS models have not yet come into general use.

GOMS↗

NASA NTRS1976

Implementation on a nonlinear concrete cracking algorithm in NASTRAN

A computer code for the analysis of reinforced concrete structures was developed using NASTRAN as a basis. Nonlinear iteration procedures were developed for obtaining solutions with a wide variety of loading sequences. A direct access file system was used to save results at each load step to restart within the solution module for further analysis. A multi-nested looping capability was implemented to control the iterations and change the loads. The basis for the analysis is a set of mutli-layer plate elements which allow local definition of materials and cracking properties.

Herting, D. N.↗

NASA NTRS1998

Nested Quantum Search and NP-Complete Problem

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order square-root of d, where d is the dimension of the search space, whereas any classical algorithm scales as O(d).

NP-complete problems quantum search algorithm tree↗

NASA NTRS2016

Assessment of Preconditioner for a USM3D Hierarchical Adaptive Nonlinear Method (HANIM) (Invited)

Enhancements to the previously reported mixed-element USM3D Hierarchical Adaptive Nonlinear Iteration Method (HANIM) framework have been made to further improve robustness, efficiency, and accuracy of computational fluid dynamic simulations. The key enhancements include a multi-color line-implicit preconditioner, a discretely consistent symmetry boundary condition, and a line-mapping method for the turbulence source term discretization. The USM3D iterative convergence for the turbulent flows is assessed on four configurations. The configurations include a two-dimensional (2D) bump-in-channel, the 2D NACA 0012 airfoil, a three-dimensional (3D) bump-in-channel, and a 3D hemisphere cylinder. The Reynolds Averaged Navier Stokes (RANS) solutions have been obtained using a Spalart-Allmaras turbulence model and families of uniformly refined nested grids. Two types of HANIM solutions using line- and point-implicit preconditioners have been computed. Additional solutions using the point-implicit preconditioner alone (PA) method that broadly represents the baseline solver technology have also been computed. The line-implicit HANIM shows superior iterative convergence in most cases with progressively increasing benefits on finer grids.

Pandya, Mohagna J.↗

NASA NTRS1997

Modeling of Rolling Element Bearing Mechanics: Computer Program Updates

The Rolling Element Bearing Analysis System (REBANS) extends the capability available with traditional quasi-static bearing analysis programs by including the effects of bearing race and support flexibility. This tool was developed under contract for NASA-MSFC. The initial version delivered at the close of the contract contained several errors and exhibited numerous convergence difficulties. The program has been modified in-house at MSFC to correct the errors and greatly improve the convergence. The modifications consist of significant changes in the problem formulation and nonlinear convergence procedures. The original approach utilized sequential convergence for nested loops to achieve final convergence. This approach proved to be seriously deficient in robustness. Convergence was more the exception than the rule. The approach was changed to iterate all variables simultaneously. This approach has the advantage of using knowledge of the effect of each variable on each other variable (via the system Jacobian) when determining the incremental changes. This method has proved to be quite robust in its convergence. This technical memorandum documents the changes required for the original Theoretical Manual and User's Manual due to the new approach.

Ryan, S. G.↗

NASA NTRS1988

Newton solution of inviscid and viscous problems

The application of Newton iteration to inviscid and viscous airfoil calculations is examined. Spatial discretization is performed using upwind differences with split fluxes. The system of linear equations which arises as a result of linearization in time is solved directly using either a banded matrix solver or a sparse matrix solver. In the latter case, the solver is used in conjunction with the nested dissection strategy, whose implementation for airfoil calculations is discussed. The boundary conditions are also implemented in a fully implicit manner, thus yielding quadratic convergence. Complexities such as the ordering of cell nodes and the use of a far field vortex to correct freestream for a lifting airfoil are addressed. Various methods to accelerate convergence and improve computational efficiency while using Newton iteration are discussed. Results are presented for inviscid, transonic nonlifting and lifting airfoils and also for laminar viscous cases.

Venkatakrishnan, V.↗

NASA NTRS1993

Nested Krylov methods and preserving the orthogonality

Recently the GMRESR inner-outer iteraction scheme for the solution of linear systems of equations was proposed by Van der Vorst and Vuik. Similar methods have been proposed by Axelsson and Vassilevski and Saad (FGMRES). The outer iteration is GCR, which minimizes the residual over a given set of direction vectors. The inner iteration is GMRES, which at each step computes a new direction vector by approximately solving the residual equation. However, the optimality of the approximation over the space of outer search directions is ignored in the inner GMRES iteration. This leads to suboptimal corrections to the solution in the outer iteration, as components of the outer iteration directions may reenter in the inner iteration process. Therefore we propose to preserve the orthogonality relations of GCR in the inner GMRES iteration. This gives optimal corrections; however, it involves working with a singular, non-symmetric operator. We will discuss some important properties, and we will show by experiments that, in terms of matrix vector products, this modification (almost) always leads to better convergence. However, because we do more orthogonalizations, it does not always give an improved performance in CPU-time. Furthermore, we will discuss efficient implementations as well as the truncation possibilities of the outer GCR process. The experimental results indicate that for such methods it is advantageous to preserve the orthogonality in the inner iteration. Of course we can also use iteration schemes other than GMRES as the inner method; methods with short recurrences like GICGSTAB are of interest.

Desturler, Eric↗

NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗

NASA NTRS2019

Optimal Control within the Context of Multidisciplinary Design, Analysis, and Optimization

Multidisciplinary design, analysis and optimization involves modeling the interactions of complex systems across a variety of disciplines. The optimization of such systems can be a computationally expensive exercise with multiple levels of nested nonlinear solvers running under an optimizer.The application of optimal control in project development often involves performing trajectory optimization for fixed vehicle designs or parametric sweeps across some key vehicle properties.This information is then relayed to the subsystem design teams who update their designs and relay some bulk characteristics back to the trajectory optimization procedure.This iteration is then repeated until the design closes.However, with increasing interest in more tightly coupled systems, such as electric and hybrid-electric aircraft propulsion and boundary layer ingestion, this process is prone to ignore subtle coupling between vehicle subsystem designs and vehicle operation on a given mission.Integrating trajectory optimization into a tightly coupled multidisciplinary design procedure can be computationally prohibitive, depending on the complexity of the subsystem analyses and the optimal control technique applied.To address these issues a new optimal control software tool, Dymos, has been developed.Dymos is built upon NASA's OpenMDAO software and can leverage its capabilities to efficiently compute gradients for the optimization and optimize complex models in parallel on distributed memory systems.This report provides some explanation into the numerical methods employed in Dymos and provides several use cases that demonstrate its performance on traditional optimal control problems and improvements ino techniques have been used extensively in recent decades to solve a variety of optimal control problems, typically in the form of aerospace vehicle trajectory optimization.

pseudospectral↗