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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

An algorithm for a generalization of the Richardson extrapolation process

The paper presents a recursive method, designated the W exp (m)-algorithm, for implementing a generalization of the Richardson extrapolation process. Compared to the direct solution of the linear sytems of equations defining the extrapolation procedure, this method requires a small number of arithmetic operations and very little storage. The technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying a d-transformation to power series. In the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced, for solving the same problem. A FORTRAN program for the W exp (m)-algorithm is also appended.

Ford, William F.↗
NASA NTRS1988

Recursive algorithms for vector extrapolation methods

Three classes of recursion relations are devised for implementing some extrapolation methods for vector sequences. One class of recursion relations can be used to implement methods like the modified minimal polynomial extrapolation and the topological epsilon algorithm; another allows implementation of methods like minimal polynomial and reduced rank extrapolation; while the remaining class can be employed in the implementation of the vector E-algorithm. Operation counts and storage requirements for these methods are also discussed, and some related techniques for special applications are also presented. Included are methods for the rapid evaluations of the vector E-algorithm.

Ford, William F.↗
NASA NTRS1988

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, and free surface flows. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, Avram↗
NASA NTRS1990

Efficient Implementation of Minimal Polynomial and Reduced Rank Extrapolation Methods

The minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) are two effective techniques that have been used in accelerating the convergence of vector sequences, such as those that are obtained from iterative solution of linear and nonlinear systems of equation. Their definitions involve some linear least squares problems, and this causes difficulties in their numerical implementation. Timewise efficient and numerically stable implementations for MPE and RRE are developed. A computer program written in FORTRAN 77 is also appended and applied to some model problems.

Sidi, Avram↗
NASA NTRS1990

Convergence acceleration for vector sequences and applications to computational fluid dynamics

Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.

Sidi, Avram↗
NASA NTRS1992

Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods

Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.

Sidi, Avram↗
NASA NTRS1992

Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations

The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.

Sidi, Avram↗
NASA NTRS1992

Rational approximations from power series of vector-valued meromorphic functions

Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.

Sidi, Avram↗
NASA NTRS1993

Efficient Minimum-Polynomial And Reduced-Rank Extrapolation

MPERRE computer program accelerates convergence of sequence of vectors by use of minimum-polynomial extrapolation (MPE) and reduced-rank extrapolation (RRE). Effective in accelerating convergences of such sequences of vectors as those obtained from iterative solution of systems of linear and nonlinear equations. In conjunction with various iterative techniques, successfully employed in finite-difference solution of large-scale elliptic boundary-value problems and problems in computational-fluid-dynamics. Only input required is sequence of vectors, convergence of which is accelerated. Program economical and easy to use. Written in FORTRAN 77.

Sidi, Avram↗
NASA NTRS1993

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗
NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations That Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗
NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations that Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗
NASA NTRS2008

The Liquid Nitrogen System for Chamber A: A Change from Original Forced Flow Design to a Natural Flow (Thermo Siphon) System

NASA Johnson Space Center (JSC) in Houston is currently supplementing its 20K helium refrigeration system to meet the new requirements for testing the James Web Space Telescope in the environmental control Chamber-A (65 dia x 120 high) in Building 32. The new system is required to meet the various operating modes which include a high 20K heat load, a required temperature stability at the load, rapid (but controlled) cool down and warm up and bake out of the chamber. This paper will present the proposed modifications to the existing helium system(s) to incorporate the new requirements and the integration of the new helium refrigerator with the existing two 3.5KW 20K helium refrigerators. In addition, the floating pressure process control philosophy to achieve high efficiency over the operating range (40% to 100% of the refrigeration system capacity), and the required temperature stability of +/- 0.25 K at the load will be discussed. The refrigeration systems ability to naturally seek the operating conditions under various loads and thus minimizing operator involvement and the over all improvements to the system operability and the reliability will be explained.

Homan, Jonathan↗
NASA NTRS2008

Floating Pressure Conversion and Equipment Upgrades of Two 3.5kw, 20k, Helium Refrigerators

Two helium refrigerators, each rated for 3.5KW at 20K, are used at NASA s Johnson Space Center (JSC) in Building No. 32 to provide cryo-pumping within two large thermal-vacuum chambers. These refrigerators were originally commissioned in 1996. Equipment refurbishment and upgrades to the controls of these refrigerators were recently completed. This paper describes some of the mechanical and control issues that necessitated the equipment refurbishment and controls change-over. It will describe the modifications and the new process control which allows the refrigerators to take advantage of the Ganni Cycle "floating pressure" control technology. The upgrades -- the controls philosophy change-over to the floating pressure control technology and the newly refurbished equipment -- have greatly improved the performance, stability, and efficiency of these two refrigerators. The upgrades have also given the operators more information and details about the operational status of the main components (compressors, expanders etc.) of the refrigerators at all operating conditions (i.e.: at various loads in the vacuum chambers). Capabilities, configuration, and performance data pre, and post, upgrading will be presented.

Homan, Jonathan↗
NASA NTRS2013

20 K Helium Refrigeration System for NASA-JSC Chamber-A

A new 20 K helium refrigerator installed at NASA Johnson Space Center's Space Environment Simulation Laboratory (SESL) was successfully commissioned and tested in 2012. The refrigerator is used to create a deep space environment within SESL s Chamber A to perform ground testing of the James Webb Space Telescope. The chamber previously and currently still has helium cryopumping panels (CPP) and LN2 shrouds used to create Low Earth Orbit environments. Now with the new refrigerator and new helium shrouds (45 x 65 ) the chamber can create a deep space environment. The process design, system analysis, specification development, and commissioning oversight were performed by the cryogenics department at Jefferson Labs, while the contracts and system installation was performed by the ESC group at JSC. Commissioning data indicate a inverse coefficient of performance better than 70 W/W for a 18 KW load at 20 K (accounting for liquid nitrogen precooling power) that remains essentially constant down to 1/3 of this load. Even at 10 percent of the maximum capacity, the performance is better than 140 W/W at 20K. The refrigerator exceeded all design goals and demonstrated the ability to support a wide load range from 10kW at 15 K to 100 kW at 100K. The refrigerator is capable of operating at any load temperature from 15K to ambient with tight temperature stability. The new shroud (36 tons of aluminum) can be cooled from room temperature to 20 K in 24 hours. This paper will outline the process design and commissioning results.

Homan, J.↗
NASA NTRS2013

Commissioning of the Liquid Nitrogen Thermo-Siphon System for NASA-JSC Chamber-A

NASA's Space Environment Simulation Laboratory's (SESL) Chamber A, located at the Johnson Space Center in Houston Texas has recently implemented major enhancements of its cryogenic and vacuum systems. The new liquid nitrogen (LN2) thermo-siphon system was successfully commissioned in August of 2012. Chamber A, which has 20 K helium cryo-panels (or shrouds ) which are shielded by 80 K nitrogen shrouds, is capable of simulating a deep space environment necessary to perform ground testing of NASA s James Webb Space Telescope (JWST). Chamber A s previous system used forced flow LN2 cooling with centrifugal pumps, requiring 200,000 liters of LN2 to cool-down and consuming 180,000 liters per day of LN2 in steady operation. The LN2 system did not have the reliability required to meet the long duration test of the JWST, and the cost estimate provided in the initial approach to NASA-JSC by the sub-contractor for refurbishment of the system to meet the reliability goals was prohibitive. At NASA-JSC's request, the JLab Cryogenics Group provided alternative options in 2007, including a thermo-siphon, or natural flow system. This system, eliminated the need for pumps and used one tenth of the original control valves, relief valves, and burst disks. After the thermo-siphon approach was selected, JLab provided technical assistance in the process design, mechanical design, component specification development and commissioning oversight, while the installation and commissioning operations of the system was overseen by the Jacobs Technology/ESC group at JSC. The preliminary commissioning data indicate lower shroud temperatures, 70,000 liters to cool-down and less than 90,000 liters per day consumed in steady operation. All of the performance capabilities have exceeded the design goals. This paper will outline the comparison between the original system and the predicted results of the selected design option, and the commissioning results of thermo-siphon system.

Homan, J.↗
NASA NTRS2013

Commissioning of a 20 K Helium Refrigeration System for NASA-JSC Chamber A

A new 20 K helium refrigerator installed at NASA Johnson Space Center s Space Environment Simulation Laboratory (SESL) was successfully commissioned and tested in 2012. The refrigerator is used to create a deep space environment within SESL s Chamber A to perform ground testing of the James Webb Space Telescope (JWST). The chamber previously and currently still has helium cryo-pumping panels (CPP) and liquid nitrogen shrouds used to create low earth orbit environments. Now with the new refrigerator and new helium shrouds the chamber can create a deep space environment. The process design, system analysis, specification development, and commissioning oversight were performed by the cryogenics department at Jefferson Lab, while the contracts and system installation was performed by the ESC group at JSC. Commissioning data indicate an inverse coefficient of performance better than 70 W/W for a 18 kW load at 20 K (accounting for liquid nitrogen pre-cooling power) that remains essentially constant down to one third of this load. Even at 10 percent of the maximum capacity, the performance is better than 150 W/W at 20 K. The refrigerator exceeded all design goals and demonstrated the ability to support a wide load range from 10 kW at 15 K to 100 kW at 100 K. The refrigerator is capable of operating at any load temperature from 15 K to ambient with tight temperature stability. The new shroud (36 tons of aluminum) can be cooled from room temperature to 20 K in 24 hours. This paper will outline the process design and commissioning results.

Homan, J.↗
NASA NTRS2013

Commissioning of the Liquid Nitrogen Thermo-Siphon System for NASA-JSC Chamber A

NASA s Space Environment Simulation Laboratory s (SESL) Chamber A, located at the Johnson Space Center in Houston Texas has recently implemented major enhancements of its cryogenic and vacuum systems. The new liquid nitrogen (LN) thermo-siphon system was successfully commissioned in August of 2012. Chamber A, which has 20 K helium cryo-panels (or shrouds ) which are shielded by 80 K nitrogen shrouds, is capable of simulating a deep space environment necessary to perform ground testing of NASA s James Webb Space Telescope (JWST). Chamber A s previous system used forced flow LN cooling with centrifugal pumps, requiring 220,000 liters of LN to cool-down and consuming 180,000 liters per day of LN in steady operation. The LN system did not have the reliability required to meet the long duration test of the JWST, and the cost estimate provided in the initial approach to NASA-JSC by the subcontractor for refurbishment of the system to meet the reliability goals was prohibitive. At NASA-JSC s request, the JLab Cryogenics Group provided alternative options in 2007, including a thermo-siphon, or natural flow system. This system, eliminated the need for pumps and used one tenth of the original control valves, relief valves, and burst disks. After the thermo-siphon approach was selected, JLab provided technical assistance in the process design, mechanical design, component specification development and commissioning oversight, while the installation and commissioning operations of the system was overseen by the Jacobs Technology/ESC group at JSC. The preliminary commissioning data indicate lower shroud temperatures, 68,000 liters to cool-down and less than 91,000 liters per day consumed in steady operation. All of the performance capabilities have exceeded the design goals. This paper will outline the comparison between the original system and the predicted results of the selected design option, and the commissioning results of thermo-siphon system.

Homan, J.↗