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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Analytic error analysis of cross section interpolation methods in nodal diffusion codes - I : Theory

This paper discusses two cross section interpolation methods commonly found in popular nodal codes; the partial derivatives and multiple tables models. The motivation for choosing a model, and thus a case matrix structure, is a trade off between accuracy and computational cost. Due to decades of experience, there are default structures that are sufficient for current light water reactor analysis. However, this is not necessarily the case for advanced reactor designs. Therefore, it is advantageous to understand the sources of error in cross section interpolation models so that the quality of a case matrix may be improved. A mathematical framework for these models is presented in this work that provides a more rigorous connection between the nuclear engineering field's cross section interpolation methods and the broader mathematical field of function approximation. The two cross section models examined in this paper were found to utilize Lagrange interpolation and are a subset of Lagrange tensor products. Classical results of Lagrange polynomial error analysis were then applied to the partial derivative and multiple tables models to derive expressions for the total point-wise error. The analytical results classify the total error into two parts: the model form error and interpolation error. Finally, based on our observations, a better foundation for improving the quality of a case matrix is proposed. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1972

Bounded state variables and the calculus of variations

An optimal control problem with bounded state variables is transformed into a Lagrange problem by means of differentiable mappings which take some Euclidean space onto the control and state regions. Whereas all such mappings lead to a Lagrange problem, it is shown that only those which are defined as acceptable pairs of transformations are suitable in the sense that solutions to the transformed Lagrange problem will lead to solutions to the original bounded state problem and vice versa. In particular, an acceptable pair of transformations is exhibited for the case when the control and state regions are right parallelepipeds. Finally, a description of the necessary conditions for the bounded state problem which were obtained by this method is given.

Hanafy, L. M.↗
NASA NTRS1974

A macroscopic plasma Lagrangian and its application to wave interactions and resonances

The derivation of a macroscopic plasma Lagrangian is considered, along with its application to the description of nonlinear three-wave interaction in a homogeneous plasma and linear resonance oscillations in a inhomogeneous plasma. One approach to obtain the Lagrangian is via the inverse problem of the calculus of variations for arbitrary first and second order quasilinear partial differential systems. Necessary and sufficient conditions for the given equations to be Euler-Lagrange equations of a Lagrangian are obtained. These conditions are then used to determine the transformations that convert some classes of non-Euler-Lagrange equations to Euler-Lagrange equation form. The Lagrangians for a linear resistive transmission line and a linear warm collisional plasma are derived as examples. Using energy considerations, the correct macroscopic plasma Lagrangian is shown to differ from the velocity-integrated low Lagrangian by a macroscopic potential energy that equals twice the particle thermal kinetic energy plus the energy lost by heat conduction.

Peng, Y. K. M.↗
NASA NTRS1975

Quasicoordinate equations for flexible spacecraft

Equations of motion equivalent to Lagrange's equations for generalized coordinates can be formulated in quasi-coordinate form. An analysis of the resulting equations is conducted and some alternative methods are considered. It is found that Lagrange's quasi-coordinate equations produce precisely the same equations of motion for any material system as would energy from the combination of Lagrange's generalized coordinate equations and Euler's rotational equation in vector form.

Likins, P. W.↗
NASA NTRS1981

Periodic solutions about the collinear Lagrangian solution in the general problem of three bodies

The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame. The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses. Finally, the fundamental family of periodic solutions is extended by numerical integration all the way up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated.

Broucke, R.↗
NASA NTRS1996

Computational Methods for Frictional Contact With Applications to the Space Shuttle Orbiter Nose-Gear Tire

A computational procedure is presented for the solution of frictional contact problems for aircraft tires. A Space Shuttle nose-gear tire is modeled using a two-dimensional laminated anisotropic shell theory which includes the effects of variations in material and geometric parameters, transverse-shear deformation, and geometric nonlinearities. Contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the stress resultants, the generalized displacements, and the Lagrange multipliers associated with both contact and friction conditions. The contact-friction algorithm is based on a modified Coulomb friction law. A modified two-field, mixed-variational principle is used to obtain elemental arrays. This modification consists of augmenting the functional of that principle by two terms: the Lagrange multiplier vector associated with normal and tangential node contact-load intensities and a regularization term that is quadratic in the Lagrange multiplier vector. These capabilities and computational features are incorporated into an in-house computer code. Experimental measurements were taken to define the response of the Space Shuttle nose-gear tire to inflation-pressure loads and to inflation-pressure loads combined with normal static loads against a rigid flat plate. These experimental results describe the meridional growth of the tire cross section caused by inflation loading, the static load-deflection characteristics of the tire, the geometry of the tire footprint under static loading conditions, and the normal and tangential load-intensity distributions in the tire footprint for the various static vertical loading conditions. Numerical results were obtained for the Space Shuttle nose-gear tire subjected to inflation pressure loads and combined inflation pressure and contact loads against a rigid flat plate. The experimental measurements and the numerical results are compared.

Tanner, John A.↗
NASA NTRS1996

Computational methods for frictional contact with applications to the Space Shuttle orbiter nose-gear tire: Comparisons of experimental measurements and analytical predictions

A computational procedure is presented for the solution of frictional contact problems for aircraft tires. A Space Shuttle nose-gear tire is modeled using a two-dimensional laminated anisotropic shell theory which includes the effects of variations in material and geometric parameters, transverse-shear deformation, and geometric nonlinearities. Contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the stress resultants, the generalized displacements, and the Lagrange multipliers associated with both contact and friction conditions. The contact-friction algorithm is based on a modified Coulomb friction law. A modified two-field, mixed-variational principle is used to obtain elemental arrays. This modification consists of augmenting the functional of that principle by two terms: the Lagrange multiplier vector associated with normal and tangential node contact-load intensities and a regularization term that is quadratic in the Lagrange multiplier vector. These capabilities and computational features are incorporated into an in-house computer code. Experimental measurements were taken to define the response of the Space Shuttle nose-gear tire to inflation-pressure loads and to inflation-pressure loads combined with normal static loads against a rigid flat plate. These experimental results describe the meridional growth of the tire cross section caused by inflation loading, the static load-deflection characteristics of the tire, the geometry of the tire footprint under static loading conditions, and the normal and tangential load-intensity distributions in the tire footprint for the various static vertical-loading conditions. Numerical results were obtained for the Space Shuttle nose-gear tire subjected to inflation pressure loads and combined inflation pressure and contact loads against a rigid flat plate. The experimental measurements and the numerical results are compared.

Tanner, John A.↗
NASA NTRS2004

Using Redundancy To Reduce Errors in Magnetometer Readings

A method of reducing errors in noisy magnetic-field measurements involves exploitation of redundancy in the readings of multiple magnetometers in a cluster. By "redundancy"is meant that the readings are not entirely independent of each other because the relationships among the magnetic-field components that one seeks to measure are governed by the fundamental laws of electromagnetism as expressed by Maxwell's equations. Assuming that the magnetometers are located outside a magnetic material, that the magnetic field is steady or quasi-steady, and that there are no electric currents flowing in or near the magnetometers, the applicable Maxwell 's equations are delta x B = 0 and delta(raised dot) B = 0, where B is the magnetic-flux-density vector. By suitable algebraic manipulation, these equations can be shown to impose three independent constraints on the values of the components of B at the various magnetometer positions. In general, the problem of reducing the errors in noisy measurements is one of finding a set of corrected values that minimize an error function. In the present method, the error function is formulated as (1) the sum of squares of the differences between the corrected and noisy measurement values plus (2) a sum of three terms, each comprising the product of a Lagrange multiplier and one of the three constraints. The partial derivatives of the error function with respect to the corrected magnetic-field component values and the Lagrange multipliers are set equal to zero, leading to a set of equations that can be put into matrix.vector form. The matrix can be inverted to solve for a vector that comprises the corrected magnetic-field component values and the Lagrange multipliers.

Kulikov, Igor↗
NASA NTRS2013

An Exposition on the Nonlinear Kinematics of Shells, Including Transverse Shearing Deformations

An in-depth exposition on the nonlinear deformations of shells with "small" initial geometric imperfections, is presented without the use of tensors. First, the mathematical descriptions of an undeformed-shell reference surface, and its deformed image, are given in general nonorthogonal coordinates. The two-dimensional Green-Lagrange strains of the reference surface derived and simplified for the case of "small" strains. Linearized reference-surface strains, rotations, curvatures, and torsions are then derived and used to obtain the "small" Green-Lagrange strains in terms of linear deformation measures. Next, the geometry of the deformed shell is described mathematically and the "small" three-dimensional Green-Lagrange strains are given. The deformations of the shell and its reference surface are related by introducing a kinematic hypothesis that includes transverse shearing deformations and contains the classical Love-Kirchhoff kinematic hypothesis as a proper, explicit subset. Lastly, summaries of the essential equations are given for general nonorthogonal and orthogonal coordinates, and the basis for further simplification of the equations is discussed.

Nemeth, Michael P.↗
DOE OSTI2021

Optimal Control of the Energy-Saving Hybrid Hydraulic-Electric Architecture (HHEA) for Off-Highway Mobile Machines

Most off-highway constructions and agriculture equipment use hydraulics, which has unmatched power density, for power transmission and throttling as a means for control. A novel hybrid hydraulic-electric architecture (HHEA) has recently been proposed to improve efficiency for high-power machines that would have been cost-prohibitive to electrify directly. HHEA uses a set of common pressure rails (CPRs) to transmit the majority of power hydraulically and small electric motor drives to modulate that power and to achieve precise control. This article proposes a computationally efficient Lagrange multiplier method (LMM) for computing the optimal sequence of pressure rail selections to minimize energy use. This is needed to evaluate HHEA's energy-saving potential and for iterative architecture design and sizing. An interesting complication is that the cost function is not fully defined until the candidate control sequence is fully specified. This issue is dealt with by decomposing the original problem into a set of sub-problems with additional constraints that can be solved efficiently. Computational effort can be further reduced if actuators are optimized individually instead of together. However, additional steps are required to prevent the constraint functions from becoming discontinuous with respect to the Lagrange multipliers, which is necessary for meeting the constraints. Lastly, a case study of a construction machine demonstrates the efficacy of the method and shows that the HHEA reduces energy consumption by 68%-73% compared to the baseline load-sensing architecture.

Lagrange multiplier↗
DOE OSTI2026

Implementing a unified solver for nonlinearly constrained optimization

SQP and interior-point methods (also referred to as Lagrange-Newton methods) typically share key algorithmic components, such as strategies for computing descent directions and mechanisms that promote global convergence. Building on this insight, we introduce a unifying framework with eight building blocks that abstracts the workflows of Lagrange-Newton methods. We then present Uno, a modular C++ solver that implements our unifying framework and allows the automatic combination of a wide range of strategies with no programming effort from the user. Uno is meant to (1) organize mathematical optimization strategies into a coherent hierarchy; (2) offer a wide range of efficient and robust methods that can be compared for a given instance; (3) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno’s software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against state-of-the-art solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTE collection. Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno and via its C, Julia, Python, Fortran, and AMPL interfaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

On restricting first order form of gauge theories to one-loop order

Highlights: • We prove the equivalence of the first and second-order formalism. • We use Lagrange multiplier fields to limit radiative effects to one-loop order. • We prove that the quantized theory is both renormalizable and unitary. The first order form of the Yang–Mills and Einstein–Hilbert actions are quantized, and it is shown how Green’s functions computed using the first and the second order form of these theories are related. Next we show how by use of Lagrange multiplier fields (LM), radiative effects beyond one-loop order can be eliminated. This allows one to compute Green’s functions exactly without loss of unitarity. The consequences of this restriction on radiative effects are examined for the Yang–Mills and Einstein–Hilbert actions. In these two gauge theories, we find that the quantized theory is both renormalizable and unitary once the LM field is used to eliminate effects beyond one-loop order.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Restricting loop expansions in gauge theories coupled to matter

Highlights: • Lagrange multiplier fields are used to restrict radiative effects to one-loop order. • This approach is exemplified in the Yang-Mills theory coupled to a scalar matter field. • We showed that it can be used with the Einstein–Hilbert action coupled to matter. • The resulting quantum gravity theory is both renormalizable and unitary. Quantizing any model in which a Lagrange multiplier (LM) field is used to restrict field configurations to those that satisfy the classical equations of motion, leads to at most one-loop radiative corrections. This approach can be used with both the Yang–Mills (YM) and Einstein–Hilbert (EH) action; the resulting theory is both renormalizable and unitary, has a positive energy spectrum and has no negative norm states contributing to physical processes. Although this approach cannot be consistently used with scalar fields alone, scalar fields can be coupled to gauge fields so that loop effects in the gauge sector are restricted to one-loop order in a way that satisfies the usual criterion for a consistent quantum field theory. The tree-level diagrams are those of the classical theory in which the metric couples to the energy–momentum tensor.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

A priori error analysis of high-order LL* (FOSLL*) finite element methods

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are LL* methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement convergence rates. In fact, one may not even see rates proportional to the polynomial order p > 1 when the exact solution is a constant function. Here, we show that the convergence rate is limited by the regularity of an extraneous Lagrange multiplier variable which naturally appears via a saddle-point analysis. In turn, limited convergence rates appear because the regularity of this Lagrange multiplier is determined, in part, by the geometry of the domain. Numerical experiments support our conclusions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Integral boundary conditions in phase field models

Modeling the chemical, electric and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential equations often with integral boundary conditions. Employing the familiar Poisson equation describing the electric potential evolution in a material exhibiting insulator to metal transitions, we exploit a special property of such an integral boundary condition, and we properly formulate the variational problem and establish its well-posedness. Next, we compare our method with the commonly-used Lagrange multiplier method that can also handle such boundary conditions. Numerical experiments demonstrate that our new method achieves optimal convergence rate in contrast to the conventional Lagrange multiplier method. Furthermore, the linear system derived from our method is symmetric positive definite, and can be efficiently solved by Conjugate Gradient method with algebraic multigrid preconditioning.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems

We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. Finally, this work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.

42 ENGINEERING↗
DOE OSTI2022

Characterizing Tradeoffs in Memory, Accuracy, and Speed for Chemistry Tabulation Techniques

Chemistry tabulation is a common approach in practical simulations of turbulent combustion at engineering scales. Linear interpolants have traditionally been used for accessing precomputed multidimensional tables but suffer from large memory requirements and discontinuous derivatives. Higher-degree interpolants address some of these restrictions but are similarly limited to relatively low-dimensional tabulation. Artificial neural networks (ANNs) can be used to overcome these limitations but cannot guarantee the same accuracy as interpolants and introduce challenges in reproducibility and reliable training. These challenges are enhanced as the physics complexity to be represented within the tabulation increases. Here, we assess the efficiency, accuracy, and memory requirements of Lagrange polynomials, tensor product B-splines, and ANNs as tabulation strategies. We analyze results in the context of nonadiabatic flamelet modeling where higher dimension counts are necessary. While ANNs do not require structuring of data, providing benefits for complex physics representation, interpolation approaches often rely on some structuring of the table. Interpolation using structured table inputs that are not directly related to the variables transported in a simulation can incur additional query costs. This is demonstrated in the present implementation of heat losses. We show that ANNs, despite being difficult to train and reproduce, can be advantageous for high-dimensional, unstructured datasets relevant to nonadiabatic flamelet models. Furthermore we demonstrate that Lagrange polynomials show significant speedup for similar accuracy compared to B-splines.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

An Algorithmic and Software Pipeline for Very Large Scale Scientific Data Compression with Error Guarantees

Efficient data compression is becoming increasingly critical for storing scientific data because many scientific applications produce vast amounts of data. This paper presents an end-to-end algorithmic and software pipeline for data compression that guarantees both error bounds on primary data (PD) and derived data, known as Quantities of Interest (QoI).We demonstrate the effectiveness of the pipeline by compressing fusion data generated by a large-scale fusion code, XGC, which produces tens of petabytes of data in a single day. We demonstrate that the compression is conducted by setting aside computational resources known as staging nodes, and does not impact the simulation performance. For efficient parallel I/O, the pipeline uses ADIOS2, which many codes such as XGC already use for their parallel I/O. We show that our approach can compress the data by two orders of magnitude while guaranteeing high accuracy on both the PD and the QoIs. Further, the amount of resources required by compression is a few percent of the resources required by simulation while ensuring that the compression time for each stage is less than the corresponding simulation time.This pipeline consists of three main steps. The first step decomposes the data using domain decomposition into small subdomains. Each subdomain is then compressed independently to achieve a high level of parallelism. The second step uses existing techniques that guarantee error bounds on the primary data for each subdomain. The third step uses a post-processing optimization technique based on Lagrange multipliers to reduce the QoI errors for data corresponding to each subdomain. The Lagrange multipliers generated can be further quantized or truncated to increase the compression level. All of the above characteristics of our approach make it highly practical to apply on-the-fly compression while guaranteeing errors on QoIs that are critical to the scientists.

Banerjee, Tania↗