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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.

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NASA NTRS1973

The finite element method in shell stability analysis.

A development of the finite element method for thin shell instability analysis is presented, covering three principal aspects: (1) representation of shell geometry, (2) representation of element behavior, and (3) algorithmic tools for solution of the large-order systems of nonlinear algebraic equations which characterize various phases of shell instability. Two shell elements are described, an arbitrary quadrilateral and a triangle, and numerical results are presented for two widely-employed comparison problems for linear (stable) analysis. Two shell problems which include instability effects are also solved.-

Gallagher, R. H.↗
NASA NTRS1976

The method of Ritz applied to the equation of Hamilton

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Bailey, C. D.↗
NASA NTRS1985

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗
NASA NTRS1986

Bounding solutions of geometrically nonlinear viscoelastic problems

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Stubstad, J. M.↗
NASA NTRS1993

Vibration analysis of heated anisotropic plates with free edge conditions

Tabular and plotted results are presented (as a function of lamination angle and temperature) for the first two vibration frequencies and mode shapes of a heated, single-layer Boron/epoxy laminate. A Ritz formulation is used to determine the thermally induced in-plane forces and the out-of-plane vibration frequencies. This approach is formulated in terms of the out-of-plane displacement and the in-plane stress function; both the displacement and the stress function are expressed as simple algebraic polynomials modified to satisfy the essential boundary conditions. Heating is applied as a steady-state temperature distribution; material properties are considered to be temperature-dependent. To establish the correctness and effectiveness of the method, the in-plane forces are compared with Fourier series analytical results.

Locke, James E.↗
NASA NTRS2015

Projection-Based Reduced Order Modeling for Spacecraft Thermal Analysis

This paper presents a mathematically rigorous, subspace projection-based reduced order modeling (ROM) methodology and an integrated framework to automatically generate reduced order models for spacecraft thermal analysis. Two key steps in the reduced order modeling procedure are described: (1) the acquisition of a full-scale spacecraft model in the ordinary differential equation (ODE) and differential algebraic equation (DAE) form to resolve its dynamic thermal behavior; and (2) the ROM to markedly reduce the dimension of the full-scale model. Specifically, proper orthogonal decomposition (POD) in conjunction with discrete empirical interpolation method (DEIM) and trajectory piece-wise linear (TPWL) methods are developed to address the strong nonlinear thermal effects due to coupled conductive and radiative heat transfer in the spacecraft environment. Case studies using NASA-relevant satellite models are undertaken to verify the capability and to assess the computational performance of the ROM technique in terms of speed-up and error relative to the full-scale model. ROM exhibits excellent agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) along with salient computational acceleration (up to two orders of magnitude speed-up) over the full-scale analysis. These findings establish the feasibility of ROM to perform rational and computationally affordable thermal analysis, develop reliable thermal control strategies for spacecraft, and greatly reduce the development cycle times and costs.

0000↗
DOE OSTI2022

A symbolic framework to obtain mid-fidelity models of flexible multibody systems with application to horizontal-axis wind turbines

Abstract. The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.

Branlard, Emmanuel (ORCID:0000000277506128)↗
NASA NTRS1986

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Warming, Robert F.↗
NASA NTRS1995

DRS: Derivational Reasoning System

The high reliability requirements for airborne systems requires fault-tolerant architectures to address failures in the presence of physical faults, and the elimination of design flaws during the specification and validation phase of the design cycle. Although much progress has been made in developing methods to address physical faults, design flaws remain a serious problem. Formal methods provides a mathematical basis for removing design flaws from digital systems. DRS (Derivational Reasoning System) is a formal design tool based on advanced research in mathematical modeling and formal synthesis. The system implements a basic design algebra for synthesizing digital circuit descriptions from high level functional specifications. DRS incorporates an executable specification language, a set of correctness preserving transformations, verification interface, and a logic synthesis interface, making it a powerful tool for realizing hardware from abstract specifications. DRS integrates recent advances in transformational reasoning, automated theorem proving and high-level CAD synthesis systems in order to provide enhanced reliability in designs with reduced time and cost.

Bose, Bhaskar↗
NASA NTRS1997

Inherent Conservatism in Deterministic Quasi-Static Structural Analysis

The cause of the long-suspected excessive conservatism in the prevailing structural deterministic safety factor has been identified as an inherent violation of the error propagation laws when reducing statistical data to deterministic values and then combining them algebraically through successive structural computational processes. These errors are restricted to the applied stress computations, and because mean and variations of the tolerance limit format are added, the errors are positive, serially cumulative, and excessively conservative. Reliability methods circumvent these errors and provide more efficient and uniform safe structures. The document is a tutorial on the deficiencies and nature of the current safety factor and of its improvement and transition to absolute reliability.

Verderaime, V.↗
NASA NTRS1989

More accurate predictions with transonic Navier-Stokes methods through improved turbulence modeling

Significant improvements in predictive accuracies for off-design conditions are achievable through better turbulence modeling; and, without necessarily adding any significant complication to the numerics. One well established fact about turbulence is it is slow to respond to changes in the mean strain field. With the 'equilibrium' algebraic turbulence models no attempt is made to model this characteristic and as a consequence these turbulence models exaggerate the turbulent boundary layer's ability to produce turbulent Reynolds shear stresses in regions of adverse pressure gradient. As a consequence, too little momentum loss within the boundary layer is predicted in the region of the shock wave and along the aft part of the airfoil where the surface pressure undergoes further increases. Recently, a 'nonequilibrium' algebraic turbulence model was formulated which attempts to capture this important characteristic of turbulence. This 'nonequilibrium' algebraic model employs an ordinary differential equation to model the slow response of the turbulence to changes in local flow conditions. In its original form, there was some question as to whether this 'nonequilibrium' model performed as well as the 'equilibrium' models for weak interaction cases. However, this turbulence model has since been further improved wherein it now appears that this turbulence model performs at least as well as the 'equilibrium' models for weak interaction cases and for strong interaction cases represents a very significant improvement. The performance of this turbulence model relative to popular 'equilibrium' models is illustrated for three airfoil test cases of the 1987 AIAA Viscous Transonic Airfoil Workshop, Reno, Nevada. A form of this 'nonequilibrium' turbulence model is currently being applied to wing flows for which similar improvements in predictive accuracy are being realized.

Johnson, Dennis 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↗
DOE OSTI2026

Commuting embeddings for parallel strategies in non-local games

Non-local games provide a versatile framework for probing quantum correlations and for benchmarking the power of entanglement. In finite dimensions, the standard method for playing several games in parallel requires a tensor product of the local Hilbert spaces, which scales additively in the number of qubits. In this work, we show that this additive cost can be reduced by exploiting algebraic embeddings. We introduce two forms of compressions. First, when a referee selects one game from a finite collection of games at random, the game quantum strategy can be implemented using a maximally entangled state of dimension equal to the largest individual game, thereby eliminating the need for repeated state preparations. Second, we establish conditions under which several games can be played simultaneously in parallel on fewer qubits than the tensor product baseline. These conditions are expressed in terms of commuting embeddings of the game algebras. Moreover, we provide a constructive framework for building such embeddings. Using tools from Lie theory, we show that aligning the various game algebras into a common Cartan decomposition enables such a qubit reduction. Beyond the theoretical contribution, our framework casts NLGs as algebraic primitives for distributed and resource-constrained quantum computations and suggested NLGs as a comparable device-independent dimension witness.

Commuting embeddings↗
DOE OSTI2024

A fast and accurate domain decomposition nonlinear manifold reduced order model

Here, this paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM ROMs approximate the full order model (FOM) state in a nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for problems with slowly decaying Kolmogorov n-width. However, the number of NM-ROM parameters that need to be trained scales with the size of the FOM. Moreover, for “extreme-scale” problems, the storage of high-dimensional FOM snapshots alone can make ROM training expensive. To alleviate the training cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and couples them to obtain a global NM-ROM. This approach has several advantages: Subdomain NM-ROMs can be trained in parallel, involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and can be tailored to subdomain specific features of the FOM. The shallow, sparse architecture of the autoencoder used in each subdomain NM-ROM allows application of hyper-reduction (HR), reducing the complexity caused by nonlinearity and yielding computational speedup of the NM-ROM. This paper provides the first application of NM-ROM (with HR) to a DD problem. In particular, this paper details an algebraic DD reformulation of the FOM, training a NM-ROM with HR for each sub domain, and a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and a priori and a posteriori error estimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM with HR on the 2D steady-state Burgers’ equation, showing an order of magnitude improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

Lunar materials processing system integration

The theme of this paper is that governmental resources will not permit the simultaneous development of all viable lunar materials processing (LMP) candidates. Choices will inevitably be made, based on the results of system integration trade studies comparing candidates to each other for high-leverage applications. It is in the best long-term interest of the LMP community to lead the selection process itself, quickly and practically. The paper is in five parts. The first part explains what systems integration means and why the specialized field of LMP needs this activity now. The second part defines the integration context for LMP -- by outlining potential lunar base functions, their interrelationships and constraints. The third part establishes perspective for prioritizing the development of LMP methods, by estimating realistic scope, scale, and timing of lunar operations. The fourth part describes the use of one type of analytical tool for gaining understanding of system interactions: the input/output model. A simple example solved with linear algebra is used to illustrate. The fifth and closing part identifies specific steps needed to refine the current ability to study lunar base system integration. Research specialists have a crucial role to play now in providing the data upon which this refinement process must be based.

Sherwood, Brent↗
NASA NTRS1999

Solving Upwind-Biased Discretizations: Defect-Correction Iterations

This paper considers defect-correction solvers for a second order upwind-biased discretization of the 2D convection equation. The following important features are reported: (1) The asymptotic convergence rate is about 0.5 per defect-correction iteration. (2) If the operators involved in defect-correction iterations have different approximation order, then the initial convergence rates may be very slow. The number of iterations required to get into the asymptotic convergence regime might grow on fine grids as a negative power of h. In the case of a second order target operator and a first order driver operator, this number of iterations is roughly proportional to h-1/3. (3) If both the operators have the second approximation order, the defect-correction solver demonstrates the asymptotic convergence rate after three iterations at most. The same three iterations are required to converge algebraic error below the truncation error level. A novel comprehensive half-space Fourier mode analysis (which, by the way, can take into account the influence of discretized outflow boundary conditions as well) for the defect-correction method is developed. This analysis explains many phenomena observed in solving non-elliptic equations and provides a close prediction of the actual solution behavior. It predicts the convergence rate for each iteration and the asymptotic convergence rate. As a result of this analysis, a new very efficient adaptive multigrid algorithm solving the discrete problem to within a given accuracy is proposed. Numerical simulations confirm the accuracy of the analysis and the efficiency of the proposed algorithm. The results of the numerical tests are reported.

Diskin, Boris↗
NASA NTRS1999

Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach

A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.

Kiris, Cetin↗