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NASA NTRS1995

Numerical simulation of the nonlinear response of composite plates under combined thermal and acoustic loading

A time-domain study of the random response of a laminated plate subjected to combined acoustic and thermal loads is carried out. The features of this problem also include given uniform static inplane forces. The formulation takes into consideration a possible initial imperfection in the flatness of the plate. High decibel sound pressure levels along with high thermal gradients across thickness drive the plate response into nonlinear regimes. This calls for the analysis to use von Karman large deflection strain-displacement relationships. A finite element model that combines the von Karman strains with the first-order shear deformation plate theory is developed. The development of the analytical model can accommodate an anisotropic composite laminate built up of uniformly thick layers of orthotropic, linearly elastic laminae. The global system of finite element equations is then reduced to a modal system of equations. Numerical simulation using a single-step algorithm in the time-domain is then carried out to solve for the modal coordinates. Nonlinear algebraic equations within each time-step are solved by the Newton-Raphson method. The random gaussian filtered white noise load is generated using Monte Carlo simulation. The acoustic pressure distribution over the plate is capable of accounting for a grazing incidence wavefront. Numerical results are presented to study a variety of cases.

Mei, Chuh↗

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↗

NASA NTRS1977

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

A scheme for the calculation of eigenvalues in the problem of acoustic propagation in a two-dimensional duct is described. The computation method involves changing the coupled transcendental nonlinear algebraic equations into an initial value problem involving a nonlinear ordinary differential equation. The simplest approach is to use as initial values the hardwall eigenvalues and to integrate away from these values as the admittance varies from zero to its actual value with a linear variation. The approach leads to a powerful root finding routine capable of computing the transverse and axial wave numbers for two-dimensional ducts for any frequency, lining, admittance and Mach number without requiring initial guesses or starting points.

Eversman, W.↗

DOE OSTI2016

Formulation and numerical analysis of a fully-coupled dynamically deforming electromagnetic wire

Here, an electromagnetic beam model is developed for the simulation of actuated electronic textiles. The beam is solved using a nonlinear director-based kinematic description with additional temperature and electric potential fields along its length. The three fields are fully coupled by mutual dependences on the deformation, Lorenz force, back electromotive force, temperature dependent constitutive responses, and the Seebeck effect. Instead of solving Maxwell's equations in full detail, a quasistatic approximation is used to solve the electric potential in the presence of a moving material medium. The current-carrying beam approximation is used to further simplify the solution space for the potential. While this formulation alleviates the spatial and temporal discretization restrictions, the coupled problem is an index-1 semi-explicit Differential Algebraic Equation requiring special treatment. The time dependent problem is solved using different Runge-Kutta methods. Diagonally implicit Runge-Kutta methods and explicit Runge-Kutta methods using implicit solution of the electric potential problem are explored. The finite element model is implemented using the open source package FEniCS, which is able to automatically generate the linearizations of the multiphysics equations required for the implicit solutions. A model problem is constructed with which to test and analyze the physical formulation and numerical solution techniques. The time stepping methods are verified using the convergence orders of the higher-order Runge-Kutta methods. Runtime comparisons show that the explicit methods are generally more computationally efficient than the implicit schemes used for this problem. For the implicit schemes, a staggered solution is significantly faster than a monolithic solution at most time step sizes. However, at very large time steps, such as those that would be used for dynamic relaxation, the monolithic solution can be more efficient than the staggered solution.

36 MATERIALS SCIENCE↗

DOE OSTI2013

Nonassociativity, Malcev algebras and string theory

Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2000

Extended Decentralized Linear-Quadratic-Gaussian Control

A straightforward extension of a solution to the decentralized linear-Quadratic-Gaussian problem is proposed that allows its use for commonly encountered classes of problems that are currently solved with the extended Kalman filter. This extension allows the system to be partitioned in such a way as to exclude the nonlinearities from the essential algebraic relationships that allow the estimation and control to be optimally decentralized.

Carpenter, J. Russell↗

DOE OSTI2017

On performance of Krylov smoothing for fully-coupled AMG preconditioners for VMS resistive MHD

Here, this study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid (AMG) preconditioned Newton- Krylov solution approach applied to a fully-implicit variational multiscale (VMS) nite element (FE) resistive magnetohydrodynamics (MHD) formulation. In this context a Newton iteration is used for the nonlinear system and a Krylov (GMRES) method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play a critical role. Krylov smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. Three time dependent resistive MHD test cases are considered to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

An Algebraic Quantum Circuit Compression Algorithm for Hamiltonian Simulation

Quantum computing is a promising technology that harnesses the peculiarities of quantum mechanics to deliver computational speedups for some problems that are intractable to solve on a classical computer. Current generation noisy intermediate-scale quantum (NISQ) computers are severely limited in terms of chip size and error rates. Shallow quantum circuits with uncomplicated topologies are essential for successful applications in the NISQ era. In this work, based on matrix analysis, we derive localized circuit transformations to efficiently compress quantum circuits for simulation of certain spin Hamiltonians known as free fermions. The depth of the compressed circuits is independent of simulation time and grows linearly with the number of spins. The proposed numerical circuit compression algorithm behaves backward stable and scales cubically in the number of spins enabling circuit synthesis beyond O(10 3 ) spins. The resulting quantum circuits have a simple nearest-neighbor topology, which makes them ideally suited for NISQ devices.

Hamiltonian simulation↗

NASA NTRS1989

A variational approach to dynamics of flexible multibody systems

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body references frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed.

Wu, Shih-Chin↗

DOE OSTI2014

Unveiling the hidden structure of complex stochastic biochemical networks

Complex Markov models are widely used and powerful predictive tools to analyze stochastic biochemical processes. However, when the network of states is unknown, it is necessary to extract information from the data to partially build the network and estimate the values of the rates. The short-time behavior of the first-passage time distributions between two states in linear chains has been shown recently to behave as a power of time with an exponent equal to the number of intermediate states. For a general Markov model we derive the complete Taylor expansion of the first-passage time distribution between two arbitrary states. By combining algebraic methods and graph theory approaches it is shown that the first term of the Taylor expansion is determined by the shortest path from the initial state to the final state. When this path is unique, we prove that the coefficient of the first term can be written in terms of the product of the transition rates along the path. It is argued that the application of our results to first-return times may be used to estimate the dependence of rates on external parameters in experimentally measured time distributions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2015

Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions

Background: The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic-viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods. Methods: Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine-Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes the rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied. Results: Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime). Conclusions: For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide insight into the shock structure afforded by the numerical methods.

36 MATERIALS SCIENCE↗

NASA NTRS1973

Research on unsteady transonic flow theory

A two-dimensional theory is considered for the unsteady flow disturbances caused by aeroelastic deformations of a thick wing at high subsonic freestream Mach numbers, having a single, internally embedded supercritical (locally supersonic) steady flow region adjacent to the low pressure side of the wing. The theory develops a matrix of unsteady aerodynamic influence coefficients (AICs) suitable as a strip theory for aeroelastic analysis of large aspect ratio thick wings of moderate sweep, typical of a wide class of current and future aircraft. The theory derives the linearized unsteady flow solutions separately for both the subcritical and supercritical regions. These solutions are coupled together to give the requisite (wing pressure-downwash) AICs by the intermediate step of defining flow disturbances on the sonic line, and at the shock wave; these intermediate quantities are then algebraically eliminated by expressing them in terms of the wing surface downwash.

Revell, J. D.↗

DOE OSTI2012

Adaptive Algebraic Smoothers

methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.

99 GENERAL AND MISCELLANEOUS↗

DOE OSTI2015

HEMS: Home Energy Management System [SWR-15-12] - Stochastic Optimal Scheduling of Residential Appliances with Renewable Energy Sources

The HEMS algorithm schedules residential appliance operation to minimize the weighted sum of occupant discomfort, total energy cost, peak electricity consumption and carbon footprint, weighted according to occupant preferences. A centralized mixed integer linear programming (MILP) based optimization approach is taken, with the HEMS managing all appliances directly, following a Model Predictive Control (MPC) methodology which determines the optimal schedule over the user-specified scheduling horizon and repeats the optimization at configurable time intervals. The HEMS is developed on the General Algebraic Modeling System (GAMS) using the MILP solver. This system is able to effectively deal with the stochasticity in the distributed system by utilizing Monte Carlo simulation where uncertainties in future electricity price, outdoor temperature, water usage, solar and wind generation can be accounted for. The HEMS controls the operation of the heating, ventilation and air conditioning (HVAC) system, water heater, refrigerator, residential batteries, electric vehicles, standalone micro combined heat and power (CHP) generator, dishwasher, washer/dryer, pool pumps, and lighting.

Pratt, Annabelle↗

NASA NTRS1991

A second order iteration procedure for correlation of analysis frequencies with test data

A second derivative frequency modification procedure is developed to iteratively change analysis modeling parameters for the more rapid convergence of analytically determined frequencies to those obtained by structural dynamic testing. A nonlinear system of algebraic equations in the modeling parameters is derived and a rapidly converging iterative method for their solution is presented. Numerical results obtained show that the higher order formulation and nonlinear predictor-corrector solution procedure yields results which are more rapidly convergent than those obtained using only the usual linear first-order parameter modification procedure. The proposed higher order theory also connverges in some cases when the ordinary linear theory does not.

Ojalvo, I. U.↗

NASA NTRS1996

Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

Sorensen, Danny C.↗

DOE OSTI2020

Approximate Inverse Chain Preconditioner: Iteration Count Case Study for Spectral Support Solvers

As the growing availability of computational power slows, there has been an increasing reliance on algorithmic advances. However, faster algorithms alone will not necessarily bridge the gap in allowing computational scientists to study problems at the edge of scientific discovery in the next several decades. Often, it is necessary to simplify or precondition solvers to accelerate the study of large systems of linear equations commonly seen in a number of scientific fields. Preconditioning a problem to increase efficiency is often seen as the best approach; yet, preconditioners which are fast, smart, and efficient do not always exist. Following the progress of [1], we present a new preconditioner for symmetric diagonally dominant (SDD) systems of linear equations. These systems are common in certain PDEs, network science, and supervised learning among others. Based on spectral support graph theory, this new preconditioner builds off of the work of [2], computing and applying a V-cycle chain of approximate inverse matrices. This preconditioner approach is both algebraic in nature as well as hierarchically-constrained depending on the condition number of the system to be solved. Due to its generation of an Approximate Inverse Chain of matrices, we refer to this as the AIC preconditioner. We further accelerate the AIC preconditioner by utilizing precomputations to simplify setup and multiplications in the con-text of an iterative Krylov-subspace solver. While these iterative solvers can greatly reduce solution time, the number of iterations can grow large quickly in the absence of good preconditioners. Initial results for the AIC preconditioner have shown a very large reduction in iteration counts for SDD systems as compared to standard preconditioners such as Incomplete Cholesky (ICC) and Multigrid (MG). We further show significant reduction in iteration counts against the more advanced Combinatorial Multigrid (CMG) preconditioner. We have further developed no-fill sparsification techniques to ensure that the computational cost of applying the AIC preconditioner does not grow prohibitively large as the depth of the V-cycle grows for systems with larger condition numbers. Our numerical results have shown that these sparsifiers maintain the sparsity structure of our system while also displaying significant reductions in iteration counts.1 2

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Approximate Inverse Chain Preconditioner: Iteration Count Case Study for Spectral Support Solvers

97 MATHEMATICS AND COMPUTING↗