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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Discrete-Layer Piezoelectric Plate and Shell Models for Active Tip-Clearance Control

The objectives of this work were to develop computational tools for the analysis of active-sensory composite structures with added or embedded piezoelectric layers. The targeted application for this class of smart composite laminates and the analytical development is the accomplishment of active tip-clearance control in turbomachinery components. Two distinct theories and analytical models were developed and explored under this contract: (1) a discrete-layer plate theory and corresponding computational models, and (2) a three dimensional general discrete-layer element generated in curvilinear coordinates for modeling laminated composite piezoelectric shells. Both models were developed from the complete electromechanical constitutive relations of piezoelectric materials, and incorporate both displacements and potentials as state variables. This report describes the development and results of these models. The discrete-layer theories imply that the displacement field and electrostatic potential through-the-thickness of the laminate are described over an individual layer rather than as a smeared function over the thickness of the entire plate or shell thickness. This is especially crucial for composites with embedded piezoelectric layers, as the actuating and sensing elements within these layers are poorly represented by effective or smeared properties. Linear Lagrange interpolation polynomials were used to describe the through-thickness laminate behavior. Both analytic and finite element approximations were used in the plane or surface of the structure. In this context, theoretical developments are presented for the discrete-layer plate theory, the discrete-layer shell theory, and the formulation of an exact solution for simply-supported piezoelectric plates. Finally, evaluations and results from a number of separate examples are presented for the static and dynamic analysis of the plate geometry. Comparisons between the different approaches are provided when possible, and initial conclusions regarding the accuracy and limitations of these models are given.

Heyliger, P. R.↗
NASA NTRS2001

Distributed Relaxation for Conservative Discretizations

A multigrid method is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. The way to achieve this efficiency is the distributed relaxation approach. TME solvers employing distributed relaxation have already been demonstrated for nonconservative formulations of high-Reynolds-number viscous incompressible and subsonic compressible flow regimes. The purpose of this paper is to provide foundations for applications of distributed relaxation to conservative discretizations. A direct correspondence between the primitive variable interpolations for calculating fluxes in conservative finite-volume discretizations and stencils of the discretized derivatives in the nonconservative formulation has been established. Based on this correspondence, one can arrive at a conservative discretization which is very efficiently solved with a nonconservative relaxation scheme and this is demonstrated for conservative discretization of the quasi one-dimensional Euler equations. Formulations for both staggered and collocated grid arrangements are considered and extensions of the general procedure to multiple dimensions are discussed.

Diskin, Boris↗
NASA NTRS1995

On the Importance of the Dynamics of Discretizations

It has been realized recently that the discrete maps resulting from numerical discretizations of differential equations can possess asymptotic dynamical behavior quite different from that of the original systems. This is the case not only for systems of Ordinary Differential Equations (ODEs) but in a more complicated manner for Partial Differential Equations (PDEs) used to model complex physics. The impact of the modified dynamics may be mild and even not observed for some numerical methods. For other classes of discretizations the impact may be pronounced, but not always obvious depending on the nonlinear model equations, the time steps, the grid spacings and the initial conditions. Non-convergence or convergence to periodic solutions might be easily recognizable but convergence to incorrect but plausible solutions may not be so obvious - even for discretized parameters within the linearized stability constraint. Based on our past four years of research, we will illustrate some of the pathology of the dynamics of discretizations, its possible impact and the usage of these schemes for model nonlinear ODEs, convection-diffusion equations and grid adaptations.

Sweby, Peter K.↗
NASA NTRS2003

Discrete Abstractions of Hybrid Systems: Verification of Safety and Application to User-Interface Design

Human interaction with a complex control system involves the user, the automation’s discrete mode logic, and the underlying continuous dynamics of the physical system. The user-interface of such systems always displays a reduced set of information about the entire system. Designing interfaces such that all the pertinent information is available and assuring that this information is correct is important for any user-interface, but especially so for safety-critical systems such as automotive systems and autopilots. Here we describe a methodology for the analysis of hybrid control systems that incorporate user interaction, with the goal of assuring that the information provided to the user is correct. That is, the user-interface must contain all information necessary to safely complete a desired procedure or task. We begin with a hybrid system model which incorporates discrete mode logic as well as nonlinear continuous dynamics. Using a hybrid computational tool for reachability, we find the largest region of the state-space in which we can guarantee the state of the system can always remain – this is the safe region of operation. By implementing a controller for safety which arises from this computation, we mathematically guarantee that this safe region is invariant, meaning that the system will always remain within the safe region if the determined controller is used on the boundary of the safe region. Verification within a hybrid framework allows us to account for the continuous dynamics underlying the discrete representations displayed to the user. Using the computed invariant regions as discrete states, we can abstract a discrete event system from this hybrid system with safety restrictions. This abstraction can be used to determine what information must be provided on the display. Furthermore, in cases in which an interface already exists, the abstraction provides the necessary input into existing interface verification methods. We provide two examples: a car traveling through a yellow light at an intersection and an aircraft autopilot in an automatic landing/go-around maneuver. The examples demonstrate the applicability of this methodology to hybrid systems that have operational constraints we can pose in terms of safety. This methodology differs from existing work in hybrid system verification in that we directly account for the user’s interactions with the system.

Meeko Oishi↗
NASA NTRS2004

A Two-Timescale Discretization Scheme for Collocation

The development of a two-timescale discretization scheme for collocation is presented. This scheme allows a larger discretization to be utilized for smoothly varying state variables and a second finer discretization to be utilized for state variables having higher frequency dynamics. As such. the discretization scheme can be tailored to the dynamics of the particular state variables. In so doing. the size of the overall Nonlinear Programming (NLP) problem can be reduced significantly. Two two-timescale discretization architecture schemes are described. Comparison of results between the two-timescale method and conventional collocation show very good agreement. Differences of less than 0.5 percent are observed. Consequently. a significant reduction (by two-thirds) in the number of NLP parameters and iterations required for convergence can be achieved without sacrificing solution accuracy.

Desai, Prasun↗
NASA NTRS2010

Accuracy Analysis for Finite-Volume Discretization Schemes on Irregular Grids

A new computational analysis tool, downscaling test, is introduced and applied for studying the convergence rates of truncation and discretization errors of nite-volume discretization schemes on general irregular (e.g., unstructured) grids. The study shows that the design-order convergence of discretization errors can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all. The downscaling test is a general, efficient, accurate, and practical tool, enabling straightforward extension of verification and validation to general unstructured grid formulations. It also allows separate analysis of the interior, boundaries, and singularities that could be useful even in structured-grid settings. There are several new findings arising from the use of the downscaling test analysis. It is shown that the discretization accuracy of a common node-centered nite-volume scheme, known to be second-order accurate for inviscid equations on triangular grids, degenerates to first order for mixed grids. Alternative node-centered schemes are presented and demonstrated to provide second and third order accuracies on general mixed grids. The local accuracy deterioration at intersections of tangency and in flow/outflow boundaries is demonstrated using the DS tests tailored to examining the local behavior of the boundary conditions. The discretization-error order reduction within inviscid stagnation regions is demonstrated. The accuracy deterioration is local, affecting mainly the velocity components, but applies to any order scheme.

Diskin, Boris↗
NASA NTRS2012

Effects of Mesh Irregularities on Accuracy of Finite-Volume Discretization Schemes

The effects of mesh irregularities on accuracy of unstructured node-centered finite-volume discretizations are considered. The focus is on an edge-based approach that uses unweighted least-squares gradient reconstruction with a quadratic fit. For inviscid fluxes, the discretization is nominally third order accurate on general triangular meshes. For viscous fluxes, the scheme is an average-least-squares formulation that is nominally second order accurate and contrasted with a common Green-Gauss discretization scheme. Gradient errors, truncation errors, and discretization errors are separately studied according to a previously introduced comprehensive methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry, anisotropic grids typical of adapted grids, and anisotropic grids over a curved surface typical of advancing layer grids. The meshes within the classes range from regular to extremely irregular including meshes with random perturbation of nodes. Recommendations are made concerning the discretization schemes that are expected to be least sensitive to mesh irregularities in applications to turbulent flows in complex geometries.

Diskin, Boris↗
DOE OSTI2023

Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce “Blackout Diffusion”, which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

Santos, Javier E.↗
DOE OSTI2021

Solution Irregularity Remediation for Spatial Discretization Error Estimation for S N Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. Here, a metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

A hybrid meshfree discretization to improve the numerical performance of peridynamic models

Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. Additionally, it is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.

42 ENGINEERING↗
DOE OSTI2025

A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure

Thermodynamically consistent models in continuum physics, i.e. models which satisfy the first and second laws of thermodynamics, may be expressed using the metriplectic formalism. In this work, we leverage the structures underlying this modeling formalism to preserve thermodynamic consistency in discretizations of a fluid model. The procedure relies (1) on ensuring that the spatial semi-discretization retains certain symmetries and degeneracies of the Poisson and metriplectic 4-brackets, and (2) on the use of an appropriate energy conserving time-stepping method. Here, the minimally simple yet nontrivial example of a one-dimensional thermal-fluid model is treated. It is found that preservation of the requisite symmetries and degeneracies of the 4-bracket is relatively simple to ensure in Galerkin spatial discretizations, suggesting a path forward for thermodynamically consistent discretizations of more complex fluid models using more specialized Galerkin methods.

Hamiltonian structure↗
DOE OSTI2021

A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty

Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distribution of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). Here, the outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Localized Evaluation for Constructing Discrete Vector Fields

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. Here, we couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Comparative Investigation of Current-Source Inverters using SiC Discrete Devices and Power Modules

The purpose of this paper is to investigate the impact of the SiC device packages on the commutation performance characteristics of current-source inverters (CSIs). The parasitic components in the CSI current commutation loop between the two-phase legs and output capacitors have a significant impact on the high-frequency performance of the SiC devices. To meet the elevated current requirements of high-power CSIs, it is often necessary to connect multiple discrete devices in parallel which increases the current commutation loop length. The selection of compact high-power SiC MOSFET and Schottky diode modules instead of discrete devices can be highly desirable to reduce the loop inductance and improve the system performance and power density. Two CSI benchtop prototype units, one with SiC discrete devices and the other with power modules, have been designed and tested, and the performances of the two CSIs are compared. The CSI with SiC power modules significantly reduces the inverter volume and lowers the parasitic inductance by 60% and the voltage ripple amplitude by 20% compared to the CSI unit with discrete devices.

42 ENGINEERING↗
DOE OSTI2021

Efficient exascale discretizations: High-order finite element methods

Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of floating point operations to energy intensive data movement. One of the few viable approaches to achieve high efficiency in the area of PDE discretizations on unstructured grids is to use matrix-free/partially assembled high-order finite element methods, since these methods can increase the accuracy and/or lower the computational time due to reduced data motion. In this paper we provide an overview of the research and development activities in the Center for Efficient Exascale Discretizations (CEED), a co-design center in the Exascale Computing Project that is focused on the development of next-generation discretization software and algorithms to enable a wide range of finite element applications to run efficiently on future hardware. CEED is a research partnership involving more than 30 computational scientists from two US national labs and five universities, including members of the Nek5000, MFEM, MAGMA and PETSc projects. We discuss the CEED co-design activities based on targeted benchmarks, miniapps and discretization libraries and our work on performance optimizations for large-scale GPU architectures. We also provide a broad overview of research and development activities in areas such as unstructured adaptive mesh refinement algorithms, matrix-free linear solvers, high-order data visualization, and list examples of collaborations with several ECP and external applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A coupling approach for linear elasticity problems with spatially non-coincident discretized interfaces

Here we present a new method for coupled linear elasticity problems whose finite element discretization may lead to spatially non-coincident discretized interfaces. Our approach combines the classical Dirichlet–Neumann coupling formulation with a new set of discretized interface conditions obtained through Taylor series expansions. We show that these conditions ensure linear consistency of the coupled finite element solution. We then formulate an iterative solution method for the coupled discrete system and apply the new coupling approach to two representative settings for which we also provide several numerical illustrations. The first setting is a mesh-tying problem in which both coupled structures have the same Lamé parameters whereas the second setting is an interface problem for which the Lamé parameters in the two coupled structures are different.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Model Predictive Control of Discrete-Continuous Energy Systems via Generalized Disjunctive Programming

Generalized Disjunctive Programming (GDP) provides an alternative framework to model optimization problems with both discrete and continuous variables. The key idea behind GDP involves the use of logical disjunctions to represent discrete decisions in the continuous space, and logical propositions to denote algebraic constraints in the discrete space. Compared to traditional mixed-integer programming (MIP), the inherent logic structure in GDP yields tighter relaxations that are exploited by global branch and bound algorithms to improve solution quality. In this paper, we present a general GDP model for optimal control of hybrid systems that exhibit both discrete and continuous dynamics. Specifically, we use GDP to formulate a model predictive control (MPC) model for piecewise-affine systems with implicit switching logic. As an example, the GDP-based MPC approach is used as a supervisory control to improve energy efficiency in residential buildings with binary on/off, relay-based thermostats. A simulation study is used to demonstrate the validity of the proposed approach, and the improved solution quality compared to existing MIPbased control approaches.

Bhattacharya, Arnab↗
DOE OSTI2021

Foundations of variational discrete action theory

Variational wave functions and Green's functions are two important paradigms for solving quantum Hamiltonians, each having their own advantages. Here we detail the variational discrete action theory (VDAT), which exploits the advantages of both paradigms in order to approximately solve the ground state of quantum Hamiltonians. VDAT consists of two central components: the sequential product density matrix (SPD) ansatz and a discrete action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer $\mathscr{N}$, recovering many well-known variational wave functions, in addition to the exact solution for $\mathscr{N}$ = ∞. The discrete action describes all dynamical information of an effective integer time evolution with respect to the SPD. We generalize the path integral to our integer time formalism, which converts a dynamic correlation function in integer time to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism to integer time, which results in analogous but distinct mathematical structures, yielding integer time versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We prove that the SPD can be exactly evaluated in the multiband Anderson impurity model (AIM) by summing a finite number of diagrams. For the multiband Hubbard model, we prove that the self-consistent canonical discrete action approximation (SCDA), which is the integer time analog of the dynamical mean-field theory, exactly evaluates the SPD for d = ∞. VDAT within the SCDA provides an efficient yet reliable method for capturing the local physics of quantum lattice models, which will have broad applications for strongly correlated electron materials. More generally, VDAT should find applications in various many-body problems in physics.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗