Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “numerical mathematics”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 829 records · Page 46
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Nonlinear Anti-(Parity-Time) Symmetric Dimer

In the present work we propose a nonlinear anti- P T -symmetric dimer, that at the linear level has been experimentally created in the realm of electric circuit resonators. We find four families of solutions, the so-called upper and lower branches, both in a symmetric and in an asymmetric (symmetry-broken) form. We unveil analytically and confirm numerically the critical thresholds for the existence of such branches and explore the bifurcations (such as saddle-node ones) that delimit their existence, as well as transcritical ones that lead to their potential exchange of stability. We find that out of the four relevant branches, only one, the upper symmetric branch, corresponds to a spectrally and dynamically robust solution. We subsequently leverage detailed direct numerical computations in order to explore the dynamics of the different states, corroborating our spectral analysis results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Galerkin formulation of path integrals in lattice field theory

We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom (DOFs) associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the DOFs. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, this two-point function satisfies a weak propagator (or Green’s function) identity, in analogy to the continuum case, as well as a convergence estimate obtained from the standard finite element techniques. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Parameterization and Explicit Modeling of Cloud Microphysics: Approaches, Challenges, and Future Directions

Cloud microphysical processes occur at the smallest end of scales among cloud-related processes and thus must be parameterized not only in large-scale global circulation models (GCMs) but also in various higher-resolution limited-area models such as cloud-resolving models (CRMs) and large-eddy simulation (LES) models. Instead of giving a comprehensive review of existing microphysical parameterizations that have been developed over the years, this study concentrates purposely on several topics that we believe are understudied but hold great potential for further advancing bulk microphysics parameterizations: multi-moment bulk microphysics parameterizations and the role of the spectral shape of hydrometeor size distributions; discrete vs “continuous” representation of hydrometeor types; turbulence-microphysics interactions including turbulent entrainment-mixing processes and stochastic condensation; theoretical foundations for the mathematical expressions used to describe hydrometeor size distributions and hydrometeor morphology; and approaches for developing bulk microphysics parameterizations. Also presented are the spectral bin scheme and particle-based scheme (especially, super-droplet method) for representing explicit microphysics. Their advantages and disadvantages are elucidated for constructing cloud models with detailed microphysics that are essential to developing processes understanding and bulk microphysics parameterizations. Particle-resolved direct numerical simulation (DNS) models are described as an emerging technique to investigate turbulence-microphysics interactions at the most fundamental level by tracking individual particles and resolving the smallest turbulent eddies in turbulent clouds. Outstanding challenges and future research directions are explored as well.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Probabilistic partition of unity networks for high–dimensional regression problems

We explore the probabilistic partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems and propose a general framework focusing on adaptive dimensionality reduction. With the proposed framework, the target function is approximated by a mixture of experts model on a low-dimensional manifold, where each cluster is associated with a fixed-degree polynomial. We present a training strategy that leverages the expectation maximization (EM) algorithm. During the training, we alternate between (i) applying gradient descent to update the DNN coefficients; and (ii) using closed-form formulae derived from the EM algorithm to update the mixture of experts model parameters. Under the probabilistic formulation, step (ii) admits the form of embarrassingly paralleliazable weighted least-squares solves. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments of various data dimensions. Here, we also explore the proposed model in applications of quantum computing, where the PPOU-Nets act as surrogate models for cost landscapes associated with variational quantum circuits.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. Here we also show at the end an application in the framework of plasma physics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A novel framework for elastoplastic behaviour of anisotropic solids

A general framework for developing nonlinear hyperelastic/plastic constitutive laws for anisotropic solids experiencing large strains and strain rates has been developed. The proposed framework does not rely on the “a priori” known strain energy function, but instead introduces a physical decomposition of the material element into seven physically independent stress bearing mechanisms, each of which has a constitutive law in terms of internal moments described by a scalar function of a single variable. Furthermore, the model has been encoded into a combined finite-discrete element method and tested against static geomechanical test data. The numerical validation experiments show the model can reproduce plastic anisotropic behavior in both biaxial and uniaxial loading of a geomaterial.

58 GEOSCIENCES↗
DOE OSTI2020

Boundary-consistent B-spline filtering schemes and application to high-fidelity simulations of turbulence

A filtering operation, based on B-spline discretizations, is introduced to target weakly growing mesh-scale oscillations that can arise in high-fidelity turbulence simulations. This is a spectral regularization that can be described using the singular values of a banded matrix operator, with the filtering strength set by a scalar- or vector-valued penalty parameter. The penalty parameter can be specified though it can also be advantageously selected to minimize the generalized cross validation (GCV) measure of distance between the pre- and post-filtered solutions. Efficient algorithms are developed to compute both the scalar and vector penalty parameters. The B-spline filter has a sharper localization to high-wavenumber than compact or explicit filters of the same stencil width and is demonstrated for solutions of the Burgers' equation, decaying Burgers' turbulence, and compressible Navier–Stokes turbulent channel flow. Furthermore, these simulations confirm the scheme's numerical stability and ability to narrowly target the high wavenumber components of numerical solutions. An advantage over finite-difference filters is that these B-spline filters are stable on bounded domains and even preserve formal order of accuracy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Scalable preconditioning for the stabilized contact mechanics problem

We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.

58 GEOSCIENCES↗
DOE OSTI2025

A theoretical study of the upper bound of surface elevation variance in the Phillips initial stage during wind-wave generation

The resonance mechanism in the initial of wind-wave generation proposed by Phillips is a foundation of wind-wave generation theory, but a precise theoretical quantification of wave energy growth in this initial stage has not been obtained yet after more than six decades of research. In this study, we aim to address this knowledge gap by developing an analytical approach based on a novel complex analysis method to theoretically investigate the temporal evolution of the wave energy in the Phillips initial stage. We quantitatively derive and analyse the growth behaviour of the surface wave energy and obtain an analytical solution for its upper bound. Our result highlights the crucial effects of surface tension. Because the phase velocity of gravity–capillary waves has a minimal value at a critical wavenumber, gravity–capillary waves and gravity waves (which neglect surface tension) exhibit distinct resonance curve properties and wave energy growth behaviours. For gravity waves, the resonance curve extends indefinitely; for gravity–capillary waves, it either forms a finite-length curve or does not exist, depending on the wind speed. The leading-order term of the upper-bound solution of the energy of gravity waves increases linearly over time, while for gravity–capillary waves, the term increases linearly over time under strong wind conditions but remains finite under weak wind conditions. This theoretical study provides an analytical framework for the generation of wind-waves in the Phillips initial stage, which may inspire further theoretical, numerical and experimental research.

58 GEOSCIENCES↗
DOE OSTI2025

Stability analysis of the Eulerian–Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension

In this paper, we construct a novel Eulerian–Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine–Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with at least twice larger time step constraints than the classical first order Eulerian method for Burgers’ equation. Numerical experiments verify the optimality of the designed time step sizes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Randomized Sketching Algorithms for Low-Memory Dynamic Optimization

This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well suited to incorporating into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the right-hand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g., multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine-scale random fields in a hierarchical fashion from coarse-scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.

algebraic multigrid↗
DOE OSTI2024

Quantum Circuit Cutting for Classical Shadows

Classical shadow tomography is a sample-efficient technique for characterizing quantum systems and predicting many of their properties. Circuit cutting is a technique for dividing large quantum circuits into smaller fragments that can be executed more robustly using fewer quantum resources. We introduce a divide-and-conquer circuit cutting method for estimating the expectation values of observables using classical shadows. We derive a general formula for making predictions using the classical shadows of circuit fragments from arbitrarily cut circuits and provide the sample complexity analysis for the case when observables factorize across fragments. Then, we numerically show that our divide-and-conquer method outperforms traditional uncut shadow tomography when estimating high-weight observables that act non-trivially on many qubits and discuss the mechanisms for this advantage.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An Adaptive Framework for Extreme Deformation and Failure in Solids

Recent developments at Sandia in meshfree methods have delivered improved robustness in solid mechanics problems that prove difficult for traditional Lagrangian, mesh-based finite elements. Nevertheless, there remains a limitation in accurately predicting very large material deformations. It seems robust meshfree discretizations and integration schemes are necessary, but not sufficient, to close this capability gap. This state of affairs directly impacts current and future LEPs, whose simulation needs are not well met for extremely large deformation problems. We propose to use a new numerical framework, the Optimal Transportation Meshfree (OTM) method enhanced by meshfree adaptivity, as we believe that a combination of both will provide a novel way to close this capability gap.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Sierra/SolidMechanics 5.4 Capabilities in Development

This user’s guide documents capabilities in Sierra/SolidMechanics which remain “in-development” and thus are not tested and hardened to the standards of capabilities listed in Sierra/SM 5.4 User’s Guide. Capabilities documented herein are available in Sierra/SM for experimental use only until their official release. These capabilities include, but are not limited to, novel discretization approaches such as the conforming reproducing kernel (CRK) method, numerical fracture and failure modeling aids such as the extended finite element method (XFEM) and J-integral, explicit time step control techniques, dynamic mesh rebalancing, as well as a variety of new material models and finite element formulations.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

SIERRA/Aero Theory Manual - Version 5.8

SIERRA/Aero is a compressible fluid dynamics program intended to solve a wide variety compressible fluid flows including transonic and hypersonic problems. This document describes the commands for assembling a fluid model for analysis with this module, henceforth referred to simply as Aero for brevity. Aero is an application developed using the SIERRA Toolkit (STK). The intent of STK is to provide a set of tools for handling common tasks that programmers encounter when developing a code for numerical simulation. For example, components of STK provide field allocation and management, and parallel input/output of field and mesh data. These services also allow the development of coupled mechanics analysis software for a massively parallel computing environment.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Challenges for Implementing FAIR Digital Objects with High Performance Workflows

New types of workflows are being used in science that couple traditional distributed and high-performance computing (HPC) with data-intensive approaches, and orchestrate ensembles of numerical simulations and artificial intelligence (AI) models. Such workflows may use AI models to supplement computation where numerical simulations may be too computationally expensive, to automate trivial yet time consuming operations, to perform preliminary selections among intractable numbers of combinations in domains as diverse as protein binding, fine-grid climate simulations, and drug discovery.

97 MATHEMATICS AND COMPUTING↗