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DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Gaussian process hydrodynamics

Abstract We present a Gaussian process (GP) approach, called Gaussian process hydrodynamics (GPH) for approximating the solution to the Euler and Navier-Stokes (NS) equations. Similar to smoothed particle hydrodynamics (SPH), GPH is a Lagrangian particle-based approach that involves the tracking of a finite number of particles transported by a flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior ξ on the velocity field and conditioning it on the vorticity at the particle locations. Known physics (e.g., the Richardson cascade and velocity increment power laws) is incorporated into the GP prior by using physics-informed additive kernels. This is equivalent to expressing ξ as a sum of independent GPs ξ l , which we call modes, acting at different scales (each mode ξ l self-activates to represent the formation of eddies at the corresponding scales). This approach enables a quantitative analysis of the Richardson cascade through the analysis of the activation of these modes, and enables us to analyze coarse-grain turbulence statistically rather than deterministically. Because GPH is formulated by using the vorticity equations, it does not require solving a pressure equation. By enforcing incompressibility and fluid-structure boundary conditions through the selection of a kernel, GPH requires significantly fewer particles than SPH. Because GPH has a natural probabilistic interpretation, the numerical results come with uncertainty estimates, enabling their incorporation into an uncertainty quantification (UQ) pipeline and adding/removing particles (quanta of information) in an adapted manner. The proposed approach is suitable for analysis because it inherits the complexity of state-of-the-art solvers for dense kernel matrices and results in a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on the accuracy and stability. Because the proposed approach uses a Bayesian interpretation, it naturally enables data assimilation and predictions and estimations by mixing simulation data and experimental data.

Mathematics↗

DOE OSTI2023

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Microscale mechanical modeling of deformable geomaterials with dynamic contacts based on the numerical manifold method

Abstract Micromechanical modeling of geomaterials is challenging because of the complex geometry of discontinuities and potentially large number of deformable material bodies that contact each other dynamically. In this study, we have developed a numerical approach for micromechanical analysis of deformable geomaterials with dynamic contacts. In our approach, we detect contacts among multiple blocks with arbitrary shapes, enforce different contact constraints for three different contact states of separated, bonded, and sliding, and iterate within each time step to ensure convergence of contact states. With these features, we are able to simulate the dynamic contact evolution at the microscale for realistic geomaterials having arbitrary shapes of grains and interfaces. We demonstrate the capability with several examples, including a rough fracture with different geometric surface asperity characteristics, settling of clay aggregates, compaction of a loosely packed sand, and failure of an intact marble sample. With our model, we are able to accurately analyze (1) large displacements and/or deformation, (2) the process of high stress accumulated at contact areas, (3) the failure of a mineral cemented rock samples under high stress, and (4) post-failure fragmentation. The analysis highlights the importance of accurately capturing (1) the sequential evolution of geomaterials responding to stress as motion, deformation, and high stress; (2) large geometric features outside the norms (such as large asperities and sharp corners) as such features can dominate the micromechanical behavior; and (3) different mechanical behavior between loosely packed and tightly packed granular systems.

58 GEOSCIENCES↗

DOE OSTI2018

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In this work, we introduce a framework for modeling dynamic fracture problems using cohesive polygonal finite elements. Random polygonal meshes provide a robust, efficient method for generating an unbiased network of fracture surfaces. Further, these meshes have more facets per element than standard triangle or quadrilateral meshes, providing more possible facets per element to insert cohesive surfaces. This property of polygonal meshes is advantageous for the modeling of pervasive fracture. We use both Wachspress and maximum entropy shape functions to form a finite element basis over the polygons. Fracture surfaces are captured through dynamically inserted cohesive zone elements at facets between the polygons in the mesh. Contact is enforced through a penalty method that is applied to both closed cohesive surfaces and general interpenetration of two polygonal elements. Finally, several numerical examples are presented that illustrate the capabilities of the method and demonstrate convergence of solutions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

A smeared crack modeling framework accommodating multi-directional fracture at finite strains

A generic smeared crack modeling framework predicated on the deformation gradient decomposition (DGD) approach is proposed for use in dynamic fracture problems at finite strains, accommodating failure along multiple mutually orthogonal fracture planes embedded within an independently defined bulk material model. Within this constitutive framework, the traction equilibrium conditions imposed at each failure surface are used to determine the associated crack displacements stored as internal state variables. In general, the enforcement of interfacial equilibrium entails the implicit solution of a non-linear system of equations within the constitutive update procedure. However, if inertial effects arising due to the relative motion of the fractured material are incorporated within the model, the traction equilibrium conditions are shown to give rise to corresponding dynamic equations of motion governing the time-evolution of the crack opening displacements. For dynamic problems, an explicit time-integration procedure is devised to efficiently update the material state, subject to a set of internal frictionless contact constraints to prevent material inversion. Finally, the efficacy of the proposed modeling framework is investigated through several benchmark dynamic fracture problems run within the explicit finite element code DYNA3D.

42 ENGINEERING↗

DOE OSTI2025

A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations

We analyze the properties and compare the performance of several positivity limiters for spectral approximations with respect to the angular variable of linear transport equations. It is well-known that spectral methods suffer from the occurrence of (unphysical) negative spatial particle concentrations due to the fact that the underlying polynomial approximations are not always positive at the kinetic level. Positivity limiters address this defect by enforcing positivity of the polynomial approximation on a finite set of preselected points. With a proper PDE solver, they ensure positivity of the particle concentration at each step in a time integration scheme. We review several known positivity limiters proposed in other contexts and also introduce a modification for one of them. We give error estimates for the consistency of the positive approximations produced by these limiters and compare the theoretical estimates to numerical results. We then solve two benchmark problems with these limiters, make qualitative and quantitative observations about the solutions, and then compare the efficiency of the different limiters.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.

Mass conservation↗

DOE OSTI2018

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L{sup 2} space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L{sup 2} space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2026

Using Filter Methods to Guide Convergence for ADMM, with Applications to Nonnegative Matrix Factorization Problems

Nonconvex, nonlinear optimization problems arise naturally in parameter fitting and machine learning. While augmented Lagrangian methods have demonstrated robust convergence for classes of these problems, their convergence for block updates has been relatively unexplored outside of the context of the alternating direction method of multipliers (ADMM). ADMM has seen extensive use in these applications, but may exhibit uncertain convergence behavior in many practical nonconvex settings, and struggles with general nonlinear constraints. In contrast, filter methods have proved effective in enforcing convergence for sequential quadratic programming methods and interior point methods with feasibility criteria. We develop an ADMM-filter method for highly nonlinear and nonconvex problems. Here, we show convergence under mild assumptions for several types of coordinate descent schemes, and demonstrate our algorithm on nonnegative matrix factorization and completion problems in imaging and chemical spectrum analysis.

Nonconvex optimization↗

DOE OSTI2012

Intrinsic dosimetry of glass containers: A potential interrogation tool for nuclear forensics and waste management

Intrinsic dosimetry is the method of measuring total absorbed dose received by the walls of a container holding radioactive material. By considering the total absorbed dose received by a container in tandem with the physical characteristics of the radioactive material housed within that container, this method has the potential to provide law enforcement officials or waste management personnel with enhanced pathway information regarding the history of the container and its radioactive contents. We report the latest in a series of experiments designed to validate and demonstrate this newly developed tool. Thermoluminescence (TL) dosimetry was used to measure dose effects on the raw stock material of borosilicate container glass with time up to 60 days after gamma ray, x-ray, beta particle or ultraviolet irradiations at doses from 0.15 to 20 Gy. Two peaks were identified in the TL glow curve when irradiated with 60Co, a relatively unstable peak around 120°C and a more stable peak around 225°C. Signal strength of both peaks decayed with time. The minimum measurable dose using this technique was measured to be 0.15 Gy, which is roughly equivalent to a 24 hr irradiation at 1 cm from a 50 ng source of 60Co. After fading, this dose would still be observable approximately 1 year post-irradiation. Similar doses with beta particles and x-rays produced TL glow curves of lower intensity. The TL glow curves were separated into five peaks centered at approximately 120, 160, 225, 300, and 340°C. Differences in TL glow curve shape and intensity were observed for the glasses from different geographical origins. These differences can be explained by changes in the intensities of the five peaks. EPR results showed production of BOHC and E’ silica centers.

Clark, Richard A.↗

DOE OSTI2012

Xe-135 Production from Cf-252

135Xe is a good indicator that fission has occurred and is a valuable isotope that helps enforce the Comprehensive Test Ban Treaty. Due to its rather short half life and minimal commercial interest, there are no known sources where 135Xe can be purchased. Readily available standards of this isotope for calibrating collection and analytical techniques would be very useful. 135Xe can be produced in the fissioning of actinide isotopes, or by neutron capture on 134Xe. Since the neutron capture cross section of 134Xe is 3 mB, neutron capture is a low yield, though potentially useful, production route. 135Xe is also produced by spontaneous fission of 252Cf. 252Cf has a spontaneous fission rate of about 6 x 1011 s-1g-1. The cumulative yield from the spontaneous fission of 252Cf is 4.19%; and the competing neutron capture reaction that depletes 135Xe in thermal reactor systems is negligible because the neutron capture cross-section is low for fast fission neutrons. At the INL, scientists have previously transported fission products from an electroplated 252Cf thin source for the measurement of nuclear data of short-lived fission products using a technique called He-Jet collection. We have applied a similar system to the collection of gaseous 135Xe, in order to produce valuable standards of this isotope.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

DOE OSTI2016

The Materials Data Facility: Data Services to Advance Materials Science Research

With increasingly strict data management requirements from funding agencies and institutions, expanding focus on the challenges of research replicability, and growing data sizes and heterogeneity, new data needs are emerging in the materials community. The materials data facility (MDF) operates two cloudhosted services, data publication and data discovery, with features to promote open data sharing, self-service data publication and curation, and encourage data reuse, layered with powerful data discovery tools. The data publication service simplifies the process of copying data to a secure storage location, assigning data a citable persistent identifier, and recording custom (e.g., material, technique, or instrument specific)and automatically-extractedmetadata in a registrywhile the data discovery service will provide advanced search capabilities (e.g., faceting, free text range querying, and full text search) against the registered data and metadata. TheMDF services empower individual researchers, research projects, and institutions to (I) publish research datasets, regardless of size, from local storage, institutional data stores, or cloud storage, without involvement of thirdparty publishers; (II) build, share, and enforce extensible domain-specific custom metadata schemas; (III) interact with published data and metadata via representational state transfer (REST) application program interfaces (APIs) to facilitate automation, analysis, and feedback; and (IV) access a data discovery model that allows researchers to search, interrogate, and eventually build on existing published data. We describe MDF’s design, current status, and future plans.

data management↗

DOE OSTI2018

The influence of interval arithmetic on the shape of uncertainly defined domains modelled by closed curves

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C{sup 2} continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Modeling ash deposition and shedding during oxy-combustion of coal/rice husk blends at 70% inlet O2

Abstract Co-firing rice husk (RH) and coal with carbon capture using oxy-combustion presents a net carbon negative energy production opportunity. In addition, the high fusion temperature of the non-sticky, silica rich, RH can mitigate ash deposition as well as promote shedding of deposits. To identify the optimum operating conditions, fuel particle sizes, and blend ratios that minimize ash deposition, a Computational Fluid Dynamic methodology with add-on ash deposition and shedding models were employed to predict outer ash deposition and shedding rates during co-combustion of coal/RH in AIR and O 2 /CO 2 (70/30 vol%, OXY70) oxidizer compositions. After ensuring that the fly-ash particle size distributions and particle Stokes numbers near the deposition surface were accurately represented (to model impaction), appropriate models for coal ash and RH ash viscosities that were accurate in the temperature region (1200–1300 K) of interest in this study were identified. A particle viscosity and kinetic energy (PKE) based capture criterion was enforced to model the ash capture. An erosion/shedding criterion that takes the deposit melt fraction and the energy consumed during particle impact into account was also implemented. Deposition rate predictions as well as the deposition rate enhancement (OXY70/AIR) were in good agreement with measured values. While the OXY70 scenario was associated with a significant reduction (60%–70%) in flue gas velocities, it also resulted in larger fly-ash particles. As a result, the PKE distributions of the erosive RH ash were similar in both scenarios and resulted in similar shedding rates.

Energy & Fuels↗

DOE OSTI2024

Chapter 7: Learning Stable Local Volt/Var Controllers in Distribution Grids

This chapter describes a framework to synthesize provably stable local Volt/Var controllers for distributed energy resources (DERs) in power distribution grids (DGs). The goal is to control the reactive power injections of DERs to improve the system performance as quantified by a generic optimal reactive power flow (ORPF) problem. To achieve this, we jointly design for each DER the control function, which prescribes the reactive power update rule, and the equilibrium function, which approximates the ORPF solutions from local measurements of voltages and powers. We provide conditions on the equilibrium functions and the control parameters ensuring the stability of the closed-loop system. In particular, we discuss the trade-offs between each set of conditions accounting for practical considerations, like fully exploiting the DERs' generation capabilities and reducing the optimality gap. These conditions are then translated into learning constraints on the neural networks' parameters that are enforced in the training phase. We validate our framework with numerical simulations on the IEEE 37-bus network and through a comparison with an optimized version of standard piece wise linear control rules.

closed-loop asymptotic stability↗

DOE OSTI2020

Injury severity analysis of truck-involved crashes under different weather conditions

This paper investigates truck-involved crashes to determine the statistically significant factors that contribute to injury severity under different weather conditions. The analysis uses crash data from the state of Ohio between 2011 and 2015 available from the Highway Safety Information System. To determine if weather conditions should be considered separately for truck safety analyses, parameter transferability tests are conducted; the results suggest that weather conditions should be modeled separately with a high level of statistical confidence. To this end, three separate mixed logit models are estimated for three different weather conditions: normal, rain and snow. The estimated models identify a variety of statistically significant factors influencing the injury severity. Different weather conditions are found to have different contributing effects on injury severity in truck-involved crashes. Rural, rear-end and sideswipe crash parameters were found to have significantly different levels of impact on injury severity. Based on the findings of this study, several countermeasures are suggested: 1) safety and enforcement programs should focus on female truck drivers, 2) a variable speed limit sign should be used to lower speeds of trucks during rainy condition, and 3) trucks should be restricted or prohibited on non-interstates during rainy and snowy conditions. These countermeasures could reduce the number and severity of truck-involved crashes under different weather conditions.

60 APPLIED LIFE SCIENCES↗