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DOE OSTI2024

Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization

This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on prevalent applications in computational mechanics and design optimization. We leverage Matérn-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models as well as the shape and topology of optimized designs. Here we describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications, emphasizing the potential to influence these domains. The flexibility and efficiency of SPDE-based GRF generation empowers us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate design features and topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model and quantify geometric uncertainties on reconstructed submanifolds, such as the interpolated surfaces of cerebral aneurysms provided by postprocessing CT scans. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Distribution of rare earth and other critical elements in lignites from the Eocene Jackson Group, Texas

Coal is increasingly evaluated as a source of rare earth elements (REEs) in the United States to address the overreliance on imported REEs. The objective of this study was to assess the distribution of REEs in lignites from selected mining areas in the Texas Gulf Coastal Plain region. Thirty-one archived lignite and rock samples previously collected by the U.S. Geological Survey were analyzed for their rare earth element and critical mineral content. These include samples from one core (5400 and 5500 lignite horizons) and two opencast lignite mines (Gibbons Creek 3500 and 4500 horizons, and San Miguel horizons A to D) in the Eocene Jackson Group of the Texas Gulf of Mexico Coastal Plain. Some lithologies in the Gibbons Creek 3500 and 4500 lignite-bearing sections have high total rare earth, yttrium (Y), and scandium (Sc) (REYSc) values, up to 7800 ppm (ash basis) REYSc. The lignite lithologies show an enrichment in rare earths, [samarium (Sm) through gadolinium (Gd)]. The basal Gibbons Creek 3500 lignite bench shows a heavy rare earth element enrichment pattern resembling that often seen in peats through high volatile A bituminous coals. The 5500 lignite sequence, overlying the latter lignite sections, shows a light rare earth enrichment. The San Miguel lignite benches have heavy rare earth enrichments with a negative europium (Eu) anomaly.

58 GEOSCIENCES↗

DOE OSTI2025

Chemistry imaging and distribution analysis of rare earth elements in coal using LIBS and LA-ICP-MS instruments

Currently, demand for rare earth elements (REEs) increased significantly. Coal is actively evaluated as potential economic sources for extraction of REEs. Here, in this work, laser-induced breakdown spectroscopy (LIBS) was evaluated for rapid estimation of REEs content and their distribution in the natural coal samples. The results were compared with similar laser ablation–inductively coupled plasma–mass spectrometry (LA-ICP-MS) measurements. Thirteen coal samples (nine standard samples and five natural samples) were used in this study. Powder samples were pressed into pellets while coal chunks were directly ablated for data recording. Pellets of the powder standard samples were used to optimize the data acquisition system and then data recorded with this optimized system was used to identify the proper data acquisition and analysis models. After establishing the proper data acquisition system and analysis model using the standard samples, natural coal samples in powder form and their chunks were utilized to record LIBS and LA-ICP-MS spectra. Multivariate calibration models were developed using four of the natural samples, which were evaluated by predicting the REE content in the fifth sample. Principal component analysis was performed on the LIBS data obtained from the natural samples and it classified all the samples with high accuracy. Two-dimensional (2D) elemental mapping on coal chunk samples was also performed using both LIBS and LA-ICP-MS to study the distribution of REEs in the samples. The resulting elemental images and their correlations can be used to infer mineral distributions.

01 COAL, LIGNITE, AND PEAT↗

DOE OSTI2016

Partition of unity finite element method for quantum mechanical materials calculations

The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires significant nonlocal communications, which limit parallel efficiency. Real-space methods such as finite-differences (FD) and finite-elements (FE) have partially addressed both resolution and parallel-communications issues but have been plagued by one key disadvantage relative to PW: excessive number of degrees of freedom (basis functions) needed to achieve the required accuracies. In this paper, we present a real-space partition of unity finite element (PUFE) method to solve the Kohn–Sham equations of density functional theory. In the PUFE method, we build the known atomic physics into the solution process using partition-of-unity enrichment techniques in finite element analysis. The method developed herein is completely general, applicable to metals and insulators alike, and particularly efficient for deep, localized potentials, as occur in calculations at extreme conditions of pressure and temperature. Full self-consistent Kohn–Sham calculations are presented for LiH, involving light atoms, and CeAl, involving heavy atoms with large numbers of atomic-orbital enrichments. We find that the new PUFE approach attains the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the PW method. As a result, we compute the equation of state of LiH and show that the computed lattice constant and bulk modulus are in excellent agreement with reference PW results, while requiring an order of magnitude fewer degrees of freedom to obtain.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2015

Detection of rare earth elements in Powder River Basin sub-bituminous coal ash using laser-induced breakdown spectroscopy (LIBS)

We reported our preliminary results on the use of laser-induced breakdown spectroscopy to analyze the rare earth elements contained in ash samples from Powder River Basin sub-bituminous coal (PRB-coal). We have identified many elements in the lanthanide series (cerium, europium, holmium, lanthanum, lutetium, praseodymium, promethium, samarium, terbium, ytterbium) and some elements in the actinide series (actinium, thorium, uranium, plutonium, berkelium, californium) in the ash samples. In addition, various metals were also seen to present in the ash samples

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2018

Siderophile element constraints on the thermal history of the H chondrite parent body

The abundances of highly siderophile elements (HSE: Re, Os, Ir, Ru, Pt, Pd), as well as 187 Re- 187 Os and 182 Hf- 182 W isotopic systematics were determined for separated metal, slightly magnetic, and nonmagnetic fractions from seven H4 to H6 ordinary chondrites. The HSE are too abundant in nonmagnetic fractions to reflect metal-silicate equilibration. The disequilibrium was likely a primary feature, as 187 Re- 187 Os data indicate only minor open-system behavior of the HSE in the slightly and non-magnetic fractions. 182 Hf- 182 W data for slightly magnetic and nonmagnetic fractions define precise isochrons for most meteorites that range from 5.2 ± 1.6 Ma to 15.2 ± 1.0 Ma after calcium aluminum inclusion (CAI) formation. By contrast, 182W model ages for the metal fractions are typically 2–5 Ma older than the slope-derived isochron ages for their respective, slightly magnetic and nonmagnetic fractions, with model ages ranging from 1.4 ± 0.8 Ma to 12.6 ± 0.9 Ma after CAI formation. This indicates that the W present in the silicates and oxides was not fully equilibrated with the metal when diffusive transport among components ceased, consistent with the HSE data. Further, the W isotopic compositions of size-sorted metal fractions from some of the H chondrites also differ, indicating disequilibrium among some metal grains. The chemical/isotopic disequilibrium of siderophile elements among H chondrite components is likely the result of inefficient diffusion of siderophile elements from silicates and oxides to some metal and/or localized equilibration as H chondrites cooled towards their respective Hf-W closure temperatures. The tendency of 182 Hf- 182 W isochron ages to young from H5 to H6 chondrites may indicate derivation of these meteorites from a slowly cooled, undisturbed, concentrically-zoned parent body, consistent with models that have been commonly invoked for H chondrites. Overlap of isochron ages for H4 and H5 chondrites, by contrast, appear to be more consistent with shallow impact disruption models. The W isotopic composition of metal from one CR chondrite was examined to compare with H chondrite metals. In contrast to the H chondrites, the CR chondrite metal is characterized by an enrichment in 183 W that is consistent with nucleosynthetic s-process depletion. Once corrected for the correlative nucleosynthetic effect on 182 W, the 182 W model age for this meteorite of 7.0 ± 3.6 Ma is within the range of model ages of most metal fractions from H chondrites. Finally, the metal is therefore too young to be a direct nebular condensate, as proposed by some prior studies.

58 GEOSCIENCES↗

DOE OSTI2013

A 2.5D boundary element formulation for modeling damped waves in arbitrary cross-section waveguides and cavities

Highlights: •Dispersive properties of viscoelastic waveguides and cavities are computed using a regularized 2.5D BEM. •Linear viscoelasticity is introduced at the constitutive level by means of frequency dependent complex moduli. •A contour integral algorithm is used to solve the nonlinear eigenvalue problem. •The Sommerfeld radiation condition is used to select the permissible Riemann sheets. •Attenuation of surface waves in cavities approaches the attenuation of Rayleigh waves. -- Abstract: A regularized 2.5D boundary element method (BEM) is proposed to predict the dispersion properties of damped stress guided waves in waveguides and cavities of arbitrary cross-section. The wave attenuation, induced by material damping, is introduced using linear viscoelastic constitutive relations and described in a spatial manner by the imaginary component of the axial wavenumber. The discretized dispersive wave equation results in a nonlinear eigenvalue problem, which is solved obtaining complex axial wavenumbers for a fixed frequency using a contour integral algorithm. Due to the singular characteristics and the multivalued feature of the wave equation, the requirement of holomorphicity inside the contour region over the complex wavenumber plane is fulfilled by the introduction of the Sommerfeld branch cuts and by the choice of the permissible Riemann sheets. A post processing analysis is developed for the extraction of the energy velocity of propagative guided waves. The reliability of the method is demonstrated by comparing the results obtained for a rail and a bar with square cross-section with those obtained from a 2.5D Finite Element formulation also known in literature as Semi Analytical Finite Element (SAFE) method. Next, to show the potential of the proposed numerical framework, dispersion properties are predicted for surface waves propagating along cylindrical cavities of arbitrary cross-section. It is demonstrated that the attenuation of surface waves approaches asymptotically the attenuation of Rayleigh waves.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

All-electron Kohn–Sham density functional theory on hierarchic finite element spaces

In this work, a real space formulation of the Kohn–Sham equations is developed, making use of the hierarchy of finite element spaces from different polynomial order. The focus is laid on all-electron calculations, having the highest requirement onto the basis set, which must be able to represent the orthogonal eigenfunctions as well as the electrostatic potential. A careful numerical analysis is performed, which points out the numerical intricacies originating from the singularity of the nuclei and the necessity for approximations in the numerical setting, with the ambition to enable solutions within a predefined accuracy. In this context the influence of counter-charges in the Poisson equation, the requirement of a finite domain size, numerical quadratures and the mesh refinement are examined as well as the representation of the electrostatic potential in a high order finite element space. The performance and accuracy of the method is demonstrated in computations on noble gases. In addition the finite element basis proves its flexibility in the calculation of the bond-length as well as the dipole moment of the carbon monoxide molecule.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD

The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems, including the evolution of complex plasma dynamics in tokamak disruptions. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Additionally, therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies is of considerable importance. In this work, we develop a high-order stabilized finite-element algorithm for the reduced visco-resistive MHD equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the Jacobian-free Newton-Krylov (JFNK) method with a physics-based preconditioning strategy. Our preconditioning strategy is a generalization of the physics-based preconditioning methods in Chacón et al. (2002) to adaptive, stabilized finite elements. Algebraic multigrid methods are used to invert sub-block operators to achieve scalability. A parallel adaptive mesh refinement scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. Our implementation uses the MFEM framework, which provides arbitrary-order polynomials and flexible adaptive conforming and non-conforming meshes capabilities. Results demonstrate the accuracy, efficiency, and scalability of the implicit scheme in the presence of large scale disparity. The potential of the AMR approach is demonstrated on an island coalescence problem in the high Lundquist-number regime (≥ 10 7 ) with the successful resolution of plasmoid instabilities and thin current sheets.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Multi-element flow-driven spectral chaos (ME-FSC) method for uncertainty quantification of dynamical systems

The flow-driven spectral chaos (FSC) is a recently developed method for tracking and quantifying uncertainties in the long-time response of stochastic dynamical systems using the spectral approach. The method uses a novel concept called enriched stochastic flow maps as a means to construct an evolving finite-dimensional random function space that is both accurate and computationally efficient in time. In this paper, we present a multi-element version of the FSC method (the ME-FSC method for short) to tackle (mainly) those dynamical systems that are inherently discontinuous over the probability space. In ME-FSC, the random domain is partitioned into several elements, and then the problem is solved separately on each random element using the FSC method. Subsequently, results are aggregated to compute the probability moments of interest using the law of total probability. To demonstrate the effectiveness of the ME-FSC method in dealing with discontinuities and long-time integration of stochastic dynamical systems, four representative numerical examples are presented in this paper, including the Van-der-Pol oscillator problem and the Kraichnan-Orszag three-mode problem. Results show that the ME-FSC method is capable of solving problems that have strong nonlinear dependencies over the probability space, both reliably and at low computational cost.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

A face-upwinded spectral element method

Here we present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A method for bounding high-order finite element functions: Applications to mesh validity and bounds-preserving limiters

We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be used to locally bound any combination of these basis functions. This approach can be applied to any element/basis type at any approximation order, can provide local (i.e., subcell) extremum bounds to a desired level of accuracy, and can be evaluated efficiently on-the-fly in simulations. Furthermore, we show that this approach generally yields more accurate bounds in comparison to traditional methods based on convex hull properties (e.g., Bernstein polynomials). Furthermore, the efficacy of this technique is shown in applications such as mesh validity checks and optimization for high-order curved meshes, where positivity of the element Jacobian determinant can be ensured throughout the entire element, and continuously bounds-preserving limiters for hyperbolic systems, which can enforce maximum principle bounds across the entire solution polynomial.

Bounding box↗

DOE OSTI2018

A discrete element method representation of an anisotropic elastic continuum

A method for modeling cubically anisotropic elasticity within the discrete element method is presented. The discrete element method (DEM) is an approach originally intended for modeling granular materials (sand, soil, and powders); however, recent developments have usefully extended it to model stochastic mechanical processes in monolithic solids which, to date, have been assumed to be elastically isotropic. The method presented here for efficiently capturing cubic elasticity in DEM is an important prerequisite for further extending DEM to capture the influence of elastic anisotropy on the mechanical response of polycrystals, composites, etc. The system demonstrated here uses a directionally assigned stiffness in the bonds between adjacent elements and includes separate schemes for achieving anisotropy with Zener ratios greater and smaller than one. Finally, the model framework is presented along with an analysis of the accessible space of elastic properties that can be modeled and an artificial neural network interpolation scheme for mapping input parameters to model elastic behavior.

42 ENGINEERING↗

DOE OSTI2017

Insight into γ-Ni/γ'-Ni 3 Al interfacial energy affected by alloying elements

Interfacial energy (σ γ/γ' ) of the γ-Ni/γ'-Ni 3 Al interface is critical to understand and design Ni-based superalloys. Here, in the present work, a first-principles methodology with constrained relaxations has been used to study the effect of dilute alloying element (X) on σ γ/γ' of the (100) coherent interface, where the sixteen X's include Al, Co, Cr, Fe, Hf, Mo, Nb, Pd, Pt, Re, Ru, Ta, Ti, W, Y, and Zr. σ γ/γ' = 19 mJ/m 2 has been predicted for the unalloyed γ/γ' interface, agreeing well with the previous estimations. It is found that the σ γ/γ' value is the lowest when alloying element is in γ-Ni, while the addition to γ'-Ni 3 Al increases σ γ/γ' due mainly to the in-plane lattice expansion. Specifically, our calculations show that alloying elements Mo, W, and Re have the largest effect on the σ γ/γ' value by decreasing it to 4–5 mJ/m 2 when partitioned to γ-Ni; Ru and Pt are shown to increase greatly the σ γ/γ' value up to 25–28 mJ/m 2 when partitioned to γ-Ni; all ternary additions substituting for Ni in γ'-Ni 3 Al increase σ γ/γ' except for Pt; and for Ti and Ta, which segregate strongly to γ'-Ni 3 Al and substitute for Al, the σ γ/γ' values increase to 32 and 35 mJ/m 2 , respectively.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Acceleration of matrix element computations for precision measurements

The matrix element technique provides a superior statistical sensitivity for precision measurements of important parameters at hadron colliders, such as the mass of the top quark or the cross-section for the production of Higgs bosons. The main practical limitation of the technique is its high computational demand. Using the example of the top quark mass, we present two approaches to reduce the computation time of the technique by a factor of 90. First, we utilize low-discrepancy sequences for numerical Monte Carlo integration in conjunction with a dedicated estimator of numerical uncertainty, a novelty in the context of the matrix element technique. We then utilize a new approach that factorizes the overall jet energy scale from the matrix element computation, a novelty in the context of top quark mass measurements. The utilization of low-discrepancy sequences is of particular general interest, as it is universally applicable to Monte Carlo integration, and independent of the computing environment.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2015

Finite Element Based Stress Analysis of Graphite Component in High Temperature Gas Cooled Reactor Core Using Linear and Nonlinear Irradiation Creep Models

Irradiation creep plays a major role in the structural integrity of the graphite components in high temperature gas cooled reactors. Finite element procedures combined with a suitable irradiation creep model can be used to simulate the time-integrated structural integrity of complex shapes, such as the reactor core graphite reflector and fuel bricks. In the present work a comparative study was undertaken to understand the effect of linear and nonlinear irradiation creep on results of finite element based stress analysis. Numerical results were generated through finite element simulations of a typical graphite reflector.

High temperature gas cooled reactor, Finite elemen↗

DOE OSTI2020

CU-BENs: A structural modeling finite element library

The present work discusses capabilities within the finite element library CU-BENs. CU-BENs focuses on applying the finite element method to structural mechanics problems encountered within the context of inverse problems and partitioned fluid–structure interaction; thus element formulations are primarily of a structural type — truss, frame, and triangular discrete Kirchhoff theory shells. CU-BENs defaults to the skyline sparse storage scheme for the system matrix, but also supports other storage schemes when using external libraries such as LAPACK and UMFPACK. CU-BENs includes built-in nonlinear solution strategies, such as the Newton Raphson method and modified spherical arc length method, that are available within static analyses as well within the context of transient dynamic analyses involving a generalized-α implementation of the Newmark implicit time integration scheme.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Enhancing the Quantification of Critical Elements in WTE and Coal Ash via Alkaline Fusion: Superiority of Lithium Metaborate (LiBO 2 )

Ashes generated from coal combustion, as well as waste incineration, can be a potential source of critical elements necessary for the ongoing transition towards electrification and greener energy technologies. For the quantification of critical elements, traditional methods such as acid digestion are time-intensive and can fail to dissolve critical elements in refractory minerals. One potential solution is to adopt alkaline fusion for faster, total digestion. However, the role of flux choice and the subsequent digestion efficiency (DE) is unknown. Here, we report a systematic investigation on the feasibility of alkaline fusion with Lithium Metaborate (LiBO 2 ) and Lithium Tetraborate (Li 2 B 4 O 7 ) as fluxes for the digestion of two standard reference materials (BCR 176R and SRM 1633c). Our findings suggest that LiBO 2 yields higher DE values than Li 2 B 4 O 7 for several critical REEs and volatiles, such as Pb and Cd. Specifically, for REEs in SRM 1633c, the DE values with LiBO 2 are, on average, ~16 percent higher than those with Li 2 B 4 O 7 . Similarly, for Pb and Cd in BCR 176R, the DE values with LiBO 2 are ~20 percent higher than with Li 2 B 4 O 7 . These results suggest that LiBO 2 is a superior flux for rapid ash digestion.

01 COAL, LIGNITE, AND PEAT↗