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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.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

On Residual Stresses and Reference Temperatures in Thermomechanical Simulations of Photovoltaic Modules Using the Finite Element Method

Thermomechanical simulation of photovoltaic (PV) modules using the finite element method (FEM) is a useful tool to evaluate module design features in terms of structural integrity, reliability, and durability. One of the main challenges in the numerical modeling of a PV module is the incorporation of residual stresses induced by the manufacturing process. Modeling assumptions and abstractions are necessary to limit the model complexity and reduce the computational time. However, oversimplifications and incorrect assumptions can lead to erroneous numerical results. Unfortunately, much simulation work still neglects process-induced stresses. This can lead to incorrect predictions of the stress-strain history and erroneous conclusions during the design process. In this work, we review current modeling practices for incorporating process-induced stresses, and contrast numerical models that consider residual stresses with those that neglect them. We find that the simulation objective and available material properties dictate which process steps need to be modeled, and explore in depth the modeling of residual stresses induced by the lamination process. We demonstrate that a simplified cooldown procedure at the beginning of the simulation can increase the model accuracy and discuss appropriate choices for starting and reference temperatures in the finite element model.

14 SOLAR ENERGY↗
DOE OSTI2021

Time integrator agnostic charge conserving finite element PIC

Developing particle-in-cell (PIC) methods using finite element basis sets, and without auxiliary divergence cleaning methods, was a longstanding problem until recently. It was shown that if consistent spatial basis functions are used, one can indeed create a methodology that was charge conserving, albeit using a leapfrog time stepping method. While this is a significant advance, leapfrog schemes are only conditionally stable and time step sizes are closely tied to the underlying mesh. Ideally, to take full advantage of advances in finite element methods (FEMs), one needs a charge conserving PIC methodology that is agnostic to the time stepping method. This is the principal contribution of this paper. In what follows, we shall develop this methodology, prove that both charge and Gauss’ laws are discretely satisfied at every time step, provide the necessary details to implement this methodology for both the wave equation FEM and Maxwell solver FEM, and finally demonstrate its efficacy on a suite of test problems. The method will be demonstrated by single particle evolution, non-neutral beams with space-charge, and adiabatic expansion of a neutral plasma, where the Debye length has been resolved, and real mass ratios are used.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Comparative analysis of plasticity-based GND density estimation methods in crystal plasticity finite element models

In crystal plasticity finite element (CPFE) simulations, accurately quantifying geometrically necessary dislocations (GNDs) is critical for capturing strain gradients in polycrystals. We compare different methods for quantifying GNDs, all of which originate from the Nye tensor, which is computed as the curl of the plastic deformation gradient. The projection technique directly decomposes the Nye tensor onto individual screw and edge dislocation components to compute GNDs. This approach requires converting a nine-component Nye tensor into densities for a larger number of dislocation systems, a fundamentally underdetermined (non-unique) process, which is resolved using L2 minimization. In contrast, when employing CPFE analysis, one could directly compute dislocation densities on each slip system using shear gradients. Projection and slip gradient methods are compared with respect to their prediction of GNDs with changing grain size, strain, and grain neighborhoods, including multigrain junctions. Although these techniques match analytical GND densities for single slip, single crystal deformation, and are consistent with anticipated overall GND trends, we find that the GND densities from projection techniques are significantly lower than those predicted from CPFE-based slip gradients in polycrystals. A suggested improvement of only using the active dislocation systems in the projection technique almost entirely resolved this mismatch.

Crystal plasticity↗
DOE OSTI2022

Droplet formation simulation using mixed finite elements

Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Parallel projection—An improved return mapping algorithm for finite element modeling of shape memory alloys

Here, we present a novel finite element analysis of inelastic structures containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for SMAs lead to material nonlinearities, that require substantial computational effort to resolve. Finite element analysis methods, which rely on Gauss quadrature integration schemes, must solve two sets of coupled differential equations: one at the global level and the other at the local, i.e. Gauss point level. In contrast to the conventional return mapping algorithm, which solves these two sets of coupled differential equations separately using a nested Newton procedure, we propose a scheme to solve the local and global differential equations simultaneously. In the process we also derive closed-form expressions used to update the internal/constitutive state variables, and unify the popular closest-point and cutting plane methods with our formulas. Numerical testing indicates that our method allows for larger thermomechanical loading steps and provides increased computational efficiency, over the standard return mapping algorithm.

42 ENGINEERING↗
DOE OSTI2020

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L 2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H -1 -type norm, but not L 2 -coercive. Nevertheless, the L 2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L 2 norm, or by properly choosing the finite element spaces, providing implicit control of the L 2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L 2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. Finally, we believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Finite Element Method for Compressible and Turbulent Multiphase Flow Instabilities with Heat Transfer

We present a new finite element framework for modeling compressible, turbulent multiphase flows with heat transfer. For two-fluid systems with a free surface, the Volume of Fluid (VOF) method is implemented without the need for interface reconstruction, while turbulence is resolved using a dynamic Vreman large eddy simulation (LES) model. Unlike most two-phase VOF studies, which neglect heat transfer, the present approach incorporates energy transport equations within the VOF formulation to account for heat exchange, an effect particularly important in turbulent flows. Conjugate heat transfer is often challenging in finite volume methods, which require explicit specification of heat fluxes at the solid–fluid interface, limiting accuracy and predictive capability. By contrast, the finite element formulation does not require heat flux inputs, allowing more accurate and robust simulation of heat transfer between solids and fluids. The method is demonstrated through three representative cases. First, a two-fluid instability with a single-mode perturbation is simulated and validated against analytical growth rates. Second, conjugate heat transfer is examined in a high-temperature flow over a cold metal cylinder, with validation performed both quantitatively—via pressure coefficient comparisons with experimental data—and qualitatively using vector field topology. Finally, compressible spray injection and breakup are modeled, demonstrating the ability of the framework to capture interfacial dynamics and atomization under turbulent, high-speed conditions. In the compressible spray injection and breakup case, the results indicate that the finite element formulation achieved higher predictive accuracy and robustness than the finite-volume method. With the same mesh resolution, the FEM reduced the root mean square error (RMSE) and mean absolute percentage error (MAPE) from 6.96 mm and 26.0% (for the FVM) to 4.85 mm and 12.7%, respectively, demonstrating improved accuracy and robustness in capturing interfacial dynamics and heat transfer. The study also introduced vector field topology to visualize and interpret coherent flow structures and instabilities, offering insights beyond conventional scalar-field analyses.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients

In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite element framework’s discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

convection–diffusion–reaction equation↗
DOE OSTI2020

An implementation of the phase-field model based on coupled thermomechanical finite element solvers for large-strain twinning, explicit dynamic fracture and the classical Stefan problem

The implementation of a phase-field model in finite elements usually requires significant expertise and involves the development of a user element with additional degrees of freedom. An alternative implementation of the phase-field model within a thermo-mechanical finite element simulation package was presented in (Cho et al 2012 Int. J. Solids Struct. 49 1973–1992), where the phase-field variable is treated as the temperature degree of freedom. However, this approach has only been used for small strain phase-field modelling of martensitic transformations and quasistatic phase-field modelling of fracture. Here, we present a phase-field finite element implementation via the temperature degree of freedom for several additional cases from the literature: (i) the large-strain phase-field description of deformation twinning presented in (Clayton and Knap 2011 Physica D 240 841–858), (ii) phase-field description of brittle fracture with inertial effects based on the theory from (Molnár and Gravouil 2017, Finite Elem. Anal. Des. 130 27–38) and (Miehe et al 2010 Int. J. Numer. Methods Eng. 83 1273–1311) and (iii) the classical Stefan problem of solidification presented in (Mackenzie and Robertson 2002 J. Comput. Phys. 181 526–544). The last problem involves the temperature and phase-field variables as unknowns.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Smoothers for Matrix-Free Algebraic Multigrid Preconditioning of High-Order Finite Elements

We investigate smoothers for use in matrix-free algebraic multigrid (AMG) preconditioning of high-order finite element problems. These AMG preconditioners are matrix-free in the sense that they are built from a related low-order refined finite element problem whose system matrix can be much more rapidly assembled than the high-order problem. Our proposed smoother, which we call distributive relaxation, is more robust to the anisotropy present in many low-order refined meshes which feature a clustering of nodes near the boundaries between high-order finite elements. For solving the low-order refined problem, we show that this new distributive relaxation smoother possesses significantly improved performance compared to more traditional smoothers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Neural network ensembles and uncertainty estimation for predictions of inelastic mechanical deformation using a finite element method-neural network approach

The finite element method (FEM) is widely used to simulate a variety of physics phenomena. Approaches that integrate FEM with neural networks (NNs) are typically leveraged as an alternative to conducting expensive FEM simulations in order to reduce the computational cost without significantly sacrificing accuracy. However, these methods can produce biased predictions that deviate from those obtained with FEM, since these hybrid FEM-NN approaches rely on approximations trained using physically relevant quantities. In this work, an uncertainty estimation framework is introduced that leverages ensembles of Bayesian neural networks to produce diverse sets of predictions using a hybrid FEM-NN approach that approximates internal forces on a deforming solid body. The uncertainty estimator developed herein reliably infers upper bounds of bias/variance in the predictions for a wide range of interpolation and extrapolation cases using a three-element FEM-NN model of a bar undergoing plastic deformation. This proposed framework offers a powerful tool for assessing the reliability of physics-based surrogate models by establishing uncertainty estimates for predictions spanning a wide range of possible load cases.

42 ENGINEERING↗
DOE OSTI2023

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Quasi-Helmholtz decomposition, Gauss' laws and charge conservation for finite element particle-in-cell

Development of particle-in-cell (PIC) methods using finite element based methods (FEMs) have been a topic of renewed interest; this has largely been driven by (a) the ability of finite element methods to better model geometry, (b) better understanding of function spaces that are necessary to represent all Maxwell quantities, and (c) more recently, the fundamental rubrics that should be obeyed in space and time so as to satisfy Gauss' laws and the equation of continuity. In that vein, methods have been developed recently that satisfy these equations and are agnostic to time stepping methods. While this development is indeed a significant advance, it should be noted that implicit FEM transient solvers support an underlying null space that corresponds to a gradient of a scalar potential ∇Φ(r) (or t∇Φ(r) in the case of wave equation solvers). While explicit schemes do not suffer from this drawback, they are only conditionally stable with time step sizes that are mesh dependent and very small. Furthermore, the null space produces spurious charge that can corrupt the desired physics of a PIC simulation. The way to overcome this bottleneck, and indeed, satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation on a tessellation. In the re-formulation presented, we strictly satisfy the equation of continuity and Gauss' laws for both the electric and magnetic flux densities. Results illustrating the efficacy of this scheme will be demonstrated by analyzing non-neutral beams with space-charge and the adiabatic expansion of a neutral plasma with realistic parameters (Debye length and real mass ratios).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Finite Element Modeling of Spent Nuclear Fuel in Dry Storage under Seismic Excitation

Spent nuclear fuel (SNF) is currently stored at independent spent fuel storage installations (ISFSIs) at nuclear power plant sites across the US. The US Department of Energy (DOE) Spent Fuel and Waste Science and Technology (SFWST) program is currently working to understand the response of SNF during a seismic event. This investigation is part of a long-term effort by DOE to quantify the external mechanical loads on SNF. The SFWST program has recently completed campaigns evaluating SNF behavior during normal conditions of transport and in 30 cm package drop scenarios; both campaigns included a combination of testing and modeling. Likewise, the investigation into seismic response of SNF will include full-size shake table tests of a vertical storage overpack containing instrumented surrogate fuel assemblies, as well as pre- and post-test finite element modeling. This work focuses on the development of finite element models simulating the planned test configuration. The seismic response is calculated at two scales: at the package level, and at the fuel level. The package level model evaluates the behavior of the storage overpack and its contents in response to the ground motions of a representative earthquake. The motion of the package is input to the fuel level model, which consists of a detailed finite element model of a 17x17 or 16x16 pressurized water reactor (PWR) fuel assembly or of a 10x10 boiling water reactor (BWR) fuel assembly. The stress and strain on individual components such as fuel rods and guide tubes are extracted from this model. This work details the SNF response to various input ground motions, considering representative earthquakes in the Central and Eastern US under various site conditions. Also discussed is the development of a “reduced scope” finite element model of the SNF, which consists of a single fuel rod while retaining the dynamic response of the entire fuel assembly. This reduced scope model addresses the significant computational expense of modeling the SNF under long-duration input ground motions and facilitates simulating additional parameter variations of the model beyond the as-tested configuration. The modeling results provide important insight that will help to inform future testing and modeling efforts by the SFWST program to close the knowledge gap related to external loads on SNF.

Kadooka, Kevin↗
DOE OSTI2021

Modeling of High-Temperature Corrosion of Zirconium Alloys Using the eXtended Finite Element Method (X-FEM)

Oxidation modeling in modern nuclear fuel performance codes is currently limited by the lack of coupling with mechanics, thus preventing proper description of how high-temperature oxidation impacts mechanical properties. This is mostly due to the fact that the finite difference formalism adopted in corrosion models is incompatible with the direct coupling with mechanics in the finite element modeling employed in modern nuclear fuel performance codes. In this study, a physically based zirconium alloy corrosion model called the Coupled-Current Charge Compensation (C4) model, which was initially developed for operating temperature conditions, has been updated to include high-temperature corrosion in order to provide additional critical information (e.g., oxygen concentration profile) under loss-of-coolant accident (LOCA) conditions—information lacking in existing empirical models. The C4 model was implemented in the MOOSE finite-element framework developed at Idaho National Laboratory, enabling it to be used in the BISON nuclear fuel performance code based on the MOOSE framework. To precisely track the different interfaces at a relatively low computational cost, the eXtended Finite Element Method (X-FEM) was applied in MOOSE. The model’s results were compared to those of existing empirical models as well as metallographic analysis of high-temperature oxidized Zircaloy-4 coupons. Oxygen diffusivities in the a and ß phases resulting from this comparison closely agree with those found in the literature. The C4 model implemented with X-FEM in MOOSE now has the capability to accurately predict oxide, oxygen-stabilized a, and prior ß phase layer growth kinetics under isothermal exposure at high temperature (1000–1500°C). Furthermore, in contrast with the empirical models, the C4 model accounts for the finite thickness of the fuel cladding. It can predict the oxygen concentration profile evolution through the whole cladding, enabling evaluation of the remaining ductile thickness—a crucial variable for modeling the mechanical behavior of the fuel cladding under LOCA. Furthermore, this implementation allows direct coupling with mechanics, at a low computing cost, using finite-element-based nuclear fuel performance codes such as BISON.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Efficient quadrature rules for finite element discretizations of nonlocal equations

In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first showthat the “mollified” solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two- and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient finite element assembly.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A unified framework of stabilized finite element method for solving the Boltzmann transport equation

This paper presents a unified framework of stabilized finite element method for solving the Boltzmann transport equation. Unlike the traditional Petrov-Galerkin finite element method which modifies the test function to construct the stabilization term, we derive the stabilization methods from the standard Galerkin weak form with Sub-grid scale model. The basic idea of this method is to decompose the unknowns into its numerical solution and residual, with an approximation for the residual and embeds it in the Galerkin weak form to yield a stabilized variational formula. Different approximations of the residual lead to different stabilization methods, all the frequently used stabilized methods, including the Streamline Upwinding Petrov-Galerkin (SUPG) method, Galerkin/Least-Square (GLS) method, and Algebraic Sub-Grid Scale (ASGS) method can be obtained from this framework. The similarities and differences of the different approximations are compared in this paper. The numerical results show that the behaviors of the different methods area similar with the same stabilization parameter, and all these stabilized techniques can obtain a correct and stable solution. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
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↗