Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “implicit and explicit algorithms”

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 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

For decades, the Vlasov–Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. We explore a fully implicit PIC algorithm for the Vlasov–Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space–time-centered, employing particle sub-cycling and orbit-averaging. This algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. Finally, we demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D–3V.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

On numerical energy conservation for an implicit particle-in-cell method coupled with a binary Monte-Carlo algorithm for Coulomb collisions

We report conventional particle-in-cell (PIC) methods suffer from enhanced numerical heating (explicit PIC) or cooling (semi-implicit PIC) when coupled with a binary Monte-Carlo algorithm for Coulomb collisions. In this work, a fully-implicit θ-PIC scheme (with adjustable time-biasing parameter 1/2 ≤ θ ≤) is considered. The discrete change in energy of a closed system after a time step for this scheme scales with (1/2 - θ)C θ , where C θ is a positive definite quantity that depends on the frequency spectrum of the energy in the fields. Collisions lead to additional energy in the field fluctuations associated with high-frequency light waves produced by a numerical Bremsstrahlung process, which can result in a large increase in the numerical cooling rate for θ > 1/2. However, for θ = 1/2, energy is exactly conserved. The energy in the field fluctuations on long time scales agrees with that calculated using the equipartition theorem for a classical system in thermodynamic equilibrium.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Implicit fast sweeping method for hyperbolic systems of conservation laws

Implicit time-accurate methods are often used to integrate stiff problems where explicit schemes impose severe time step restrictions. This paper presents an efficient numerical framework based on the Fast Sweeping Method (FSM) for solving linear and nonlinear hyperbolic systems of conservation laws. The solution at each discrete location is computed by sweeping the numerical domain in several predetermined directions that follow the causality of the characteristic families. The use of a fractional step strategy eliminates the need for a solution selection criterion while one-sided stencils limit the number of sweeps to at most 2 d for d space dimensions. This work focuses on the first-order implicit upwind method since it constitutes the building block for high-order conservative schemes. For problems where the degree of stiffness evolves over time, implicit-explicit hybridization can be accomplished with the same algorithm by simply switching the stencil at each time level. As opposed to traditional implicit solvers, the sweeping method does not require a local time linearization of the fluxes thereby preserving the nonlinear stability properties of the original implicit scheme. It also avoids the large computational and memory requirements associated with solving large block-diagonal systems of equations. Here, a series of one- and two-dimensional test cases are presented for the inviscid Burgers' equation and the reactive Euler equations. The results indicate that the implicit FSM can allow a major reduction in the number of time steps even in the presence of discontinuous solution profiles.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

Physics constrained learning for data-driven inverse modeling from sparse observations

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that measures the discrepancy between predictions and observations in some chosen norm. This loss function often includes the PDE constraints as a penalty term when only sparse observations are available. As a result, the PDE is only satisfied approximately by the solution. However, the penalty term typically slows down the convergence of the optimizer for stiff problems. We present a new approach that trains the embedded DNNs while numerically satisfying the PDE constraints. We develop an algorithm that enables differentiating both explicit and implicit numerical solvers in reverse-mode automatic differentiation. This allows the gradients of the DNNs and the PDE solvers to be computed in a unified framework. We demonstrate that our approach enjoys faster convergence and better stability in relatively stiff problems compared to the penalty method. Furthermore, our approach allows for the potential to solve and accelerate a wide range of data-driven inverse modeling, where the physical constraints are described by PDEs and need to be satisfied accurately.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Metrics for the Complexity of Material Models

Quantitative measures are proposed for characterizing the complexity of material models used in computational mechanics. The algorithms for evaluating these metrics operate on the mathematical equations in the model rather than a code implementation and are different from software complexity measures. The metrics do not rely on a physical understanding of the model, using instead only a formal statement of the equations. A new algorithm detects the dependencies, whether explicit or implicit, between all the variables. The resulting pattern of dependencies is expressed in a set of pathways, each of which represents a chain of dependence between the variables. These pathways provide the raw data used in the metrics, which correlate with the expected ease of understanding, coding, and applying the model. Usage of the ComplexityMetrics code is described, with examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Semi-implicit method to solve compressible multiphase fluid flows without acoustic time step restrictions

A new 3D multiphase numerical capability is presented here for simulating multiphase flow regimes at all Mach numbers (M). The new method is a semi-implicit extension of the finite volume discrete equation method (DEM) of Chinnaya et al. (2004), which originally used explicit time-stepping. The capability is also developed to work with another extension of the DEM to moving grids for arbitrary Lagrangian-Eularian (ALE) methods detailed in Dunn (2011). Rather than solving all phase equations simultaneously, the DEM reduces the equations to a system of single-phase Riemann solves, where each phase has its own velocity and thermodynamic state. Exchanges between the phases are modeled through source terms accounting for the phase interactions. Furthermore, since the original multiphase scheme uses an explicit time-advancement scheme, it has time step restrictions dictated by the speed of sound, which limits the model's ability to simulate weakly compressible flows which typically need to be integrated for longer time periods. Here, we extend the current multiphase formulation by implementing a pressure-correcting step to enable implicit calculations and remove acoustic time step limitations. The new semi-implicit algorithm allows use of relatively large time steps compared to an explicit method. Validation and benefits of the new implicit time-step method are illustrated using several examples including weakly compressible flows and strong shock waves. The scheme presented here is general and may be used for a variety of applications which require capabilities for handling multiphase flow at a wide range of Mach numbers. However, the goal of this research is to simulate all stages of high energy explosions, including the shock formation (high Mach numbers) and evolution of the buoyant cloud (low Mach numbers).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Implicit-Explicit Multirate Infinitesimal GARK Methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called infinitesimal formulation for the fast time scale through definition of a sequence of modified “fast" initial-value problems that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Semi-implicit time integration of atmospheric flows with characteristic-based flux partitioning

Here, this paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge-Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO

In this work, we provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
DOE OSTI2026

thornado+FLASH-X: A Hybrid Discontinuous Galerkin–Implicit-explicit and Finite-volume Framework for Neutrino-radiation Hydrodynamics in Core-collapse Supernovae

We present neutrino-transport algorithms implemented in the toolkit for high-order neutrino-radiation hydrodynamics (thornado) and their coupling to self-gravitating hydrodynamics within the adaptive mesh refinement–based multiphysics simulation framework FLASH-X. thornado, developed primarily for simulations of core-collapse supernovae (CCSNe), employs a spectral, six-species two-moment formulation with algebraic closure and special-relativistic observer corrections accurate to $\mathcal{O}(v/c)$, and uses discontinuous Galerkin (DG) methods for phase-space discretization combined with implicit-explicit time stepping. A key development is a nonlinear neutrino–matter coupling algorithm based on nested fixed-point iteration with Anderson acceleration, enabling fully implicit treatment of collisional processes, including energy-coupling interactions such as neutrino–electron scattering and pair production. Coupling to finite-volume (FV) hydrodynamics is achieved through a hybrid DG-FV representation of the fluid variables and operator-split evolution within FLASH-X. The implementation is verified using basic transport tests with idealized opacities and relaxation and deleptonization problems with tabulated microphysics. Spherically symmetric CCSN simulations demonstrate accuracy and robustness of the coupled scheme, including close agreement with the CCSN simulation code Chimera. An axisymmetric CCSN simulation further demonstrates the viability of DG-based neutrino transport for multidimensional supernova modeling within FLASH-X. thornado’s neutrino-transport solver is GPU-enabled using OpenMP offloading or OpenACC, and all CCSN applications included in this work use the GPU implementation. Together, these results establish a foundation for future enhancements in physics fidelity, numerical algorithms, and computational performance, for increasingly realistic large-scale CCSN simulations.

Endeve, Eirik [Oak Ridge National Laboratory (ORNL↗
DOE OSTI2013

Explicit Integration of Extremely Stiff Reaction Networks: Asymptotic Methods

We show that, even for extremely stiff systems, explicit integration may compete in both accuracy and speed with implicit methods if algebraic methods are used to stabilize the numerical integration. The stabilizing algebra differs for systems well removed from equilibrium and those near equilibrium. This paper introduces a quantitative distinction between these two regimes and addresses the former case in depth, presenting explicit asymptotic methods appropriate when the system is extremely stiff but only weakly equilibrated. A second paper [1] examines quasi-steady-state methods as an alternative to asymptotic methods in systems well away from equilibrium and a third paper [2] extends these methods to equilibrium conditions in extremely stiff systems using partial equilibrium methods. All three papers present systematic evidence for timesteps competitive with implicit methods. Because explicit methods can execute a timestep faster than an implicit method, our results imply that algebraically stabilized explicit algorithms may offer a means to integration of larger networks than have been feasible previously in various disciplines.

Guidry, Mike W↗
DOE OSTI2015

Fully implicit Particle-in-cell algorithms for multiscale plasma simulation

The outline of the paper is as follows: Particle-in-cell (PIC) methods for fully ionized collisionless plasmas, explicit vs. implicit PIC, 1D ES implicit PIC (charge and energy conservation, moment-based acceleration), and generalization to Multi-D EM PIC: Vlasov-Darwin model (review and motivation for Darwin model, conservation properties (energy, charge, and canonical momenta), and numerical benchmarks). The author demonstrates a fully implicit, fully nonlinear, multidimensional PIC formulation that features exact local charge conservation (via a novel particle mover strategy), exact global energy conservation (no particle self-heating or self-cooling), adaptive particle orbit integrator to control errors in momentum conservation, and canonical momenta (EM-PIC only, reduced dimensionality). The approach is free of numerical instabilities: ω pe Δt >> 1, and Δx >> λ D . It requires many fewer dofs (vs. explicit PIC) for comparable accuracy in challenging problems. Significant CPU gains (vs explicit PIC) have been demonstrated. The method has much potential for efficiency gains vs. explicit in long-time-scale applications. Moment-based acceleration is effective in minimizing N FE , leading to an optimal algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Efficient hybrid explicit-implicit learning for multiscale problems

Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. Here, in this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. We propose a machine learning assisted splitting scheme which improves the efficiency of the scheme meanwhile preserves the accuracy. We consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

An integrated model-driven method for in-treatment upper airway motion tracking using cine MRI in head and neck radiation therapy

Purpose: For the first time, MRI-guided radiation therapy systems can acquire cine images to dynamically monitor in-treatment internal organ motion. However, the complex head and neck (H&N) structures and low-contrast/resolution of on-board cine MRI images make automatic motion tracking a very challenging task. In this study, the authors proposed an integrated model-driven method to automatically track the in-treatment motion of the H&N upper airway, a complex and highly deformable region wherein internal motion often occurs in an either voluntary or involuntary manner, from cine MRI images for the analysis of H&N motion patterns. Methods: Considering the complex H&N structures and ensuring automatic and robust upper airway motion tracking, the authors firstly built a set of linked statistical shapes (including face, face-jaw, and face-jaw-palate) using principal component analysis from clinically approved contours delineated on a set of training data. The linked statistical shapes integrate explicit landmarks and implicit shape representation. Then, a hierarchical model-fitting algorithm was developed to align the linked shapes on the first image frame of a to-be-tracked cine sequence and to localize the upper airway region. Finally, a multifeature level set contour propagation scheme was performed to identify the upper airway shape change, frame-by-frame, on the entire image sequence. The multifeature fitting energy, including the information of intensity variations, edge saliency, curve geometry, and temporal shape continuity, was minimized to capture the details of moving airway boundaries. Sagittal cine MR image sequences acquired from three H&N cancer patients were utilized to demonstrate the performance of the proposed motion tracking method. Results: The tracking accuracy was validated by comparing the results to the average of two manual delineations in 50 randomly selected cine image frames from each patient. The resulting average dice similarity coefficient (93.28% ± 1.46%) and margin error (0.49 ± 0.12 mm) showed good agreement between the automatic and manual results. The comparison with three other deformable model-based segmentation methods illustrated the superior shape tracking performance of the proposed method. Large interpatient variations of swallowing frequency, swallowing duration, and upper airway cross-sectional area were observed from the testing cine image sequences. Conclusions: The proposed motion tracking method can provide accurate upper airway motion tracking results, and enable automatic and quantitative identification and analysis of in-treatment H&N upper airway motion. By integrating explicit and implicit linked-shape representations within a hierarchical model-fitting process, the proposed tracking method can process complex H&N structures and low-contrast/resolution cine MRI images. Future research will focus on the improvement of method reliability, patient motion pattern analysis for providing more information on patient-specific prediction of structure displacements, and motion effects on dosimetry for better H&N motion management in radiation therapy.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2013

A Global, Multi-Resolution Approach to Regional Ocean Modeling

In this collaborative research project between Pennsylvania State University, Colorado State University and Florida State University, we mainly focused on developing multi-resolution algorithms which are suitable to regional ocean modeling. We developed hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two ignorantly different scales. We studied the e ects of spatial simplicial meshes on the stability and the conditioning of fully discrete approximations. We also studies adaptive nite element method (AFEM) based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent gradient recovery. Some of these techniques are now being used by geoscientists(such as those at LANL).

58 GEOSCIENCES↗
DOE OSTI2020

Clustering-Based Predictive Analytics to Improve Scientific Data Discovery

Given the sheer volume of scientific data archived within the data-intensive projects at the US Department of Energy's Oak Ridge National Laboratory, finding precisely what data we are looking for may not be a trivial task; conversely, we may also miss a more prominent data product. To address such issues, we propose improving the data discovery system and using data analytics methods to comprehend what specific users might be interested in based on their physiological state, search patterns, and past data usage history. This work's primary goal is to prune the complexity, increase the visibility of popular data products, and direct users toward the data that best meet their needs. The proposed algorithm constructs a user profile based on the user's explicit or implicit interactions with the system, such as items they are currently looking at on-site and the key metadata mappings related to the data set. The pattern is then used to build a training data set, which will help find relevant data to recommend to the user.

Devarakonda, Ranjeet↗