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Results for “bounds preservation”

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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At least 19 records
All sourcesNASA NTRSDOE OSTI
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 OSTI2024

Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagrangian–Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. Here, the efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

On differentiable local bounds preserving stabilization for Euler equations

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

42 ENGINEERING↗
DOE OSTI2024

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

In Dzanic (2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Finally, some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Bound-preserving finite element approximations of the Keller–Segel equations

We report this paper aims to develop numerical approximations of the Keller–Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear and can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and nonblowup phenomena. In the blowup setting, we identify a locking phenomenon that relates the L ∞ (Ω)-norm to the L 1 (Ω)-norm limiting the growth of the singularity when supported on a macroelement.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Multi-material ALE remap with interface sharpening using high-order matrix-free finite element methods

The arbitrary Lagrangian-Eulerian (ALE) technique involves remapping field quantities from a Lagrangian mesh to an optimized mesh in a conservative, accurate and bounds-preserving manner. For methods based on arbitrary order finite elements, as described in a reference, material volume fractions are advected in pseudo-time using flux-corrected transport (FCT) without any form of interface reconstruction. In practice, this can lead to excessive propagation of small volume fractions throughout the domain. In addition, this method requires assembly of a global advection matrix to compute the bounds-preserving low-order FCT solution. In this work, we introduce a new approach for ALE remap using a high-order matrix-free technique which incorporates a flux modification to sharpen material interfaces in a conservative manner. Our approach begins with computing a bounds-preserving low-order solution to the ALE remap equations at the element level. We then compute a sharp interface solution (not guaranteed to be bounds-preserving) which comes from solving an augmented version of the ALE remap equations with a conservative flux modification which acts to sharpen material volume fractions based on their gradients and transport directions. Using the sharp interface solution, we make global corrections to the bounds-preserving solution while maintaining preservation of bounds. By blending with the sharpened solution at the global level we are able to globally conserve mass without hindering the remap pseudo-time step. This new interface-aware ALE remap method is based entirely on partial assembly techniques where globally assembled matrix operators are no longer needed, resulting in a globally matrix-free FCT method for multi-material, multi-field ALE remap with high performance on GPU architectures. We present results of our new remap method on 1D, 2D and 3D benchmarks and describe the algorithmic tailoring for GPU architectures that was developed.

Vargas, Arturo [Lawrence Livermore National Labora↗
DOE OSTI2022

Adaptive clipping‐and‐redistribution algorithms for bounded and conservative high‐order interpolations applied to discontinuous and reactive flows

Abstract A new adaptive clipping‐and‐redistribution method is presented which provides bounds‐preservation for multidimensional interpolation in the context of high‐order finite‐volume discretizations with adaptive mesh refinement (AMR). The underlying finite‐volume method (FVM) for the computational fluid dynamics applications is fourth‐order accurate for smooth solutions and utilizes AMR for computational efficiency in solving multiscale problems involving turbulence and combustion. High‐order interpolation between different AMR levels is required. However, this operation often leads to numerical issues because combustion species must have physical bounds preserved. The present study overcomes two major challenges in the development of the high‐order interpolation method. First, the method needs to be bound‐preserving near extrema or discontinuities to prevent the emergence of unphysical oscillations while maintaining fourth‐order accuracy in smooth flows. Second, the method needs to satisfy the conservation requirement in multiple dimensions, particularly in the context of curvilinear coordinate transformations. Additionally, the method is designed to be localized and computationally inexpensive. The new interpolation scheme is demonstrated by solving reacting flows, which are extremely sensitive to unphysical overshoots in conserved quantities. The test problems are shock‐induced ‐ combustion and a ‐air flame in a practical bluff‐body combustor. Results show the method prevents new extrema near discontinuities while maintaining high‐order accuracy in smooth regions. In particular, the method is extremely beneficial for combustion with stiff chemistry. With the proposed new method, even if flame fronts cross AMR interfaces or new grids are created in the vicinity of the flame, solution stability is retained.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen-Cahn type of equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Conservative remapping of material-dependent fields between possibly misaligned material regions

In this work, we propose an interpolation or remapping algorithm of material-dependent fields on polyhedral meshes where any source or target cell contains only one material. It is conservative and it preserves sharp material boundaries on the target mesh, even if the source and target regions delineating the same material are slightly misaligned. If those material regions are aligned, then the algorithm is also linearity-preserving and bounds-preserving. For a given material, it consists of a conservative field reconstruction on a target mesh part from a source mesh part associated with that material, followed by a repair step in case of misaligned boundaries. No assumption is made regarding the topology of the input meshes

36 MATERIALS SCIENCE↗
DOE OSTI2025

Analysis of the SiMPL Method for Density-Based Topology Optimization

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides pointwise bound-preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [D. Kim et al., Struct. Multidiscip. Optim., 68 (2025), 74]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. Here, we refer to the new algorithm as the SiMPL method (pronounced “simple”), which stands for Sigmoidal Mirror descent with a Projected Latent variable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A positivity-preserving and conservative intersection-distribution-based remapping algorithm for staggered ALE hydrodynamics on arbitrary meshes

In this study, we introduce new intersection-distribution-based remapping tools for indirect staggered arbitrary Lagrangian-Eulerian (ALE) simulations of multi-material shock hydrodynamics on arbitrary meshes. In addition to conserving momentum and total energy, the three-stage remapper proposed in this work preserves non-negativity of the internal energy. At the first stage, we construct slope-limited piecewise-linear reconstructions of all conserved quantities on zones of the source mesh and perform intersection-based remap to obtain bound-preserving zonal quantities on the target mesh. At the second stage, we define bound-preserving nodal quantities of the staggered ALE discretization as convex combinations of corner quantities. The nodal internal energy is corrected in a way which keeps it non-negative, while providing exact conservation of total energy. At the final stage, we distribute the non-negative nodal internal energy to corners, zones and materials using non-negative weights. Proofs of positivity preservation are provided for each stage. This work is a natural extension of our paper [14] in which a similar intersection-distribution-based remapping procedure was employed. The original version used a nodal kinetic energy fix which did not provably ensure positivity preservation for the zonal internal energy after the final distribution stage. The new algorithm cures this potential drawback by using ‘coordinated’ limiters for piecewise-linear reconstructions, remapping the internal energy to nodes and correcting it before redistribution. The effectiveness of the new nodal fix is illustrated by numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Conservative high-order data transfer method on generalized polygonal meshes

A conservative data transfer (remap) between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) hydrodynamics simulations. High-order numerical methods for ALE simulations require both high-order (curvilinear) meshes and high-order remap algorithms. Here we develop a conservative and bounds-preserving method for accurate remapping of discrete fields on generalized polygonal meshes with curvilinear edges. The properties of the proposed method are studied theoretically and numerically for various (smooth and non-smooth) mesh deformations and discrete fields that represent smooth and discontinuous functions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Geometric remapping of particle distributions in the Discrete Element Model for Sea Ice (DEMSI v0.0)

Abstract. A new sea ice dynamical core, the Discrete Element Model for Sea Ice (DEMSI), is under development for use in coupled Earth system models. DEMSI is based on the discrete element method, which models collections of ice floes as interacting Lagrangian particles. In basin-scale sea ice simulations the Lagrangian motion results in significant convergence and ridging, which requires periodic remapping of sea ice variables from a deformed particle configuration back to an undeformed initial distribution. At the resolution required for Earth system models we cannot resolve individual sea ice floes, so we adopt the sub-grid-scale thickness distribution used in continuum sea ice models. This choice leads to a series of hierarchical tracers depending on ice fractional area or concentration that must be remapped consistently. The circular discrete elements employed in DEMSI help improve the computational efficiency at the cost of increased complexity in the effective element area definitions for sea ice cover that are required for the accurate enforcement of conservation. An additional challenge is the accurate remapping of element values along the ice edge, the location of which varies due to the Lagrangian motion of the particles. In this paper we describe a particle-to-particle remapping approach based on well-established geometric remapping ideas that enforces conservation, bounds preservation, and compatibility between associated tracer quantities, while also robustly managing remapping at the ice edge. One element of the remapping algorithm is a novel optimization-based flux correction that enforces concentration bounds in the case of nonuniform motion. We demonstrate the accuracy and utility of the algorithm in a series of numerical test cases.

58 GEOSCIENCES↗
DOE OSTI2020

Intersection-distribution-based remapping between arbitrary meshes for staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics

In this paper, we present a new intersection-distribution-based remapping method between arbitrary polygonal meshes for indirect staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics. All cell-centered material quantities are conservatively remapped using intersections between the Lagrangian (old, source) mesh and the rezoned (new, target) mesh. The new nodal masses are obtained by conservative distribution of all material masses in each new cell to the cell's corners and then collecting those corner masses at new nodes. This distribution is done using a local constrained optimization approach for each cell in the new mesh. In order to remap nodal momentum we first define cell-centered momentum for each cell in the old mesh, conservatively remap this to the new mesh and then conservatively distribute the new zonal momentum to each cell's bounding nodes, again using local constrained optimization. Our method also conserves total energy by applying a new nodal kinetic energy correction that relies on a process similar to that used for remapping nodal mass and momentum. Cell-centered kinetic energy is computed, conservatively remapped and then distributed to nodes. The discrepancy between this conservatively remapped and actual nodal kinetic energy is then conservatively distributed to the internal energies of the materials in the cells surrounding each node. Unlike conventional cell-based corrections of this type, this new nodal kinetic energy correction has not been observed to drive material internal energy negative in any of our testing. Unlike flux based remapping, our new intersection-distribution method can be applied to remapping between source and target meshes that are arbitrarily different, which provides superior flexibility in the rezoning strategy. Our method is accurate, essentially conservative and essentially bounds preserving.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Modeling Material Interfaces with the Six-Equation Model in FLEXO

FLEXO is a multiphysics code developed at Sandia National Laboratories for predictive simulation of pulsed power target physics with extended magnetohydrodynamics modeling. Given this application space, FLEXO must be able to perform high-fidelity simulations of pulsed-power systems that include multiple materials. This work presents the augmentation of FLEXO with a six-equation model for multimaterial flow. We design and analyze discretizations of the six-equation model, emphasizing desirable properties for robust multimaterial simulation. We also present solution limiting, bounds preservation, and pressure equilibration/relaxation techniques that enhance the robustness of the overall multimaterial scheme. We present a suite of test problems to document these new capabilities of the FLEXO code.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Portage: A Modular Data Remap Library for Multiphysics Applications on Advanced Architectures

Portage is a scalable and extensible remap library for numerical simulations. It supports state-of-the-art remap schemes for meshes and particles in 2D and 3D up to a second-order accuracy. Portage ensures critical properties such as local/global conservation and bounds preservation for mesh remap. It enables multi-material field remap through a dedicated plugin, and leverages the hybrid parallelism exposed by advanced architectures using multi-processing and multi-threading.

97 MATHEMATICS AND COMPUTING↗
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↗