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All sources NASA NTRS DOE 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↗

NASA NTRS2003

Revisiting the Least-squares Procedure for Gradient Reconstruction on Unstructured Meshes

The accuracy of the least-squares technique for gradient reconstruction on unstructured meshes is examined. While least-squares techniques produce accurate results on arbitrary isotropic unstructured meshes, serious difficulties exist for highly stretched meshes in the presence of surface curvature. In these situations, gradients are typically under-estimated by up to an order of magnitude. For vertex-based discretizations on triangular and quadrilateral meshes, and cell-centered discretizations on quadrilateral meshes, accuracy can be recovered using an inverse distance weighting in the least-squares construction. For cell-centered discretizations on triangles, both the unweighted and weighted least-squares constructions fail to provide suitable gradient estimates for highly stretched curved meshes. Good overall flow solution accuracy can be retained in spite of poor gradient estimates, due to the presence of flow alignment in exactly the same regions where the poor gradient accuracy is observed. However, the use of entropy fixes has the potential for generating large but subtle discretization errors.

Mavriplis, Dimitri J.↗

NASA NTRS1982

Surface contour of a biaxially strained, doubly curved antenna reflective mesh

The elastic membrane equations which describe the deflection of a biaxially tensioned reflective mesh from an idealized surface were developed and solved. The conditions of equilibrium of the forces acting on a membrane element furnish equations which may be used to solve a variety of problems. Configurations considered include continuous, as well as incremental mesh attachment to structural members to form shapes which are approximately parabolic or spherical.

Saylor, A. V.↗

NASA NTRS1985

Examination of the Circle Spline Routine

The Circle Spline routine is currently being used for generating both two and three dimensional spline curves. It was developed for use in ESCHER, a mesh generating routine written to provide a computationally simple and efficient method for building meshes along curved surfaces. Circle Spline is a parametric linear blending spline. Because many computerized machining operations involve circular shapes, the Circle Spline is well suited for both the design and manufacturing processes and shows promise as an alternative to the spline methods currently supported by the Initial Graphics Specification (IGES).

Dolin, R. M.↗

DOE OSTI2024

A high-order Shifted Interface Method for Lagrangian shock hydrodynamics

Here, we present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Curvature Induced Modifications of Chirality and Magnetic Configuration in Perpendicular Films

Designing curvature in three-dimensional (3D) magnetic nanostructures enables controlled manipulation of local energy landscapes, allowing for the modification of noncollinear spin textures relevant for next-generation spintronic devices. In this study, we experimentally investigate 3D magnetization textures in a Co/Pd multilayer film, exhibiting strong perpendicular magnetic anisotropy (PMA), deposited onto curved Cu nanowire meshes with diameters as small as 50 nm and lengths of several microns. Utilizing magnetic soft X-ray nanotomography, we achieve reconstructions of 3D magnetic domain patterns at approximately 30 nm spatial resolution. This approach provides detailed information on both the orientation and magnitude of magnetization within the film. Our results reveal that interfacial anisotropy in the Co/Pd multilayers drives the magnetization toward the local surface normal. In contrast to typical labyrinth domains observed in planar films, the presence of curved nanowires significantly alters the domain structure, with domains preferentially aligning along the nanowire axis in close proximity, while adopting random orientations farther away. We report direct experimental observation of a curvature-induced Dzyaloshinskii-Moriya interaction (DMI), which is quantified to be approximately one-third of the intrinsic DMI in Co/Pd stacks. The curvature induced DMI enhances stability of Néel-type domain walls. These experimental observations are further supported by micromagnetic simulations. Altogether, our findings demonstrate that introducing curvature into magnetic nanostructures provides a powerful strategy for tailoring complex magnetic behaviors, paving the way for the design of advanced 3D racetrack memory and neuromorphic computing devices.

Raftrey, David↗

DOE OSTI2026

High-Order Mesh r-Adaptivity with Tangential Relaxation and Guaranteed Mesh Validity

High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations [1], but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE) [2, 3]. Here, we extend this framework to address two key gaps in the literature for highorder mesh 𝑟-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from 𝑟-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.

Mathematics and Computing↗

NASA NTRS2003

Applications of Space-Filling-Curves to Cartesian Methods for CFD

The proposed paper presents a variety novel uses of Space-Filling-Curves (SFCs) for Cartesian mesh methods in 0. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, most are applicable on general body-fitted meshes -both structured and unstructured. We demonstrate the use of single O(N log N) SFC-based reordering to produce single-pass (O(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations. Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 512 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 10% of ideal even with only around 50,000 cells in each subdomain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with O(max(M,N)) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for finite-difference-based gradient design methods.

Aftosmis, Michael J.↗

NASA NTRS2003

Applications of Space-Filling-Curves to Cartesian Methods for CFD

This paper presents a variety of novel uses of space-filling-curves (SFCs) for Cartesian mesh methods in CFD. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, many are applicable on general body-fitted meshes-both structured and unstructured. We demonstrate the use of single theta(N log N) SFC-based reordering to produce single-pass (theta(N)) algorithms for mesh partitioning, multigrid coarsening, and inter-mesh interpolation. The intermesh interpolation operator has many practical applications including warm starts on modified geometry, or as an inter-grid transfer operator on remeshed regions in moving-body simulations Exploiting the compact construction of these operators, we further show that these algorithms are highly amenable to parallelization. Examples using the SFC-based mesh partitioner show nearly linear speedup to 640 CPUs even when using multigrid as a smoother. Partition statistics are presented showing that the SFC partitions are, on-average, within 15% of ideal even with only around 50,000 cells in each sub-domain. The inter-mesh interpolation operator also has linear asymptotic complexity and can be used to map a solution with N unknowns to another mesh with M unknowns with theta(M + N) operations. This capability is demonstrated both on moving-body simulations and in mapping solutions to perturbed meshes for control surface deflection or finite-difference-based gradient design methods.

Aftosmis, M. J.↗

NASA NTRS1999

Inlet Spillage Drag Predictions Using the AIRPLANE Code

AIRPLANE (Jameson/Baker) is a steady inviscid unstructured Euler flow solver. It has been validated on many HSR geometries. It is implemented as MESHPLANE, an unstructured mesh generator, and FLOPLANE, an iterative flow solver. The surface description from an Intergraph CAD system goes into MESHPLANE as collections of polygonal curves to generate the 3D mesh. The flow solver uses a multistage time stepping scheme with residual averaging to approach steady state, but R is not time accurate. The flow solver was ported from Cray to IBM SP2 by Wu-Sun Cheng (IBM); it could only be run on 4 CPUs at a time because of memory limitations. Meshes for the four cases had about 655,000 points in the flow field, about 3.9 million tetrahedra, about 77,500 points on the surface. The flow solver took about 23 wall seconds per iteration when using 4 CPUs. It took about eight and a half wall hours to run 1,300 iterations at a time (the queue limit is 10 hours). A revised version of FLOPLANE (Thomas) was used on up to 64 CPUs to finish up some calculations at the end. We had to turn on more communication when using more processors to eliminate noise that was contaminating the flow field; this added about 50% to the elapsed wall time per iteration when using 64 CPUs. This study involved computing lift and drag for a wing/body/nacelle configuration at Mach 0.9 and 4 degrees pitch. Four cases were considered, corresponding to four nacelle mass flow conditions.

Thomas, Scott D.↗

NASA NTRS1998

Self-Avoiding Walks over Adaptive Triangular Grids

In this paper, we present a new approach to constructing a "self-avoiding" walk through a triangular mesh. Unlike the popular approach of visiting mesh elements using space-filling curves which is based on a geometric embedding, our approach is combinatorial in the sense that it uses the mesh connectivity only. We present an algorithm for constructing a self-avoiding walk which can be applied to any unstructured triangular mesh. The complexity of the algorithm is O(n x log(n)), where n is the number of triangles in the mesh. We show that for hierarchical adaptive meshes, the algorithm can be easily parallelized by taking advantage of the regularity of the refinement rules. The proposed approach should be very useful in the run-time partitioning and load balancing of adaptive unstructured grids.

Heber, Gerd↗

NASA NTRS1998

Simulation of Conformal Spiral Slot Antennas on Composite Platforms

During the course of the grant, we wrote and distributed about 12 reports and an equal number of journal papers supported fully or in part by this grant. The list of reports (title & abstract) and papers are given in Appendices A and B. This grant has indeed been instrumental in developing a robust hybrid finite element method for the analysis of complex broadband antennas on doubly curved platforms. Previous to the grant, our capability was limited to simple printed patch antennas on mostly planar platforms. More specifically: (1) mixed element formulations were developed and new edge-based prisms were introduced; (2) these elements were important in permitting flexibility in geometry gridding for most antennas of interest; (3) new perfectly matched absorbers were introduced for mesh truncations associated with highly curved surfaces; (4) fast integral algorithms were introduced for boundary integral truncations reducing CPU time from O(N-2) down to O(N-1.5) or less; (5) frequency extrapolation schemes were developed for efficient broadband performance evaluations. This activity has been successfully continued by NASA researchers; (6) computer codes were developed and extensively tested for several broadband configurations. These include FEMA-CYL, FEMA-PRISM and FEMA-TETRA written by L. Kempel, T. Ozdemir and J. Gong, respectively; (7) a new infinite balun feed was designed nearly constant impedance over the 800-3000 MHz operational band; (8) a complete slot spiral antenna was developed, fabricated and tested at NASA Langley. This new design is a culmination of the projects goals and integrates the computational and experimental efforts. this antenna design resulted in a U.S. patent and was revised three times to achieve the desired bandwidth and gain requirements from 800-3000 MHz.

Volakis, J. L.↗

NASA NTRS1989

Unsteady density-current equations for highly curved terrain

New nonlinear partial differential equations containing terrain curvature and its rate of change are derived that describe the flow of an atmospheric density current. Unlike the classical hydraulic-type equations for density currents, the new equations are valid for two-dimensional, gradually varied flow over highly curved terrain, hence suitable for computing unsteady (or steady) flows over arbitrary mountain/valley profiles. The model assumes the atmosphere above the density current exerts a known arbitrary variable pressure upon the unknown interface. Later this is specialized to the varying hydrostatic pressure of the atmosphere above. The new equations yield the variable velocity distribution, the interface position, and the pressure distribution that contains a centrifugal component, often significantly larger than its hydrostatic component. These partial differential equations are hyperbolic, and the characteristic equations and characteristic directions are derived. Using these to form a characteristic mesh, a hypothetical unsteady curved-flow problem is calculated, not based upon observed data, merely as an example to illustrate the simplicity of their application to unsteady flows over mountains.

Sivakumaran, N. S.↗

DOE OSTI2026

Non-conformal interface-cohesive modeling with the shifted boundary method

The accurate simulation of boundary- and interface-dominated problems on complex geometries remains challenging when boundary- or interface-fitted meshes are difficult to generate, particularly for curved boundaries, polycrystalline microstructures, and dense interface networks. The Shifted Boundary Method (SBM) alleviates this meshing burden by shifting the enforcement of boundary conditions from the true boundary to a nearby surrogate boundary and recovering the effect of the true boundary through geometric correction terms, thereby enabling standard finite element spaces on non-boundary-fitted meshes. In this report, we develop a general shiftedboundary and shifted-interface framework within the open-source MOOSE framework. We first present a general SBM implementation for complex geometries on non-boundary-fitted meshes. We then adopt the Shifted Interface Method (SIM) for internal interfaces and develop a unified shifted-interface treatment in which the interface law is enforced on a surrogate interface and the effect of the true interface is recovered through shifted jumps, fluxes, and tractions. This perspective brings scalar thermal-contact and vector-valued cohesive-zone mechanics into a single framework, the latter realized as the Shifted Cohesive Zone Method (SCZM) and coupled with history-dependent constitutive models from NEML2. We further extend the MOOSE mesh infrastructure to support cohesive-zone calculations on distributed meshes. The framework is verified and demonstrated through three progressive studies: Poisson’s equation on a smoothed starshaped domain, a manufactured thermal-contact problem on a non-interface-fitted mesh, and a two-dimensional polycrystalline representative volume element combining crystal plasticity with cohesive grain-boundary interfaces. Across these studies, the shifted formulations reproduce boundary- and interface-fitted reference solutions with high fidelity, indicating that the proposed framework provides an accurate and efficient route to boundary- and interface-dominated simulations on arbitrary geometries without requiring fitted meshes.

Yang, Cheng-Hau↗

NASA NTRS1993

Massively parallel computation of RCS with finite elements

One of the promising combinations of finite element approaches for scattering problems uses Whitney edge elements, spherical vector wave-absorbing boundary conditions, and bi-conjugate gradient solution for the frequency-domain near field. Each of these approaches may be criticized. Low-order elements require high mesh density, but also result in fast, reliable iterative convergence. Spherical wave-absorbing boundary conditions require additional space to be meshed beyond the most minimal near-space region, but result in fully sparse, symmetric matrices which keep storage and solution times low. Iterative solution is somewhat unpredictable and unfriendly to multiple right-hand sides, yet we find it to be uniformly fast on large problems to date, given the other two approaches. Implementation of these approaches on a distributed memory, message passing machine yields huge dividends, as full scalability to the largest machines appears assured and iterative solution times are well-behaved for large problems. We present times and solutions for computed RCS for a conducting cube and composite permeability/conducting sphere on the Intel ipsc860 with up to 16 processors solving over 200,000 unknowns. We estimate problems of approximately 10 million unknowns, encompassing 1000 cubic wavelengths, may be attempted on a currently available 512 processor machine, but would be exceedingly tedious to prepare. The most severe bottlenecks are due to the slow rate of mesh generation on non-parallel machines and the large transfer time from such a machine to the parallel processor. One solution, in progress, is to create and then distribute a coarse mesh among the processors, followed by systematic refinement within each processor. Elimination of redundant node definitions at the mesh-partition surfaces, snap-to-surface post processing of the resulting mesh for good modelling of curved surfaces, and load-balancing redistribution of new elements after the refinement are auxiliary steps expected to result in a robust low i/o system for very large finite element problems.

Parker, Jay↗

NASA NTRS2002

Simulation of an Impact Test of the All-Composite Lear Fan Aircraft

An MSC.Dytran model of an all-composite Lear Fan aircraft fuselage was developed to simulate an impact test conducted at the NASA Langley Research Center Impact Dynamics Research Facility (IDRF). The test was the second of two Lear Fan impact tests. The purpose of the second test was to evaluate the performance of retrofitted composite energy-absorbing floor beams. A computerized photogrammetric survey was performed to provide airframe geometric coordinates, and over 5000 points were processed and imported into MSC.Patran via an IGES file. MSC.Patran was then used to develop the curves and surfaces and to mesh the finite element model. A model of the energy-absorbing floor beams was developed separately and then integrated into the Lear Fan model. Structural responses of components such as the wings were compared with experimental data or previously published analytical data wherever possible. Comparisons with experimental results were used to guide structural model modifications to improve the simulation performance. This process was based largely on qualitative (video and still camera images and post-test inspections) rather than quantitative results due to the relatively few accelerometers attached to the structure.

Stockwell, Alan E.↗

NASA NTRS2004

A Six-Node Curved Triangular Element and a Four-Node Quadrilateral Element for Analysis of Laminated Composite Aerospace Structures

Mathematical development and some computed results are presented for Mindlin plate and shell elements, suitable for analysis of laminated composite and sandwich structures. These elements use the conventional 3 (plate) or 5 (shell) nodal degrees of freedom, have no communicable mechanisms, have no spurious shear energy (no shear locking), have no spurious membrane energy (no membrane locking) and do not require arbitrary reduction of out-of-plane shear moduli or under-integration. Artificial out-of-plane rotational stiffnesses are added at the element level to avoid convergence problems or singularity due to flat spots in shells. This report discusses a 6-node curved triangular element and a 4-node quadrilateral element. Findings show that in regular rectangular meshes, the Martin-Breiner 6-node triangular curved shell (MB6) is approximately equivalent to the conventional 8-node quadrilateral with integration. The 4-node quadrilateral (MB4) has very good accuracy for a 4-node element, and may be preferred in vibration analysis because of narrower bandwidth. The mathematical developments used in these elements, those discussed in the seven appendices, have been applied to elements with 3, 4, 6, and 10 nodes and can be applied to other nodal configurations.

Martin, C. Wayne↗

NASA NTRS1985

Diffraction Analysis of Mesh Deployable Reflector Antennas

A formulation and many representative numerical results for mesh reflector antennas are presented. The reflection coefficient matrix for the prescribed mesh configuration was determined and the local coordinate system of the mesh cells at each point on the curved reflector surface was accentuated. A novel strip aperture model was used to formulate the transmission coefficient matrix for a variety of mesh cell configurations. Numerical data are tailored to the dimensions of a conceptually designed land mobile satellite system (LMSS) which employs a large mesh deployable offset parabolic antenna. Results are shown for an offset parabolic reflector with mesh surfaces similar to the mesh surface of tracking and data relay satellite system (TDRSS).

Rahmat-Samii, Y.↗