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DOE OSTI2024

Numerical integration in the virtual element method with the scaled boundary cubature scheme

Abstract The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Chin, Eric B.↗

NASA NTRS1990

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Chakravarthy, Sukumar R.↗

NASA NTRS1973

Two-dimensional laminar incompressible separated flow past airfoils

A method is proposed to treat the problem of steady, two-dimensional, laminar, incompressible high Reynolds number separated flow past thin airfoils. An integral form of the boundary layer equations with interaction is used and the interaction between the inviscid and viscous flow fields is provided for by use of a thin-airfoil integral. Documentation of the attempts at obtaining a solution is presented. A survey of the current state-of-the-art of problems involving viscous-inviscid interactions in flow fields with separation is given.

Plotkin, A.↗

NASA NTRS1985

Massive separation and dynamic stall on a cusped trailing-edge airfoil

The cross-over from a predominantly attached two-dimensional flow to the bluff body form of separation is modeled via the interacting boundary layer approximation. The initial breakdown of the predominantly attached flow on a cusped trailing edge airfoil is examined using the Hilbert integral form of the unsteady interacting boundary layer equations. In addition, an interacting boundary layer technique is developed for calculating bluff body separation. This new model eliminates the severe scaling problems associated with bluff body separation through the use of a realistic inviscid eddy model, based on the infinite eddy Kirchhoff free-streamline description of separation. Brief consideration is given to the cross-over from bluff body separation to a predominantly attached flow, the extension to finite eddies and cascade flows, and the possible coupling with full, or parabolized, Navier-Stokes calculations.

Rothmayer, A. P.↗

NASA NTRS1981

Two-dimensional integrating matrices on rectangular grids

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

Lakin, W. D.↗

DOE OSTI2024

A novel conjugative transposon carrying an autonomously amplified plasmid

Tetracyclines serve as broad-spectrum antibiotics to treat bacterial infections. The discovery of new tetracycline resistance genes has led to new questions about the underlying mechanisms of resistance, gene transfer, and their relevance to human health. We tracked changes in the abundance of a 55-kbp conjugative transposon (CTn214) carrying tetQ, a tetracycline resistance gene, within a Bacteroides fragilis metagenome-assembled genome derived from shotgun sequencing of microbial DNA extracted from the ileal pouch of a patient with ulcerative colitis. The mapping of metagenomic reads to CTn214 revealed the multi-copy nature of a 17,044-nt region containing tetQ in samples collected during inflammation and uninflamed visits. B. fragilis cultivars isolated from the same patient during periods of inflammation harbored CTn214 integrated into the chromosome or both a circular, multi-copy, extrachromosomal region of the CTn214 containing tetQ and the corresponding integrated form. The tetracycline-dependent mechanism for the transmission of CTn214 is nearly identical to a common conjugative transposon found in the genome of B. fragilis (CTnDOT), but the autonomously amplified nature of a circular 17,044-nt region of CTn214 that codes for tetQ and the integration of the same sequence in the linear chromosome within the same cell is a novel observation. Genome and transcriptome sequencing of B. fragilis cultivars grown under different concentrations of tetracycline and ciprofloxacin indicates that tetQ in strains containing the circular form remains actively expressed regardless of treatment, while the expression of tetQ in strains containing the linear form increases only in the presence of tetracycline.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS1973

Error analysis of multi-conic techniques

A general error analysis of three recently developed multi-conic methods of three-body trajectory integration has been carried out. Single-step error functions for position and velocity have been derived as Taylor series in powers of the time step and also in integral form. These error functions are used to investigate the relative accuracy of the three methods in various regions of the earth-moon space and to provide a method of variable step size control for the trajectory integration procedure. Numerical results are used to compare the multi-step performance of the methods for both large and small step sizes.

D'Amario, L. A.↗

NASA NTRS1990

Three-dimensional simulation of vortex breakdown

The integral form of the complete, unsteady, compressible, three-dimensional Navier-Stokes equations in the conservation form, cast in generalized coordinate system, are solved, numerically, to simulate the vortex breakdown phenomenon. The inviscid fluxes are discretized using Roe's upwind-biased flux-difference splitting scheme and the viscous fluxes are discretized using central differencing. Time integration is performed using a backward Euler ADI (alternating direction implicit) scheme. A full approximation multigrid is used to accelerate the convergence to steady state.

Kuruvila, G.↗

NASA NTRS1993

Analytical evaluation of a surface integral expressing the coupling between interior and exterior volumes in a FE-IE approach

Recently an approach which combines the finite element technique and an integral equation to determine the fields scattered by inhomogeneous bodies of complicated shape has been proposed. Basically, a mathematical surface which encloses the scatterers is introduced, thus dividing the space into an interior and an exterior volume, in which the finite element technique and an integral equation for EM scattering, respectively, are applied. The integral equation is set up for the tangential components of the fields at the surface, while the interior volume the unknowns are the total fields. Continuity of the tangential fields at the boundary, as required by Maxwell's equations, is imposed, thus coupling the two methods to obtain a consistent solution. The coupling term is expressed by a surface integral formed by the dot product of a FE basis function and an IE testing function, or viceversa. By choosing the boundary to be a surface of revolution and by making a convenient selection of IE basis (testing) functions, it is possible to evaluate the integrals analytically on surfaces such as curved triangles, curved quadrilaterals and curved pentagons. We will illustrate the salient steps involved in setting up and carrying out these integrals and discuss what class of basis (testing) functions and analytic surfaces of revolution they are applicable to. Analytic calculations offer the advantage of better accuracy than purely numerical ones, and, when combined with them, often shed light on issues of numerical convergence and limiting values. Furthermore, they may reduce computation time and storage requirements.

Zuffada, C.↗

NASA NTRS1976

Compressible laminar boundary layers with large acceleration and cooling

The paper extends an approximate prediction method involving the integral form of the momentum equation to deduce the flow quantities of interest when compressibility effects become important and heat transfer may occur. The approximate method is applicable to a two-dimensional laminar boundary layer on an impermeable surface of negligible curvature for a specified integral momentum equation. The proposed approximate method involves the use of similar solutions in conjunction with the integral momentum equation for isentropic freestream flow of a perfect gas by assuming that viscosity is proportional to stagnation condition and Prandtl number is unity. The study pertains to larger values of acceleration (acceleration parameter to 20 rather than 2) than previously considered to account for rapidly accelerating flows such as in supersonic nozzles. A discussion of some applications confirms the advantages of the method in establishing better confidence in the predictions.

Back, L. H.↗

NASA NTRS1980

Structural members, method and apparatus

A method and apparatus for fabricating a structural member such as truss from flexible sheet material in compacted form are disclosed. A number of generally tubular columns are progressively formed from the sheet material and deployed generally parallel to one another. Adjacent pairs of the columns are interconnected by respective side members, each of which is comprised of a strip of the sheet material. The sheet material is fastened together by self-attaching fasteners integrally formed from the sheet material of the columns and side members themselves.

Kinzler, J. A.↗

NASA NTRS1984

Freezing of Martian streams under climatic conditions

The valley networks of Mars are widely believed to have formed at a time when climatic conditions on the planet were significantly different from those that currently prevail. This view arises from the following observations: (1) the valleys form integrated branching networks which suggests fluid drainage, and water is the most plausible fluid, (2) the present atmosphere contains only minute amounts of water, (3) the networks appear to be more akin to terrestrial valleys that are eroded by streams of modest discharges than features that form by catastrophic floods, and (4) small streams of water will rapidly freeze under present climatic conditions. Climatic conditions at the time of formation of the valleys are studied based on the assumption that they were cut by running water.

Carr, M. H.↗

NASA NTRS1982

Asymptotic Expansions and Estimates for the Capillary Problem

The asymptotic properties for the small Bond number B of the equilibrium capillary interface interior to a circular cylindrical tube vertically dipped in an infinite reservoir of liquid are discussed. (The Bond number B is a dimensionless parameter which is the ratio of gravitational to capillary forces.) The formal expansion in powers of B of the solution to the differential equation describing the equilibrium surface (as can be obtained by standard perturbation methods) is proved to be truly asymptotic -- to all orders and uniformly in the variable and parameter gamma, the contact angle. Sequences of general estimates, in closed form, from above and from below, are also given for the solution and related functions. The M-th term in these sequences are asymptotically exact to order m. An idiosyncrasy of the problem, crucial in obtaining these estimates, is the absolute monotonicity of the structural function of the system in integral form.

Brulois, F. P.↗

NASA NTRS1971

On some problems in a theory of thermally and mechanically interacting continuous media

Using a linearized theory of thermally and mechanically interacting mixture of linear elastic solid and viscous fluid, we derive a fundamental relation in an integral form called a reciprocity relation. This reciprocity relation relates the solution of one initial-boundary value problem with a given set of initial and boundary data to the solution of a second initial-boundary value problem corresponding to a different initial and boundary data for a given interacting mixture. From this general integral relation, reciprocity relations are derived for a heat-conducting linear elastic solid, and for a heat-conducting viscous fluid. An initial-boundary value problem is posed and solved for the mixture of linear elastic solid and viscous fluid. With the aid of the Laplace transform and the contour integration, a real integral representation for the displacement of the solid constituent is obtained as one of the principal results of the analysis.

Lee, Y. M.↗

NASA NTRS2009

Management of Service Projects in Support of Space Flight Research

Goal:To provide human health and performance countermeasures, knowledge, technologies, and tools to enable safe, reliable, and productive human space exploration . [HRP-47051] Specific Objectives: 1) Develop capabilities, necessary countermeasures, and technologies in support of human space exploration, focusing on mitigating the highest risks to human health and performance. 2) Define and improve human spaceflight medical, environmental, and human factors standards. 3) Develop technologies that serve to reduce medical and environmental risks, to reduce human systems resource requirements (mass, volume, power, data, etc.) and to ensure effective human-system integration across exploration systems. 4) Ensure maintenance of Agency core competencies necessary to enable risk reduction in the following areas: A. Space medicine B. Physiological and behavioral effects of long duration spaceflight on the human body C. Space environmental effects, including radiation, on human health and performance D. Space "human factors" [HRP-47051]. Service projects can form integral parts of research-based project-focused programs to provide specialized functions. Traditional/classic project management methodologies and agile approaches are not mutually exclusive paradigms. Agile strategies can be combined with traditional methods and applied in the management of service projects functioning in changing environments. Creative collaborations afford a mechanism for mitigation of constrained resource limitations.

Love, J.↗

DOE OSTI2025

A hereditary integral, transient network approach to modeling permanent set and viscoelastic response in polymers

An efficient numerical framework is presented for modeling viscoelasticity and permanent set of polymers. It is based on the hereditary integral form of transient network theory, in which polymer chains belong to distinct networks each with different natural equilibrium states. Chains continually detach from previously formed networks and reattach to new networks in a state of zero stress. The free energy of these networks is given in terms of the deformation gradient relative to the configuration at which the network was born . A decomposition of the kernel for various free energies allows for a recurrence relationship to be established, bypassing the need to integrate over all time history. The technique is established for both highly compressible and nearly incompressible materials through the use of neo-Hookean, Blatz–Ko, Yeoh, and Ogden-Hill material models. Multiple examples are presented showing the ability to handle rate-dependent response and residual strains under complex loading histories.

Finite element method↗

NASA NTRS1986

Finite volume solultion of the compressible boundary layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicitly to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and very simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗

NASA NTRS1986

Finite volume solution of the compressible boundary-layer equations

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Loyd, B.↗