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At least 433 records · Page 24

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NASA NTRS1975

NASTRAN solutions of problems described by simultaneous parabolic differential equations

NASTRAN solution techniques are shown for a numerical analysis of a class of coupled vector flow processes described by simultaneous parabolic differential equations. To define one physical problem type where equations of this form arise, the differential equations describing the coupled transfers of heat and mass in mechanical equilibrium with negligible mass average velocity are presented and discussed. Also shown are the equations describing seepage when both electrokinetic and hydrodynamic forces occur. Based on a variational statement of the general problem type, the concepts of scalar transfer elements and parallel element systems are introduced. It is shown that adoptation of these concepts allows the direct use of NASTRAN's existing Laplace type elements for uncoupled flow (the heat transfer elements) for treating multicomponent coupled transfer. Sample problems are included which demonstrate the application of these techniques for both steady-state and transient problems.

Mason, J. B.↗

NASA NTRS1977

A fourth-order box method for solving the boundary layer equations

A fourth order box method for calculating high accuracy numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations is presented. The method is the natural extension of the second order Keller Box scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary layer equations. Numerical results for high accuracy test cases show the method to be significantly faster than other higher order and second order methods.

Wornom, S. F.↗

NASA NTRS1978

Two-dimensional computer code for design assessment of SRM aft closure environment

A general interpolants method for constructing numerical analogs of the partial differential equations of continuum mechanics and which combines the best features of both finite element and finite difference methods was used in a two dimensional axisymmetric analysis of the space shuttle solid motor aft closure region flowfield. The numerical technique used is discussed as well as the results of the analysis. A steady state solution for the submerged nozzle nose region indicates the development of an area gas flow recirculation near the propellant burning surface boundary and in the region of the underside of the nozzle nose. This recirculation region remains spatially fixed for successive iterations (greater than 800) of the solution.

Freeman, J. A.↗

DOE OSTI2022

Niowave: Flow measurement development - FY2022

It is critical to maintain proper flow rate of lead bismuth eutectic (LBE) for the operation of the Niowave converter. The high temperature of the LBE (> 200 °C) makes it difficult to purchase simple off the shelf flow meters. One method that can be used to measure the flow rate indirectly, is to use a venturi, with a known throat diameter and differential pressure measurement across the throat. LANL designed the venturi and after numerous iterations with vendors, narrowed down a differential pressure transducer to use with the venturi.

43 PARTICLE ACCELERATORS↗

NASA NTRS1983

Adaptive grids generated by elliptic systems

It is pointed out that a finite difference grid which moves with the solution of the partial differential equation being solved can improve the accuracy and efficiency of a numerical algorithm. This technique is particularly advantageous in the solution of problems involving boundary layers or shocks where a poorly chosen grid may give a numerical solution which is useless because of poor resolution or extreme oscillations. The present investigation is concerned with the development of a scheme which does not excessively distort the grid. The grid generation algorithm is based on the numerical solution of a system of elliptic differential equations. Holst and Brown (1981) have used a preliminary solution to move points on the boundary of the physical region and then resolved the problem on a new grid generated by an elliptic system. In the current investigation, the grid movement and the solution will develop simultaneously. The solution is used to modify the generating equations, thereby controlling the grid point distributions.

Mastin, C. W.↗

NASA NTRS1993

Some aspects of high-order numerical solutions of the linear convection equation with forced boundary conditions

A six-stage low-storage Runge-Kutta time-marching method is presented and shown to be an efficient method for use with high-accuracy spatial difference operators for wave propagation problems. The accuracy of the method for inhomogeneous ordinary differential equations is demonstrated through numerical solutions of the linear convection equation with forced boundary conditions. Numerical experiments are presented simulating a sine wave and a Gaussian pulse propagating into and through the domain. For practical levels of mesh refinement corresponding to roughly ten points per wavelength, the six-stage Runge-Kutta method is more accurate than the popular fourth-order Runge-Kutta method. Further numerical experiments are presented which show that the numerical boundary scheme at an inflow boundary can be a significant source of error when high-accuracy spatial discretizations are used.

Zingg, D. W.↗

NASA NTRS1993

A multi-level solution algorithm for steady-state Markov chains

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing significant reductions in computation time, often an order of magnitude or more, relative to the Gauss-Seidel and optimal SOR algorithms for a variety of test problems. The multi-level method is compared and contrasted with the iterative aggregation-disaggregation algorithm of Takahashi.

Horton, Graham↗

NASA NTRS1981

Numerical model of the moon's rotation

The differential equations for the Euler angles that define the moon's orientation with respect to an inertial coordinate frame have been integrated numerically. This numerical model of the moon's rotation, when used in conjunction with current lunar-orbit and earth-rotation models, fits lunar laser ranging observations over a five-year period within 28 cm (rms). The present model is also compared with the numerical model of Williams (1975) and Eckhardt's (1981) semianalytical model.

Cappalo, R. J.↗

DOE OSTI2026

Mono-mix strategy enables comparative proteomics of a cross-kingdom microbial symbiosis

Cross-kingdom microbial symbioses, such as those between algae and bacteria, are key players in biogeochemical cycles. The molecular changes during initiation and establishment of symbiosis are of great interest, but quantitatively monitoring such changes can be challenging, particularly when the microorganisms differ greatly in size or are intimately associated. Here, we analyze output from label-free, data-dependent acquisition (DDA) LC-MS/MS proteomics experiments investigating the well-studied interaction between the alga Chlamydomonas reinhardtii and the heterotrophic bacterium Mesorhizobium japonicum. We found that detection of bacterial proteins decreased in coculture by 50% proteome-wide due to the abundance of algal proteins. As a result, standard differential expression analysis led to numerous false-positive reports of significantly downregulated proteins, where it was not possible to distinguish meaningful biological responses to symbiosis from artifacts of the reduced protein detection in coculture relative to monoculture. We show that data normalization alone does not eliminate the impact of altered detection on differential expression analysis of the cross-kingdom symbiosis. We assessed two additional strategies to overcome this methodological artifact inherent to DDA proteomics. In the first, we combined algal and bacterial monocultures at a relative abundance that mimicked the coculture, creating a “mono-mix” control to which the coculture could be compared. This approach enabled comparable detection of bacterial proteins in the coculture and the monoculture control. In the second strategy, we enhanced detection of lowly abundant bacterial proteins by using sample fractionation upstream of LC-MS/MS analysis. When these simple approaches were combined, they allowed for meaningful comparisons of nearly 10,000 algal proteins and over 4,000 bacterial proteins in response to symbiosis by DDA. They successfully recovered expected changes in the bacterial proteome in response to algal coculture, including upregulation of sugar-binding proteins and transporters. They also revealed novel proteomic responses to coculture that guide hypotheses about algal-bacterial interactions.

Dupuis, Sunnyjoy [University of California, Berkel↗

DOE OSTI2020

Numerical integrators for Lagrangian oceanography

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

58 GEOSCIENCES↗

NASA NTRS1974

Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies

A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected region containing any number of arbitrarily shaped bodies is presented. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system. Numerical solutions for the lifting and nonlifting potential flow about Joukowski and Karman-Trefftz airfoils using this coordinate system generation show excellent comparison with the analytic solutions. The application to fields with multiple bodies is illustrated by a potential flow solution for multiple airfoils.

Thompson, J. F.↗

NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Chang, S. C.↗

NASA NTRS1972

A theory of supercritical wing sections, with computer programs and examples.

Mathematical methods for the design of supercritical wings, which depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics, are developed. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. The mathematical theory is described, and a manual for users of the programs is provided. Numerical examples are given and computational results are discussed, and the computer programs themselves are listed. The analysis routine can be used to ascertain whether the profiles behave well at off-design conditions, or to smooth coordinates and obtain a desirable shape more quickly when perfectly shockless flow is not essential.

Bauer, F.↗

NASA NTRS1983

Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.

Durgun, K.↗

NASA NTRS1994

Dynamics of Numerics and CFD

This lecture attempts to illustrate the basic ideas of how the recent advances in nonlinear dynamical systems theory (dynamics) can provide new insights into the understanding of numerical algorithms used in solving nonlinear differential equations (DEs). Examples will be given of the use of dynamics to explain unusual phenomena that occur in numerics. The inadequacy of the use of linearized analysis for the understanding of long time behavior of nonlinear problems will be illustrated, and the role of dynamics in studying the nonlinear stability, accuracy, convergence property and efficiency of using time- dependent approaches to obtaining steady-state numerical solutions in computational fluid dynamics (CFD) will briefly be explained.

Yee, H. C.↗

NASA NTRS1973

A numerical solution for thermoacoustic convection of fluids in low gravity

A finite difference numerical technique for solving the differential equations which describe thermal convection of compressible fluids in low gravity are reported. Results of one-dimensional calculations are presented, and comparisons are made to previous solutions. The primary result presented is a one-dimensional radial model of the Apollo 14 heat flow and convection demonstration flight experiment. The numerical calculations show that thermally induced convective motion in a confined fluid can have significant effects on heat transfer in a low gravity environment.

Spradley, L. W.↗

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↗

NASA NTRS1989

A stable numerical solution method in-plane loading of nonlinear viscoelastic laminated orthotropic materials

In response to the tremendous growth in the development of advanced materials, such as fiber-reinforced plastic (FRP) composite materials, a new numerical method is developed to analyze and predict the time-dependent properties of these materials. Basic concepts in viscoelasticity, laminated composites, and previous viscoelastic numerical methods are presented. A stable numerical method, called the nonlinear differential equation method (NDEM), is developed to calculate the in-plane stresses and strains over any time period for a general laminate constructed from nonlinear viscoelastic orthotropic plies. The method is implemented in an in-plane stress analysis computer program, called VCAP, to demonstrate its usefulness and to verify its accuracy. A number of actual experimental test results performed on Kevlar/epoxy composite laminates are compared to predictions calculated from the numerical method.

Gramoll, K. C.↗