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NASA NTRS1993

Thermoviscoplastic analysis of fibrous periodic composites using triangular subvolumes

The nonlinear viscoplastic behavior of fibrous periodic composites is analyzed by discretizing the unit cell into triangular subvolumes. A set of these subvolumes can be configured by the analyst to construct a representation for the unit cell of a periodic composite. In each step of the loading history, the total strain increment at any point is governed by an integral equation which applies to the entire composite. A Fourier series approximation allows the incremental stresses and strains to be determined within a unit cell of the periodic lattice. The nonlinearity arising from the viscoplastic behavior of the constituent materials comprising the composite is treated as fictitious body force in the governing integral equation. Specific numerical examples showing the stress distributions in the unit cell of a fibrous tungsten/copper metal matrix composite under viscoplastic loading conditions are given. The stress distribution resulting in the unit cell when the composite material is subjected to an overall transverse stress loading history perpendicular to the fibers is found to be highly heterogeneous, and typical homogenization techniques based on treating the stress and strain distributions within the constituent phases as homogeneous result in large errors under inelastic loading conditions.

Walker, Kevin P.↗

NASA NTRS1980

Numerical analysis of natural convection in two-dimensional square and circular containers in low gravity

A numerical study of natural convection in circular cylinder and square enclosures shows that the analytic low Rayleigh number theory of previous investigators is valid for Rayleigh numbers up to 1000. For a Rayleigh number of 5000, steady state values of maximum fluid velocity differ by 20 percent. This deviation between analytic theory and numerical results increases for higher Rayleigh numbers. In addition, the low Rayleigh number theory is shown to be valid for higher Rayleigh numbers for a portion of the transient phase before significant deviation becomes apparent. It is also shown that square shaped experimental configurations may be analytically approximated with good accuracy by circular cylinders of equal cross sectional area for the prediction of convection velocities and flow patterns at low Rayleigh number.

Robertson, S. J.↗

NASA NTRS1987

Current-driven magnetohydrodynamic thermal instabilities in sheared fields

Approximate analytic solutions are sought for the dispersion relation for the MHD stability of magnetized medium in current-driven filamentation modes such as those observed in the solar atmosphere. The magnetic field is assumed to have a self-consistent sheared equilibrium structure. The analysis is carried out in the small wavenumber regime, where shear length is similar to the mode wavelength. Instability is found to depend on the ratio between the thermal and magnetic diffusivities, i.e., the Prandtl number, which identifies the unstable transverse wavenumbers. The instability conditions are expressed in an algebraic equation amenable to numerical solution. Results are provided from use of the model to determine the maximum growth rate and typical scale lengths of instabilities in a precoronal atmosphere and the lower transition region.

Bodo, G.↗

NASA NTRS1993

Modeling of composite beams and plates for static and dynamic analysis

The main purpose of this research was to develop a rigorous theory and corresponding computational algorithms for through-the-thickness analysis of composite plates. This type of analysis is needed in order to find the elastic stiffness constants for a plate and to post-process the resulting plate solution in order to find approximate three-dimensional displacement, strain, and stress distributions throughout the plate. This also requires the development of finite deformation plate equations which are compatible with the through-the-thickness analyses. After about one year's work, we settled on the variational-asymptotical method (VAM) as a suitable framework in which to solve these types of problems. VAM was applied to laminated plates with constant thickness in the work of Atilgan and Hodges. The corresponding geometrically nonlinear global deformation analysis of plates was developed by Hodges, Atilgan, and Danielson. A different application of VAM, along with numerical results, was obtained by Hodges, Lee, and Atilgan. An expanded version of this last paper was submitted for publication in the AIAA Journal.

Hodges, Dewey H.↗

NASA NTRS1992

Rapid time variations and radiative instabilities of astrophysical masers

The time-varying intensities are obtained for astrophysical masers that are radiatively unstable. Numerical integrations of the time-dependent, nonlinear equations of radiative transfer are performed with the usual approximation of a linear maser. At long times after changes in the physical conditions, the intensity of maser radiation reaches an asymptotic behavior and oscillates permanently in these idealized calculations with a period that is related to the length of the maser divided by the speed of light. The intensity varies by more than a factor of 10. These intensities depend upon the same four parameters as we originally found to determine the regime for radiative instabilities based on a stability analysis of the steady state. A detailed comparison is made between the predictions of the stability analysis and the time variations of the intensities. Calculations are performed for interacting pairs as well as isolated, individual masers.

Scappaticci, Gerardo A.↗

NASA NTRS1997

Efficient Evaluation and Optimization of Structural Reliability

Use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect. But efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures. This research addresses the modern approximation concepts and their utility in structural reliability estimation. Emphasis is placed on building higher order approximations using intervening variables to closely represent the nonlinear limit state functions. Several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method. Many of the structural problems are modeled and simulated using the finite element methods (FEM) for obtaining a detailed structural response. FEM is a computationally intensive numerical procedure with a large number of degrees of freedom. With an increase in the complexity of the structural model and the multidisciplinary nature of analyses, the number of limit states and their computation increases very significantly. The requirement of frequent repetitive analysis of the structure hinders the utility of any tool and discourages that approach from being used. One particular tool which faces this difficulty is the computation of the structural reliability under parametric uncertainties.

Grandhi, Ramana V.↗

NASA NTRS1994

Navier-Stokes Flowfield Simulation of Boeing 747-200 as Platform for SOFIA

Steady and unsteady viscous, three-dimensional flowfields are calculated using a thin layer approximation of Navier-Stokes equations in conjunction with Chimera overset grids. The finite-difference numerical scheme uses structured grids and a pentadiagonal flow solver called "OVERFLOW". The configuration of Boeing 747-200 has been chosen as one of configurations to be used as a platform for the SOFIA (Stratospheric Observatory For Infrared Astronomy). Initially, the steady flowfield of the full aircraft is calculated for the clean configuration (without a cavity to house telescope). This solution is then used to start the unsteady flowfield of a configuration containing cavity housing the observation telescope and its peripheral units. Analysis of unsteady flowfield in the cavity and its influence on the tail empennage, as well as the noise due to turbulence and optical quality of the flow are the main focus of this study. For the configuration considered here, the telescope housing cavity is located slightly downstream of the portwing. The entire flow-field is carefully constructed using 45 overset grids and consists of nearly 4 million grid points. All the computations axe done at one freestream flow condition of M(sub infinity) = 0.85, alpha = 2.5deg, and a Reynolds of Re = 1.85x10deg

Srinivasan, G.R.↗

NASA NTRS1994

Numerical study of the effects of icing on viscous flow over wings

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the outer zone, the unsteady full-potential equation (FPE) is solved. In the inner zone, the Navier-Stokes equations are solved using a diagonal form of an alternating-direction implicit (ADI) approximate factorization procedure. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic-based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are less than 60 percent of that required for a full-blown Navier-Stokes analysis for steady flow applications and about 60 percent of the Navier-Stokes CPU times for unsteady flows in non-vector processing machines. Applications of the method are presented for a rectangular NACA 0012 wing in low subsonic steady flow at moderate and high angles of attack, and for an F-5 wing in steady and unsteady subsonic and transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to Navier-Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier-Stokes equations can be retained with a significant savings in computational time.

Sankar, L. N.↗

NASA NTRS1993

Numerical and experimental investigation of the bending response of thin-walled composite cylinders

A numerical and experimental investigation of the bending behavior of six eight-ply graphite-epoxy circular cylinders is presented. Bending is induced by applying a known end-rotation to each end of the cylinders, analogous to a beam in bending. The cylinders have a nominal radius of 6 inches, a length-to-radius ratio of 2 and 5, and a radius-to-thickness ratio of approximately 160. A (+/- 45/0/90)S quasi-isotropic layup and two orthotropic layups, (+/- 45/0 sub 2)S and (+/- 45/90 sub 2)S, are studied. A geometrically nonlinear special-purpose analysis, based on Donnell's nonlinear shell equations, is developed to study the prebuckling responses and gain insight into the effects of non-ideal boundary conditions and initial geometric imperfections. A geometrically nonlinear finite element analysis is utilized to compare with the prebuckling solutions of the special-purpose analysis and to study the buckling and post buckling responses of both geometrically perfect and imperfect cylinders. The imperfect cylinder geometries are represented by an analytical approximation of the measured shape imperfections. Extensive experimental data are obtained from quasi-static tests of the cylinders using a test fixture specifically designed for the present investigation. A description of the test fixture is included. The experimental data are compared to predictions for both perfect and imperfect cylinder geometries. Prebuckling results are presented in the form of displacement and strain profiles. Buckling end-rotations, moments, and strains are reported, and predicted mode shapes are presented. Observed and predicted moment vs. end-rotation relations, deflection patterns, and strain profiles are illustrated for the post buckling responses. It is found that a geometrically nonlinear boundary layer behavior characterizes the prebuckling responses. The boundary layer behavior is sensitive to laminate orthotropy, cylinder geometry, initial geometric imperfections, applied end-rotation, and non-ideal boundary conditions. Buckling end-rotations, strains, and moments are influenced by laminate orthotropy and initial geometric imperfections. Measured buckling results correlate well with predictions for the geometrically imperfect specimens. The postbuckling analyses predict equilibrium paths with a number of scallop-shaped branches that correspond to unique deflection patterns. The observed postbuckling deflection patterns and measured strain profiles show striking similarities to the predictions in some cases. Ultimate failure of the cylinders is attributed to an interlaminar shear failure mode along the nodal lines of the postbuckling deflection patterns.

Fuchs, J. P.↗

NASA NTRS1972

Transient-forced convection film boiling on an isothermal flat plate.

Development of a new approach for the solution of transient-forced convection film boiling on an isothermal flat plate using the boundary layer model. The similarity variables are used to convert the governing partial differential equations to ordinary ones. The results of numerical solutions of these ordinary equations indicate that the transient process can be classified as one-dimensional conduction, intermediate, and the steady-state regions. The time required for the one-dimensional conduction and the time necessary to attain a steady-state condition are obtained. The use of local similarity approximations for the intermediate regime facilitates prediction of complete boundary layer growth. Using the ratio of time at any instant to the steady-state time as abscissa, the curves representing the boundary layer growth can be merged into a single mean curve within 5%. Further, the analysis shows that the average rate of heat transfer during transient is 50 to 100% higher than those at steady state. The average rate of vapor convected away is 10 to 15% lower than at steady state while the average rate of accumulation to form the vapor layer is 1 to 14 times larger.

Nagendra, H. R.↗

NASA NTRS1979

Mean-spherical model for soft potentials - The hard core revealed as a perturbation

The mean-spherical approximation for fluids is extended to treat the case of dense systems interacting via soft potentials. The extension takes the form of a generalized statement concerning the behavior of the direct-correlation function c(r) and the radial-distribution function g(r). From a detailed analysis that views the hard-core portion of a potential as a perturbation on the whole, a specific model is proposed which possesses analytic solutions for both Coulomb and Yukawa potentials, in addition to certain other remarkable properties. A variational principle for the model leads to a relatively simple method for obtaining numerical solutions.

Rosenfeld, Y.↗

NASA NTRS1994

An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws

This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

Bey, Kim S.↗

NASA NTRS1993

Hybrid Analytical Technique for Nonlinear Vibration Analysis of Thin-Walled Beams

A two-step hybrid analytical technique is presented for the nonlinear vibration analysis of thin-walled beams. The first step involves the generation of various-order perturbation functions using the Linstedt-Poincare perturbation technique. The second step consists of using the perturbation functions as coordinate (or approximation) functions and then computing both the amplitudes of these functions and the nonlinear frequency of vibration via a direct variational procedure. The analytical formulation is based on a form of the geometrically nonlinear beam theory with the effects of in-plane inertia, rotatory inertia, and transverse shear deformation included. The effectiveness of the proposed technique is demonstrated by means of a numerical example of thin-walled beam with a doubly symmetric I-section. The solutions obtained using a single-spatial mode were compared with those obtained using multiple-spatial modes. The standard of comparison was taken to be the frequencies obtained by the direct integration/fast Fourier transform (FFT) technique. The nonlinear frequencies obtained by the hybrid technique were shown to converge to the corresponding ones obtained by the direct integration/fast Fourier transform (FFT) technique well beyond the range of applicability of the perturbation technique. The frequencies and total strain energy of the beam were overestimated by using a single-spatial mode.

Noor, Ahmed K.↗

NASA NTRS1994

The Relationship Between Constraint and Ductile Fracture Initiation as Defined by Micromechanical Analyses

The use of small laboratory specimens to predict the integrity of large, complex structures relies on the validity of single parameter fracture mechanics. Unfortunately, the constraint loss associated with large scale yielding, whether in a laboratory specimen because of its small size or in a structure because it contains shallow flaws loaded in tension, can cause the breakdown of classical fracture mechanics and the loss of transferability of critical, global fracture parameters. Although the issue of constraint loss can be eliminated by testing actual structural configurations, such an approach can be prohibitively costly. Hence, a methodology that can correct global fracture parameters for constraint effects is desirable. This research uses micromechanical analyses to define the relationship between global, ductile fracture initiation parameters and constraint in two specimen geometries (SECT and SECB with varying a/w ratios) and one structural geometry (circumferentially cracked pipe). Two local fracture criteria corresponding to ductile fracture micromechanisms are evaluated: a constraint-modified, critical strain criterion for void coalescence proposed by Hancock and Cowling and a critical void ratio criterion for void growth based on the Rice and Tracey model. Crack initiation is assumed to occur when the critical value in each case is reached over some critical length. The primary material of interest is A516-70, a high-hardening pressure vessel steel sensitive to constraint; however, a low-hardening structural steel that is less sensitive to constraint is also being studied. Critical values of local fracture parameters are obtained by numerical analysis and experimental testing of circumferentially notched tensile specimens of varying constraint (e.g., notch radius). These parameters are then used in conjunction with large strain, large deformation, two- and three-dimensional finite element analyses of the geometries listed above to predict crack initiation loads and to calculate the associated (critical) global fracture parameters. The loads are verified experimentally, and microscopy is used to measure pre-crack length, crack tip opening displacement (CTOD), and the amount of stable crack growth. Results for A516-70 steel indicate that the constraint-modified, critical strain criterion with a critical length approximately equal to the grain size (0.0025 inch) provides accurate predictions of crack initiation. The critical void growth criterion is shown to considerably underpredict crack initiation loads with the same critical length. The relationship between the critical value of the J-integral for ductile crack initiation and crack depth for SECT and SECB specimens has been determined using the constraint-modified, critical strain criterion, demonstrating that this micromechanical model can be used to correct in-plane constraint effects due to crack depth and bending vs. tension loading. Finally, the relationship developed for the SECT specimens is used to predict the behavior of circumferentially cracked pipe specimens.

Panontin, Tina L.↗

NASA NTRS1972

Numerical methods for solving the Boltzmann equation (a review)

The methods are reviewed which are utilized in principal attempts to obtain the numerical solution or modeling of the Boltzmann equation over a broad range of Knudsen numbers. The primary methods considered are the Monte Carlo and the discrete velocities methods. The conculsions drawn from the analysis include the following: (1) The Monte Carlo methods are not well suited in the area of small Knudsen numbers. (2) Among the Monte Carlo methods, the Bird method appears to be the most attractive, since it is more directly related to the Boltzmann equation. (3) The deterministic methods, which include the discrete ordinate technique, offer great possibilities but require exceedingly large computer times. (4) The use of approximating equations in combination with the discrete velocities method will possibly improve computation time and reduce the required memory volume.

Limar, Y. F.↗

NASA NTRS1985

Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies

The particular problems investigated in the present study arise from nonlinear transient heat conduction. One of two types of nonlinearities considered is related to a material temperature dependence which is frequently needed to accurately model behavior over the range of temperature of engineering interest. The second nonlinearity is introduced by radiation boundary conditions. The finite element equations arising from the solution of nonlinear transient heat conduction problems are formulated. The finite element matrix equations are temporally discretized, and a nonlinear iterative solution algorithm is proposed. Algorithms for solving the linear problem are discussed, taking into account the form of the matrix equations, Gaussian elimination, cost, and iterative techniques. Attention is also given to approximate factorization, implementational aspects, and numerical results.

Winget, J. M.↗

NASA NTRS1992

High-frequency techniques for RCS prediction of plate geometries

The principal-plane scattering from perfectly conducting and coated strips and rectangular plates is examined. Previous reports have detailed Geometrical Theory of Diffraction/Uniform Theory of Diffraction (GTD/UTD) solutions for these geometries. The GTD/UTD solution for the perfectly conducting plate yields monostatic radar cross section (RCS) results that are nearly identical to measurements and results obtained using the Moment Method (MM) and the Extended Physical Theory of Diffraction (EPTD). This was demonstrated in previous reports. The previous analysis is extended to bistatic cases. GTD/UTD results for the principal-plane scattering from a perfectly conducting, infinite strip are compared to MM and EPTD data. A comprehensive overview of the advantages and disadvantages of the GTD/UTD and of the EPTD and a detailed analysis of the results from both methods are provided. Several previous reports also presented preliminary discussions and results for a GTD/UTD model of the RCS of a coated, rectangular plate. Several approximations for accounting for the finite coating thickness, plane-wave incidence, and far-field observation were discussed. Here, these approximations are replaced by a revised wedge diffraction coefficient that implicitly accounts for a coating on a perfect conductor, plane-wave incidence, and far-field observation. This coefficient is computationally more efficient than the previous diffraction coefficient because the number of Maliuzhinets functions that must be calculated using numerical integration is reduced by a factor of 2. The derivation and the revised coefficient are presented in detail for the hard polarization case. Computations and experimental data are also included. The soft polarization case is currently under investigation.

Balanis, Constantine A.↗

NASA NTRS1992

Analytical multiple scattering correction to the Mie theory: Application to the analysis of the lidar signal

The contribution of the multiple scattering to the lidar signal is dependent on the optical depth tau. Therefore, the radar analysis, based on the assumption that the multiple scattering can be neglected is limited to cases characterized by low values of the optical depth (tau less than or equal to 0.1) and hence it exclude scattering from most clouds. Moreover, all inversion methods relating lidar signal to number densities and particle size must be modified since the multiple scattering affects the direct analysis. The essential requests of a realistic model for lidar measurements which include the multiple scattering and which can be applied to practical situations follow. (1) Requested are not only a correction term or a rough approximation describing results of a certain experiment, but a general theory of multiple scattering tying together the relevant physical parameter we seek to measure. (2) An analytical generalization of the lidar equation which can be applied in the case of a realistic aerosol is requested. A pure analytical formulation is important in order to avoid the convergency and stability problems which, in the case of numerical approach, are due to the large number of events that have to be taken into account in the presence of large depth and/or a strong experimental noise.

Flesia, C.↗