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NASA NTRS1990

Parallel processors and nonlinear structural dynamics algorithms and software

Techniques are discussed for the implementation and improvement of vectorization and concurrency in nonlinear explicit structural finite element codes. In explicit integration methods, the computation of the element internal force vector consumes the bulk of the computer time. The program can be efficiently vectorized by subdividing the elements into blocks and executing all computations in vector mode. The structuring of elements into blocks also provides a convenient way to implement concurrency by creating tasks which can be assigned to available processors for evaluation. The techniques were implemented in a 3-D nonlinear program with one-point quadrature shell elements. Concurrency and vectorization were first implemented in a single time step version of the program. Techniques were developed to minimize processor idle time and to select the optimal vector length. A comparison of run times between the program executed in scalar, serial mode and the fully vectorized code executed concurrently using eight processors shows speed-ups of over 25. Conjugate gradient methods for solving nonlinear algebraic equations are also readily adapted to a parallel environment. A new technique for improving convergence properties of conjugate gradients in nonlinear problems is developed in conjunction with other techniques such as diagonal scaling. A significant reduction in the number of iterations required for convergence is shown for a statically loaded rigid bar suspended by three equally spaced springs.

Belytschko, Ted↗

NASA NTRS1990

Nonlinear Modeling of Joint Dominated Structures

The development and verification of an accurate structural model of the nonlinear joint-dominated NASA Langley Mini-Mast truss are described. The approach is to characterize the structural behavior of the Mini-Mast joints and struts using a test configuration that can directly measure the struts' overall stiffness and damping properties, incorporate this data into the structural model using the residual force technique, and then compare the predicted response with empirical data taken by NASA/LaRC during the modal survey tests of the Mini-Mast. A new testing technique, referred to as 'link' testing, was developed and used to test prototype struts of the Mini-Masts. Appreciable nonlinearities including the free-play and hysteresis were demonstrated. Since static and dynamic tests performed on the Mini-Mast also exhibited behavior consistent with joints having free-play and hysteresis, nonlinear models of the Mini-Mast were constructed and analyzed. The Residual Force Technique was used to analyze the nonlinear model of the Mini-Mast having joint free-play and hysteresis.

Chapman, J. M.↗

NASA NTRS1989

Nonlinear evolution of oblique waves on compressible shear layers

The effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams are considered. The analysis shows that mean temperature nonuniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integrodifferential equation which bears some resemblance to the Landau equation, in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. Inviscid solutions always end in a singularity at a finite downstream distance, but viscosity can eliminate this singularity for certain parameter ranges.

Goldstein, M. E.↗

NASA NTRS1989

Nonlinear evolution of interacting oblique waves on two-dimensional shear layers

The effects of critical layer nonlinearity are considered on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. The numerical solutions always end in a singularity at a finite downstream distance.

Goldstein, M. E.↗

NASA NTRS1990

Nonlinear response of unidirectional boron/aluminum

Experimental results obtained for unidirectional boron/aluminum subjected to combined loading using off-axis tension, compression and Iosipescu shear specimens are correlated with a nonlinear micromechanics model. It is illustrated that the nonlinear response in the principal material directions is markedly influenced by the different loading modes and different ratios of the applied stress components. The observed nonlinear response under pure and combined loading is discussed in terms of initial yielding, subsequent hardening, stress-interaction effects and unloading-reloading characteristics. The micromechanics model is based on the concept of a repeating unit cell representative of the composite-at-large and employs the unified theory of Bodner and Partom to model the inelastic response of the matrix. It is shown that the employed micromechanics model is sufficiently general to predict the observed nonlinear response of unidirectional boron/aluminum with good accuracy.

Pindera, M.-J.↗

NASA NTRS1990

Nonlinear analysis of compressively loaded linked-plate structures

A method is presented for the geometrically nonlinear analysis of the equilibrium behavior of a general class of compressively loaded, prismatic, linked-plate structures. The semi-analytical method accounts for geometric shape imperfections, and allows the use of laminated composite plate properties. Buckling eigenfunctions are used as displacement shape functions, and the associated second-order displacement fields are incorporated. The nonlinear plate equilibrium equations are expressed in terms of the assumed displacement form, and the Galerkin method is used to obtain a set of algebraic equations which are nonlinear in the modal amplitudes and the load parameter. Buckling eigensolutions for the class of structures under consideration can be obtained from the VIPASA computer code; thus, subject to some additional restrictions in loading and plate anisotropy, structural configurations which can be analyzed with regard to buckling response by VIPASA can be analyzed with regard to geometrically nonlinear response using the new method.

Stoll, Frederick↗

NASA NTRS1990

Probabilistic analysis of a materially nonlinear structure

A probabilistic finite element program is used to perform probabilistic analysis of a materially nonlinear structure. The program used in this study is NESSUS (Numerical Evaluation of Stochastic Structure Under Stress), under development at Southwest Research Institute. The cumulative distribution function (CDF) of the radial stress of a thick-walled cylinder under internal pressure is computed and compared with the analytical solution. In addition, sensitivity factors showing the relative importance of the input random variables are calculated. Significant plasticity is present in this problem and has a pronounced effect on the probabilistic results. The random input variables are the material yield stress and internal pressure with Weibull and normal distributions, respectively. The results verify the ability of NESSUS to compute the CDF and sensitivity factors of a materially nonlinear structure. In addition, the ability of the Advanced Mean Value (AMV) procedure to assess the probabilistic behavior of structures which exhibit a highly nonlinear response is shown. Thus, the AMV procedure can be applied with confidence to other structures which exhibit nonlinear behavior.

Millwater, H. R.↗

NASA NTRS1990

Critical-layer nonlinearity in the resonance growth of three-dimensional waves in boundary layers

The nonlinear interactions of a triad of initially linear stability waves are addressed. The triad consisted of a single two-dimensional mode at a given frequency and two oblique modes with equal and opposite spanwise wave numbers. The oblique waves were at half the frequency and streamwise wave number of the two-dimensional mode. Attention was focused on the boundary-layer transition at low frequencies and high Reynolds numbers. A five-zoned structure and low-frequency scaling were used to derive the nonlinear-interaction equations. The initial nonlinear development of the waves was analyzed; the results indicated that the two-dimensional wave behaves according to linear theory. Nonlinear interactions caused exponential-of-an-exponential growth of the oblique modes. This resonant amplification of the subharmonic depended on the initial amplitude of the two-dimensional wave and on the initial phase angle between the two-dimensional wave and the oblique waves. The resonant growth of the oblique modes was more pronounced at lower frequencies than at higher frequencies. The results are in good agreement with experimental results and offer explanations of the observed process.

Mankbadi, Reda R.↗

NASA NTRS1991

Nonlinear compensation techniques for magnetic suspension systems

In aerospace applications, magnetic suspension systems may be required to operate over large variations in air-gap. Thus the nonlinearities inherent in most types of suspensions have a significant effect. Specifically, large variations in operating point may make it difficult to design a linear controller which gives satisfactory stability and performance over a large range of operating points. One way to address this problem is through the use of nonlinear compensation techniques such as feedback linearization. Nonlinear compensators have received limited attention in the magnetic suspension literature. In recent years, progress has been made in the theory of nonlinear control systems, and in the sub-area of feedback linearization. The idea is demonstrated of feedback linearization using a second order suspension system. In the context of the second order suspension, sampling rate issues in the implementation of feedback linearization are examined through simulation.

Trumper, David L.↗

NASA NTRS1991

Nonlinear Instability of Hypersonic Flow past a Wedge

The nonlinear stability of a compressible flow past a wedge is investigated in the hypersonic limit. The analysis follows the ideas of a weakly nonlinear approach. Interest is focussed on Tollmien-Schlichting waves governed by a triple deck structure and it is found that the attached shock can profoundly affect the stability characteristics of the flow. In particular, it is shown that nonlinearity tends to have a stabilizing influence. The nonlinear evolution of the Tollmien-Schlichting mode is described in a number of asymptotic limits.

Seddougui, Sharon O.↗

NASA NTRS1990

Modelling and computationally efficient time domain linear equalisation of nonlinear bandlimited QPSK satellite channels

The problem of modeling and equalization of a nonlinear satellite channel is considered. The channel is assumed to be bandlimited and exhibits both amplitude and phase nonlinearities. In traditional models, computations are usually performed in the frequency domain and solutions are based on complex numerical techniques. A discrete time model is used to represent the satellite link with both uplink and downlink white Gaussian noise. Under conditions of practical interest, a simple and computationally efficient time-domain design technique for the minimum mean square error linear equalizer is presented. The efficiency of this technique is enhanced by the use of a fast and simple iterative algorithm for the computation of the autocorrelation coefficients of the output of the nonlinear channel. Numerical results on the evaluations of bit error probability and other relevant parameters needed in the design and analysis of a nonlinear bandlimited QPSK system demonstrate the simplicity and computational efficiency of the proposed approach.

Konstantinides, K.↗

NASA NTRS1991

Nonlinear disturbances in a hypersonic laminar boundary layer

Three separate but similar experiments on the growth of naturally-occurring instability waves in the laminar boundary-layers on sharp cones at hypersonic Mach numbers have been conducted. Each provided clear evidence that the theoretically-predicted second mode of instability was responsible for high-amplitude wave trains observed to prevail upstream of boundary-layer transition to turbulence. However, each also seemed to reveal the presence of an additional instability not accounted for by the linear theory. Here, examination is made of the tape-recorded hot-wire anemometer signals of one experiment on a sharp cone at Mach 8 for evidence of nonlinearity, the finding of which would explain the presence of the additional mode as a consequence of harmonic generation. Several approaches for identification of the residual effects of nonlinearity are described and utilized. Also, a simplified model describing certain fluctuation characteristics has been developed. Altogether, the evidence of nonlinear wave development is found to be strong. Quantitative comparisons of linear theory to experiments must be made with caution when nonlinearity is present in the experiment.

Kimmel, Roger L.↗

NASA NTRS1991

Special class of nonlinear damping models in flexible space structures

A special class of nonlinear damping models is investigated in which the damping force is proportional to the product of positive integer or the fractional power of the absolute values of displacement and velocity. For a one-degree-of-freedom system, the classical Krylov-Bogoliubov 'averaging' method is used, whereas for a distributed system, both an ad hoc perturbation technique and the finite difference method are employed to study the effects of nonlinear damping. The results are compared with linear viscous damping models. The amplitude decrement of free vibration for a single mode system with nonlinear models depends not only on the damping ratio but also on the initial amplitude, the time to measure the response, the frequency of the system, and the powers of displacement and velocity. For the distributed system, the action of nonlinear damping is found to reduce the energy of the system and to pass energy to lower modes.

Hu, Anren↗

NASA NTRS1990

Nonlinear spatial evolution of externally excited instability waves in free shear layers

The effects of critical-layer nonlinearity on spatially growing instability waves on shear layers between parallel streams are considered. In the two-dimensional incompressible case, the flow in the critical layer is governed by a nonequilibrium, nonlinear vorticity equation. The initial exponential growth of the instability wave is converted into algebraic growth during the streamwise 'aging' of the critical layer into a quasi-equilibrium state. This leads to a next stage of evolution where the instability-wave growth is affected by both mean-flow-divergence and nonlinear effects and is eventually converted to decay. A uniformly valid composite formula for the instability wave amplitude, accounting for both nonparallel and nonlinear effects, is given and compared to experimental results. The inviscid solutions to these equations always end in a singularity at finite downstream distance.

Hultgren, L. S.↗

NASA NTRS1991

A geometric nonlinear degenerated shell element using a mixed formulation with independently assumed strain fields

A mixed formulation is chosen to overcome deficiencies of the standard displacement-based shell model. Element development is traced from the incremental variational principle on through to the final set of equilibrium equations. Particular attention is paid to developing specific guidelines for selecting the optimal set of strain parameters. A discussion of constraint index concepts and their predictive capability related to locking is included. Performance characteristics of the elements are assessed in a wide variety of linear and nonlinear plate/shell problems. Despite limiting the study to geometric nonlinear analysis, a substantial amount of additional insight concerning the finite element modeling of thin plate/shell structures is provided. For example, in nonlinear analysis, given the same mesh and load step size, mixed elements converge in fewer iterations than equivalent displacement-based models. It is also demonstrated that, in mixed formulations, lower order elements are preferred. Additionally, meshes used to obtain accurate linear solutions do not necessarily converge to the correct nonlinear solution. Finally, a new form of locking was identified associated with employing elements designed for biaxial bending in uniaxial bending applications.

Graf, Wiley E.↗

NASA NTRS1992

Nonlinear stability and control study of highly maneuverable high performance aircraft, phase 2

This research should lead to the development of new nonlinear methodologies for the adaptive control and stability analysis of high angle-of-attack aircraft such as the F18 (HARV). The emphasis has been on nonlinear adaptive control, but associated model development, system identification, stability analysis and simulation is performed in some detail as well. Various models under investigation for different purposes are summarized in tabular form. Models and simulation for the longitudinal dynamics have been developed for all types except the nonlinear ordinary differential equation model. Briefly, studies completed indicate that nonlinear adaptive control can outperform linear adaptive control for rapid maneuvers with large changes in alpha. The transient responses are compared where the desired alpha varies from 5 degrees to 60 degrees to 30 degrees and back to 5 degrees in all about 16 sec. Here, the horizontal stabilator is the only control used with an assumed first-order linear actuator with a 1/30 sec time constant.

Mohler, R. R.↗

NASA NTRS1992

Fully nonlinear Goertler vortices in constricted channel flows and their effect on the onset of separation

The development of fully nonlinear Goertler vortices in high Reynolds number flow in a symmetrically constricted channel is investigated. Attention is restricted to the case of 'strongly' constricted channels considered by Smith and Daniels (1981) for which the scaled constriction height is asymptotically large. Such flows are known to develop a Goldstein singularity and subsequently become separated at some downstream station past the point of maximum channel constriction. It is shown that these flows can support fully nonlinear Goertler vortices, of the form elucidated by Hall and Lakin (1988), for constrictions which have an appreciable region of local concave curvature upstream of the position at which separation occurs. The effect on the onset of separation due to the nonlinear Goertler modes is discussed. A brief discussion of other possible nonlinear states which may also have a dramatic effect in delaying (or promoting) separation is given.

Denier, James P.↗

NASA NTRS1990

An adaptive grid algorithm for one-dimensional nonlinear equations

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Gutierrez, William E.↗