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All sourcesNASA NTRSDOE OSTI
NASA NTRS2014

Adaptive Shape Functions and Internal Mesh Adaptation for Modelling Progressive Failure in Adhesively Bonded Joints

Enhanced finite elements are elements with an embedded analytical solution that can capture detailed local fields, enabling more efficient, mesh independent finite element analysis. The shape functions are determined based on the analytical model rather than prescribed. This method was applied to adhesively bonded joints to model joint behavior with one element through the thickness. This study demonstrates two methods of maintaining the fidelity of such elements during adhesive non-linearity and cracking without increasing the mesh needed for an accurate solution. The first method uses adaptive shape functions, where the shape functions are recalculated at each load step based on the softening of the adhesive. The second method is internal mesh adaption, where cracking of the adhesive within an element is captured by further discretizing the element internally to represent the partially cracked geometry. By keeping mesh adaptations within an element, a finer mesh can be used during the analysis without affecting the global finite element model mesh. Examples are shown which highlight when each method is most effective in reducing the number of elements needed to capture adhesive nonlinearity and cracking. These methods are validated against analogous finite element models utilizing cohesive zone elements.

computational mechanics↗
NASA NTRS1989

Viscous flow in simple curved gaps. I - An asymptotic theory. II - Viscous stress and shape function

The present asymptotic theory for generalized incompressible two-dimensional steady flow in curved channels has been constructed in the limit when gas thickness approaches zero with its lateral dimensions fixed; successive asymptotic solution terms are analytically generated by quadratures. In the second part of this work, the curvature of the gap treated is arbitrary. It is established that each term in the series solution of velocity and pressure is the product of a scale factor and a universal shape functions. Various interaction modes between the volume rate-of-flow, curvature, and its variations, are identified and quantitatively characterized.

Fan, D.-N.↗
NASA NTRS2009

Modeling and Simulation of Variable Mass, Flexible Structures

The advent of the new Ares I launch vehicle has highlighted the need for advanced dynamic analysis tools for variable mass, flexible structures. This system is composed of interconnected flexible stages or components undergoing rapid mass depletion through the consumption of solid or liquid propellant. In addition to large rigid body configuration changes, the system simultaneously experiences elastic deformations. In most applications, the elastic deformations are compatible with linear strain-displacement relationships and are typically modeled using the assumed modes technique. The deformation of the system is approximated through the linear combination of the products of spatial shape functions and generalized time coordinates. Spatial shape functions are traditionally composed of normal mode shapes of the system or even constraint modes and static deformations derived from finite element models of the system. Equations of motion for systems undergoing coupled large rigid body motion and elastic deformation have previously been derived through a number of techniques [1]. However, in these derivations, the mode shapes or spatial shape functions of the system components were considered constant. But with the Ares I vehicle, the structural characteristics of the system are changing with the mass of the system. Previous approaches to solving this problem involve periodic updates to the spatial shape functions or interpolation between shape functions based on system mass or elapsed mission time. These solutions often introduce misleading or even unstable numerical transients into the system. Plus, interpolation on a shape function is not intuitive. This paper presents an approach in which the shape functions are held constant and operate on the changing mass and stiffness matrices of the vehicle components. Each vehicle stage or component finite element model is broken into dry structure and propellant models. A library of propellant models is used to describe the distribution of mass in the fuel tank or Solid Rocket Booster (SRB) case for various propellant levels. Based on the mass consumed by the liquid engine or SRB, the appropriate propellant model is coupled with the dry structure model for the stage. Then using vehicle configuration data, the integrated vehicle model is assembled and operated on by the constant system shape functions. The system mode shapes and frequencies can then be computed from the resulting generalized mass and stiffness matrices for that mass configuration. The rigid body mass properties of the vehicle are derived from the integrated vehicle model. The coupling terms between the vehicle rigid body motion and elastic deformation are also updated from the constant system shape functions and the integrated vehicle model. This approach was first used to analyze variable mass spinning beams and then prototyped into a generic dynamics simulation engine. The resulting code was tested against Crew Launch Vehicle (CLV-)class problems worked in the TREETOPS simulation package and by Wilson [2]. The Ares I System Integration Laboratory (SIL) is currently being developed at the Marshall Space Flight Center (MSFC) to test vehicle avionics hardware and software in a hardware-in-the-loop (HWIL) environment and certify that the integrated system is prepared for flight. The Ares I SIL utilizes the Ares Real-Time Environment for Modeling, Integration, and Simulation (ARTEMIS) tool to simulate the launch vehicle and stimulate avionics hardware. Due to the presence of vehicle control system filters and the thrust oscillation suppression system, which are tuned to the structural characteristics of the vehicle, ARTEMIS must incorporate accurate structural models of the Ares I launch vehicle. The ARTEMIS core dynamics simulation models the highly coupled nature of the vehicle flexible body dynamics, propellant slosh, and vehicle nozzle inertia effects combined with mass and flexible body properties that vary significant with time during the flight. All forces that act on the vehicle during flight must be simulated, including deflected engine thrust force, spatially distributed aerodynamic forces, gravity, and reaction control jet thrust forces. These forces are used to excite an integrated flexible vehicle, slosh, and nozzle dynamics model for the vehicle stack that simulates large rigid body translations and rotations along with small elastic deformations. Highly effective matrix math operations on a distributed, threaded high-performance simulation node allow ARTEMIS to retain up to 30 modes of flex for real-time simulation. Stage elements that separate from the stack during flight are propagated as independent rigid six degrees of freedom (6DOF) bodies. This paper will present the formulation of the resulting equations of motion, solutions to example problems, and describe the resulting dynamics simulation engine within ARTEMIS.

Tobbe, Patrick A.↗
NASA NTRS1989

Derivation of a Tappered p-Version Beam Finite Element

A tapered p-version beam finite element suitable for dynamic applications is derived. The taper in the element is represented by allowing the area moments of inertia to vary as quartic polynomials along the length of the beam, and the cross-sectional area to vary as a quadratic polynomial. The p-version finite-element characteristics are implemented through a set of polynomial shape functions. The lower-order shape functions are identical to the classical cubic and linear shape functions normally associated with a beam element. The higher-order shape functions are a hierarchical set of polynomials that are integrals of orthogonal polynomials. Explicit expressions for the mass and stiffness matrices are presented for an arbitrary value of p. The element has been verified to be numerically stable using shape functions through 22nd order.

Hinnant, Howard E.↗
NASA NTRS1995

Geometric analysis of wing sections

This paper describes a new geometric analysis procedure for wing sections. This procedure is based on the normal mode analysis for continuous functions. A set of special shape functions is introduced to represent the geometry of the wing section. The generators of the NACA 4-digit airfoils were included in this set of shape functions. It is found that the supercritical wing section, Korn airfoil, could be well represented by a set of ten shape functions. Preliminary results showed that the number of parameters to define a wing section could be greatly reduced to about ten. Hence, the present research clearly advances the airfoil design technology by reducing the number of design variables.

Chang, I.-CHUNG↗
NASA NTRS1998

Input Shaping to Reduce Solar Array Structural Vibrations

Structural vibrations induced by actuators can be minimized using input shaping. Input shaping is a feedforward method in which actuator commands are convolved with shaping functions to yield a shaped set of commands. These commands are designed to perform the maneuver while minimizing the residual structural vibration. In this report, input shaping is extended to stepper motor actuators. As a demonstration, an input-shaping technique based on pole-zero cancellation was used to modify the Solar Array Drive Assembly (SADA) actuator commands for the Lewis satellite. A series of impulses were calculated as the ideal SADA output for vibration control. These impulses were then discretized for use by the SADA stepper motor actuator and simulated actuator outputs were used to calculate the structural response. The effectiveness of input shaping is limited by the accuracy of the knowledge of the modal frequencies. Assuming perfect knowledge resulted in significant vibration reduction. Errors of 10% in the modal frequencies caused notably higher levels of vibration. Controller robustness was improved by incorporating additional zeros in the shaping function. The additional zeros did not require increased performance from the actuator. Despite the identification errors, the resulting feedforward controller reduced residual vibrations to the level of the exactly modeled input shaper and well below the baseline cases. These results could be easily applied to many other vibration-sensitive applications involving stepper motor actuators.

Doherty, Michael J.↗
NASA NTRS1987

Fixed order compensator design based on frequency shaped cost functionals

This paper treats the design of fixed order compensators using frequency shaped cost functionals. The formulation is done in an output feedback setting which exploits an observer canonical form to represent the compensator dynamics. The major advantages of the design approach are that the order of the compensator is fixed by the design process, and it is not necessary to realize the frequency shaping dynamics as an integral part of the compensator design. This avoids the two-step approach of full state feedback design followed by the design of an observer, which in the case of a full order observer requires a compensator of order n + nf where n is the order of the plant and nf is the order of the frequency shaping realization. The formulation also precludes the use of direct feedback of the plant output, thus reducing the effect of sensor noise and improving the robustness to high frequency unmodeled dynamics. A structural vibration model describing the fast dynamics of a lightweight flexible arm is used as an example to illustrate the design procedure.

Calise, Anthony J.↗
NASA NTRS1988

Time-domain transient elastodynamic analysis of 3-D solids by BEM

The BEM algorithm developed by Banerjee and Ahmad (1985) and Banerjee et al. (1986) for the analysis of three-dimensional time-dependent elastodynamics problems is extended and refined to improve the accuracy of the internal stress predictions. A quadratic spatial shape function and a linear temporal shape function are employed, and the complete numerical implementation (as part of BEST3D) is outlined. Applications of the improved BEM to typical problems are presented graphically and briefly characterized. Included are spherical cavities subjected to sudden radial expansion, triangular or rectangular pulses of radial pressure, or ramp loading; a circular loaded area on a half space; and a flexible square plate foundation on an elastic half space.

Ahmad, S.↗
NASA NTRS2013

Large-Deformation Displacement Transfer Functions for Shape Predictions of Highly Flexible Slender Aerospace Structures

Large deformation displacement transfer functions were formulated for deformed shape predictions of highly flexible slender structures like aircraft wings. In the formulation, the embedded beam (depth wise cross section of structure along the surface strain sensing line) was first evenly discretized into multiple small domains, with surface strain sensing stations located at the domain junctures. Thus, the surface strain (bending strains) variation within each domain could be expressed with linear of nonlinear function. Such piecewise approach enabled piecewise integrations of the embedded beam curvature equations [classical (Eulerian), physical (Lagrangian), and shifted curvature equations] to yield closed form slope and deflection equations in recursive forms.

Deflection equations↗
NASA NTRS2012

New Vary-Chap Profile of the Topside Ionosphere Electron Density Distribution for use with the IRI Model and the GIRO Real-Time Data

A new Vary-Chap function is introduced for the empirical modeling of the electron density N(h) profile in the topside ionosphere that uses a shape function S(h) in the generalized Chapman function. The Vary-Chap profile extends the bottomside profile that is specified by the IRI model or measured by the Global Ionospheric Radio Observatory (GIRO) to the altitude of the ISIS-2 satellite. Some 80,000 topside profiles, measured by the topside sounder on the ISIS-2 satellite were analyzed, and the shape function S(h) was calculated for each profile. A parameterized function S*(h), composed of two sub-functions S1(h) and S2(h), is fitted to the measured S(h) profile using three free parameters. At altitudes just above the F2 layer peak height hmF2, the shape function S1 controls S(h), and at greater altitudes S2 controls S(h). The height of the intersection of S1 and S2 is defined as the transition height h(sub T) indicating the transition from an O(+) to an H(+)-dominated profile shape. The observed transition heights range from approx.500 km to 800 km.

Nsumei, Patrick↗
NASA NTRS1975

Spectral characteristics of digital phase-modulated signals

The power spectral density of two-level digital phase modulated signals is determined using various polar pulse shaping functions, the only restriction being that the pulses are independent and do not overlap. Rectangular pulses and pulses having finite rise and decay times are considered. It is shown that a substantial improvement in terms of minimizing spectral occupancy on a power basis over a given frequency band can be achieved by using linear and raised cosine pulse shaping functions. Further, the rate of decrease of the power spectral density is given for the asymptotic limit as f goes to infinity. The use of these and possibly other well chosen pulse shaping functions can then provide an additional aid to overall efficient spectrum utilization and management.

Mcgregor, D. N.↗
NASA NTRS2011

An Estimate of the Size and Shape of Sunspot Cycle 24 Based on its Early Cycle Behavior using the Hathaway-Wilson-Reichmann Shape-Fitting Function

On the basis of 12-month moving averages (12-mma) of monthly mean sunspot number (R), sunspot cycle 24 had its minimum amplitude (Rm = 1.7) in December 2008. At 12 mo past minimum, R measured 8.3, and at 18 mo past minimum, it measured 16.4. Thus far, the maximum month-to-month rate of rise in 12-mma values of monthly mean sunspot number (AR(t) max) has been 1.7, having occurred at elapsed times past minimum amplitude (t) of 14 and 15 mo. Compared to other sunspot cycles of the modern era, cycle 24?s Rm and AR(t) max (as observed so far) are the smallest on record, suggesting that it likely will be a slow-rising, long-period sunspot cycle of below average maximum amplitude (RM). Supporting this view is the now observed relative strength of cycle 24?s geomagnetic minimum amplitude as measured using the 12-mma value of the aa-geomagnetic index (aam = 8.4), which also is the smallest on record, having occurred at t equals 8 and 9 mo. From the method of Ohl (the inferred preferential association between RM and aam), one predicts RM = 55 +/- 17 (the ?1 se prediction interval) for cycle 24. Furthermore, from the Waldmeier effect (the inferred preferential association between the ascent duration (ASC) and RM) one predicts an ASC longer than 48 mo for cycle 24; hence, maximum amplitude occurrence should be after December 2012. Application of the Hathaway-Wilson-Reichmann shape-fitting function, using an RM = 70 and ASC = 56 mo, is found to adequately fit the early sunspot number growth of cycle 24.

Wilson, Robert M.↗
NASA NTRS1987

A finite element computational method for high Reynolds number laminar flows

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables are interpolated using complete quadratic shape functions, and the pressure is interpolated using linear shape functions which are defined on a triangular element for the two-dimensional case and on a tetrahedral element for the three-dimensional case. The triangular element and the tetrahedral element are contained inside the complete bi- and tri-quadratic elements for velocity variables for two and three dimensional cases, respectively, so that the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow of Reynolds numbers 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorably with the finite difference computational results and/or experimental data available. It was found that the present method can capture the delicate pressure driven recirculation zones, that the method did not yield any spurious pressure modes, and that the method requires fewer grid points than the finite difference methods to obtain comparable computational results.

Kim, Sang-Wook↗
NASA NTRS1988

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for high Reynolds number laminar flows

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions and the pressure was interpolated using linear shape functions. For the two dimensional case, the pressure is defined on a triangular element which is contained inside the complete biquadratic element for velocity variables; and for the three dimensional case, the pressure is defined on a tetrahedral element which is again contained inside the complete tri-quadratic element. Thus the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow for Reynolds number of 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorable with those of the finite difference methods as well as experimental data available. A finite elememt computer program for incompressible, laminar flows is presented.

Kim, Sang-Wook↗
NASA NTRS1988

A fine grid finite element computation of two-dimensional high Reynolds number flows

A velocity-pressure-integrated mixed-interpolation Galerkin finite-element computation of the Navier-Stokes equations using the grids is presented. The velocity variables are interpolated using complete quadratic shape functions, and the pressure was interpolated using linear shape functions defined on a triangular element which is contained inside the quadratic element for velocity variables. Comprehensive computational results for a cavity flow at Reynolds number Re = 400-10,000 and a laminar backward-facing step flow at Re = 100-900 are presented. Many high-Re flows involve convection-dominated motion as well as diffusion-dominated motion in the flow domain. The computational results for both of the fluid motions compared favorably with high-accuracy finite-difference computational results and/or experimental data.

Kim, Sang-Wook↗
NASA NTRS1989

Neural networks for data compression and invariant image recognition

An approach to invariant image recognition (I2R), based upon a model of biological vision in the mammalian visual system (MVS), is described. The complete I2R model incorporates several biologically inspired features: exponential mapping of retinal images, Gabor spatial filtering, and a neural network associative memory. In the I2R model, exponentially mapped retinal images are filtered by a hierarchical set of Gabor spatial filters (GSF) which provide compression of the information contained within a pixel-based image. A neural network associative memory (AM) is used to process the GSF coded images. We describe a 1-D shape function method for coding of scale and rotationally invariant shape information. This method reduces image shape information to a periodic waveform suitable for coding as an input vector to a neural network AM. The shape function method is suitable for near term applications on conventional computing architectures equipped with VLSI FFT chips to provide a rapid image search capability.

Gardner, Sheldon↗