SEARCH · Engineering Papers

Results for “Step”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,495 records · Page 83

All sources NASA NTRS DOE OSTI

NASA NTRS1993

Measurement of polycyclic aromatic hydrocarbon (PAHs) in interplanetary dust particles

We report here the first definitive measurements of specific organic molecules (polycyclic aromatic hydrocarbons (PAH's)) in interplanetary dust particles (IDP's). An improved version of the microbeam-two-step laser mass spectrometer was used for the analysis. Two IDP's gave similar mass spectra showing an abundance of PAH's. Control samples, including particles of probable terrestrial origin from the same stratospheric collector, gave either null results or quite different spectra. We conclude that the PAH's are probably indigenous to the IDP's and are not terrestrial contaminants. The instrument used to study the particles is a two-step laser mass spectrometer. Constituent neutral molecules of the sample are first desorbed with a pulsed infrared laser beam focussed to 40 micrometers. In the second step, PAH's in the desorbed plume are preferentially ionized by a pulsed UV laser beam. Resulting ions produced by resonant absorption are extracted into a reflectron time-of-flight mass spectrometer. This instrument has high spatial resolution, high ion transmission, unlimited mass range, and multichannel detection of all ion masses from a single laser shot.

Clemett, S. J.↗

NASA NTRS1993

Venus gravity

The anomalous gravity field of Venus shows high correlation with surface features revealed by radar. We extract gravity models from the Doppler tracking data from the Pioneer Venus Orbiter (PVO) by means of a two-step process. In the first step, we solve the nonlinear spacecraft state estimation problem using a Kalman filter-smoother. The Kalman filter was evaluated through simulations. This evaluation and some unusual features of the filter are discussed. In the second step, we perform a geophysical inversion using a linear Bayesian estimator. To allow an unbiased comparison between gravity and topography, we use a simulation technique to smooth and distort the radar topographic data so as to yield maps having the same characteristics as our gravity maps. The maps presented cover 2/3 of the surface of Venus and display the strong topography-gravity correlation previously reported. The topography-gravity scatter plots show two distinct trends.

Reasenberg, Robert D.↗

NASA NTRS1993

Diffractive optics fabricated by direct write methods with an electron beam

State-of-the-art diffractive optics are fabricated using e-beam lithography and dry etching techniques to achieve multilevel phase elements with very high diffraction efficiencies. One of the major challenges encountered in fabricating diffractive optics is the small feature size (e.g. for diffractive lenses with small f-number). It is not only the e-beam system which dictates the feature size limitations, but also the alignment systems (mask aligner) and the materials (e-beam and photo resists). In order to allow diffractive optics to be used in new optoelectronic systems, it is necessary not only to fabricate elements with small feature sizes but also to do so in an economical fashion. Since price of a multilevel diffractive optical element is closely related to the e-beam writing time and the number of etching steps, we need to decrease the writing time and etching steps without affecting the quality of the element. To do this one has to utilize the full potentials of the e-beam writing system. In this paper, we will present three diffractive optics fabrication techniques which will reduce the number of process steps, the writing time, and the overall fabrication time for multilevel phase diffractive optics.

Kress, Bernard↗

NASA NTRS1993

The quantum dynamics of electronically nonadiabatic chemical reactions

Considerable progress was achieved on the quantum mechanical treatment of electronically nonadiabatic collisions involving energy transfer and chemical reaction in the collision of an electronically excited atom with a molecule. In the first step, a new diabatic representation for the coupled potential energy surfaces was created. A two-state diabatic representation was developed which was designed to realistically reproduce the two lowest adiabatic states of the valence bond model and also to have the following three desirable features: (1) it is more economical to evaluate; (2) it is more portable; and (3) all spline fits are replaced by analytic functions. The new representation consists of a set of two coupled diabatic potential energy surfaces plus a coupling surface. It is suitable for dynamics calculations on both the electronic quenching and reaction processes in collisions of Na(3p2p) with H2. The new two-state representation was obtained by a three-step process from a modified eight-state diatomics-in-molecules (DIM) representation of Blais. The second step required the development of new dynamical methods. A formalism was developed for treating reactions with very general basis functions including electronically excited states. Our formalism is based on the generalized Newton, scattered wave, and outgoing wave variational principles that were used previously for reactive collisions on a single potential energy surface, and it incorporates three new features: (1) the basis functions include electronic degrees of freedom, as required to treat reactions involving electronic excitation and two or more coupled potential energy surfaces; (2) the primitive electronic basis is assumed to be diabatic, and it is not assumed that it diagonalizes the electronic Hamiltonian even asymptotically; and (3) contracted basis functions for vibrational-rotational-orbital degrees of freedom are included in a very general way, similar to previous prescriptions for locally adiabatic functions in various quantum scattering algorithms.

Truhlar, Donald G.↗

NASA NTRS1993

Time-accurate Navier-Stokes calculations with multigrid acceleration

A numerical scheme to solve the unsteady Navier-Stokes equations is described. The scheme is implemented by modifying the multigrid-multiblock version of the steady Navier-Stokes equations solver, TLNS3D. The scheme is fully implicit in time and uses TLNS3D to iteratively invert the equations at each physical time step. The design objective of the scheme is unconditional stability (at least for first- and second-order discretizations of the physical time derivatives). With unconditional stability, the choice of the time step is based on the physical phenomena to be resolved rather than limited by numerical stability which is especially important for high Reynolds number viscous flows, where the spatial variation of grid cell size can be as much as six orders of magnitude. An analysis of the iterative procedure and the implementation of this procedure in TLNS3D are discussed. Numerical results are presented to show both the capabilities of the scheme and its speed up relative to the use of global minimum time stepping. Reductions in computational times of an order of magnitude are demonstrated.

Melson, N. Duane↗

NASA NTRS1993

Extending substructure based iterative solvers to multiple load and repeated analyses

Direct solvers currently dominate commercial finite element structural software, but do not scale well in the fine granularity regime targeted by emerging parallel processors. Substructure based iterative solvers--often called also domain decomposition algorithms--lend themselves better to parallel processing, but must overcome several obstacles before earning their place in general purpose structural analysis programs. One such obstacle is the solution of systems with many or repeated right hand sides. Such systems arise, for example, in multiple load static analyses and in implicit linear dynamics computations. Direct solvers are well-suited for these problems because after the system matrix has been factored, the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions. On the other hand, iterative solvers in general are ill-suited for these problems because they often must restart from scratch for every different right hand side. In this paper, we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides. Basically, we formulate the overall problem as a series of minimization problems over K-orthogonal and supplementary subspaces, and tailor the preconditioned conjugate gradient algorithm to solve them efficiently. The resulting solution method is scalable, whereas direct factorization schemes and forward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structural problems, and highlight its potential to outperform forward and backward substitutions on parallel computers. As an example, we show that for a linear structural dynamics problem with 11640 degrees of freedom, every time-step beyond time-step 15 is solved in a single iteration and consumes 1.0 second on a 32 processor iPSC-860 system; for the same problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.

Farhat, Charbel↗

NASA NTRS1993

On the large eddy simulation of turbulent flows in complex geometry

Application of the method of Large Eddy Simulation (LES) to a turbulent flow consists of three separate steps. First, a filtering operation is performed on the Navier-Stokes equations to remove the small spatial scales. The resulting equations that describe the space time evolution of the 'large eddies' contain the subgrid-scale (sgs) stress tensor that describes the effect of the unresolved small scales on the resolved scales. The second step is the replacement of the sgs stress tensor by some expression involving the large scales - this is the problem of 'subgrid-scale modeling'. The final step is the numerical simulation of the resulting 'closed' equations for the large scale fields on a grid small enough to resolve the smallest of the large eddies, but still much larger than the fine scale structures at the Kolmogorov length. In dividing a turbulent flow field into 'large' and 'small' eddies, one presumes that a cut-off length delta can be sensibly chosen such that all fluctuations on a scale larger than delta are 'large eddies' and the remainder constitute the 'small scale' fluctuations. Typically, delta would be a length scale characterizing the smallest structures of interest in the flow. In an inhomogeneous flow, the 'sensible choice' for delta may vary significantly over the flow domain. For example, in a wall bounded turbulent flow, most statistical averages of interest vary much more rapidly with position near the wall than far away from it. Further, there are dynamically important organized structures near the wall on a scale much smaller than the boundary layer thickness. Therefore, the minimum size of eddies that need to be resolved is smaller near the wall. In general, for the LES of inhomogeneous flows, the width of the filtering kernel delta must be considered to be a function of position. If a filtering operation with a nonuniform filter width is performed on the Navier-Stokes equations, one does not in general get the standard large eddy equations. The complication is caused by the fact that a filtering operation with a nonuniform filter width in general does not commute with the operation of differentiation. This is one of the issues that we have looked at in detail as it is basic to any attempt at applying LES to complex geometry flows. Our principal findings are summarized.

Ghosal, Sandip↗

NASA NTRS1993

A finite-element-based perturbation model for the rotordynamic analysis of shrouded pump impellers: Part 2: User's guide

This report describes the computational steps involved in executing a finite-element-based perturbation model for computing the rotor dynamic coefficients of a shrouded pump impeller or a simple seal. These arise from the fluid/rotor interaction in the clearance gap. In addition to the sample cases, the computational procedure also applies to a separate category of problems referred to as the 'seal-like' category. The problem, in this case, concerns a shrouded impeller, with the exception that the secondary, or leakage, passage is totally isolated from the primary-flow passage. The difference between this and the pump problem is that the former is analytically of the simple 'seal-like' configuration, with two (inlet and exit) flow-permeable stations, while the latter constitutes a double-entry / double-discharge flow problem. In all cases, the problem is that of a rotor clearance gap. The problem here is that of a rotor excitation in the form of a cylindrical whirl around the housing centerline for a smooth annular seal. In its centered operation mode, the rotor is assumed to give rise to an axisymmetric flow field in the clearance gap. As a result, problems involving longitudinal or helical grooves, in the rotor or housing surfaces, go beyond the code capabilities. Discarding, for the moment, the pre- and post-processing phases, the bulk of the computational procedure consists of two main steps. The first is aimed at producing the axisymmetric 'zeroth-order' flow solution in the given flow domain. Detailed description of this problem, including the flow-governing equations, turbulence closure, boundary conditions, and the finite-element formulation, was covered by Baskharone and Hensel. The second main step is where the perturbation model is implemented, with the input being the centered-rotor 'zeroth-order' flow solution and a prescribed whirl frequency ratio (whirl frequency divided by the impeller speed). The computational domain, in the latter case, is treated as three dimensional, with the number of computational planes in the circumferential direction being specified a priori. The reader is reminded that the deformations in the finite elements are all infinitesimally small because the rotor eccentricity itself is a virtual displacement. This explains why we have generically termed the perturbation model the 'virtually' deformable finite-element category. The primary outcome of implementing the perturbation model is the tangential and radial components, F(sub theta)(sup *) and F(sub r)(sup *) of the fluid-exerted force on the rotor surface due to the whirling motion. Repetitive execution of the perturbation model subprogram over a sufficient range of whirl frequency ratios, and subsequent interpolation of these fluid forces, using the least-square method, finally enable the user to compute the impeller rotor dynamic coefficients of the fluid/rotor interaction. These are the direct and cross-coupled stiffness, damping, and inertia effects of the fluid/rotor interaction.

Baskharone, Erian A.↗

NASA NTRS1994

An improved method of measuring tropospheric NO2, NO3, HO2, and RO2 by Matrix Isolation and Electronic Spin Resonance (MIESR)

The MIESR method consists of two steps (1) collection of the radicals present in the ambient air at 77K in a polycrystalline D2O matrix and (2) identification and quantification of the different radicals in the laboratory by Electron Spin Resonance spectroscopy. In step (1), the sampling efficiency for sampling NO2 and RO2 was determined to be greater than or equal to 95 percent, with a measured accuracy of plus or minus 5 percent. In step (2), after collection, the samples are maintained at 77 K and spectra are recorded in the laboratory using a standard 9.5 GHz ESR system (Varian E-line). About 50 individual scans of each spectrum are recorded and digitally averaged in order to improve the signal-to-noise ratio. The ESR-spectra are analyzed with a recently developed numerical procedure which was demonstrated to allow speciation of NO2, NO3, HO2, CH3C(O)O2, and the sum of the alkylperoxy radicals. The detection limit is 5ppt for HO2, RO2, and NO2 and 3ppt for NO3 due to its narrower ESR-linewidth.

Source record↗

NASA NTRS1994

Implementing Software Safety in the NASA Environment

Until recently, NASA did not consider allowing computers total control of flight systems. Human operators, via hardware, have constituted the ultimate safety control. In an attempt to reduce costs, NASA has come to rely more and more heavily on computers and software to control space missions. (For example. software is now planned to control most of the operational functions of the International Space Station.) Thus the need for systematic software safety programs has become crucial for mission success. Concurrent engineering principles dictate that safety should be designed into software up front, not tested into the software after the fact. 'Cost of Quality' studies have statistics and metrics to prove the value of building quality and safety into the development cycle. Unfortunately, most software engineers are not familiar with designing for safety, and most safety engineers are not software experts. Software written to specifications which have not been safety analyzed is a major source of computer related accidents. Safer software is achieved step by step throughout the system and software life cycle. It is a process that includes requirements definition, hazard analyses, formal software inspections, safety analyses, testing, and maintenance. The greatest emphasis is placed on clearly and completely defining system and software requirements, including safety and reliability requirements. Unfortunately, development and review of requirements are the weakest link in the process. While some of the more academic methods, e.g. mathematical models, may help bring about safer software, this paper proposes the use of currently approved software methodologies, and sound software and assurance practices to show how, to a large degree, safety can be designed into software from the start. NASA's approach today is to first conduct a preliminary system hazard analysis (PHA) during the concept and planning phase of a project. This determines the overall hazard potential of the system to be built. Shortly thereafter, as the system requirements are being defined, the second iteration of hazard analyses takes place, the systems hazard analysis (SHA). During the systems requirements phase, decisions are made as to what functions of the system will be the responsibility of software. This is the most critical time to affect the safety of the software. From this point, software safety analyses as well as software engineering practices are the main focus for assuring safe software. While many of the steps proposed in this paper seem like just sound engineering practices, they are the best technical and most cost effective means to assure safe software within a safe system.

Wetherholt, Martha S.↗

NASA NTRS1994

Theory and computation of optimal low- and medium-thrust transfers

This report describes the current state of development of methods for calculating optimal orbital transfers with large numbers of burns. Reported on first is the homotopy-motivated and so-called direction correction method. So far this method has been partially tested with one solver; the final step has yet to be implemented. Second is the patched transfer method. This method is rooted in some simplifying approximations made on the original optimal control problem. The transfer is broken up into single-burn segments, each single-burn solved as a predictor step and the whole problem then solved with a corrector step.

Chuang, C.-H.↗

NASA NTRS1994

Distributed parameter modeling of repeated truss structures

A new approach to find homogeneous models for beam-like repeated flexible structures is proposed which conceptually involves two steps. The first step involves the approximation of 3-D non-homogeneous model by a 1-D periodic beam model. The structure is modeled as a 3-D non-homogeneous continuum. The displacement field is approximated by Taylor series expansion. Then, the cross sectional mass and stiffness matrices are obtained by energy equivalence using their additive properties. Due to the repeated nature of the flexible bodies, the mass, and stiffness matrices are also periodic. This procedure is systematic and requires less dynamics detail. The first step involves the homogenization from a 1-D periodic beam model to a 1-D homogeneous beam model. The periodic beam model is homogenized into an equivalent homogeneous beam model using the additive property of compliance along the generic axis. The major departure from previous approaches in literature is using compliance instead of stiffness in homogenization. An obvious justification is that the stiffness is additive at each cross section but not along the generic axis. The homogenized model preserves many properties of the original periodic model.

Wang, Han-Ching↗

NASA NTRS1994

Application of an Integrated Methodology for Propulsion and Airframe Control Design to a STOVL Aircraft

An advanced methodology for integrated flight propulsion control (IFPC) design for future aircraft, which will use propulsion system generated forces and moments for enhanced maneuver capabilities, is briefly described. This methodology has the potential to address in a systematic manner the coupling between the airframe and the propulsion subsystems typical of such enhanced maneuverability aircraft. Application of the methodology to a short take-off vertical landing (STOVL) aircraft in the landing approach to hover transition flight phase is presented with brief description of the various steps in the IFPC design methodology. The details of the individual steps have been described in previous publications and the objective of this paper is to focus on how the components of the control system designed at each step integrate into the overall IFPC system. The full nonlinear IFPC system was evaluated extensively in nonreal-time simulations as well as piloted simulations. Results from the nonreal-time evaluations are presented in this paper. Lessons learned from this application study are summarized in terms of areas of potential improvements in the STOVL IFPC design as well as identification of technology development areas to enhance the applicability of the proposed design methodology.

Garg, Sanjay↗

NASA NTRS1994

Nonlinear Asymptotic Integration Algorithms for One-dimensional Autonomous Dissipative First-order Odes

Nonlinear asymptotic integrators are applied to one-dimensional, nonlinear, autonomous, dissipative, ordinary differential equations. These integrators, including a one-step explicit, a one-step implicit, and a one- and two-step midpoint algorithm, are designed to follow the asymptotic behavior of a system approaching a steady state. The methods require that the differential equation be written in a particular asymptotic form. This is always possible for a one-dimensional equation with a globally asymptotic steady state. In this case, conditions are obtained to guarantee that the implicit algorithms are well defined. Further conditions are determined for the implicit methods to be contractive. These methods are all first order accurate, while under certain conditions the midpoint algorithms may also become second order accurate. The stability of each method is investigated and an estimate of the local error is provided.

Haslach, Henry W., Jr.↗

NASA NTRS1994

Advances in polycrystalline thin-film photovoltaics for space applications

Polycrystalline, thin-film photovoltaics represent one of the few (if not the only) renewable power sources which has the potential to satisfy the demanding technical requirements for future space applications. The demand in space is for deployable, flexible arrays with high power-to-weight ratios and long-term stability (15-20 years). In addition, there is also the demand that these arrays be produced by scalable, low-cost, high yield, processes. An approach to significantly reduce costs and increase reliability is to interconnect individual cells series via monolithic integration. Both CIS and CdTe semiconductor films are optimum absorber materials for thin-film n-p heterojunction solar cells, having band gaps between 0.9-1.5 ev and demonstrated small area efficiencies, with cadmium sulfide window layers, above 16.5 percent. Both CIS and CdTe polycrystalline thin-film cells have been produced on a laboratory scale by a variety of physical and chemical deposition methods, including evaporation, sputtering, and electrodeposition. Translating laboratory processes which yield these high efficiency, small area cells into the design of a manufacturing process capable of producing 1-sq ft modules, however, requires a quantitative understanding of each individual step in the process and its (each step) effect on overall module performance. With a proper quantification and understanding of material transport and reactivity for each individual step, manufacturing process can be designed that is not 'reactor-specific' and can be controlled intelligently with the design parameters of the process. The objective of this paper is to present an overview of the current efforts at MMC to develop large-scale manufacturing processes for both CIS and CdTe thin-film polycrystalline modules. CIS cells/modules are fabricated in a 'substrate configuration' by physical vapor deposition techniques and CdTe cells/modules are fabricated in a 'superstrate configuration' by wet chemical methods. Both laser and mechanical scribing operations are used to monolithically integrate (series interconnect) the individual cells into modules. Results will be presented at the cell and module development levels with a brief description of the test methods used to qualify these devices for space applications. The approach and development efforts are directed towards large-scale manufacturability of established thin-film, polycrystalline processing methods for large area modules with less emphasis on maximizing small area efficiencies.

Lanning, Bruce R.↗

NASA NTRS1995

Absorption of water and lubricating oils into porous nylon

Oil and water absorption from air into sintered porous nylon can be described by infiltration into the pores of the material. This process can be modeled by a diffusion-like mechanism. For water absorption, we find a formal diffusion coefficient of 1.5 x 10(exp -4)sq cm/min when the nylon is initially dry. The diffusion coefficient is 4 x 10(exp -6)sq cm/min when the nylon is oil-impregnated prior to air exposure. In a 52% RH atmosphere, dry nylon absorbs 3% w/w water, and oil-impregnated nylon absorbs 0.6% w/w water. For oil absorption there are three steps: (1) surface absorption and infiltration into (2) larger and (3) smaller pores. Surface absorption is too fast to be measured in these experiments. The diffusion coefficient for the second step is 6 x 10(exp -4)sq cm/min for SRG-60 oil into dry nylon and 4 x 10(exp -4)sq cm/min for air-equilibrated nylon. The diffusion coefficient for the third step is about 1 x 10(exp -6)sq cm/min for both cases. The total amount of oil absorbed is 31% w/w. The interaction between water and nylon is not as strong as that between water and cotton-phenolic: oil can replace water, and only a small amount of water can enter previously oil-impregnated nylon.

Bertrand, P. A.↗

NASA NTRS1993

A time-accurate finite volume method valid at all flow velocities

A finite volume method to solve the Navier-Stokes equations at all flow velocities (e.g., incompressible, subsonic, transonic, supersonic and hypersonic flows) is presented. The numerical method is based on a finite volume method that incorporates a pressure-staggered mesh and an incremental pressure equation for the conservation of mass. Comparison of three generally accepted time-advancing schemes, i.e., Simplified Marker-and-Cell (SMAC), Pressure-Implicit-Splitting of Operators (PISO), and Iterative-Time-Advancing (ITA) scheme, are made by solving a lid-driven polar cavity flow and self-sustained oscillatory flows over circular and square cylinders. Calculated results show that the ITA is the most stable numerically and yields the most accurate results. The SMAC is the most efficient computationally and is as stable as the ITA. It is shown that the PISO is the most weakly convergent and it exhibits an undesirable strong dependence on the time-step size. The degenerated numerical results obtained using the PISO are attributed to its second corrector step that cause the numerical results to deviate further from a divergence free velocity field. The accurate numerical results obtained using the ITA is attributed to its capability to resolve the nonlinearity of the Navier-Stokes equations. The present numerical method that incorporates the ITA is used to solve an unsteady transitional flow over an oscillating airfoil and a chemically reacting flow of hydrogen in a vitiated supersonic airstream. The turbulence fields in these flow cases are described using multiple-time-scale turbulence equations. For the unsteady transitional over an oscillating airfoil, the fluid flow is described using ensemble-averaged Navier-Stokes equations defined on the Lagrangian-Eulerian coordinates. It is shown that the numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The calculated streaklines are in very good comparison with the experimentally obtained smoke picture. The calculated turbulent viscosity contours show that the transition from laminar to turbulent state and the relaminarization occur widely in space as well as in time. The ensemble-averaged velocity profiles are also in good agreement with the measured data and the good comparison indicates that the numerical method as well as the multipletime-scale turbulence equations successfully predict the unsteady transitional turbulence field. The chemical reactions for the hydrogen in the vitiated supersonic airstream are described using 9 chemical species and 48 reaction-steps. Consider that a fast chemistry can not be used to describe the fine details (such as the instability) of chemically reacting flows while a reduced chemical kinetics can not be used confidently due to the uncertainty contained in the reaction mechanisms. However, the use of a detailed finite rate chemistry may make it difficult to obtain a fully converged solution due to the coupling between the large number of flow, turbulence, and chemical equations. The numerical results obtained in the present study are in good agreement with the measured data. The good comparison is attributed to the numerical method that can yield strongly converged results for the reacting flow and to the use of the multiple-time-scale turbulence equations that can accurately describe the mixing of the fuel and the oxidant.

Kim, S.-W.↗

NASA NTRS1995

Effect of mesh distortion on the accuracy of transverse shear stresses and their sensitivity coefficients in multilayered composites

A study is made of the effect of mesh distortion on the accuracy of transverse shear stresses and their first-order and second-order sensitivity coefficients in multilayered composite panels subjected to mechanical and thermal loads. The panels are discretized by using a two-field degenerate solid element, with the fundamental unknowns consisting of both displacement and strain components, and the displacement components having a linear variation throughout the thickness of the laminate. A two-step computational procedure is used for evaluating the transverse shear stresses. In the first step, the in-plane stresses in the different layers are calculated at the numerical quadrature points for each element. In the second step, the transverse shear stresses are evaluated by using piecewise integration, in the thickness direction, of the three-dimensional equilibrium equations. The same procedure is used for evaluating the sensitivity coefficients of transverse shear stresses. Numerical results are presented showing no noticeable degradation in the accuracy of the in-plane stresses and their sensitivity coefficients with mesh distortion. However, such degradation is observed for the transverse shear stresses and their sensitivity coefficients. The standard of comparison is taken to be the exact solution of the three-dimensional thermoelasticity equations of the panel.

Noor, Ahmed K.↗