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A general solution to the jump-discontinuous two-point boundary value problem (TPBVP) of the minimum model error is presented. Its existence and uniqueness are then proved.
The fastest known algorithms for the solution of a large elliptic boundary value problem on a massively parallel hypercube all require O(log(n)) floating point operations and O(log(n)) distance-1 communications, if massively parallel is defined to mean a number of processors proportional to the size n of the problem. The Totally Parallel Multilevel Algorithm (TPMA) that has, as special cases, four of these fast algorithms is described. These four algorithms are Parallel Superconvergent Multigrid (PSMG), Robust Multigrid, the Fast Fourier Transformation (FFT) based Spectral Algorithm, and Parallel Cyclic Reduction. The algorithm TPMA, when described recursively, has four steps: (1) project to a collection of interlaced, coarser problems at the next lower level; (2) apply TPMA, recursively, to each of these lower level problems, solving directly at the lowest level; (3) interpolate these approximate solutions to the finer grid, and to verage them to form an approximate solution on this grid; and (4) refine this approximate solution with a defect-correction step, using a local approximate inverse. Choice of the projection operator (P), the interpolation operator (Q), and the smoother (S) determines the class of problems on which TPMA is most effective. There are special cases in which the first three steps produce an exact solution, and the smoother is not needed (e.g., constant coefficient operators).
The stability characteristics of various compact fourth and sixth order spatial operators are assessed using the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semi-discrete Initial Boundary Value Problem (IBVP). These results are then generalized to the fully discrete case using a recently developed theory of Kreiss. In all cases, favorable comparisons are obtained between the G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability is then sharpened to include only those spatial discretizations that are asymptotically stable. It is shown that many of the higher order schemes which are G-K-S stable are not asymptotically stable. A series of compact fourth and sixth order schemes, which are both asymptotically and G-K-S stable for the scalar case, are then developed.
Some ideas for using hodograph theory, mapping techniques and methods of characteristics to formulate typical aerodynamic design boundary value problems are developed. The inverse method of characteristics is shown to be a fast tool for design of transonic flow elements as well as supersonic flows with given shock waves.
Examined here is the effect of fiber and interfacial layer morphologies on thermal fields in metal matrix composites (MMCs). A micromechanics model based on an arbitrarily layered concentric cylinder configuration is used to calculate thermal stress fields in MMCs subjected to spatially uniform temperature changes. The fiber is modelled as a layered material with isotropic or orthotropic elastic layers, whereas the surrounding matrix, including interfacial layers, is treated as a strain-hardening, elastoplastic, von Mises solid with temperature-dependent parameters. The solution to the boundary-value problem of an arbitrarily layered concentric cylinder under the prescribed thermal loading is obtained using the local/global stiffness matrix formulation originally developed for stress analysis of multilayered elastic media. Examples are provided that illustrate how the morphology of the SCS6 silicon carbide fiber and the use of multiple compliant layers at the fiber/matrix interface affect the evolution of residual stresses in SiC/Ti composites during fabrication cool-down.
A method for estimating an unknown domain in elliptic boundary value problems is considered. The problem is formulated as an inverse problem of integral equations of the second kind. A computational method is developed using a splice collocation scheme. The results can be applied to the inverse problem of impedance computed tomography (ICT) for image reconstruction.
The micromechanical constitutive theory is used to examine the nonlinear behavior of continuous-fiber-reinforced metal-matrix composite structures. Effective lamina constitutive relations based on the Abouli micromechanics theory are presented. The inelastic matrix behavior is modeled by the unified viscoplasticity theory of Bodner and Partom. The laminate constitutive relations are incorporated into a first-order deformation plate theory. The resulting boundary value problem is solved by utilizing the finite element method. Attention is also given to computational aspects of the numerical solution, including the temporal integration of the inelastic strains and the spatial integration of bending moments. Numerical results the nonlinear response of metal matrix composites subjected to extensional and bending loads are presented.
A tool which supports integrated assessment of air-breathing hypersonic vehicle trajectory management and control strategy has been developed by Hattis and Malchow (1991). Using a two-point boundary value problem solution technique, the tool can construct an entire near-minimum-fuel trajectory and desired control strategy from takeoff to orbit, while satisfying physically derived inequality constraints and while achieving efficient propulsive mode phasing. A strategy is also formulated to construct preliminary trajectory and control history representations with less computational burden than required for the overall flight profile assessment. A tabulated example hypersonic vehicle model has been used to demonstrate the integrated analysis methodology. To assure good numerical behavior when the algorithm uses tabulated model data, a multidimensional cubic spline curve smoothing routine has been developed. A dynamic-pressure-constrained, near-fuel-optimal trajectory was computed to provide examples of information that the methodology can provide. Previously unspecified propulsive discontinuities were located, flight regimes demanding rapid attitude changes were identified, available and closed-loop controller authority was ascertained, and inadequacies in vehicle and specific subsystem model representations were determined.
A theoretical broadband coupling-loss factor is developed analytically for use in the statistical energy analysis (SEA) of a shaft-bearing-plate system. The procedure is based on the solution of the boundary-value problem at the plate-bearing interface and incorporates a bearing-stiffness matrix developed by the authors. Three examples are utilized to illustrate the SEA incorporating the coupling-loss factor including: (1) a shaft-bearing-plate system; (2) a plate-cantilevered beam; and (3) a circular-shaft-bearing plate. The coupling-loss factor in the case of the thin plate-cantilevered beam is found to be more accurate than that developed by Lyon and Eichler (1964). The coupling-loss factor is described for the bearing system and extended to describe the mean-square vibratory response of a rectangular plate. The proposed techniques are of interest to the study of vibration and noise in rotating machinery such as gearboxes.
A new algorithm for computing one-dimensional acoustic waves in a pipe is demonstrated by solving the acoustic equations as an initial-boundary-value problem. Conventional dissipation-free second-order finite difference methods suffer severe phase distortion for grids with less that about ten mesh points per wavelength. Using the signal generation by a piston in a duct as an example, transient acoustic computations are presented using a new compact three-point algorithm which allows about 60 percent fewer mesh points per wavelength. Both pulse and harmonic excitation are considered. Coupling of the acoustic signal with the pipe resonant modes is shown to generate a complex transient wave with rich harmonic content.
This paper addresses the fracture resistance mechanism in fiber reinforced ceramics, and focuses attention on the specific effects associated with nonlinear nature of the fiber pullout mechanism. The model is based on a consideration of the representative boundary value problem typical for the bridging process. The theoretical solution includes an accurate account of the nonlinear fiber matrix friction. The developed approach allows consideration of several types of nonlinear fiber pullout-force dependence. The distinct features of the nonlinear process demonstrate that, contrary to the linear case, the universal fracture resistance curves cannot be developed in cases with significant nonlinear contribution in the fiber friction law. The resulting resistance curves depend strongly on the absolute values of the matrix fracture toughness. On the other hand, these distinct patterns may be used for identification of the particular friction law and determination of the friction parameters.
MPERRE computer program accelerates convergence of sequence of vectors by use of minimum-polynomial extrapolation (MPE) and reduced-rank extrapolation (RRE). Effective in accelerating convergences of such sequences of vectors as those obtained from iterative solution of systems of linear and nonlinear equations. In conjunction with various iterative techniques, successfully employed in finite-difference solution of large-scale elliptic boundary-value problems and problems in computational-fluid-dynamics. Only input required is sequence of vectors, convergence of which is accelerated. Program economical and easy to use. Written in FORTRAN 77.
This paper describes the prediction of noise radiation from supersonic elliptic jets. The noise is associated with the large scale structures in the jet mixing layer. These structures are described as instability waves. The local characteristics of the instability waves are determined from a compressible, linear, analysis. The jet mean velocity and density are described in elliptic cylindrical coordinates. The local eigensolution for the instability waves is determined from a finite difference solution of the non-separable boundary value problem. This inner solution which is formulated in terms of the method of multiple scales is matched with the radiated field using the method of matched asymptotic expansions. The form of the far-field directivity is derived. Predictions are presented for the noise radiation by the several modes of instability in the elliptic jet. The radiated field is not axisymmetric and certain modes radiate strongly in the directions of the major and minor axes of the jet. The extension of the present work to other geometries and flow fields is discussed.
The utility of a recently developed analytical micromechanics model for the response of metal matrix composites under thermal loading is illustrated by comparison with the results generated using the finite-element approach. The model is based on the concentric cylinder assemblage consisting of an arbitrary number of elastic or elastoplastic sublayers with isotropic or orthotropic, temperature-dependent properties. The elastoplastic boundary-value problem of an arbitrarily layered concentric cylinder is solved using the local/global stiffness matrix formulation (originally developed for elastic layered media) and Mendelson's iterative technique of successive elastic solutions. These features of the model facilitate efficient investigation of the effects of various microstructural details, such as functionally graded architectures of interfacial layers, on the evolution of residual stresses during cool down. The available closed-form expressions for the field variables can readily be incorporated into an optimization algorithm in order to efficiently identify optimal configurations of graded interfaces for given applications. Comparison of residual stress distributions after cool down generated using finite-element analysis and the present micromechanics model for four composite systems with substantially different temperature-dependent elastic, plastic, and thermal properties illustrates the efficacy of the developed analytical scheme.
The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.
An analytical study of interlaminar stresses in unsymmetrically laminated plates is presented. The study examines the linear elastic, large deflection response of square laminated composite plates subjected to uniform thermal loading. Both cross-ply and angle-ply, symmetric and unsymmetric, laminates are examined to evaluate the effects of mismatch between adjacent layers in elastic constants and coefficients of thermal expansion. A geometrically nonlinear kinematic description is used to predict the large out-of-plane (transverse) deflections. The nonlinear, three-dimensional boundary value problems are formulated from elasticity theory and approximate solutions are determined using the finite element method. A global/local analysis procedure is used to obtain improved free edge stress predictions. For the laminates and loading conditions considered, the results indicate that the out-of-plane deflections of the unsymmetric laminates reduce interlaminar shear stresses. These deflections also reduce interlaminar normal stresses in some laminates and increase these stresses for others. The results indicate that as the out-of-plane deflections become large, the differences in interlaminar stress predictions between linear and nonlinear theory can become quite large.
A method is presented for the calculation of the flutter speed of a uniform wing carrying an arbitrarily placed concentrated mass. The method, an extension of recently published work by Goland and Luke, involves the solution of the differential equations of motion of the wing at flutter speed and therefore does not require the assumption of specific normal modes of vibration. The order of the flutter determinant to be solved by this method depends upon the order of the system of differential equations and not upon the number of modes of vibration involved. The differential equations are solved by operational methods, and a brief discussion of operational methods as applied to boundary-value problems is included in one of two appendixes. A comparison is made with experiment for a wing with a large eccentrically mounted weight and good agreement is obtained. Sample calculations are presented to illustrate the method; and curves of amplitudes of displacement, torque, and shear for a particular case are compared with corresponding curves computed from the first uncoupled normal modes.
A theoretical study is described of the aerodynamic characteristics at small angle of attack of a thin, double-wedge profile in the range of supersonic flight speed in which the bow wave is detached. The analysis is carried out within the framework of the transonic (nonlinear) small-disturbance theory, and the effects of angle of attack are regarded as a small perturbation on the flow previously calculated at zero angle. The mixed flow about the front half of the profile is calculated by relaxation solution of a suitably defined boundary-value problem for transonic small-disturbance equation in the hodograph plane (i.e., the Tricomi equation). The purely supersonic flow about the rear half is found by an extension of the usual numerical method of characteristics. Analytical results are also obtained, within the framework of the same theory, for the range of speed in which the bow wave is attached and the flow is completely supersonic.