Buckling of Segments of Toroidal Shells
Nonlinear differential equations of equilibrium and buckling equations for segments of toroidal shells having Gaussian curvature
BUCKLING↗
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Theory of elastic shells in the reference state.
Nonlinear field equations of elastic shell in terms of reference state subjected to strain displacement
SHELL THEORY↗
Multigrid Methods for Fully Implicit Oil Reservoir Simulation
In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.
Molenaar, J.↗
Three-dimensional vibration analysis of a uniform beam with offset inertial masses at the ends
Analysis of a flexible beam with displaced end-located inertial masses is presented. The resulting three-dimensional mode shape is shown to consist of two one-plane bending modes and one torsional mode. These three components of the mode shapes are shown to be linear combinations of trigonometric and hyperbolic sine and cosine functions. Boundary conditions are derived to obtain nonlinear algebraic equations through kinematic coupling of the general solutions of the three governing partial differential equations. A method of solution which takes these boundary conditions into account is also presented. A computer program has been written to obtain unique solutions to the resulting nonlinear algebraic equations. This program, which calculates natural frequencies and three-dimensional mode shapes for any number of modes, is presented and discussed.
Robertson, D. K.↗
An efficient model for coupling structural vibrations with acoustic radiation
The scattering of an incident wave by a flexible panel is studied. The panel vibration is governed by the nonlinear plate equations while the loading on the panel, which is the pressure difference across the panel, depends on the reflected and transmitted waves. Two models are used to calculate this structural-acoustic interaction problem. One solves the three dimensional nonlinear Euler equations for the flow-field coupled with the plate equations (the fully coupled model). The second uses the linear wave equation for the acoustic field and expresses the load as a double integral involving the panel oscillation (the decoupled model). The panel oscillation governed by a system of integro-differential equations is solved numerically and the acoustic field is then defined by an explicit formula. Numerical results are obtained using the two models for linear and nonlinear panel vibrations. The predictions given by these two models are in good agreement but the computational time needed for the 'fully coupled model' is 60 times longer than that for 'the decoupled model'.
Frendi, Abdelkader↗
Critical problems of computational aeroacoustics
The aeroacoustics of rigid boundaries is discussed. Lighthill gave a formulation of this problem in which he showed that the sources of the acoustic field were quadrupole in nature. We have preferred a different formulation of the problem, in which the quadrupoles are sources for a nonlinear wave equation, as opposed to the linear one used by Lighthill. This is given here in a figure which also gives further details of a solution procedure for the Euler equations appropriate for the aeroelastic problem and motivated by the analysis of Crow. In accordance with our formulation, we contend that an algorithm for accurate solutions of an inhomogeneous nonlinear wave equation is of prime importance.
Rogers, Joel C. W.↗
A note on Volterra integral equations and topological dynamics
Topological dynamics theory applied to nonlinear Volterra equations
Miller, R. K.↗
Variable mesh methods for differential equations
Variable mesh multistep methods for numerical solution of nonlinear differential equations
Van Wyk, R.↗
An initial value method for the Ambarzumian integral equation.
It is shown that a general class of nonlinear integral equations may be transformed into a Cauchy system. That this leads to an effective numerical scheme is demonstrated by solving the Ambarzumian integral equation. The new method does not involve successive approximations or series expansions.
Buell, J.↗
Chaotic /strange/ and periodic behavior in instability saturation by the oscillating two-stream instability
The nonlinear Schroedinger equation with linear growth and damping is truncated to three waves. The resulting system of nonlinear ordinary differential equations describes the excitation of linearly damped waves by the oscillating two-stream instability driven by a linearly unstable pump wave. This system represents a simple model for the nonlinear saturation of a linearly unstable wave. The model is examined analytically and numerically as a function of the dimensionless parameters of the system. It is found that the model can exhibit a wealth of characteristic dynamical behavior including stationary equilibria, Hopf bifurcations to periodic orbits, period doubling bifurcations, chaotic solutions characteristic of a strange attractor, tangent bifurcations from chaotic to periodic solutions, transient chaos, and hysteresis. Many of these features are shown to be explainable on the basis of one-dimensional maps. In the case of chaotic solutions, evidence for the presence of a strange attractor is provided by demonstrating Cantor set-like structure (i.e., scale invariance) in the surface of section.
Russell, D. A.↗
Postbuckling of orthotropic composite plates loaded in compression
The nonlinear large deflection equations of von Karman are written for 'specially' orthotropic plates. The equations are then manipulated to determine the parameters required to establish postbuckling behavior. It is found that only two new parameters are needed beyond those required for buckling. By assuming trigonometric functions in one direction, the plate equations are converted into ordinary nonlinear differential equations which are solved numerically using a two point boundary problem solver that makes use of Newton's method. The postbuckling behavior is obtained for simply supported and clamped, long, rectangular, orthotropic plates covering the complete range of dimensions and material properties.
Stein, M.↗
Aeroelastic equations of motion of a Darrieus vertical-axis wind-turbine blade
The second-degree nonlinear aeroelastic equations of motion for a slender, flexible, nonuniform, Darrieus vertical-axis wind turbine blade which is undergoing combined flatwise bending, edgewise bending, torsion, and extension are developed using Hamilton's principle. The blade aerodynamic loading is obtained from strip theory based on a quasi-steady approximation of two-dimensional incompressible unsteady airfoil theory. The derivation of the equations has its basis in the geometric nonlinear theory of elasticity and the resulting equations are consistent with the small deformation approximation in which the elongations and shears are negligible compared to unity. These equations are suitable for studying vibrations, static and dynamic aeroelastic instabilities, and dynamic response. Several possible methods of solution of the equations, which have periodic coefficients, are discussed.
Kaza, K. R. V.↗
Finite element analysis of geometrically nonlinear problems
Recent developments in geometrically nonlinear finite element analysis are reviewed. Following a discussion of the theoretical bases for the construction of finite element equations for geometrically nonlinear analysis, the algorithmic tools for solving the resulting nonlinear equations of the complete system are described. Separate consideration is given to preinstability nonlinear analysis, the calculation of limit points, bifurcation points, and to postbifurcation analysis.
Gallagher, R. H.↗
A note on Stratonovich's equation for the conditional probability distribution of the state of a nonlinear system, with application to nonlinear filtering
Analysis of equation for conditional probability distribution of state of nonlinear system and application to nonlinear filtering
STATE EQUATION↗
Pulsatile instability in rapid directional solidification - Strongly-nonlinear analysis
In the rapid directional solidification of a dilute binary alloy, analysis reveals that, in addition to the cellular mode of Mullins and Sekerka (1964), there is an oscillatory instability. For the model analyzed by Merchant and Davis (1990), the preferred wavenumber is zero; the mode is one of pulsation. Two strongly nonlinear analyses are performed that describe this pulsatile mode. In the first case, nonequilibrium effects that alter solute rejection at the interface are taken asymptotically small. A nonlinear oscillator equation governs the position of the solid-liquid interface at leading order, and amplitude and phase evolution equations are derived for the uniformly pulsating interface. The analysis provides a uniform description of both subcritical and supercritical bifurcation and the transition between the two. In the second case, nonequilibrium effects that alter solute rejection are taken asymptotically large, and a different nonlinear oscillator equation governs the location of the interface to leading order. A similar analysis allows for the derivation of an amplitude evolution equation for the uniformly pulsating interface. In this case, the bifurcation is always supercritical. The results are used to make predictions about the characteristics of solute bands that would be frozen into the solid.
Merchant, G. J.↗
Continuum Fatigue Damage Modeling for Critical Design, Control, and Fault Prognosis
This paper develops a simplified continuum (continuous with respect to time, stress, etc.) fatigue damage model for use in critical design, Life Extending Control and fault prognosis. The work is based on the local strain cyclic damage modeling method. New nonlinear explicit equation forms of cyclic damage in terms of stress amplitude are derived to facilitate the continuum modelling. Stress based continuum models are derived. Extension to plastic strain-strain rate models is also presented. Progress toward a non-zero mean stress based is presented. Also new nonlinear explicit equation forms in terms of stress amplitude are derived for this case. Application of the various models to design, control, and fault prognosis is considered.
Lorenzo, Carl F.↗
The construction of the thermoelasticity equations on the basis of contemporary notions of nonlinear continua mechanics
The fundamental concepts of the theory of simple thermomechanical materials are applied to a thermoelastic medium. Nonlinear coupled thermoelasticity equations are derived, with free energy and heat flux approximated by polynomials of deformation and temperature invariants. Details of the derivation procedure are presented.
Kilchinskaya, G. A.↗
Linear and nonlinear propagation of water wave groups
Results are presented from a study of the evolution of waveforms with known analytical group shapes, in the form of both transient wave groups and the cloidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schroedinger equation. The waveforms were generated in a long wind-wave tank of the Canada Centre for Inland Waters. It was found that the low-amplitude transients behaved as predicted by the linear theory and that the cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schroedinger equation. Some of the results did not fit into any of the available theories for waves on water, but they provide important insight on how actual groups of waves propagate and on higher-order effects for a transient waveform.
Pierson, W. J., Jr.↗