The variational method 2 - Perturbation theory
Variational method to approximate solutions of perturbation equations, perturbation analyses, and use of perturbation theory to infer variational principle
Epstein, S. T.↗
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Variational method 3 - Steady state time dependent perturbation theory
Steady state time dependent perturbation theory and variational principles
Epstein, S. T.↗
Variational method in the stability analysis of nonconservative problems
Nonconservative problems motion stability analysis based on restricted admissible displacements, establishing two variational principles
Shieh, R.-C.↗
The analysis of nonlinear panel flutter and response under random excitation or nonlinear aerodynamic loading
Nonlinear panel flutter for random excitation and linear/nonlinear aerodynamic loading, using Rayleigh-Ritz approximation to Hamilton variational principle
Eastep, F. E.↗
Large strain, elasto-plastic finite element analysis
Two dimensional structures large strain elastoplastic analysis by finite element method, using variational principles to derive equilibrium equations
Evensen, D. A.↗
An approximate three-dimensional theory of plates with application to crack problems
Elastic plate uniform extension with rectangular crack by three dimensional bending theory, using variational principle
Hartranft, R. J.↗
On certain approximations in the finite- element method
Finite element method approximations outside variational principle in elasticity problems, examining convergence to exact solution
Mclay, R. W.↗
Analysis of nonlinear panel flutter and response under random excitation or nonlinear aerodynamic loading
Nonlinear panel flutter analysis and response under random excitation or nonlinear aerodynamic loading, using Rayleigh-Ritz approximation to Hamilton variational principle
Eastep, F. E.↗
Variational formulation of relativistic fluid thermodynamics
Relativistic fluid thermodynamics for compressible fluid reversible adiabatic flow, using variational principle to derive stress-energy tensor
Schmid, L. A.↗
Dynamic characteristics of a variable-mass flexible missile: Dynamics of a two-stage variable-mass flexible rocket
The dynamic characteristics of two-stage slender elastic body were investigated. The first stage, containing a solid-fuel rocket, possesses variable mass while the second stage, envisioned as a flexible case, contains packaged instruments of constant mass. The mathematical formulation was in terms of vector equations of motion transformed by a variational principle into sets of scalar differential equations in terms of generalized coordinates. Solutions to the complete equations were obtained numerically by means of finite difference techniques. The problem has been programmed in the FORTRAN 4 language and solved on an IBM 360/50 computer. Results for limited cases are presented showing the nature of the solutions.
Meirovitch, L.↗
Two-dimensional finite-element temperature variance analysis
The finite element method is extended to thermal analysis by forming a variance analysis of temperature results so that the sensitivity of predicted temperatures to uncertainties in input variables is determined. The temperature fields within a finite number of elements are described in terms of the temperatures of vertices and the variational principle is used to minimize the integral equation describing thermal potential energy. A computer calculation yields the desired solution matrix of predicted temperatures and provides information about initial thermal parameters and their associated errors. Sample calculations show that all predicted temperatures are most effected by temperature values along fixed boundaries; more accurate specifications of these temperatures reduce errors in thermal calculations.
Heuser, J. S.↗
Void growth in an elastic-plastic medium.
The uniaxial deformation of an elastic-plastic medium containing a doubly periodic square array of circular cylindrical voids is studied under plane-strain conditions. Both the effects of geometrical nonlinearities resulting from large deformation and physical nonlinearities arising from plastic material behavior are included in formulating the problem. A variational principle is used as the basis for implementing a finite-element solution. Results are obtained for the change in void shape and size under increasing overall strain, the overall tensile behavior of the material with voids, and the development of the plastic zone about a void.
Needleman, A.↗
Theorems satisfied by variational wave functions in scattering theory
Various theorems in the theory of spinless particle scattering by a real central potential are derived directly from the Kohn variational principle.
Epstein, S. T.↗
Transfer phenomena in reacting gas mixtures
In considering transfer phenomena in reacting gas mixtures, the following topics are discussed: (1) dynamics of paired collisions; (2) Enskog method for solving the Boltzmann equation, taking into account inelastic collisions; (3) use of the variation principle for approximate solutions; (4) computations of inelastic bracketed expressions; (5) calculation of collision integrals; and (6) formulas for transfer coefficients.
Alekseyev, B. V.↗
Solution of coupled and singular perturbation methods using duality theory.
Dual variational techniques developed by Chan and Leininger (1972) are summarized, and duality theory in the form of the Complementary Variational Principle is employed to provide a suboptimal measure for the singular and epsilon-coupled perturbation methods proposed by Kokotovic and Cruz. The suboptimal measure is independent of any a priori knowledge of the optimal solution, thereby providing an absolute estimate of the performance loss rather than an estimate relative to the unknown optimal solution.
Chan, W. L.↗
Finite element formulations for problems of large elastic-plastic deformation
An Eulerian finite element formulation is presented for problems of large elastic-plastic flow. The method is based on Hill's variational principle for incremental deformations, and is suited to isotropically hardening Prandtl-Reuss materials. The formulation is given in a manner which allows any conventional finite element program, for "small strain" elasticplastic analysis, to be simply and rigorously adapted to problems involving arbitrary amounts of deformation and arbitrary levels of stress in comparison to plastic deformation moduli. The method is applied to a necking bifurcation analysis of a bar in plane-strain tension. A unified general formulation of finite element equations, both Lagrangian and Eulerian, for large deformations, with arbitrary choice of the conjugate stress and strain measures, and a discussion is given of other proposed formulations for elastic-plastic finite element analysis at large strain.
Mcmeeking, R. M.↗
A finite element solution algorithm for the Navier-Stokes equations
A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame.
Baker, A. J.↗
An improved numerical process for solution of solid mechanics problems
This paper gives an overview of the development and status of an improved numerical process for the solution of solid mechanics problems. The proposed process uses a mixed formulation with the fundamental unknowns consisting of both stress and displacement parameters. The problem is formulated either by means of first-order partial differential equations or in a variational form by using a Hellinger-Reissner-type mixed variational principle. For presentation purposes, the components of a numerical process are characterized and the criteria for an ideal process are outlined. Commonly used finite-difference and finite-element procedures are examined in the light of these criteria and it is shown that they fall short in a number of ways. The proposed numerical process, on the other hand, satisfies most of the optimality criteria and appears to be particularly suited for use with the forthcoming generation computers.
Noor, A. K.↗