Complementary variational principles for steady heat conduction with mixed boundary conditions

Summary New stationary, maximum and minimum principles associated with the boundary value problem of steady heat conduction with general boundary conditions are derived in a unified manner from the theory of complementary variational principles. One of the results contains the Brand-Lahey [3] stationary principle as a special case.     

On steady plane magnetohydrodynamic flows with orthogonal magnetic and velocity fields

Hodograph transformation is employed to obtain a partial differential equation of second order, which is exploited to obtain solutions for plane viscous incompressible flows with orthogonal magnetic and velocity fields. This approach is illustrated through three examples.     

A new variational approach to the delamination problem

A new theory is developed which allows analysis, from a different point of view, of the problem of delamination of an angle-ply laminate [θ/−θ/−θ/θ] under uniform extension. This theory is based on a mixed variational principle which takes into account all the important variables causing the delamination and agrees closely with the numerical solution of Pipes and Pagano. A BASIC version of this analytical solution is implemented on a Tektronix 4050 microcomputer.     

A hybrid element method for steady linearized free-surface flows

It is first shown that the two-dimensional linearized ship wave problem can be recast as the sum of a radiation and a diffraction problem for simple harmonic waves. Each problem can be solved by a hybrid element method (HEM) where conventional finite elements are used near the body and analytical solutions are used in the remaining infinite regions (super-elements). Variational principles which incorporate the matching conditions between regular and super-elements as natural conditions are derived. Numerical examples are presented. The theoretical aspects for extending the above ideas to a three-dimensional ship wave problems are also described.     

A constructive approach to nonlocal variational mechanics

A novel approach to nonlocal variational problems is presented using Balakrishnan's epsilon technique. The minimum conditions are derived for the variational problem bearing restricted implicit nonlocality, and for the general problem with nonlocal constraints. The epsilon technique set up provides a simple direct method for constructing the extended Lagrangian function, and yields a constructive method for introducing the Lagrange multipliers.     

A variational formulation of three-dimensional steady water waves

We are concerned with a three-dimensional layer of a perfect and irrotational liquid of density one (e.g. water) in a gravity field pointing vertically downwards. The layer is of infinite depth and bounded above by a free surface. Babenko developed a variational theory of steady periodic two-dimensional water waves without surface tension, and the aim of this paper is to extend Babenko's formulation to three-dimensional steady periodic waves.     

On variational approaches to steady state heat conduction

Summary Several aspects of the variational approach to steady state heat conduction are examined.     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

The systematic construction of liapunov functionals in the linear stability of conservative steady flows

The linearized equations of motion for small perturbations about a state of steady flow of a conservative dynamical system, when expressed in appropriate variables, admit a class of explicit constants of motion. The construction of Liapunov functionals from these constants of motion is discussed, with applications to plane parallel flow of an inviscid incompressible fluid and plane parallel flow of an inviscid, compressible, electrically charged fluid. Bounds on the $\text{L}{}_2\text{-}\text{norms}$ of certain system variables are obtained.     

A numerical study of steady plane granular chute flows using the Jenkins–Savage model and its extension

The granular flow model proposed by Jenkins and Savage and extended by us is used here to construct numerical solutions of steady chute flows thought to be typical of granular flow behaviour. We present the governing differential equations and discuss the boundary conditions for two flow cases: (i) a fully fluidized layer of granules moving steadily under rapid shear and (ii) a fluidized bottom-near bed covered by a rigid slab of gravel in steady motion under its own weight. The boundary value problem is transformed into a dimensionless form and the emerging system of non-linear ordinary differential equations is numerically integrated. Singularities at the free surface and (in one case) also at an unknown point inside the solution interval make the problem unusual. Since the non-dimensionalization is performed with the maximum particle concentration and the maximum velocity, which are both unknown, these two parameters also enter the formulation of the problem through algebraic equations. The two-point boundary value problem is solved with the aid of the shooting method by satisfying the boundary conditions at the end of the soluton interval and these normalizing conditions by means of a minimization procedure. We outline the numerical scheme and report selective numerical results. The computations are the first performed with the exact equations of the Jenkins–Savage model; they permit delineation of the conditions of applicability of the model and thus prove to be a useful tool for the granular flow modeller.     

A variational multiscale method for particle-cloud tracking in turbomachinery flows

We present a computational method for simulation of particle-laden flows in turbomachinery. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking method. We focus on induced-draft fans used in process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. The particle-laden flow causes material wear on the fan blades, degrading their aerodynamic performance, and therefore accurate simulation of the flow would be essential in reliable computational turbomachinery analysis and design. The turbulent-flow nature of the problem is dealt with a Reynolds-Averaged Navier–Stokes model and Streamline-Upwind/Petrov–Galerkin/Pressure-Stabilizing/Petrov–Galerkin stabilization, the particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence–particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We propose a closure model utilizing the scale separation feature of the variational multiscale method, and compare that to the closure utilizing the eddy viscosity model. We present computations for axial- and centrifugal-fan configurations, and compare the computed data to those obtained from experiments, analytical approaches, and other computational methods.     

Derivation and transformation of variational principles with emphasis on inverse and hybrid problems in fluid mechanics: a systematic approach

Summary A systematic approach to the derivation of variational principles (VPs) from the partial differential equations of fluid mechanics is suggested herein, consisting essentially of two major lines: (1) establishing a first VP via reversed deduction followed by extending it successively to a family of subgeneralized VPs via a series of transformations, and (2) vice versa. Full advantage is taken of four powerful means — the functional variation with variable domain, the natural boundary/initial condition (BC/IC), the Lagrange multiplier, and the artificial interface. The occurrence of three kinds of variational crisis is demonstrated and methods for their removal are suggested. This approach has been used with great success in establishing VP-families in fluid mechanics with special attention to inverse and hybrid problems of flow in a rotating system.     

A spin-vorticity relation for unidirectional plane flows of micropolar fluids

The field equations of micropolar fluid theory are applied to consider transient unidirectional plane flows. The relative angular velocity is introduced and maximum and minimum principles of parabolic partial differential equations are utilized to establish a spin-vorticity relation which is valid for very general boundary and initial conditions on spin.     

A variational approach to a circular hyperelastic membrane problem

Summary The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight.It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.     

Recent developments in parametrized variational principles for mechanics

This paper gives an introduction to the formulation of parametrized variational principles (PVPs) in mechanics. This is complemented by more advanced material describing selected recent developments in hybrid and nonlinear variational principles. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations and natural boundary conditions. The theory of single-field PVPs, based on gauge functions, is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multified PVPs are more interesting from both theoretical and practical standpoints. The two-dimensional Poisson equation is used to present, in a tutorial fashion, the formulation of parametrized mixed functionals. This treatment is then extended to internal interfaces, which are useful in treatment of discontinuities, subdomain linkage and construction of parametrized hybrid functionals. This is followed by a similar but more compact treatment of three-dimensional classical elasticity, and a parametrization of nonlinear hyperelasticity.The research work in Parametrized Variational Principles has been supported by the Air Force Office of Scientific Research under Grant F49620-87-C-0074, NASA Langley Research Center under Grant NAG1-756, NASA Lewis Research Center under Grant NAG3-1273, and the National Science Foundation under Grant ASC-9217394.Dedicated to the 10th anniversary of Computational Mechanics     

A variational approach to magnetoelastic buckling problems for systems of superconducting tori

A variational principle for the magnetoelastic stability problem of superconductors is constructed. Independently, a pair of integral equations is derived, from which the initial and the perturbed field can be computed. The integral equations are solved for in-plane buckling of a slender pair of concentric tori, and out-of-plane buckling of a slender pair of equal coaxial tori. By using the variational principle, it is shown that both cases can become unstable when the currents on the two tori are equally directed, and the pertinent buckling values are calculated. The thus obtained buckling values are compared with the results of an alternative, mathematically less rigorous, method. A good correspondence between the two methods is found (at least as long as the two tori are not too near).     

A discrete singularity formulation for solving steady,fully developed duct flows

A very effective numerical formulation is proposed to solve approximately the problem of steady, fully developed laminar flows of incompressible fluids in straight ducts of arbitrary, simply connected cross-sections. The method is an application of the discrete singularity technique, with a finite number of singularities located outside the domain concerned. To illustrate the usefulness of the method, some examples of computations are also presented.