Admissible Boundary Conditions Research Articles

Systems of Conservation Laws 2: Geometric Structures, Oscillations and Mixed Problems

7R4. Systems of Conservation Laws 2: Geometric Structures, Oscillations and Mixed Problems. - D Serre (Ecole Normale Superieure, Lyon, France). Cambridge UP, Cambridge, UK. 2000. 269 pp. ISBN 0-521-63330-3. $74.95. Reviewed by Y Horie (Los Alamos Natl Lab, Group X-7, MS D413, Los Alamos NM 87545).Conservation laws arise in many areas of continuum physics or mechanics and are described by certain types of nonlinear partial differential equations. Solving these PDEs for non-dissipative systems with appropriate initial and boundary values is known as the Cauchy problem or the mixed problem. This two-volume set of books is concerned with the modern mathematical treatment of the mixed problem. According to the author, the coverage of the book is comparable to that of Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer, Berlin, 1984), but the subject matters are dealt with “with more detail, but less animation.” Though important, the phenomenon of relaxation and kinetic formulation are not included. Volume 1 dealt with the fundamentals, including such topics as hyperbolicity, Lax shock, the Glimm scheme, and viscosity solution. The second volume, consisting of Chapters 8 through 15, is concerned with questions such as: the convergence of the viscosity method, stability for the Cauchy problem, admissible boundary conditions, and the multi-dimensional mixed problem. But the author cautions that “the materials in the second volume represent a significant part of the active research in the last fifteen years in the field, and are intended more for research workers and less for students.” Whether dealing with fundamentals or the state-of-the art research, this book is definitely written for post-graduate students and active researchers in mathematics. All the basic ideas are discussed fully, but for nonspecialists guiding from well-versed instructors may be helpful to wade through the materials. The readers are expected to have a good mathematical background in contemporary functional analysis. Keywords such as hyperbolic systems, shock waves, viscosity, and initial and boundary value problems are familiar to physical scientists and engineers, but the questions treated in the book are primarily mathematical: existence, uniqueness, convergence, stability, and consistency, etc. The book, however, has something illuminating to offer even to those who may not have the appetite or requisite mathematics to follow theorems and lemmas in depth. It shows what it means to think with the equations from the mathematical point of view. Half-baked knowledge is always dangerous, but this reviewer, who is not a mathematician, was still able to gain new appreciation of and fresh insight into the kinds of questions he does not normally worry about. Good examples are the treatment of stability types and admissible boundary conditions on a finite domain in mathematical terms. His formal training ends at the level of classic books by such people as Courant and Friedrichs, Whitham, and Courant and Hilbert, but he could not help wonder at the mathematical transformation of the questions one used to think intuitively about on a physical basis. At the same time, as marked by Courant (Methods of Mathematical Physics, Volume 1), one fears an increasing gap between contemporary mathematics and the interest of many scientists and engineers. Systems of Conservation Laws 2 is primarily for researchers in mathematics who are interested in the current state of knowledge and students in advanced graduate courses in PDEs. The book is also recommended to non-mathematicians as a means of getting a broad brush view of the conservation equations, nonlinear wave propagation, and shock waves from the mathematical point of view. The book is definitely recommended for university libraries.

BUCKLING OF LAMINATED COMPOSITE PLATE UNDER BIAXIAL LOADING CONDITIONS

The buckling behavior of laminated composite plates subjected to biaxial loading conditions was examined using theoretical, numerical and experimental approaches. For experimental work, a special plate specimen was developed. Deflection characteristics using different boundary conditions of the postbuckled plates were also experimentally investigated to obtain admissible analytical deflection functions and boundary conditions for the theoretical and numerical analyses. Non-linear finite element analysis (N-FEA) was carried out for the two types of boundary conditions, i.e., uniform pressure and uniform end-shortening, to approximate the actual boundary conditions of the plate in experiments. Comparing experimental results with the two N-FEA results for the two different boundary conditions, it was found that results using N-FEA with the uniform pressure loading condition agreed better with experimental results. Effects of flexural stiffness ratio, ζ, induced from the stacking sequence variation, were inve...

On the frequency response of moderately thick simply supported rectangular plates with arbitrary lamination

A here-to-fore unavailable analytical solution to the boundary-value problem of free vibration response of arbitrarily laminated moderately thick rectangular plates, subjected to an admissible boundary condition is presented. A novel boundary continuous solution approach is employed to solve the highly coupled governing partial differential equations that arise from the implementation of the Yang-Norris-Stavsky (1966) theory incorporating first-order shear deformation, and rotary and in-plane inertias. The numerical results presented for various parametric effects to study first lowest seven eigenvalues, and corresponding eigenvectors, can be capitalized as bench-marks for a future comparison. An eight-node isoparametrics shear-flexible element is considered to validate its performance with respect to the present solutions.

On the classification of bulk and boundary conformal field theories

The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy's equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties.

Anti-symmetric angle-ply laminated thick cylindrical panels

Hither-to-fore an unavailable analytical solution (strong form or differential form) to the boundary-value problem of advanced fiber reinforced anti-symmetric angle-ply laminated cylindrical panels of rectangular planform subjected to static loadings is presented. A variationally consistent higher shell theory is utilized that generates five highly coupled fourth order partial differential equations in five unknowns, three displacements, and two rotations. A boundary continuous double Fourier series-based solution functions is assumed to solve such equations in conjunction with the admissible boundary conditions. The numerical results constitute the study of convergence of displacements, in-plane forces, and moments; and spatial variations of them presented in the form of contour plotting for various parametric effects. These, hither-to unavailable solutions, and analytically obtained numerical results should serve as bench-marks for future comparisons of popular approximate methods (integral form or weak form), such as finite element, boundary element, finite difference, Galerkin approach, Rayleigh-Ritz, collocation, least-squares methods, and experimental results, etc.

Saint-Venant's Principle for a Piezoelectric Body

For a semi-infinite piezoelectric body with uniform or variable cross section we establish the exponential decay of the internal energy relative to the portion of the body beyond a distance x3 from the base: Saint-Venant's principle. All possible combinations of physically admissible boundary conditions are examined. The stabilizing effect of the electric field on the decay length is shown. Finally the total internal energy is estimated in terms of the data, making use also of the Korn inequality.

BENDING OF A SIMPLY SUPPORTED RECTANGULAR PLATE WITH ARBITRARY LAMINATION

SUMMARY An unavailable analytical solution to a simply supported rectangular plate with arbitrary laminations is presented. The deformation formulation of the plate characterized by Kirchhoff theory, suitable for thin plates, results in three highly coupled partial differential equations. A double Fourier series approach has been developed to solve these equations in conjunction with the admissible boundary conditions. The accuracy of the solution approach is ascertained by studying the convergence nature of displacement and moment numerically. Numerical results are compared with the available first order shear deformation-based finite element solutions. The numerical results thus presented should serve as bench-marks for future comparisons.

AN EXPLICIT METHOD FOR THE FREE VIBRATIONS OF NON-CIRCULAR CYLINDRICAL PANELS

A method is presented for the determination of the dynamic characteristics of non-circular cylindrical panels, based on Flügge or Donnell type theories. The panels are simply supported along the curved edges, while at the straight edges they are supported by any type of admissible boundary conditions. The components of displacement are expanded in Fourier series in the axial direction and power series in the circumferential direction. The solution is determined by employing a recursive scheme in conjuction with the method of successive approximations. This approach leads to a characteristic equation in which the non-dimensional frequency appears in explicit form. The method is applied to catenary cylindrical panels. Numerical results are presented which describe the variation of the non-dimensional frequencies with respect to the geometric parameters. A comparison with circular panels is presented for different types of boundary conditions.

On boundary value problems of moderately thick shallow cylindrical panels with arbitrary laminations

An analytical (strong form) solution to the boundary value problem of moderately thick shallow cylindrical panels, with arbitrary laminations is presented. A double Fourier series approach is used to solve five second-order highly coupled partial differential equations that arise from the shallow shell formulation based on popular Donnell-Mushtari-Vlasov shell theory, and the first-order shear deformation-based through-thickness theory. An admissible boundary condition is considered to obtain numerical results that constitute the study of convergence of displacements, and moments; and spatial variations of them presented in the form of contour plotting for various parametric effects. These, hitherto unavailable, analytically obtained numerical results should serve as base-line solutions for future comparisons of popular approximate methods such as finite element, finite difference, Galerkin approach, Rayleigh-Ritz method, collocation method, least-squares method, and experimental results, for the case of arbitrary laminations.

A novel approach to the solution of shear flexible rectangular plates with arbitrary laminations

A novel analytical solution to shear flexible rectangular plates with arbitrary laminations is presented. The deformation formulation of the plate characterized by the equivalent single layer theory that considers the effects of transverse shear deformations, suitable for moderately thick plates, results in highly coupled five second-order partial differential equations in five unknowns. A novel boundary continuous double Fourier series solution approach has been developed to solve these coupled equations in conjunction with the admissible boundary conditions. The analytical results presented should serve as bench-marks for future comparisons of such popular approximate methods as finite element, finite difference, Rayleigh-Ritz, Galerkin, and collocation methods.

Fourier solution to higher order theory based laminated shell boundary-value problem

A complete (i.e., particular as well as complementary ) Fourier solution to the boundary-value problem of static response under transverse load of a general cross-ply thick doubly curved panel of rectangular planform is presented. A boundary-discontinuous double Fourier series approach is used to solve a system of five partial differential equations, generated by a higher order shear deformation theory-based shell analysis, with the SS2-type of simply supported boundary condition prescribed at all four edges. Unlike the conventional Navier and Levy type approaches which can only provide particular solutions, the present method is general enough to provide the complete (particular as well as the complementary) solution for any arbitrary combination of admissible boundary conditions with almost equal ease. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of deflections and moments of cross-ply spherical panels and also by comparison with the available first-order shear deformation theory- and classical lamination theory-based analytical solutions. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to lamination, boundary constraint, and thickness and curvature effects, as well as their interactions.

Analysis of moderately-thick and deep shells subjected to thermal and mechanical loadings

An analytical solution to a boundary value problem of moderately-thick and deep doubly curved shells of rectangular planform subjected to thermal and mechanical loadings is presented. Sanders' kinematic relations that incorporate transverse shear deformations in moderately deep shell theory are considered in the shell formulations, yielding five coupled second order partial differential equations in five unknowns displacements. These are then solved in conjunction with admissible boundary conditions. The numerical results presented herein should serve as bench-mark solutions for future comparisons.

Stress and Displacement of Thick Doubly Curved Laminated Shells

A discrete-layer, high-order theory for the stress and displacement analyses of thick, doubly-curved laminated shells is presented. The displacements of the shell in the theory are assumed to be layer-by-layer high-order polynomial functions through the shell thickness. The displacement continuity conditions at the interface between layers are imposed as constraints and are introduced into the potential energy functional by Lagrange multipliers. A set of governing equations and the admissible boundary conditions are given on the basis of the theory by applying the generalized variational principle. The analytical solutions of cross-ply doubly-curved shells with shear diaphragm supports are obtained by using the Fourier series expansion method. They are then compared with the 3D elasticity solutions and the analytical solutions obtained from other laminated shell theories. The present theory indicates very close agreement with 3D elasticity solutions.

Analysis of a simply supported plate with symmetric angle-ply laminations

A new analytical solution to a moderately thick simply supported rectangular plate with symmetric angle-ply laminations is presented. The plate is subjected to a uniformly distributed transverse load. The Resissner-Mindlin theory that incorporates transverse shear deformations into plate formulations characterizing the moderately thick behavior is considered. The plate deformation behavior for bending is defined by three highly coupled second-order partial differential equations in three unknowns. These equations, in conjunction with the admissible boundary conditions, are solved using a powerful numerical tool: a displacement-based double Fourier series approach. Two sets of solution functions are selected, where one set shows a complete continuity at the boundaries, while the other set fails to do so. The numerical results that study convergence of transverse displacement and moments, and variations of them for various parametric effects, should serve as baseline solutions for future comparisons of such popular approximate numerical techniques as the finite element and finite difference methods, as well as experimental results.

Effect of boundary constraint on the frequency response of moderately thick flat laminated panels

A boundary-discontinuous Fourier-type solution to the problem of free vibration of arbitrarily laminated moderately thick rectangular plates, subjected to general admissible boundary conditions, is presented. The YNS (Yang-Norris-Stavsky) theory incorporating first-order shear deformation, and rotary and in-plane inertias, is used in the formulation, resulting in a system of five highly coupled linear second-order partial differential equations (PDEs) with constant coefficients. For the purpose of illustration, the SSI and C4 boundary conditions are selected here because they provide the upper and lower bounds for elastically supported flat panels used in helicopter rotor blades, aircraft wings and other applications. Numerical results presented here include (i) comparison with the corresponding available FSDT-based boundary-continuous displacement-type Fourier and Galerkin solutions, and (ii) study of the effects of thickness, aspect ratio and fiber orientation angle on the normalized natural frequencies.

A local high-order deformable theory for thick laminated cylindrical shells

A local high-order deformable theory for the bending analysis of thick laminated cylindrical shells is developed here. The shell displacements in this theory are assumed to be high-order polynomial functions layer-by-layer through the shell thickness. The displacement continuity constraints at the interface between layers are introduced into the potential energy functional by the Lagrange multiplier method. A set of governing equations and admissible boundary conditions based on this modified potential energy functional are derived. The present analytical solutions of cross-ply circular cylindrical shells with shear diaphragm supports are determined by using the Fourier series expansion method. The present analytical solutions are compared with the 3-D elasticity solutions and the analytical solutions obtained from other laminated cylindrical shell theories. This reveals that the present results agree very closely with the 3-D elasticity solutions.

Boundary Conditions for the Psi Equations

This paper examines admissible boundary conditions for the quasigeostrophic psi equations that govern ageostrophic motion. Both the essential (Dirichlet-type) and natural (Neumann-type) admissible boundary conditions are derived from the variational formulations of the psi equations. Different ways of extracting the admissible boundary values from observational data are proposed. 21 refs.

Mathematical and numerical modelling of shallow water flow

This paper deals with shallow water equations. We discuss the mathematical model, the admissible boundary conditions, some popular numerical methods in the specialized literature, as well as we propose new approaches based on fractional step and finite element methods.

A new local high-order laminate theory

A new local high-order deformable theory of laminated composite/sandwich plates is presented here. The displacement fields of each discrete layer were assumed in the present theory to be of a high-order polynomial series through layer-thickness. The displacement and traction continuity conditions at the interface between layers and the traction conditions at the outer surfaces were imposed as the constraint conditions, and introduced into the potential energy functional by the Lagrange multiplier method. The equations of motion and admissible boundary conditions were given on the basis of the present theory by using the generalized variational principle. Pagano's 3-D elasticity solutions of generally rectangular laminated composite/sandwich plates, fully simply supported, subjected to transverse sinusoidal loading were used for assessment of the present theory and other theories discussed in previous literature. The present theory was found to agree very closely with 3-D elasticity solutions.

Zur wellenmechanischen Behandlung des Wasserstoff-Atoms nach E. Schrödinger

It is well-known that the radial Schrödinger equation for hydrogen-like atoms and zero angular momentum is in the limit-circle case at zero. The problem of finding the eigenvalues for all the mathematically admissible boundary conditions was first solved by Rettich in 1943. In the present paper an alternative treatment is given which is more in line with ideas from the calculus of variations.