Self-adjoint Boundary Value Problems Research Articles

Comparisons of crystal hardening laws in multiple slip

Abstract This paper brings together and concisely reviews results from recent analytical investigations on single crystals (variously done alone or with students) in which predictions from different theoretical hardening laws are contrasted and compared with experimental studies. Finitely deforming f.c.c. crystals in both constrained and unconstrained multiple-slip configurations are considered. Four crystal hardening laws are given prominence. Two of these belong to a class of theories in which the physical hardening moduli relating rates-of-change of critical strengths (in the 24 crystallographically equivalent slip systems) to slip-rates are taken as symmetric. These are G.I. Taylor's classic isotropic hardening rule (proposed in 1923), which is almost universally adopted in the metallurgical literature for various approximate analyses of single and poly-crystal deformation, and a 2-parameter modification of Taylor's rule that has an empirical basis in the qualitative features of experimentally determined latent hardening in single slip. The other two hardening laws featured here belong to a class of theories that were introduced in 1977 by this author. This class requires the above moduli to be nonsymmetric and explicity dependent upon the current stress state in such a manner that the following consequences are assured. (1) The deformation-dependent hardening of latent slip systems necessarily develops anisotropically if there is relative rotation of gross material and underlying crystal lattice. (2) The theories admit self-adjoint boundary value problems for crystalline aggregates, hence a variational formulation. (The fact that symmetric physical hardening moduli do not permit variational formulations of polycrystalline problems was shown at the 1972 Warsaw Symposium.) The two members of this class considered here are the original (and simplest p possible) theory of rotation-dependent anisotropy, which was proposed by this author in 1977 and commonly has been referred to as the “simple theory,” and a modification of this theory introduced in 1982 by Peirce, Asaro and Needleman that lies between Taylor's rule and the simple theory in its predictions for finitely deforming f.c.c. crystals. (In a series of five papers during 1977–1979, the simple theory was shown to universally account for the experimental phenomenon of “overshooting” in single slip in both f.c.c. and b.c.c. crystals.) Theoretical results from the various hardening rules are contrasted and compared with finite strain experiments in the metallurgical literature. Both tensile-loaded crystals in 4, 6 and 8-fold symmetry orientations and compressively loaded crystals under conditions of channel die constraint are treated. A postulate of minimum plastic work introduced in 1981 plays a prominent role in the theoretical analyses, in many cases providing a unique solution to the slip system inequalities and deformation constraints (where applicable). The rather remarkable ability of the simple theory to reconcile diverse qualitative features of both constrained and unconstrained finited deformation of f.c.c. crystals is demonstrated. Finally, conditions for total loading (all systems active) in 6-fold symmetry are investigated, and certain concepts regarding the selection of active systems under prescribed straining are critically assessed.

Multigrid Methods for Variational Problems: Further Results

In an earlier paper [SIAM J. Numer. Anal., 19 (1982), pp. 924–929], we developed a convergence theory for a class of multigrid methods applied to positive definite self-adjoint differential boundary value problems, where the multigrid process used W-cycling and Richardson’s iteration. In the present paper, we extend this theory to include V-cycling and more general relaxation schemes such as Jacobi and (arbitrarily ordered) Gauss-Seidel.

An adaptive, multi-level method for elliptic boundary value problems

Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions ofR2. It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. In this work we describe the method and present some numerical results and comparisons.

On computation of eigenvalues for differential equations by Newton’s method

Professor Zeev Nehari, in the last months before his untimely death, had been analyzing and developing a new algorithm for computing eigenvalues of selfadjoint boundary value problems of arbitrary order. Apparently, his main goals were Theorems 3.1 and 3.2 below (which incidentally yield the eigenvalues of the given problem, as well as those of a related problem). Unfortunately, as far as we know, Professor Nehari has not left a proof of the basic Lemma 3.1 which presumably was to be based on § 2 \S 2 or related results. It seems, however, worthwhile to publish this paper since the results of § § 1 \S \S 1 and 2 2 are complete and of independent interest and since the question of the validity of Lemma 3.1 is also of interest. While the presentation in § 1 \S 1 and 2 2 is in most respects complete, there are a few places where some minor clarifications by selected footnotes were felt to be desirable. Also, Professor Nehari’s original incomplete § 3 \S 3 has been replaced by a revised and abbreviated version.

An extension of Mercer's theorem to a class of non-symmetric integral equations

In this paper it is shown that certain properties of selfadjoint boundary value problems with positive eigenvalues, like the countability of the discrete eigenvalues and the expand-ability of the problem's Green function, are presevred if the problem is perturbed and becomes non-selfadjoint through an additional term involving a positive, continuously varying parameter. This is so, if the parameter remains small enough and does not transgress the so called stability domain of the problem.

On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems

It has been known for several years that the expected value ⟨ λ 1 ⟩ \left \langle {{\lambda _1}} \right \rangle of the smallest eigenvalue of a self-adjoint positive definite random Sturm-Liouville boundary value problem satisfies the relation ⟨ λ 1 ⟩ ≤ μ 1 \left \langle {{\lambda _1}} \right \rangle \le {\mu _1} where μ 1 {\mu _1} is the smallest eigenvalue of the corresponding deterministic problem obtained by replacing each random coefficient by its mean. It has been an open question whether similar inequalities are valid for the higher eigenvalues. The answer is negative, as shown by the counterexample given in this note.

Removable singularities for generalized Prüfer transformations

A recent generalization of the classical Prüfer transformation to fourth order differential equations encountered singularities which limited such transformations to specific classes of self-adjoint boundary value problems. This paper shows that these singularities are removable and extends such transformations to the general self-adjoint equation of order 2 n.

Two-sided estimates in unilateral elasticity

The paper presents an extension to unilateral problems of the classical method of bounding (above and below) the solutions of linear self-adjoint boundary value problems. Using this extension the solution of the general unilateral problem in linear elasticity is bounded in energy by two suitable defined admissible states belonging to two complementary convex sets.

Accelerating Convergence of Eigenfunction Expansions

A general procedure is presented for accelerating the convergence of eigenfunction expansions associated with selfadjoint boundary-value problems. The results obtained reduce, in special cases, to certain well-known methods of acceleration, including the Lanczos representation. The generality of the procedure allows the user to take advantage of structural properties of the expanded function.

On Mixed Finite Element Approximations

A theory of mixed finite element approximations of a class of linear self-adjoint boundary value problems is developed which involves splitting a problem of the type $T^ * Tu = f$ into a pair of canonical equations, $Tu = {\bf v}$, $T^ * v = f$. Consistency and stability of Galerkin/finite element approximations of the pair is described, convergence criteria are established, and error estimates are derived For a number of important cases.

A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

The classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some "physical" situations. A trial function is constructed using a modified quintic smooth Hermite space $$\tilde H_0^{(3)} (\pi )$$ , in order to meet some desired regularity conditions for the approximate solution. A collocation technique is used to reduce the amount of computational work. Known convergence properties for the projection method are recalled which, in this particular case, are illustrated by a series of numerical experiments.

Accelerating convergence of eigenfunction expansions

A general procedure is presented for accelerating the convergence of eigenfunction expansions associated with selfadjoint boundary-value problems. The results obtained reduce, in special cases, to certain well-known methods of acceleration, including the Lanczos representation. The generality of the procedure allows the user to take advantage of structural properties of the expanded function.

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.

Bifurcation in singular self-adjoint boundary value problems

Bifurcation in singular self-adjoint boundary value problems

Zur Regularität Selbstadjungierter Randwertaufgaben

In this note we derive a necessary requirement for the self-adjointness of a boundary value problem of ordinary differential equations, which involves only the leading coefficients of the normalized boundary conditions. As application it is shown that every self-adjoint boundary value problem of even order (and thus especially every real self-adjoint boundary value problem) is regular. The proof of this fact depends on the evaluation of a sort of generalized Vandermonde's determinant.

Self-adjoint boundary value problems with infinitely many boundary points

The system $$Ly = y' + Py = f,\sum\limits_{i = 0}^m {A_i y(a_i ) = 0} $$ is disvussed when it is self-adjoint under a transformation T. In an appropriate Hilbert space its eigenfunction expansion is derived.

Spectral theory for a singular linear system of Hamiltonian differential equations

In 1910 Weyl [9] inaugurated the modern theory of singular self-adjoint differential operators, considering, in particular, second-order differential operators with real coefficients. Since then, his results have been generalized by various authors to the case of arbitrary even order self-adjoint differential equations with real coefficients. Also, in 1954 Coddington [2] treated the case of an arbitrary n-th order self-adjoint differential equation with complex coefficients, He obtained the Parseval equality and spectral expansion associated with the singular case directly, through the consideration of such a problem as a limiting case of corresponding self-adjoint two-point boundary-value problems on compact subintervals of the reals. Using the same general concept, Coddington and Levinson [3] derived a Green’s function, which in turn enabled them to obtain the spectral matrix for a singular problem involving a linear homogeneous n-th order differential operator, and then proceeded to derive the Parseval equality and spectral expansions. Recently Brauer [I] treated similar problems which involve a definitely self-adjoint vector differential operator, of the sort that has been treated by Reid [6] and others. He obtained the Green’s matrix and spectral matrix through the use of the spectral theorem and the theory of direct integrals. The main purpose of the present paper is to employ the general method

Linear self-adjoint multipoint boundary value problems and related approximation schemes

Linear self-adjoint multipoint boundary value problems are investigated. The case of the homogeneous equation is shown to lead to spline solutions, which are then utilized to construct a Green's function for the case of homogeneous boundary conditions. An approximation scheme is described in terms of the eigen-functions of the inverse of the Green's operator and is shown to be optimal in the sense of then-widths of Kolmogorov. Convergence rates are given and generalizations to more general boundary value problems are discussed.

Adjoint and Self-Adjoint Boundary Value Problems with Interface Conditions

In this paper the class of adjoint and in particular self-adjoint boundary value problems associated with ordinary linear differential equations is extended to include problems which may have discontinuities in the solution or some of its derivatives at a finite number of interior points.

Self-adjoint multi-point boundary value problems

Self-adjoint multi-point boundary value problems