Computation of thermal residual stresses in optic fibers

We propose an analytic method of determining all the components of the residual stress tensor in optic fibers. The method is based on solving a plane problem for a cylindrical structure with singular stresses. We obtain closed-form solutions of the problem in the case of a piecewise-constant distribution of free deformations that model the presence of inclusions in the fiber with different values of the thermal coefficient of expansion. We also consider inclusions with cross sections in the shape of a circle, a central ellipse and a central annular sector. We describe the results obtained on this basis in the computation of residual thermal stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 79–83.     

The inverse conditionally correct problem of determining the residual stresses in compound welded shells of revoltuion

We solve the inverse conditionally correct problem of recovering the complete picture of the residual stressed state for a compound shell welded from two parts, one cylindrical and the othe conical. We apply the partial values of the stresses obtained experimentally by the method of photoelasticity. We also apply the numerical method of spline-collocations. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 130–134.     

Determination of the residual stresses in the welded joints of glass structures

We propose a computational model for determining the residual stresses in a welded glass structure taking account of the properties of the formation of residual stresses in glass. The problem is solved in displacements using Galerkin's method in conjunction with a finite-element model. A numerical solution is obtained for the axisymmetric case. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 131–134.     

A Green's function method of determining the residual stresses in cylindrical glass shells

We propose a way of modifying the mathematical model of the nondestructive theoretical-experimental method of determining the residual stresses for a specific collection of technological conditions for welding two cylindrical glass shells. The method is based on the general theory of boundary-value problems for ordinary differential equations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 69–73.     

New convergence results on the global GMRES method for diagonalizable matrices

In the present paper, we give some new convergence results of the global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the Frobenius norm of the residual. We also consider the case of normal matrices and we propose new expressions for the norm of the residual.     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

Polynomial quasisolutions of linear second-order differential-difference equations

We consider the scalar linear second-order differential-difference equation with delay {fx159-01}. This equation is investigated by the method of polynomial quasisolutions based on the representation of an unknown function in the form of a polynomial {ie159-01}. Upon the substitution of this polynomial in the original equation, the residual Δ(t) = O(t ^N−1) appears. An exact analytic representation of this residual is obtained. We show the close connection between a linear differential-difference equation with variable coefficients and a model equation with constant coefficients, the structure of whose solution is determined by the roots of the characteristic quasipolynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 140–152, January, 2008.     

Optimization of zonal thermal treatment of cylindrical component parts by moving concentrated heat sources

We propose a numerical-analytic method of determining the optimal heating regimes and stress-strain state of steplike cylindrical shells to guarantee the maximally admissible length of the annealing zones in the cirumferential direction and the number of these zones in the absence of residual strains and stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 65–69.     

Automorphic lifts of prescribed types

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.     

The residual nilpotence of verbal wreath products

We obtain necessary and sufficient conditions for residual nilpotence of verbal wreath products. We study also residual nilpotence of the groups of the formF/V(N), whereF is a free groupN◃F, andV(N) is a verbal subgroup ofN. Dedicated to the memory of Professor Brian Hartley Partially supported by NSF Grant DMS9404178.     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.     

The axisymmetric problem of determining the residual stresses in thick plates

We propose a method of determining the residual (technological) stresses in structural elements that can be regarded as plates in a computational model. The initial data are the equations of mechanics for bodies with initial stresses and experimental information obtained using nondestructive physical testing.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 113–119.     

Implicit residual error estimators for the coupling of finite elements and boundary elements

We consider a symmetric Galerkin method for the coupling of finite elements and boundary elements for elliptic problems with a monotone operator in the finite element domain. We derive an a posteriori error estimator which involves the solution of equilibrated local Neumann problems in the finite element domain and requires computation of a residual term on the coupling interface. Finally, we discuss a similar approach for a coupling with Signorini contact conditions on the interface. Copyright © 1999 John Wiley & Sons, Ltd.     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity

The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes‐type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi‐optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001     

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017     

The continued fraction methods for the solution of systems of linear equations

A class of iterative methods is presented for the solution of systems of linear equationsAx=b, whereA is a generalm ×n matrix. The methods are based on a development as a continued fraction of the inner product (r, r), wherer=b-Ax is the residual. The methods as defined are quite general and include some wellknown methods such as the minimal residual conjugate gradient method with one step.     

Preconditioners for the p-version of the Galerkin method for a coupled finite element/boundary element system

We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log² p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log³ p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998     

Iterative methods for solvingAx=b,GMRES/FOM versus QMR/BiCG

We study the convergence of the GMRES/FOM and QMR/BiCG methods for solving nonsymmetric systems of equationsAx=b. We prove, in exact arithmetic, that any type of residual norm convergence obtained using BiCG can also be obtained using FOM but on a different system of equations. We consider practical comparisons of these procedures when they are applied to the same matrices. We use a unitary invariance shared by both methods, to construct test matrices where we can vary the nonnormality of the test matrix by variations in simplified eigenvector matrices. We used these test problems in two sets of numerical experiments. The first set of experiments was designed to study effects of increasing nonnormality on the convergence of GMRES and QMR. The second set of experiments was designed to track effects of the eigenvalue distribution on the convergence of QMR. In these tests the GMRES residual norms decreased significantly more rapidly than the QMR residual norms but without corresponding decreases in the error norms. Furthermore, as the nonnormality ofA was increased, the GMRES residual norms decreased more rapidly. This led to premature termination of the GMRES procedure on highly nonnormal problems. On the nonnormal test problems the QMR residual norms exhibited less sensitivity to changes in the nonnormality. The convergence of either type of procedure, as measured by the error norms, was delayed by the presence of large or small outliers and affected by the type of eigenvalues, real or complex, in the eigenvalue distribution ofA. For GMRES this effect can be seen only in the error norm plots.In honor of the 70th birthday of Ted RivlinThis work was supported by NSF grant GER-9450081.