Formulation Of Boundary Value Problem Research Articles

A rate-dependent analysis and experiment on the sheet-stretching process

A rate-dependent elasto-plastic finite-element formulation of boundary-value problems has been developed for the large deformation sheet-stretching process. The formulation used is based on the power form constitutive equation for the stress-strain-strain rate relation of the material. The principle of virtual work rate applied towards a Lagrangian reference system was adopted. The resulting non-linear equilibrium equations have been solved through usage of an incremental method with an r- minimum technique. Experimental tests were performed, the results of which have exhibited a great deal of agreement with numerical simulations. The method proposed here has been shown to be quite powerful, reliable and could be modified for solving problems related to other manufacturing processes.

A weakly singular formulation of traction and tangent derivative boundary integral equations in three dimensional elasticity

Regularized forms of the traction and tangent derivative boundary integral equations are derived for three dimensional elasticity. Kernels of the resulting equations contain only weak singularities and thus are amenable to the ordinary numerical treatment required by weakly singular integrals. The hypersingular and strongly singular kernels of the displacement gradient representation are regularized independently, through identities of the fundamental solution and its various derivatives, before the integral equations are formed. The stress or traction equations provide an alternate formulation of boundary value problems in the same boundary variables as the displacement boundary integral equations, and are useful for problems where the displacement equations are deficient, such as in crack problems. Alternatively, the tangent derivative equations explicitly introduce surface displacement derivatives and are thus completely independent equations that may be used simultaneously with the displacement or traction equations.

Analysis of single and coupled microstrip lines on anisotropic substrates using differential matrix operators and the spectral-domain method

A differential matrix operator technique is presented to simplify the formulation of boundary-value problems for open millimeter-wave integrated circuits that use anisotropic substrates. The spectral-domain method is applied to analyze the propagation characteristics of single and coupled microstrip lines printed on anisotropic substrates whose properties are described by both ( in ) and ( mu ) tensors. In addition to considering the permittivity and permeability as a uniaxial or biaxial tensor, the effects of coordinate misalignment between the principal axes of ( in ) and those of the structure in the transverse plane are also included. It is shown that the misalignment in ( in ) and the presence of the ( mu ) tensor have a significant effect on the dispersive properties of these two structures. >

Stagnation point flow—a free boundary value problem formulation

The two dimensional laminar flow of a visous, incompressible fluid near a stagnation point is considered. The equations of motion governing the flow are reformulated as a free boundary value problem. The latter is solved numerically using the shooting method with the invocation of Newton's iterative scheme for finding the missing initial condition and the location of the free boundary. Much improved values of the skin-friction at the plate are obtained. Further it is shown that the improvement of the accuracy is dependent on the value of the free boundary. The formulation also eliminates the possibility of obtaining the extraneous solutions which do not satisfy the asymptotic boundary condition.

A Galerkin method for Stefan problems

Stefan problems are described by a parabolic partial-differential equation, along with two boundary conditions on a moving boundary, which is to be determined as part of the solution. The purpose of this paper is to develop a finite-element method for the solution of a one-dimensional Stefan problem. First, a coordinate transformation is used to transform the changing physical domain into a fixed computational domain. Then, the weak or Galerkin formulation of the initial boundary value problem is used to reduce it to a system of initial-value problems in ordinary differential equations. Several finite-difference marching methods are used to solve the resulting initial-value problems. The method is developed and illustrated using the Stefan problem concerning the heat transfer in an ice-water medium. The computational results are in very good agreement with the results produced earlier by several authors.

The solution of riemann–hilbert boundary value problem for elliptic complex equations of first order in the sobolev space

In this paper, we give a new proper formulation of Riemann-Hilbert boundary value problem for non-linear elliptic complex equations of first order in a multiply connected domain and establish the r...

The integral equation formulation of mixed finite time-dependent elastic problems

This paper establishes through a concrete example that the integral equation formulation of time-dependent mixed boundary value problems can be extended for problems in the theory of elasticity. To this end, the method applied to the resulting integral equation is the one begun by Cherski [1] and evolved by Eckhardt and El Sheikh [2] still further to solve intial mixed boundary value problems. New relationships between the fundamental coefficients characterizing the technique are obtained. These simplify the procedures and reveal more of the nature of the method.

Effective numerical algorithms for the solution of algebraic systems arising in spectral methods

This work addresses the algorithmical aspects of spectral methods for elliptic equations. We focus on the vectorization properties of widely used algorithms (e.g., direct factorization methods and the Richardson method) as well as other algorithms which are less popular within the spectral community (e.g., GMRES, CGS and Bi-CGSTAB, the latter two are variants of the conjugate gradient method for nonsymmetric systems). The GMRES, CGS and Bi-CGSTAB generally perform better than direct factorization and the Richardson methods. We show that the spectral collocation approximation in the weak form for boundary value problems with Neumann conditions is more accurate and easier to precondition than the usual strong form. We also introduce two diagonal preconditioners that dramatically reduce the condition number of the spectral matrix. Finally, we address the issue of domain decomposition algorithms, and show several results concerning accuracy and convergence of iteration-by-subdomain procedures. The numerical calculations have been performed on the CRAY X/MP-EA, the CRAY Y-MP 8/432, the IBM 3090/200S VF and NCUBE2 mod. 6401 with 16 processors.

Solvability of the Vall�e-Poussin nonlinear boundary-value problems

We consider the solvability of a boundary-value problem of form (1), (2) in the class of nonlinearities of f, for which the traditional restrictions of the Lifschitz-conditions type are replaced by concavity (or generalized concavity) conditions with respect to specially chosen variables.

Investigation of the formulation of the boundary-value problem of the local theory of elastoplastic processes

For a complex stressed state, a three-term defining relation [1, 2] is used which implies that the five-dimensional stress, stress rate and strain rate vectors are coplunar. On the hypothesis of local definiteness [3], the two coefficients occurring in the three-term relation are taken as functions of three functionals of the process—the stress intensity, the length of the arc of the deformation path and the angle of approach. For bounded derivatives of these two functions with respect to each argument, conditions securing a correct formulation of the static boundary-value problem in terms of rates of each instant of the elastoplastic process are determined. A formulation is given of the quasistatie global boundary-value problem for the whole process. It is proved that the operator of the global problem, an operator of the variational calculus [4], is positive definite, strictly monotonic in the main and possesses the ( S) i -property [5]. Using the theorem of Leray and Lions [4], it is shown that a generalized solution exists. It is proved that the global solution is unique and continuously dependent on the external loads. For the step method, using discretization of the process with respect to the load parameter, and iterational methods (of the type of SN-EVM method [2]), convergence of the approximate solutions to the exact solution of the global problem is proved.

Formulation of boundary-value problems for the dynamics of two-dimensional systems with moving loads and fastenings

A load or fastening is moving in a two-dimensional system without separation. The interdependent dynamical behaviour of these two elements is investigated. The Hamilton variational principle is used to formulate a self-consistent boundary-value problem which correctly incorporates the forces of interaction in the moving contact, including those due to the relative motion and wave pressure. The equations of energy and momentum transport are derived. It is shown that the intermediary through which the vibrational energy of the two-dimensional system is converted into the kinetic energy of the one-dimensional object is the wave pressure force. As an example, a boundary-value problem is formulated for the motion of a beam along a Kirchhoff model plate.

Algorithms for the solution of internal variable problems in plasticity

A variational formulation of the quasistatic boundary value problem of plasticity is presented. An internal variable formulation is used, and the evolution or flow law is expressed in terms of the dissipation function; this form is a dual representation of the more conventional form, in which the flow law is written in terms of the yield function. Finite dimensional approximations of the variational problem are derived, and algorithms for the solution of such approximations are discussed. A generalised midpoint rule approximation in time is used, and algorithms of predictor-corrector type are presented and discussed; we show that the consistent moduli arising in the algorithms are symmetric. Finally, a stability analysis is carried out; it is shown that the algorithm is nonlinearly stable for 1 2 ≤ α ≤ 1 where α is the midpoint rule parameter. The measure of stability used is B-stability, a measure exploited in a recent study of return mapping algorithms by Simo and Govindjee.

Regularized integral representation of thermoelastic stresses

The present paper deals with the boundary integral formulation of boundary value problems in general coupled thermoelasticity. A regularized integral representation of thermoelastic stresses is derived in which the strongest singularity does not exceed that involved in the integral representation of displacements. The boundary integral equations are written in a form which is free of Cauchy principal value integrals. The formulation possesses pure boundary character.

Goal programming solutions for heat and mass transfer problems—a feasibility study

The feasibility of utilizing goal programming, a multicriteria optimization technique, in solving heat and mass transfer problems is investigated. The heart of the idea is that approximate solutions will in general not be able to satisfy all conditions, such as boundary and initial conditions, at the same time as minimizing the error at the collocation points. This paper describes the formulation of the nonlinear boundary value problem as a preemptive goal programming model via the use of hard and soft constraints (or stiff and weak springs) in linear programming (or mechanics). The technique thus allows controlled weakening and prioritizes the different conditions. The preemptive priority structure of goal programming enables it to reduce the number of trial functions required to yield a satisfactory solution, since it does not need to simultaneously satisfy the interior collocation and boundary points as in the orthogonal collocation method. In addition, the technique enables the number of undetermined parameters to be less than the number of collocation points. Examples are provided to highlight the methodology.

A variational principle for gradient plasticity

We elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by Aifantis and co-workers. The generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher-order spatial gradients of the equivalent plastic strain in the yield condition. First it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands. Due to the presence of higher-order gradients, additional boundary conditions for the equivalent plastic strain are required. This question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work. We demonstrate here that upon assuming a certain type of additional boundary conditions, the structural symmetries of the gradient-dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain. The variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method. Explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given.

A new method for the calculation of magnetic fields

Abstract Magnetic fields in systems with unsaturated yokes can be favourably calculated by means of the boundary element method. This method consists in the formulation and numerical solution of integral equations for the potentials or field sources on the material surfaces. The different formulations of boundary value problems are finally always reduced to the task of solving a Fredholm equation with logarithmically singular integral kernel. A new technique is developed for the efficient and fast solution of such problems. It combines the Galerkin method of approximation — using interpolation kernels or modified Lagrange polynomials as test functions — with a partly analytical evaluation of the integral kernel. This technique, so far demonstrated for electrostatics, can now also be applied to magnetic systems with finite permeability of the yoke material. The feasibility of the method is demonstrated for some unconventional forms of magnetic lenses and for deflection systems. A further generalization is the field calculation in configurations of several round electrodes or pole pieces which are not aligned on a common optic axis but have arbitrary positions and orientations in space. The field calculation can be performed by an iterative solution of a sequence of coupled two-dimensional problems, which is much faster than a general three-dimensional procedure. Moreover, a hybrid technique for the field calculation in systems with saturated yokes, and finally a technique for studying small deviations from symmetries are briefly sketched.

Corner singularities between free surfaces and open boundaries

We consider the solution of the Stokes problem at a corner between a free surface and an inflow or outflow boundary. A formal asymptotic solution for the dominant contribution to the streamfunction near the corner is derived. We give a heuristic discussion of the relevance of the nature of the corner singularity to the formulation of well-posed boundary value problems.

Numerical solution of two-point boundary value problems

This paper presents an original formulation of two-point boundary value and eigenvalue problems expressed as a system of first-order equations. The fundamental difference between the new method and other methods based on a first-order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first-order equations, which are hence never regarded as an initial value problem. The consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations. Furthermore, the global character of the integral conditions (nonlocality) combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by Wilkinson. The properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability. The proposed method is conceptually simpler and numerically more convenient than existing initial value methods, while still retaining all the advantages of a formulation based on a first-order system.

On the equivalence of integral formulations for certain exterior and interior boundary value problems in mechanics

A procedure is given which relates various linear boundary value problems. It is shown that a relations exists between certain ‘exterior’ and ‘interior’ problems. For computational purposes an ‘interior’ problem is more desirable. The method is restricted by the requirement that the Neumann function, or its equivalent, be known for the problem being considered.

Parallel solution of Fredholm integral equations of the second kind by orthogonal polynomial expansions

Given a Fredholm integral equation of the second kind, the Nyström method based on an n-point Gaussian quadrature rule and the perturbed Galerkin method in which inner products and integral transforms are approximated using the Gaussian rule, are equivalent in the sense that the Nyström interpolant and the orthogonal polynomial expansion of degree n − 1> agree at the n quadrature nodes. The perturbed collocation method, using the n quadrature nodes as collocation points, in which integral transforms are approximated using the Gaussian rule, is also equivalent. A practical implication is that one can set up and solve the simple n × n linear system corresponding to the Nyström method, and then easily transform the result to that of the perturbed Galerkin method, in order to economically gain the advantages of the orthogonal expansion. This approach appears particularly useful for the numerical solution on multiprocessor systems of the integral formulation of certain two-point boundary value problems. The paper analyzes the relationship among numerical methods, it establishes the weighted least-squares convergence of the orthogonal expansions, and it describes corresponding parallel algorithms.