Main Boundary Value Problems Research Articles

Error analysis of the H1 gradient method for shape-optimization problems of continua

We present an error estimation for the H1 gradient method, which provides numerical solutions to the shape-optimization problem of the domain in which a boundary value problem is dened. The main result is that if second-order elements are used for the solutions of the main and adjoint boundary value problems to evaluate the shape derivative, and the rstorder elements are used for the solution of domain variation in the boundary value problem of the H1 gradient method, then we obtain rst-order convergence of the solution of the domain variation with respect to the size of the nite elements.

О числе решений в задачах со спектральным параметром для уравнений с разрывными операторами

In a real reflexive Banach space we consider a problem on existence of solutions to a problem with a spectral parameter for equations with discontinuous operators. By the variational approach we obtain theorems on the number of the solutions to the consi- dered problems. As an application, we consider main boundary value problems for elliptic equations with a spectral parameter and discontinuous nonlinearities.

A strong discontinuity approach on multiple levels to model solids at failure

This paper presents a modification of the well established strong discontinuity approach to model failure phenomena in solids by extending it to multiple levels. This is achieved by the resolution of the overall problem to be solved into a main boundary value problem and identified sub-domains based on the concepts of domain decomposition. The initiation of those sub-domains is based on the detection of failure onset within finite elements of the main boundary value problem which takes place at the process zone in front of the propagating cracks. Those sub-domains are subsequently adaptively discretized during run-time and comprise the so called sub-boundary value problem to be solved simultaneously with the main boundary value problem. To model failure, only the sub-elements of those sub-boundary value problems are treated by the strong discontinuity approach which, depending on their state of stress, may develop strong discontinuities to be understood as jumps in the displacement field to model cracks and shear bands. Due to its resolution into many sub-elements, the single finite element of the main boundary value problem can therefore simulate a single propagating strong discontinuity arising in quasi-static problems as well as the propagation of multiple propagating strong discontinuities arising for simulations of crack branching in brittle materials undergoing dynamic failure. Whereas the advantages of the strong discontinuity approach in the form of its efficiency by statically condensing out the degrees of freedom related to the failure zone as well as its applicability to use standard displacement based, mixed, and enhanced formulations for the underlying finite element are kept, new challenges arise due to its proposed modification. Firstly, the solutions of the different sub-boundary value problems must be transferred to the main boundary value problem, which is achieved in this work based on concepts of domain decomposition. Secondly, since multiple strong discontinuities might propagate over the boundaries of the sub-boundary value problem, the applied boundary conditions must take into account the appearance of possible jumps in the displacement fields arising from the solution of the sub-boundary value problem itself. It is shown that for single propagating cracks arising in problems of quasi-static failure only minor differences are obtained through the proposed modification. For the simulation of solids undergoing dynamic fracture the modification allows though to predict the onset of crack branching without the need for any artificial crack branching criterion. A close agreement with experiments of the simulation results in terms of micro- and macro branching in addition to studying certain key parameters like critical velocity, dynamic stress intensity factor, and the strain energy release rate at branching is found.

Bifurcation problems for equations of elliptic type with discontinuous nonlinearities

We consider the problem of the existence of semiregular solutions to the main boundary-value problems for second-order equations of elliptic type with a spectral parameter and discontinuous nonlinearities. A variational method is used to obtain the theorem on the existence of solutions and properties of the “separating” set for the problems under consideration. The results obtained are applied to the Goldshtik problem.

Calculation of the relative decrease in the stiffness of a damaged annular sector from analysis of the displacement field

The possibility for diagnostics of a breathing crack type damage in a long cylindrical shell is examined. The damage is modeled by a notch that alters the cross-sectional geometry. The method of conformal mappings is used to solve the first main boundary-value problem of elasticity for domains in the form of an annular sector with and without a notch. The solutions are analyzed and compared to estimate the relative decrease in the stiffness of the shell because of the opening of the crack

Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.

Homogenization of a boundary value problem with mixed type of boundary conditions in a thick junction

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

The study of the main boundary value problems for one degenerate elliptic equation by the method of potentials

The study of the main boundary value problems for one degenerate elliptic equation by the method of potentials

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type.

Solution of Boundary-Value Problems of the Dynamics of a Liquid in Horizontal Cylindrical Cavities with Partitions

We propose methods for the construction of approximate solutions of main boundary-value problems. These methods enable one to determine frequencies and attached masses of an ideal liquid in a moving horizontal cylinder that has an arbitrary symmetric cross section and structural elements in the form of longitudinal edges-partitions.

THE MAIN HASEMAN TYPE BOUNDARY VALUE PROBLEM FOR METAANALYTIC FUNCTION IN THE CASE OF CIRCULAR DOMAINS

In the presented work a main Haseman type boundary value problem is investigated in the class of metaanalytic functions in the case of circular domains. With the help of Schwartz equation for circumference the initial problem can be reduced to the totality of two ordinary Haseman problems for analytic functions. Besides, the picture of solvability of the problem depending on values of indexes of coefficients of boundary value conditions is is investigated in this work.

Differential operators defining a solution of an elliptic-type equation

We construct differential operators Kg(z), Lg(z), $$Kg(z),\;Lg(z),\,\,M\overline {f(z)} ,$$ , and $$N\overline {f(z)} $$ such that they map arbitrary holomorphic functions in a simply connected domain D in the complex plane z=x+iy into regular solutions of the equation $$W_{ \approx \bar \approx } + A(z,\bar z)W_{\bar \approx } + B(z,z)W = 0.$$ We give examples of applications of the constructed differential operators to a solution of the main boundary-value problems of mathematical physics. Bibliography: 1 title.

The boundary equation method in the third initial boundary value problem of the theory of elasticity. Part 1. Existence theorems

AbstractThis work deals with four variants of boundary equations appearing if the third main initial boundary value problem is solved with the help of elastic retarded potentials. The solvability of these equations is proved, and the smoothness of their solutions researched.

Iterative methods of solving main boundary value problems for second-order quasilinear elliptic equations in complexly shaped domains

Article Iterative methods of solving main boundary value problems for second-order quasilinear elliptic equations in complexly shaped domains was published on January 1, 1992 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 7, issue 4).

The best possible bounds for the convergence of the semimixed finite element method for the main boundary-value problems of the theory of shallow shells in polygonal regions

Best possible bounds are derived for the rate of convergence of a version of the finite element method for the main boundary-value problems of the theory of flat shells in a polygonal region. Approximation theorems are proved for the Sobolev weight spaces, making it possible to derive bounds with minimal assumptions on the smoothness of the solution. Recommendations are made concerning the choice of the degrees of the approximating splines, depending on the smoothness of the solution.

A generalization of the schwarz alternating method to an arbitrary number of subdomains

We give a generalization of the Schwarz alternating method to an arbitrary number of subdomains. As applications, we prove that the method converges for the main second-order boundary value problems, as well as for the system of linear elasticity. For the Dirichlet problems it is proved that the method converges geometrically and some numerical examples are given.