Regular boundary

Algebraic methods - in these techniques calculation of grid coordinates is based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. 

Using Theorem 1.26, we can consider the Green formulae in domains with regular boundaries, which are useful in the sequel. 

Again, let c i be a bounded domain with a smooth boundary T and Tc C H be a smooth orientable two-dimensional surface with a regular boundary. We assume that Tc can be extended in such a way that the domain fl is divided into two parts with Lipschitz boundaries. The surface Tc can be described parametrically... 

Exercise. The one-step process (VI.4.8) has at n = 0 a natural boundary in the sense of VI.5. However, the corresponding Fokker-Planck equation (VIII.5.3) has a regular boundary (in the present sense). Show that the reflecting condition has to be imposed to obtain agreement with the discrete M-equation. [Compare XIII.3.]... 

Exercise. Find for the general Smoluchowski equation (5.1) between two regular boundaries L, R the condition that two mixed boundary conditions at L and R are compatible. 

It is known that the regular boundary layer over a smooth surface turns out to be selfsimilar [380, 566], The velocity distributions stretching from the EPR interface z - 1 till the boundary layer border z = d2(x) were also inspected for such a property. It turns out that the numerical profiles of the longitudinal velocity can be brought together if... 

For the statement of boundary conditions we begin with regular boundaries and consider the finite-difference systems shown in Fig. 4.10(a) for a corner node and a side node. We consider three types of boundary conditions specified temperature, specified heat flux, and heat transfer to ambient. 

Regular Boundary Integral Equations for Stress Analysis". 

A Regular Boundary Method Using Non-conforming Elements for Potential Problems in 3 D . Proceedings of the 4th Int, Sem. on BEM 1982, pp. 112-126, Springer-Verlag, 1982. 

Conventional BEM [5] is used to solve the elasticity problems with a regular boundary S. For the problems with fractures S a modification of the conventional BEM - the Dual BEM is suggested in [4]. In DBEM the Displacements Boundary Integral Equation (DBIE) is solved on the regular boundary and the Traction Boundary Integral Equation (TBIE) is solved on the fracture boundary. To solve the elasticity problem near the fracture in an infinite elastic media a modification of DBEM is developed in the present paper. 

We consider elastic insulators in the framework of quasi-magnetostatics. Of necessity all equations are written down in the material description so that all boundaries are fixed even for nonlinear strains. The equations consist of the equations of motion. Maxwell s equations and the associated boundary conditions together with the nonlinear, coupled, stress and magnetic-induction contitutive equations. For a material domain of regular boundary 5Dq equipped with unit outward normal components N, we have [5] ... 

The problems already mentioned at the solvent/vacuum boundary, which always exists regardless of the size of the box of water molecules, led to the definition of so-called periodic boundaries. They can be compared with the unit cell definition of a crystalline system. The unit cell also forms an "endless system without boundaries" when repeated in the three directions of space. Unfortunately, when simulating hquids the situation is not as simple as for a regular crystal, because molecules can diffuse and are in principle able to leave the unit cell. 

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

The crack shape is defined by the function -ip. This function is assumed to be fixed. It is noteworthy that the problems of choice of the so-called extreme crack shapes were considered in (Khludnev, 1994 Khludnev, Sokolowski, 1997). We also address this problem in Sections 2.4 and 4.9. The solution regularity for biharmonic variational inequalities was analysed in (Frehse, 1973 Caffarelli et ah, 1979 Schild, 1984). The last paper also contains the results on the solution smoothness in the case of thin obstacles. As for general solution properties for the equilibrium problem of the plates having cracks, one may refer to (Morozov, 1984). Referring to this book, the boundary conditions imposed on crack faces have the equality type. In this case there is no interaction between the crack faces. 

Obviously, the domain Q can be constructed in different ways. Nevertheless, in any case one of the inclusions T c 7, T c 7 will be valid, and (2.144) will take place. The exact form of the boundary conditions on T, was obtained in Section 2.4. We omit the derivation of these conditions here. All we want to do is to discuss briefly their general form in connection with the subsequent regularity result. These conditions are as follows. Let... 

In general, the above boundary conditions hold provided that (2.141) is fulfilled and the solution is quite regular. In fact, some part of the boundary conditions can be considered as holding in the strong sense without any additional assumptions on regularity. In particular, as proved in Section 2.4,... 

The subscripts D,j denote the integration over the domain D and the boundary 7, respectively. Note that the boundary dfl of is a combination of the sets r,F, r. The formulae (3.15), (3.16) hold true for the domain despite the absence of regularity of dfl. To verify this we can extend the graph F, so that the domain is divided into two parts. For each part the formulae (3.15), (3.16) are valid, hence the statement follows. We should note at this point that the external normals on F, F have opposite directions. 

We prove the existence of the solution and state additional qualitative properties - in particular, a solution regularity near the crack faces and near the external boundary. The results of this section are obtained in (Khludnev, 1997c). 

The above boundary conditions hold provided that the solution rj = (x, ) is sufficiently smooth. We shall use only a part of the conditions (3.69), (3.70) to prove the solution regularity. 

We prove the solvability of the problem. We also find boundary conditions holding on the crack faces and having the form of a system of equations and inequalities and establish some enhanced regularity properties for the solution near the points of the crack. Some other results on thermoelasic problems can be found in (Gilbert et al., 1990 Zuazua, 1995). 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

The considered problem is formulated as a variational inequality. In general, the equations (3.140)-(3.142) hold in the sense of distributions. In addition to (3.143), complementary boundary conditions will be fulfilled on F, X (0,T). The exact form of these conditions is given at the end of the section. The assumption as to sufficient solution regularity requires the variational inequality to be a corollary of (3.140)-(3.142), the initial and all boundary conditions. The relationship between these two problem formulations is discussed in Section 3.4.4. We prove an existence of the solution in Section 3.4.2. In Section 3.4.3 the main result of the section concerned with the cracks of minimal opening is established. 

We prove the existence of solutions for the three-dimensional elastoplastic problem with Hencky s law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two-dimensional crack (see Brokate, Khludnev, 1998). It is shown in particular that the variational solution satisfies all boundary conditions. 

Proof. Using elliptic regularization and the penalty approach, we construct an auxiliary problem which approximates (5.6)-(5.9). Its solution will depend on two positive parameters a and d which are related to the elliptic regularization and to the penalty approach, respectively. We will obtain a solution a, u by passing to the limit as a, (5 —> 0. So, consider the following boundary value problem in fl... 

Remark. The specific choice of bijki as the inverse of the Uijki for the elliptic regularization appears to be natural, since in the case of pure elastic (with K = [I/ (R)] , respectively p a) = 0), the boundary condition (5.16) reduces to (5.9). However, the proof of Theorem 5.1 works with any other choice of bijki as long as requirements of symmetry, boundedness and coercivity are met. 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

Proof. We consider a parabolic regularization of the problem approximating (5.68)-(5.72). The auxiliary boundary value problem will contain two positive parameters a, 5. The first parameter is responsible for the parabolic regularization and the second one characterizes the penalty approach. Our aim is first to prove an existence of solutions for the fixed parameters a, 5 and second to justify a passage to limits as a, d —> 0. A priori estimates uniform with respect to a, 5 are needed to analyse the passage to the limits, and we shall obtain all necessary estimates while the theorem of existence is proved. 

The dependence of solutions to (5.79)-(5.82) on the parameters a, 5 is not indicated at this step in order to simplify the formulae. Note that boundary conditions (5.81) do not coincide with (5.71) the conditions (5.81) can be viewed as a regularization of (5.71) connected with the proposed regularization of the equilibrium equations (5.68). Also, the artificial initial condition for o is introduced. 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

Kinderlehrer D., Nirenberg L., Spruck J. (1979) Regularity in elliptic free boundary problems. II Equations of higher order. Ann. Scuola Norm. Sup. Pisa 6, 637-687. 

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