Convergence of solutions

Consider the limit case corresponding to = 0 in (2.130). The restriction obtained in such a way describes approximately a mutual nonpenetration of the crack faces. Note that in reality a complete account of the thickness implies the dependence of the energy functional on . This dependence is as follows (Vol mir, 1972)  

Moreover, in this case m w),t w),eij W) should also depend on e. In spite of this, in this section the parameter c is equal to 1 in the formula (2.150) just as in the previous subsections. Thus, the case c = 0 in (2.130), in fact, means both the approximate description of the nonpenetration condition and a fixed thickness. Hence, in the case under consideration a solution should satisfy the following restriction  

Herewith the problem of minimizing H over the set is equivalent to the variational inequality 

Let the set be the same as in Section 2.5.2. Consider the optimal control problem 

instead of precise nonpenetration condition (2.130) we consider the approximate condition (2.152) in this subsection. In application this approach is interesting since it is easier to find the solutions of (2.153) as compared to (2.131). In particular, it is possible to find solutions of (2.153) by using the penalty operator relative to the restriction (2.152). The displacements W and w are uncoupled in (2.153), and one can write down two variational inequalities for finding W and w, respectively. Meanwhile, when the optimal control problem (2.154) is solved, the solution (p depends on the pair (W, w), which actually means the coupling of W and w. The problem is to prove the solution proximity of (2.134), (2.131) and (2.154), (2.153), as — 0. 

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We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

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. 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

In this section we consider the model of a shallow shell analysed in the previous section and prove the convergence of solutions provided that the length of the boundary crack tends to zero. 

In the book, two- and three-dimensional bodies, 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, thermoelastic, elastoplastic. The book gives a new outlook on the crack problem, displays new methods of studying the problems and proposes new models for cracks in elastic and nonelastic bodies satisfying physically suitable nonpenetration conditions between crack faces. 

New material by name resin was created with specified density (1140 kg/m ) and viscosity (0.60 kg/m.s) along with the existing air (density 1.225 kg/cc, viscosity (1.7894e-5 kg/m.s). Multiphase VOF option was selected under the model option, air was defined as the primary phase and resin was set as secondary phase. Gravity (-9.81 m/s ) was activated in the operating conditions panel in the z direction, density of air (1.225 kg/cc) was specified under variable density parameter for better convergence of solution. Mixed mode (both for air and resin) boundary conditions for the inlet (pressure inlet, 0 pascal) and outlet (pressure outlet, -97325 pascals in z direction) were set. Mixed mode fabric permeability (viscous resistance, 1/m ) and fluid porosity (1-fabric porosity) were defined for the fabric. No slip boundary conditions were set (default) for the walls for both the phases. For resin phase, 1 was set under the volume fraction for inlet and 0 was set for back-flow volume fraction for outlet boundary conditions. 

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. 

A difficulty with the energy conserving method (6), in general, is the solution of the corresponding nonlinear equations [6]. Here, however, using the initial iterate (q + A p , p ) for (q +i, p +i), even for large values of a we did not observe any difficulties with the convergence of Newton s method. 

Theorem 2.14. From the sequence x = of solutions of the problem (2.116) one can choose a subsequence, still denoted byto, such that as 5 0 the convergence (2.119) takes place and, moreover, the limiting function satisfies (2.123). 

Here and below we emphasize the dependence of the objective functional on 5, because later we shall investigate the convergence of the solutions of problem (2.189) as 5 —> 0. 

In this section we deal with the simplified nonpenetration condition of the crack faces considered in the previous section. We formulate the model of a plate with a crack accounting for only horizontal displacements and construct approximate equations using penalty and iterative methods. The convergence of these solutions is proved and its application to the onedimensional problem is discussed. Analytical solutions for the model of a bar with a cut are obtained. The results of this section can be found in (Kovtunenko, 1996c, 1996d). 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

Kovtunenko V.A. (1994a) Convergence of the variational inequality solutions for a plate contacting a nonregular obstacle. Diff. Eqs. 30 (3), 488-492 (in Russian). 

Mosco U. (1969) Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (4), 510-585. 

The equation in cell B1 is copied into cells Cl though El. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. 

Finally, to ensure convergence of this algorithm from poor starting points, a step size Ot is chosen along the search direction so that the point at the next iteration z = z- + Ctd) is closer to the solution of the... 

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