Variational boundary value problem

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

Kovtunenko V.A. (1998) Variational and boundary value problems with a friction on the internal boundary. Siberian Math. J. 39 (5), 1060-1073 (in Russian). 

VVe say that a scheme is stable with respect to coefficients (costable) if a solution of the boundary-value problem has slight variations under small perturbations of the scheme coefficients. In order to avoid misunderstanding, we focus our attention on the scheme with coefficients a, d, ip... 

As we have seen, an external plane wave can excite resonances of a particle, which leads to significant variation in fluorescence intensity. A fluorescent molecule located in or near a particle can also excite the resonances of the particle. This can be modeled by again considering the molecule as a classical point dipole and obtaining the fields due to the dipole from the solution to the boundary value problem. 

In summary, formulation of (27) with appropriate placement of finite elements works well for parameter optimization problems. In the next subsection, however, we consider additional difficulties when control profiles are introduced. Stated briefly, the reason for these difficulties lies in the nature of the discretized variational conditions of (16). As shown in Logsdon and Biegler (1989), optimality conditions for parameter optimization problems take the form of two point boundary value problems. For optimal control... 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

The Galerkin finite element spatial discretization of the boundary value problem is formulated as an equivalent weak or variational form to the strong form of the boundary value problem. 

From the perspective of variational principles, the idea embodied in the finite element approach is similar to that in other schemes identify that particular linear combination of basis functions that is best in a sense to be defined below. The approximate solution of interest is built up as linear combinations of basis functions, and the crucial question becomes how to formulate the original boundary value problem in terms of the relevant expansion coefficients. 

This is the Kirkwood-Buckingham relation useful for analysis of boundary value problems. For example, Equation (2.4) follows from Equation (3.17) and the variational inequality for the ground state wavefunctions for Dirichlet problems in regions ST and 12. One can find some generalizations of this relation in [54,55]. 

The idea of using functions of the form frjr for approximate solutions of the boundary value problems has been used in a large number of papers. For example, this idea was used in [70] for the one-dimensional Schrodinger equation to prove the Hull-Julius relation (4.21) and to study the confined IlJ molecule. The direct use of the Kirkwood-Buckingham relation for variational calculations of the hydrogen atom in a half space was... 

V.I. Pupyshev, A.V. Scherbinin, N.F. Stepanov, The Kirkwood-Buckingham variational method and the boundary value problems for the molecular Schrodinger equation, J. Math. Phys. 38 (11) (1997) 5626-5633. 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

The mathematical modelling of the T-H-M phenomena uses initial - boundary value problems for differential or variational equations involving the physical principles. We shall assume that these problems are discretized by the finite element or similar methods. What we want to point out is that the numerical solution can be computationally very expensive due to... 

Extending Hamilton s variational principle to piezoelectric media gives an equivalent description of the above boundary value problem (BVP) ... 

It was demonstrated in the text that the number of equations to be solved using the method of finite difference depends on the type of boundary conditions. For example, if the functional values are specified at two end points of a boundary value problem, the number of equations to be solved is N — 1. On the other hand, if the functional value is specified at only one point and the other end involves a first derivative, then the number of equations is N. This homework problem will show that the variation in the number of equations can be avoided if the locations of the discrete point are shifted as... 

Indirect or variational approaches are based on Pontryagin s maximum principle [8], in which the first-order optimality conditions are derived by applying calculus of variations. For problems without inequality constraints, the optimality conditions can be written as a set of DAEs and solved as a two-point boundary value problem. If there are inequality path constraints, additional optimality conditions are required, and the determination of entry and exit points for active constraints along the integration horizon renders a combinatorial problem, which is generally hard to solve. There are several developments and implementations of indirect methods, including [9] and [10]. 

We employ a method of numerical continuation which has been earlier developed into a software tool for analysis of spatiotemporal patterns emerging in systems with simultaneous reaction, diffusion and convection. As an example, we take a catalytic cross-flow tubular reactor with first order exothermic reaction kinetics. The analysis begins with determining stability and bifurcations of steady states and periodic oscillations in the corresponding homogeneous system. This information is then used to infer the existence of travelling waves which occur due to reaction and diffusion. We focus on waves with constant velocity and examine in some detail the effects of convection on the fiiont waves which are associated with bistability in the reaction-diffusion system. A numerical method for accurate location and continuation of front and pulse waves via a boundary value problem for homo/heteroclinic orbits is used to determine variation of the front waves with convection velocity and some other system parameters. We find that two different front waves can coexist and move in opposite directions in the reactor. Also, the waves can be reflected and switched on the boundaries which leads to zig-zag spatiotemporal patterns. 

The independent variation of ip and rj vanish at arbitrary temporal values t and in Hamilton s principle (3.3) yielding the following scaled boundary value problem ... 

Several methods have been proposed for predicting the stress state at the interface, which can then be used to estimate the bond strength. The shear lag method has received extensive treatment by several investigators. This method determines the interface shear stress concentration at the end of the fiber as well as shear stress variation along the fiber. Additional methods include the Lame solution for a shrink fit, classical elasticity boundary value problems, and finite-element analysis. 

The Chapman-Enskog approximation method leads, at all stages, to hydrod5mamic equations which are first order in the time and can therefore be solved subject to given initial conditions. This procedure by which the boundary-value problem is converted into an initial-value problem is, from the mathematical point of view, somewhat mysterious. It appears likely that the procedure will converge only for processes whose scale of variation is of the order of, or less than, the mean free path (or other characteristic length). The reduction to an initial-value problem would then be impossible for rapidly varying processes. 

To outline a finite element analysis approach, we will formulate a boundary value problem, transform it into a weak or variational form, and obtain discretized finite element equations. We begin with the equations of equihhrium that are written in the deformed configuration [1] ... 

---