Optimal shape of a strongest inverted column

By using Pontryagin's maximum principle we determine the shape of the strongest column positioned in a constant gravity field, simply supported at the lower end and clamped at upper end (with the possibility of axial sliding). It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and two first integrals are constructed. These integrals lead to an a priori estimate of the value of one the missing initial condition and to the reduction of the order of the system. The optimal shape of a column is determined by numerical integration.     

The strongest intrinsic meaning of sequential-evaluation convergence

For classical Banach sequence spaces c₀(X), l_∞(X) and lp(X) (0<p<+∞) we have found the strongest intrinsical meanings of their β-duals, and two basic convergence results are established in the β-duals.     

The strongest monotone degree condition for n-connectedness of a graph

It is shown that Bondy's degree condition for n-connectedness of a graph is the strongest monotone degree condition for forcibly n-connected graphic sequences.     

Bilevel linear programming with ambiguous objective function of the follower

Bilevel linear optimization problems are the linear optimization problems with two sequential decision steps of the leader and the follower. In this paper, we focus on the ambiguity of coefficients of the follower in his objective function that hinder the leader from exactly calculating the rational response of the follower. Under the assumption that the follower’s possible range of the ambiguous coefficient vector is known as a certain convex polytope, the leader can deduce the possible set of rational responses of the follower. The leader further assumes that the follower’s response is the worst-case scenario to his objective function, and then makes a decision according to the maximin criteria. We thus formulate the bilevel linear optimization problem with ambiguous objective function of the follower as a special kind of three-level programming problem. In our formulation, we show that the optimal solution locates on the extreme point and propose a solution method based on the enumeration of possible rational responses of the follower. A numerical example is used to illustrate our proposed computational method.     

The behavior of the QR-factorization algorithm with column pivoting

A bound on the performance of QR-factorization with column pivoting is derived and two classes of matrices are constructed for which the bound is sharp or asymptotically sharp.     

The optimal shape of a pipe

In shape optimization, recently the question arose, whether or not the cylindrical pipe has the optimal shape for the transport of an incompressible fluid. In this short note, a proof will be presented that a cylindrical pipe with Poiseuille’s flow inside indeed is optimal for the transportation of an incompressible fluid under the criterion “energy dissipated by the fluid.” The proof reduces the problem to the minimization of a two-dimensional Dirichlet’s integral. This simpler problem can be solved with a symmetrization argument.     

Towards an example of a nonconvex monotone follower control problem

A one-dimensional monotone follower control problem with a nonconvex Lagrangian is considered. The control problem consists in tracking a standard Wiener process by an adapted nondecreasing process starting at 0. The verification theorem for the problem is presented. The optimal control and the value function are explicitly defined. For some values of parameters of the problem, it is shown that the value function belongs to C ². An interesting feature of the optimally controlled state process is that for some initial states it has jumps at times other than the inital time. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

The optimum shape of a bending beam

The problem of minimizing the static deflection of an elastic beam of variable cross-section and fixed volume in the case of free supported and rigidly clamped ends is considered. In the first case it is proved that the solutions obtained earlier, based on the necessary conditions for an extremum, satisfy the sufficient conditions. In the case of clamped ends, which is of the most interest from the point of view of applications, it is proved that the optimum solutions must necessarily have points inside the solution range in which the distribution of the beam thicknesses degenerates to zero (“internal hinges”). A qualitative, analytical and numerical analysis of this phenomenon is given. In particular, in the case of clamped ends for a class of point loads, analytical solutions for which the beam splits into two cantilevers are obtained.     

The shape of a figure-eight under the curve shortening flow

Research supported in part by an NSF Postdoctoral Research Fellowship     

Optimal shape analysis of a column structure under various loading conditions by using differential transform method (DTM)

The effects of loading on the optimal shape of an Euler-Bernoulli column is investigated by considering four loading conditions which are mainly classified as eccentric compressive and follower type. The governing equations obtained from the structural stability condition of the column are used as a constraint to determine the minimum value of the volume by applying Hamilton principle. In the preceding task, the analysis is presented in step-by-step manner. The calculations are carried out by using differential transform method (DTM) which is a seminumerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the non-linear constrained governing equations are reduced to recurrence relations and related boundary conditions are transformed into a set of algebraic equations. The optimal distribution of cross-sectional area along column-length is obtained. Then, the volume of such column is calculated and compared to that of the uniform column which is also stable under given loading. The results obtained revealed out that DTM is a quite powerful solution technique for optimal shape analysis of a column structure.     

The dependence of the jump in the critical follower force for a viscoelastic bar on the form of the non-linear internal viscosity

The transverse oscillations of a viscoelastic bar under the action of a follower force are considered. The constitutive relation is the non-linear relation between the stress, strain and strain rate, where the non-linear part is described by a homogeneous form of the fifth degree. A numerical-analytical investigation of the stability showed that the three values of the critical follower force, corresponding to each of the three coefficients of viscosity available in the model, are independent of these coefficients and change in comparison with the case of a cubic viscosity.     

Contact interaction of composite shells,subjected to follower loads,with a rigid convex foundation

Based on a 7-parameter shell model, a numerical algorithm is developed for solving the contact problem for a multilayered composite shell lying on a rigid convex foundation, which is subjected to a follower pressure and undergoes arbitrarily large rotations. A new geometrically exact solid shell element is formulated, which permits one to solve the nonlinear deformation problem for thin-walled composite structures under unilateral contact constraints by using a small number of load steps. The calculation of a homogeneous ring and an angle-ply toroidal shell interacting with plane and cylindrical foundations is considered.     

The vibrations of a semi-infinite strip under the action of a discontinuous load

We consider the problem of the symmetric deformation of a semi-infinite strip to whose face a force load is applied having an infinite integrable discontinuity. We establish the asymptotic behavior of the unknowns in the study of the corresponding boundary-value problem using the method of superposition. This analysis made it possible to perform a correct truncation of the system of integro-differential equations and obtain reliable quantitative results. Using the numerical results we analyze the behavioral peculiarities of the excitation coefficients of inhomogeneous waves as functions of the frequency of vibration of the semi-infinite strip. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 54–60.