Solution Of Stochastic Problems Research Articles

Reliability analysis of geogrid material with random nonlinear viscoelastic characteristics

Introduction. The behavior in the course of a time of geogrid material with random nonlinear viscoelastic characteristics under tension is analysed. Parameters of viscoelasticity are represented in form of Gaussian random vector. The components of this vector are taken from experimental data. Aim of the research. The objective of this research is the analysis of influence of different factors (value of applied load and the application of load in the form of random value instead of dead one, number of realizations, change of given level of strain) on providing of needed service life of geogrid material with given reliability level. Here reliability is interpreted as function of probability of non-failure. The first crossing of some given level by random strain is considered as a failure. The strain value corresponding to yield limit of geogrid material is accepted as the given level of longitudinal strain. Methods. The realizations of Gaussian random vector of viscoelastic parameters of material with given correlation matrix were imitated by means of linear transformation method. Results. It is demonstrated that longitudinal strain is Gaussian nonstationary random process which stochastic analysis can be made on base of 10 000 realizations. The dependencies on time of mathematical expectation and standard deviation of random longitudinal strain as well as function of probability of non-failure are found. Conclusion. It is shown that durability estimation found on base of the deterministic problem solution is overestimated in comparison with stochastic problem solution if the condition of given service life providing with some reliability level is set up.

Optimistic Planning for the Near-Optimal Control of General Nonlinear Systems with Continuous Transition Distributions

Optimistic planning is an optimal control approach from artificial intelligence, which can be applied in receding horizon. It works for very general nonlinear dynamics and cost functions, and its analysis establishes a tight relationship between computation invested and near-optimality. However, there is no optimistic planning algorithm that searches for closed-loop solutions in stochastic problems with continuous transition distributions. Such transitions are essential in control, where they arise e.g. due to continuous disturbances. Existing algorithms only search for open-loop input sequences, which are suboptimal. We therefore propose a closed-loop algorithm that discretizes the continuous transition distribution into sigma points, and call it sigma-optimistic planning. Assuming the error introduced by sigma-point discretization is bounded, we analyze the solution returned, showing that it is near-optimal. The algorithm is evaluated in simulation experiments, where it performs better than a state-of-the-art open-loop planning technique; a certainty-equivalence approach also works well.

A comparison of solution methods for the assortment problem

The problem of economically cutting a required bill of material out of standard size stock pieces is shared by the metal, leather, glass, electronic, shipbuilding, and lumber industries. One of the problems in this area is the assortment problem; that is, determining the best combination of stock sizes to keep in inventory to cut a single known (deterministic) bill of material. This problem includes determining the dimensions or the stock pieces as well as the number of unique sizes to stock. In this research a comparison is made between three common heuristic methods for solution of the assortment problem and the introduction of the application of optimization homotopy to the assortment problem, a method developed for the solution of stochastic problems. It is found that while utilizing a specific combination of nesting and sequencing algorithms, the three heuristic methods work very well for simpler problems, and as the complexity of the problem increases optimization homotopy becomes more advantageous.

On the open-loop solution of linear stochastic optimal control problems

We consider open-loop solutions of linear stochastic optimal control problems with constraints on control variables and probabilistic constraints on state variables. It is shown that this problem reduces to an equivalent linear deterministic optimal control problem with similar constraints and with a new criterion to minimize. Concavity or convexity is preserved. Hence, the machinery available for solving deterministic optimal control problems can be used to get an open-loop solution of the stochastic problem. The convex case is investigated and a bound on the difference between closed-loop and open-loop optimal costs is given.

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.