Description of approach to estimating survivability of complex structures under repeated impacts of high accuracy (part 2)

Purpose. The paper describes main concepts and definitions, survivability indices, methods used to estimate survivability in different external and internal conditions of application of technical systems, including the studies in the field of structural survivability obtained 30 years ago within the frames of the Soviet school of sciences. An attempt is made to overcome different understanding of technical survivability, which has been formed by now in a number of industrial directions - shipping, aviation, communication networks, energy systems, in industries of defense. Besides, the problem is discussed in relation to the establishing of the continuity between technical survivability and global system resilience. Technical survivability is understood in two basic meanings: a) as a property of a system to resist to negative impacts; b) as a property of a system to recover its operability after a failure or accident caused by external reasons. This article also describes the relation between structural survivability, when the logic of system operability is binary and described by a logical function of operability, and functional survivability, when the system operation is described by a criterion of functional efficiency. Thus, a system failure is a fall in the level of its efficiency lower than the value predetermined in advance. Methods. Technical system is considered as a controlled cybernetic system installed with specialized survivability aids (SA). Logical and probabilistic methods and results of combinatorial theory of random placements are used in the analysis. It is supposed that: a) negative external impacts (N1) are occasional and single-shot (one impact affects one element); b) each element of the system has binary logic (operability - failure) and zero resistance, i.e. it is for sure affected by one impact. Henceforth this assumption is generalized for the r-time N1 and L-resistant elements.Besides, the work describes the variants of non-point models when a system’s part or entire system are exposed to a group specialized affection. It runs about the variants of combination of reliability and survivability, when both external and internal failures are analyzed. Results. Different variants of affection and functions of survivability of technical systems are reproduced. It has been educed that these distributions are based on simple and generalized Morgan numbers, as well as Stirling numbers of the second kind that can be reestablished on the basis of simplest recurrence relations. If the allowances of a mathematical model are generalized for the case when there are n of r-time negative external impacts and L- resistant elements, the generalized Morgan numbers which participate in the estimate of the affection law, are defined based о nthe theory of random placements, in the course of n-tuple differentiation of a generator polynomial. In this case it is not possible to establish recurrence relation among generalized Morgan numbers. It is shown that, under uniform allowances for a survivability model (equally resistant elements of the system, equally probable negative external impacts) in the core of relations for the function of system survivability, regardless of the affection law, there is a vector of structure redundancy F(u), where и is the number of affected elements, F(u) is the number of operable states of the technical system under и failures. Conclusion. Point survivability models are a perfect tool to perform an express-analysis of structural complex systems and to obtain approximate estimates of survivability functions. Simplest allowances of structural survivability can be generalized for the case when the logic of system operability is not binary, but is specified by the level of the system efficiency. In this case we should speak about functional survivability. Computational complexity PNP of the task of survivability estimation does not make it possible to solve it by the simplest enumeration of states of the technical system and variants of negative external impacts, it is necessary to look for the ways to egress from the blind enumeration, by transformation of the system operability function and its decomposition, as well. Development and implementation of survivability property into a technical system should be conducted with consideration of the property which is assured in biological and social systems.

PART 2. Multivariate calculations

This paper is a closing article to the first one [1] and it reproduces multivariate calculations by the procedure described in the references. Computational complexity of the task of survivability estimation and the ways to overcome this problem are discussed. We also deal with a passing from structural survivability to the tasks of functional survivability, establishing a conceptual joint between technical survivability and mobilization resilience in economy.

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