Engineering computation under uncertainty – Capabilities of non-traditional models
Introduction
The central issue in computational engineering disciplines is the realistic numerical modeling of physical and mechanical phenomena and processes. This is the basis from which to derive reliable predictions regarding the behavior, performance, and safety of engineering systems. A major challenge currently is the replacement of full-scale tests by numerical simulations in order to achieve economic, as well as societal benefits. For example, the considerable expenses for the production and tests of prototypes in the vehicle industry or in aeronautics may be reduced or avoided by applying methods of virtual design. These developments are pursued throughout the entire field of civil engineering, mechanical engineering, materials science, and beyond; see [1]. Their usefulness and potential substantially depend upon the reliability of computer-generated predictions.
The numerical models and parameters must be specified according to the underlying nature in order to obtain reliable computation results. An appropriate processing of all quantities ensures that this nature is reflected in the results. That is, the crucial point is an adequate modeling and processing of the available information. In engineering, available information is frequently not certain or precise but rather imprecise, diffuse, fluctuating, incomplete, fragmentary, vague, ambiguous, dubious, or linguistic and may possess a data-based, expert-specified, objective, or subjective background. The information basis usually consists of plans, drawings, measurements, observations, experiences, expert knowledge, codes and standards, and so on. Furthermore, influences from human mistakes and errors in manufacturing, from the use and maintenance of the constructions, and from on-going boundary and environmental condition changes have to be taken into consideration. These phenomena may be summarized by the collective term uncertainty.
The uncertainty of the available information impedes the specification of certain models and precise parameter values without an artificial introduction of unwarranted information. Thus, the uncertainty present must be described by an appropriate mathematical model in accordance with the underlying real-world information and processed through numerical computations. Shortcomings, in this regard, may lead to biased computational results with an unrealistic accuracy and, therefore, lead to wrong decisions with the potential for associated serious consequences. A pertinent discussion under the aspect of robustness and vulnerability of structures is provided in [2]. As the uncertain computational results represent the basis for deriving engineering decisions, modeling and processing of uncertain quantities are clearly of vital importance[3]. An attempt is thus made in this paper to concentrate on this issue with a discussion of numerical options and capabilities. It is demonstrated for a selection of uncertainty models that sophisticated and powerful numerical procedures are available to generate uncertain computational results. For the subsequent decision making under uncertainty and associated utilities, we refer to the extensive discussion in engineering literature; see [4], [5], [6], [7], [8].
The problem of selecting an appropriate uncertainty model can only be solved by analyzing the sources of the uncertainty in each particular case; the underlying reality dictates the model.
A popular classification of uncertainty, with respect to its sources, distinguishes between aleatory and epistemic uncertainty; see [3], [9], [10]. Even though this classification is not categorical in absolute terms, it provides a convenient distinction between irreducible uncertainty as a property of the system associated with fluctuations/variability – referred to as aleatory – and reducible uncertainty as a property of the analysts associated with a lack of knowledge – referred to as epistemic. This motivates the interpretation of aleatory uncertainty as stochastic uncertainty which results from an underlying pure random experiment such as drawing a sample from a concrete mixture. Epistemic uncertainty may then be considered as subjective uncertainty which may be initiated by the range of causes that defy pure probabilistic modeling. These causes comprise, for example, a lack of information, which impedes the specification of a unique probabilistic model and underlying generation schemes of observations that deviate from a pure random nature.
Aleatory uncertainty is primarily associated with objectivity; whereas, epistemic uncertainty may be comprised of substantial amounts of both objectivity and subjectivity – separately or simultaneously. Consequently, aleatory uncertainty can be modeled and processed appropriately with the aid of pure probabilistic methods. An extension of probabilistic modeling to a particular class of epistemic uncertainty problems is achieved with the concept of subjective probability including Bayesian theory. This concept ascribes a subjective perception – besides the traditional frequentative basis – to the model of probability consistent with the axiomatic understanding of probability. In many cases in engineering practice, however, not even subjective probabilistic information is available. Examples are uncertain quantities for which mere bounds or linguistic expressions are known. A probabilistic modeling would then introduce unwarranted information in the form of a distribution function that is totally unjustified. Even the assumption of a uniform distribution ascribes a certain probabilistic regularity and, thus, much more information than is given by just bounds for a quantity. In additional frequent cases, the knowledge about the fluctuations of the structural parameters is very limited so that a clear probabilistic specification of their associated uncertainty is impossible, and the capability for the development of reliable predictions, in terms of probability, is fundamentally limited. This pertains to severe uncertainty from a whole spectrum of origins. The most prominent examples are extreme incidents such as terrorist attacks, as well as socio-economic changes and societal aspects. These possess considerable relevance to risk management of structural systems. Another problematic issue is the modeling of strongly non-stationary processes to predict exceptional environmental conditions such as extreme snow loads or wind loads that are affected by changes in global and local climate, thereby, defying a clear specification of future non-stationarities. Further examples include the evaluation of very limited test data obtained under dubious conditions used to predict the behavior of novel materials, the consideration of subjective influences such as the use of a structure for an unintended purpose or beyond its initially planned life-time after a reconstruction, and the modeling of changing technical boundary conditions such as limits for traffic loads specified in regulations that underlie a permanent development. Moreover, for design purposes, it is of essential importance to not only consider reliably predictable parameter fluctuations but also account for mere vaguely predictable or even unpredictable fluctuations, which are initialized, for instance, by subjective decisions. This applies, in particular, to investigations in the early design stage.
Consequently, epistemic uncertainty generally requires further specific models oriented to particular characteristics of the uncertainty associated with the available information. These uncertainty models are constituted on a non-probabilistic or on a mixed probabilistic/non-probabilistic mathematical basis.
Based upon this context, it becomes clear that particular attention must be paid to the phenomenon of uncertainty. The quantification, processing, and evaluation of uncertainty have already become engineering tasks of great importance and interest. Both advancements in probabilistic mechanics and the development of novel methods for dealing with uncertainty, including the associated computational techniques, are seen as a research topic for the next decade [1], [3].
While traditional probabilistic methods are already well-established and largely recognized as applicable to real-world problems [11], [12], [13], approaches of non-traditional uncertainty modeling still appear scattered in engineering literature and must stand up against premature reservations and criticism. Non-traditional uncertainty models are frequently criticized as having considerably limited and inappropriate capabilities in modeling and processing uncertain quantities, as well as substantial weaknesses in representing the meaning of uncertain quantities. They are accused of lacking sophisticated and numerically efficient algorithms for processing uncertain quantities that are needed, for instance, to analyze large structures. Furthermore, controversial discussions within the scientific community frequently impute a competition between different uncertainty models and allow non-traditional uncertainty models appear as non-systematic. Consequently, the actual potential of non-traditional uncertainty models has remained widely concealed or misconceived. Therefore, applications and tests in civil and mechanical engineering practice are still rare. The developments in non-probabilistic and mixed probabilistic/non-probabilistic uncertainty modeling have, however, recently attracted increasing attention, and attempts have been made to initiate some bridging of understanding between traditional and non-traditional uncertainty modeling [14], [15]. The collective and systematic presentations include theoretical, as well as practical, components and demonstrate the capabilities of non-traditional uncertainty models by means of engineering challenge problems [16]. Further, it has been revealed that the capabilities of non-traditional methods go beyond a realistic modeling and processing of epistemic uncertainty. These provide useful new insights during the process of engineering analysis and design, in particular, with new features for a systematic and extended sensitivity analysis [17], [18], [19], [20], [21], [22].
A key conclusion from recent developments is to consider the variety of uncertainty models as an entity in which the models exist in parallel with their specific characteristics and benefits. This provides a suitable basis for an optimum model choice in each particular case.
From this unifying perspective, the present capabilities of non-probabilistic and mixed probabilistic/non-probabilistic uncertainty modeling are reviewed in this paper within the context of engineering challenges. This responds to the criticism of non-traditional uncertainty models which imputes a considerable limitation and inappropriateness in modeling, processing, and representation of the meaning of uncertain quantities, as well as a lack of sophisticated and numerically efficient algorithms to solve industry-sized problems. Numerical concepts for processing the associated quantities such as intervals, fuzzy sets, and imprecise random quantities, are discussed with respect to their computational features, as well as to the quality of the computational results and to numerical efficiency. This review is provided for under the following two major aspects:
- (1)
The non-traditional uncertainty models are not treated as replacements or competitors, with respect to traditional probabilistics, but as supplementary elements which can be very useful in various cases.
- (2)
The selection of an uncertainty model from the variety of choices is problem specific and depends on the available information in each particular case.
Section snippets
Non-probabilistic and mixed uncertainty modeling in a general context
The developments in modeling, processing, and evaluating uncertainty using non-probabilistic and mixed probabilistic/non-probabilistic concepts range from mathematical fundamentals to engineering applications. Novel mathematical concepts are adopted in engineering theory and materialized into numerical techniques for engineering computations that are eventually tested and demonstrated via practical engineering examples.
Basic mathematical concepts with detailed and comprehensive explanations may
Interval modeling and fuzzy methods
A very popular and elementary non-probabilistic uncertainty model is the interval, which is – in uncertainty modeling – generally applied in its closed formIntervals represent an appropriate model to mathematically describe uncertainty in those cases where only a possible value range between crisp bounds xl and xr is known for the uncertain quantity and no additional information concerning variations, fluctuations, value frequencies, preferences, etc. between interval
Imprecise probabilities
Engineering problems are frequently affected by uncertainty with characteristics that fall between the traditional probabilistic model and non-probabilistic uncertainty models. That is, though statistical information is generally available, a pure probabilistic model cannot be formulated with certainty due to a lack of trustworthiness of the data or a lack of pertinent information. The statistical data may be imprecise, diffuse, rare, fragmentary, or vague; result from incomplete measurement
Conclusions
The presented review of non-probabilistic and mixed probabilistic/non-probabilistic uncertainty models provides some coarse insight into the expansive variety of developments in this field with an association to engineering computation. The emphasized capabilities and potential of selected approaches clearly show that the criticism of non-traditional uncertainty models is generally unjustified. It is elucidated that non-traditional uncertainty models are the key to solving practical engineering
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