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The work was funded by the Belgian federal government through a contract with the federal Science Policy Office, and by the Flemish regional government through a contract with FWO Vlaanderen (Fund for Scientific Research). David Moens is a research fellow of the Research Foundation – Flanders (FWO - Vlaanderen). The research by Hilde De Gersem is funded by the Flemish regional government through a fellowship with FWO Vlaanderen. Part of the work was also funded by the European Commission, through the Marie-Curie Research and Training Network "Maduse".

The emerging non-probabilistic approaches are redefining the landscape for non-deterministic FE analysis. It is the aim of this paper to give insight into the possible useful applications of these approaches, referring to the generally accepted and widely adopted probabilistic approach.

It is first shown that a clear distinction can be made between different sorts of non-deterministic properties in a numerical model. The existing classification of uncertainties and variabilities is further subdivided in certain variabilities, uncertain variabilities and invariable uncertainties. Based on these different types of non-determinism, the applicability of the different non-deterministic concepts is analysed. Different sources of uncertainty are reviewed, and it is concluded that the probabilistic approach remains the most interesting to tackle problems that are subject to complete and objective probabilistic influences. However, in the presence of uncertain quantities that require subjective information in order to be described numerically, the interval and fuzzy approach become increasingly interesting. Especially for uncertainties, the fuzzy concept is very appropriate because of its implicit subjective nature.

The reliability of a product is defined as the likelihood that it will successfully fulfil its intended task over a predefined period in time under specific environmental conditions. Numerical reliability analysis based on probabilistic analysis is very popular because when realistic data is used, it can give a clear indication of the likelihood of failure of the analysed structure. As such, it can be usefully applied in an economical product analysis taking into account the cost associated with failure. This probabilistic reliability analysis is broadly applied and already incorporated in generally accepted design specifications in civil engineering. However, its application in mechanical engineering is far less standardised. This is mainly due to the plenitude of different mechanical products, which all require a different amount of reliability. Hence, there are very few standards for reliability in mechanical design. Each product designer applies rules which are based on experience rather than on general engineering standards.

Section "Discussion of introduction of non-determinism in the engineering process" has shown that more information on a product becomes available as design decisions are taken. The evolution of non-determinism in a typical design process as described above is illustrated in figure 5. The numerical prediction of the actual design quality improves over the design process. In the early stages, the non-determinism in the numerically predicted design quality is mainly driven by model uncertainties, whereas in later stages, variability becomes more important. This figure also indicates the evolution of the numerical concepts that are most appropriate for the dominant class of the occurring non-determinism.

This section focusses on a number of practical non-deterministic analysis types that concern a design engineer. In order to evaluate the possibilities of the non-probabilistic approaches in specific applications, references will be made to the corresponding probabilistic treatment of the non-determinism. In this discussion, only the application of the stand-alone numerical concepts of probabilistic, interval and fuzzy analysis is considered.

In all phases of engineering analysis of a technical component over its product lifetime, all decisions are crisp. They are either yes/no decisions or they involve the specification of precise values or a range of values. Availability of data is usually insufficient to support a statistical interpretation of design decisions. Generally speaking, the interval or the fuzzy concepts is most appropriate for technical analysis.

Statistical interpretation may be relevant in three out of the six stages that are identified in section "Overview of stages in the engineering process":

Uncertainties and variabilities play a role in each of the phases listed in section "Overview of stages in the engineering process". Throughout the discussion, the example of a truck chassis will be presented for the purpose of illustration.

1. definition of product specifications, design data and load cases: This phase includes the establishment of a list of input design data (product requirements) and conditions of utilisation of the product. These specifications are often rather general. For the purpose of technical analysis however, numerical data are required, preferably with a maximum degree of precision. It is common to include a margin of safety. However, this margin should not be too large, to avoid unnecessary overconservatism and uneconomical design. Several factors complicate the specification of precise data. Design data are still uncertain because some design parameters will be specified only during the subsequent process of design refinement. Specifications may be imprecise as several design variants of a product should be realised with a maximum degree of commonality. Component commonality is desired to reduce the number of components that are produced by a company and to simplify maintenance. On the other hand, commonality reduces the options for optimisation of a product.
   As far as technical design specifications are concerned, several requirements should be met: strength of the product, static and dynamic stiffness, fatigue life, ... . The technical analysis that is required to verify these properties are very different, and further, these qualities depend on different product properties. Strength and fatigue life are the result of local design details, and as such, the data that are required to verify these properties are available only later in the analysis.
   The availability of design specifications depends very much on the industrial sector. In several areas design criteria are well established, based on many years of expertise with similar designs in previous cases. Design standards exist in the civil engineering sector, covering a wide spectrum of load cases in nominal and exceptional conditions, and these standards are even extensively documented. Other specific sectors of industry such as aircraft structures, pressure vessels, hoisting equipment also have well established design criteria. Standards are defined by independent normalisation and standardisation bodies, and insurance companies verify the correct application of standards before an insurance agreement is signed. In most other sectors of industry however, generally accepted standards are not available and each manufacturer has to decide for himself on the criteria to be applied. The determination of specifications is a rather delicate compromise between operational safety and economical design. Almost all consumer products are in this category.
   In a limited number of industrial products, probability of failure is prescribed. An example is space industry, with a prescribed value on the reliability of launch vehicles. This value is however theoretical as the actual failure rate of launches does not match the prescribed numbers.
   For the case of the truck, there are many product specifications, such as the type of load that the truck should transport, the maximum design load, its mission profile, the maximum dimensions, ... The mission profile may be very different~: long haul for transcontinental transport on motorways, medium to short haul such as for concrete mixer trucks, very short range with very frequent stops and starts such as for garbage collection. The determination of loads is based to a large extent on experience with previous models. The load history can be measured by instrumentation of an existing vehicle. Typical mission profiles can be deduced, and used for later design development. Standards for trucks cover only part of the design requirements.
   This phase in the design exercise should be concluded with a set of requirements that is as concise as possible, if relevant including statistical data.
   
2. definition of preliminary design and initial analysis: after basic design requirements are formulated, one or more initial concepts are proposed for the newly designed product. Comparative design analysis typically uses so-called concept models that represent the global characteristics of the product without details. A correct focus on parameters that drive the design lay-out is crucial in this stage. For the sake of effective product design in subsequent phases, it is important that crucial design decisions are taken as early as possible. The pressure on design and development departments in companies to shorten product design cycles grows continuously. Unfortunately, complete product design data are usually not yet available at that stage, and data imprecisions have to be taken into account. Conservatism is absolutely required, yet without being excessive.
   In the truck case, the concept model consists of discrete elements representing flexible components and discrete masses such as the engine and the fuel tank. The size and the filling percentage of the fuel tank (and sometimes also its position) being uncertain, the analysis has to take into account a relevant range of parameter settings.
   Relatively few people are typically involved with this phase of design, and experience shows that most companies have only few experts who are qualified to define relevant inputs. The concept of subjective probability is therefor not applicable here. On the other hand, this design phase is concluded with a preliminary yet clear definition of the product concept. In the truck case, primary structural members can now be specified.
3. design refinement, leading to final design: after the product concept is established, product design should be gradually refined. As design activities proceed --- often in separate design teams with different responsibilities --- more and more design data become available. Numerical models are refined, and detailed design analysis becomes feasible. The result set of the analysis grows likewise, and each output quantity is subject to verification of design criteria. Not only global criteria but also local criteria can now be verified, possibly including a safety factor. Each criterion is expressed as an inequality, and the degree by which it is fulfilled is unspecified and thus uncertain.
   In the truck case detailed design includes the determination of details such as the position of holes and joints, the type of joints (welded, bolted, each with their inherent uncertainties), sizes of secondary structural members, ... The number of details is so large that it is sometimes impossible to include them all into a numerical model, inevitably increasing the uncertainty on the product behaviour. This statement is especially true for local design details that typically affect local product response, such as fatigue life.
   This design phase is concluded with a concise complete set of design specifications, including all design details.
4. production process definition: after nominal design parameters are specified, the entire process of production and assembly should be outlined. However, each step in the production process has its own range of accuracy that can be achieved. It depends on the quality of the specific production machine on which the component is manufactured and on the skills of the machine operator. The required accuracy is specified by the so-called geometrical tolerances. A tolerance is a min-max range and each measure should be verified to be within that range. The specification of a production process translates into the definition of a parameter range. However, on the production machine, nominal values are set.
   For the truck case, nominal machine parameters have to be set on all machines for cutting, sawing, punching, drilling, milling, folding, grinding, ... This list contains precise, unique data, to be used at the production machinery of the manufacturer. A list of tolerances should be added specifying the ranges on geometrical properties.
5. actual production and quality control: With the machine parameters set in the previous phase, actual production can be started. The result of production operations on each individual product is then subjected to some kind of quality control. However, even with a precise setting of machine parameters, the properties of each individual product are never identical. A quality control procedure is then required to verify if each product meets the standards. Different levels of quality control are used: no control on semi-finished products, implicit verification by obvious deficiencies, a predefined sampling procedure, full quality control on each individual product, extensive qualification and acceptance tests.
   The results of quality control is interpreted in two different ways. At the level of each \textit{individual} product, the verification of quality leads to either acceptance of the product or to rejection and scrapping. This decision is of a yes/no type. At the level of a complete batch of production, a quality distribution can be established expressing which percentage of products meets a desired quality level.
   In some production facilities, non-conformancy procedures can be used, when a product exhibits an acknowledged deficiency, and the cost of scrapping is considered too high, it may be preferred to rework it and adapt the design to the observed deficiency. A new design analysis is required in that case. The result of this procedure is a yes/no decision.
6. operations in service conditions: A well-designed product usually behaves well in normal service conditions. However, unanticipated incidents may inflict damage on the product. If such an incident occurs, the operator usually verifies the extent of damage and he decides if the product should be repaired. Most mechanical systems further exhibit some kind of wear or damage accumulation (e.g. due to fatigue) over their economical lifetime. If the extent of wear or damage becomes such that nominal operation of the product gets dangerous or unreliable, it is common to replace the wear-sensitive or damaged components by new ones. This decision may be based on different criteria: preplanned after some fixed period of utilisation, by continuous monitoring of the performance of the product, or by more or less incidental observation of a deficiency. In the first or the second case, this decision may be prepared by previous expertise that is gathered after careful examination of the operational performance of previous examples of a similar product. The decision to repair a component is a yes/no decision.

The engineering process typically consists of a number of successive phases:

1. definition of product specifications and design data, and definition of load cases and load case combinations
2. definition of a preliminary design and initial analysis of its feasibility
3. gradual design refinement and improvement, and specification of design details, concluded with the definition of final design
4. definition of the production process
5. production startup and quality control
6. operations of the product in service conditions
   

Each of these phases considers a number of inputs, some of which may be uncertain. Each of these phases is concluded with a decision.

Over the entire life of a technical product many sources of non-determinism may be relevant. This section describes briefly the process, and it identifies the phases when the uncertainties and variabilities play a role. The case of a product designed to withstand mechanical loads is taken as an example, but other cases are similar.