Approaches for assessment of non-determinism in different stages of finite element analysis

by David Moens, Hilde De Gersem, Dirk Vandepitte, and Wim Desmet

This paper discusses different types of non-deterministic parameters that may be relevant in different stages of engineering analysis. The entire cycle of development of a product is considered, and it is shown that the relevance of methods is different in different stages of the development cycle.

A classification of different types of non-deterministic properties is presented. Based on the nature of these different classes of model properties, it is discussed to what degree each of these fits in the framework of either a probabilistic or a non-probabilistic concept.

Probabilistic methods are applicable in later stages of development, when a sufficiently large database of product data has been gathered. Probabilistic approaches are perfectly suited for conditions when the product is already in service. Possibilistic analysis on the other hand is best suited for application in cases when the data set about the product at hand is still incomplete.

 

Introduction

Engineering design is the activity of design and development of technical products. A technical product is built to fulfil a well specified function under more or less well prescribed conditions of utilisation. The complexity of modern technical products tends to increase systematically, increasing the need for thorough design analysis. This process consists of a number of analysis verifications on a virtual product. A common procedure for design verification is finite element analysis, a numerical method for the simulation of the effect of mechanical or thermal loads on a product. As most product parameters are undetermined in the initial phases of design, a range of non-deterministic properties have to be taken into account. This paper discusses the effects of non-determinism on engineering analysis using the finite element method (FE).

Over the past decade, the probabilistic FE analysis has gained a large popularity in the area of engineering analysis with non-deterministic parameters. Recently, a number of non-probabilistic approaches for non-deterministic analysis are emerging. The Interval FE (IFE) analysis is based on the interval concept for the description of non-deterministic model properties, and so far has been studied only on an academic level [1-3]. The Fuzzy FE (FFE) analysis is basically an extension of the IFE analysis, and has been studied in a number of specific research domains: static structural analysis [4,5], dynamic analysis [6-8], geotechnical engineering [9,10], multi-body kinematics [11], steady-state analysis of rolling [12], analysis of smart structures [13] and analysis of fibre-reinforced composite materials [14]. The numerical procedures developed for the non-probabilistic approaches are all strongly influenced by the specific properties of the analysed physical phenomenon, and only academic examples with very limited size and complexity are considered.

The non-probabilistic approaches broaden the possibilities, but simultaneously they complicate the choice for the analyst. Therefore, it is important to study to what extent they can be an alternative in the areas where probabilistic analysis has become the standard. The growing interest for non-probabilistic methods for non-deterministic numerical analysis mainly originates from criticism on the credibility of probabilistic analysis when it is based on limited information. Especially when extremely high reliabilities are analysed based on numerical models, design engineers often remain very sceptic regarding the trustworthiness of the numerical predictions. The recent development of the non-probabilistic approaches stems from the argumentation that this lack of credibility is always present in probabilistic analysis results, but generally remains unaccounted for. It is argued that the non-probabilistic concepts could be more appropriate to model certain types of non-deterministic information, resulting in a better representation of the simulated non-deterministic physical behaviour. Also, it is believed that a full probabilistic description of a non-deterministic event is not always required. Especially in early design stages, when objective probabilistic information often is not available, non-probabilistic concepts are considered to be of great value. It is the aim of this paper to critically review this argumentation, and to study to what extent the non-probabilistic methods can be considered as useful alternatives to the existing probabilistic approach.

General aspects of non-determinism in numerical analysis

(this section is only a container of subsections)

Sources of numerical non-determinism in general FE modelling

Definitions

In literature, the use of the terminology error, uncertainty and variability is not unambiguous. Different researchers apply the same terminology but the meaning attached to these is rather inconsistent. This necessitates a profound clarification of the terminology for each publication which treats uncertainties. This work does not propose a new terminology, but applies the terminology proposed by Oberkampf [15]. Some additional nuances are, however, necessary in order to enable clear distinction between probabilistic and non-probabilistic quantities in the remainder of this chapter.

The term variability covers the variation which is inherent to the modelled physical system or the environment under consideration. Generally, this is described by a distributed quantity defined over a range of possible values. The exact value is known to be within this range, but it will vary from unit to unit or from time to time. Ideally, objective information on both the range and the likelihood of the quantity within this range is available. Some literature refers to this variability as aleatory uncertainty or irreducible uncertainty, referring to the fact that even when all information on the particular property is available, the quantity cannot be deterministically determined.

An uncertainty is a potential deficiency in any phase or activity of the modelling process that is due to lack of knowledge. The word potential stresses that the deficiency may or may not occur. This definition basically states that uncertainty is caused by incomplete information resulting from either vagueness, nonspecificity or dissonance [16]. Vagueness characterises information which is imprecisely defined, unclear or indistinct. It is typically the result of human opinion on unknown quantities ("the density of this material is around $x$ "). Nonspecificity refers to the availability of a number of different models that describe the same phenomenon. The larger the number of alternatives, the larger the nonspecificity. Dissonance refers to the existence of conflicting evidence of the described phenomenon, for instance when there is evidence that a quantity belongs to disjoint sets. Possibly, limited objective information is available, for instance when a range of possible values is known. In most cases, however, information on uncertainties is subjective and based on some expert opinion. Others in literature refer to this uncertainty as reducible, epistemic or subjective uncertainty.
An error is defined as a recognisable deficiency in any phase of modelling or simulation that is not due to lack of knowledge. The fact that the error is recognisable states that it should be identifiable through examination, and as such is not caused by lack of knowledge. This means that the error could be avoided by an alternative approach which is known to be more accurate, but which is possibly limited in practical applicability by computational cost or other practical considerations. A further distinction between acknowledged and unacknowledged errors is possible. Errors will not be considered further in this paper.

Figure 1 summarises the definitions in this section with their main characteristics in the context of the FE methodology.

Occurrence of variabilities, uncertainties and errors in the FE procedure
Figure 1: Occurrence of variabilities, uncertainties and errors in the FE procedure

 

Discussion and extension of the definitions

The above definitions of uncertainty and variability are fairly straightforward and comprehensible. However, they are not mutually exclusive, since a variability could be subject to lack of knowledge when information on its range or likelihood within the range is missing. This is for instance the case for every design dimension subject to tolerances, but without further specification of manufacturing process or supplier. The tolerances represent the bounds on the feasible domain, but there is no information on the likelihood of the possible values within these bounds. Consequently, because there is a lack of knowledge, such a variability is also an uncertainty. It is referred to here as an uncertain variability. Some vague knowledge may be available ("the mean value is approximately $x$ ") but also nonspecificity may play an important role in the uncertainty, for instance in choosing an appropriate model to describe a random quantity. Opposed to the uncertain variability, a certain variability refers to a variability the range and likelihood of which are exactly known.

On the other hand, it appears logical to state that every property in a numerical model corresponding to a physical quantity is a variability, since it will eventually have a range of possible values and a likelihood inside this range in the physical model. This argumentation implies that all uncertainties are also variabilities. In practice however, the majority of model properties are implemented as constant deterministic values in the numerical model. Though they are subject to variation, the influence of their variability on the analysis result is considered to be negligible. Often, uncertainties refer to a possible lack of knowledge in these deterministic properties. This type of uncertainty is referred to as invariable uncertainty. Note that invariable in this case does not mean that the property cannot change over different analyses. According to the definition of uncertainty, it will change when additional information is gathered that decreases the amount of uncertainty. The invariable uncertainties typically occur in model properties for model parts that are difficult to describe numerically, but considered constant in the final physical product (connections, damping, ...). Other examples are design properties which have negligible variability but which are not defined exactly in an early design stage. Figure 2 gives a graphical illustration of the proposed subdivision of the definitions for uncertainty and variability.

The group of variabilities may be further subdivided into two categories. Inter-sample variability is the property of a population of nominally identical realisations of a particular product, with each individual element of the population possibly exhibiting scatter. Intra-sample variability is a property of one particular realisation --- of which other realisations possibly exist --- that exhibits one or more properties that may change over time, due to temperature differences, ageing, ...

Classification of variabilities and uncertainties in numerical modelling
Figure 2: Classification of variabilities and uncertainties in numerical modelling

 

 

 

Numerical concepts for non-deterministic numerical modelling

Historically, the introduction of the non-probabilistic approaches for non-deterministic analysis has initiated a profound discussion in literature. On one side, some claim that the probabilistic approach is only a subcategory of a more universal non-probabilistic approach. Therefore, the latter would represent a more unified approach for non-deterministic analysis. On the other side, some argue that probabilistic methods are able to model anything the non-probabilistic approach can. The goal of this section is not to choose either side in this discussion, but merely to review the applicability of the non-probabilistic concepts from an objective point of view. Therefore, for each concept, its compatibility with the definitions of uncertainty and variability in the previous section is discussed. This discussion focusses on the ability to objectively represent the available information. In order to enable a critical review of the capabilities of the non-probabilistic concepts, this section first starts with a brief discussion on the main features of the probabilistic concept in the framework of uncertainty and variability modelling.

Evidence Theory [31] can be regarded as a generalisation that covers both the probabilistic as well as the non-probabilistic approaches for non-deterministic analysis, although the operations and inference rules in these theories are completely different. Evidence theory is regarded as a universal approach that can handle combinations of both probabilistic and non-probabilistic information in a single analysis. Its practical application is based on the availability of information in the form of basic belief assignments. The practical value of this approach therefore depends by large on the availability of the information in this form. While this theory is gaining interest in recent literature, its practical application in real-life engineering has yet to be proven. Therefore, it is left out of the current discussion. The following paragraphs briefly discuss the probabilistic concept, the interval concept and the fuzzy concept as stand-alone tools for the representation of non-deterministic parameters in engineering analysis. Finally, some hybrid non-deterministic numerical modelling concepts are briefly discussed.

 

Basic properties of the probabilistic concept

In the classical frequentist application of the probabilistic concept, the goal of a numerical property description is to define a domain of possible values this property can adopt, and to give information on the frequency of occurrence of the numerical values in this domain. This is typically done by defining a probability density function over the domain of possible values.

Extensive literature exists on the subject of probability theory, treating a vast variety of PDFs and their applicability for description of random quantities. An overview of these can be found in [17] and [18].

In most available non-deterministic FEM software codes, the probabilistic concept is applied to describe both variabilities and uncertainties in a model. This is mainly due to the fact that there exists a large number of numerical analysis procedures based exclusively on probabilistic input quantities. Therefore, every non-deterministic quantity in a model is readily replaced by a probabilistic quantity by introducing an appropriate PDF*. However, the probabilistic model does not necessarily represent the available objective information. For the study of the applicability of the probabilistic model, distinction between certain variabilities, uncertain variabilities and invariable uncertainties is necessary.

It is clear that the probabilistic concept is most appropriate to represent certain variabilities, since in the frequentist interpretation, the probabilistic description using a PDF* is completely consistent with the definition of a variability as in section "Sources of numerical non-determinism in general FE modelling". The information on the range and the likelihood of a certain variability can be unambiguously incorporated in the PDF*. Furthermore, the probabilistic outcome of the analysis will give an indication of the actual expected frequency of occurrence of the analysed phenomenon. It is, however, important that all information is available in order for the model to realistically represent the variability. For instance, if more than one variable property is present in the model, the correlation between the different variabilities might play an important role in the probabilistic analysis. Ideally, the joint PDF* describing the likelihood and interdependence of all non-deterministic model properties is available. Since this is almost never the case, the probabilistic description of variability interdependence is generally limited to some moments of low order. Often, when cross correlations are unknown, the variabilities are assumed to be independent of one another.

For uncertain variabilities, a representation by a single random quantity is generally not sufficient. Engineering scientist Freudenthal [19] who was one of the pioneers of probabilistic methods in engineering states that "... ignorance of the cause of variation does not make such variation random.". By this, he means that when crucial information on a variability is missing, it is not good practice to model it as a probabilistic quantity represented by a single random PDF*. On the contrary, in this case it is mandatory to apply a number of different probabilistic models to examine the effect of the chosen PDF* on the result. For instance, when the range of the variability is known but the information on the likelihood is missing, all possible PDFs over the range should be taken into consideration in the analysis. The analyst will generally select only a few probabilistic models which he considers consistent with the limited available information or most appropriate to obtain as much knowledge as possible on the result.

Most often, invariable uncertainties are represented by random quantities in probabilistic analysis. As such, the analyst tries to express his lack of knowledge of the property. This means that some PDF* is chosen which to the knowledge of the analyst represents best the uncertain nature of the quantity, but which is not based on available objective information. It is clear that in this case, the information contained in the random quantity does not represent the actual variation of the quantity in the final product, since by definition, the invariable uncertainties are considered to be constant. The random quantity in this case merely represents the presumed likelihood that a model parameter will adopt a value. As such, the lack of knowledge is filled by subjective information provided by the analyst, expressed in the form of a PDF*. This is sometimes referred to as a subjective PDF*. In this context, Bayesian methods are becoming increasingly popular for the modelling of subjective uncertainty. The main advantage of using the probabilistic approach for subjective uncertainty modelling is that the available probabilistic procedures can be readily applied for the analysis. It should be kept in mind, however, that the main strength of the Bayesian approach is its capability of incorporating objective information that becomes gradually available. When this is not the case, the Bayesian approach remains a fully subjective representation of reality.

At this point, it is very important to emphasise the consequences of the difference in the use of the probabilistic concept for variabilities on the one hand, and invariable uncertainties on the other hand. The former represents inter-sample or intra-sample variability for the final product, while the latter clearly may not be interpreted in this sense. Consequently, when interpreting the results of a probabilistic analysis based on both uncertainties and variabilities, it is imperative to distinguish between the different meanings attached to both. Though this may seem straightforward, neglecting this distinction is a very common mistake in probabilistic uncertainty analysis.

 

Basic properties of the interval concept

Recent developments in interval arithmetics are mainly based on the work of Moore [20], who introduced interval vectors and matrices and the first non-trivial applications. By definition, an interval scalar consists of a single continuous domain. The range is bounded by a lower and an upper bound. Combining different interval numbers is generally done by simply combining all component intervals independently. This means that all entries are implicitly assumed to be mutually independent quantities. This has very important consequences for the use of the interval concept in FE analysis since there is generally a strong dependency between FE matrix coefficients and right hand side coefficients in FE analysis. Neglecting this dependency results in the implicit introduction of conservatism into the analysis.

The information represented by an interval object depends on the type of modelled non-deterministic quantity. Also here, distinction between certain variabilities, uncertain variabilities and invariable uncertainties is necessary.

For certain variabilities, the input interval objects are derived from the support of the corresponding input PDFs. Consequently, the result of an interval analysis only represents the actual range of the variable outcome of the analysis. The available information on the likelihood inside the range is lost, which is an important disadvantage. Especially for a variability with a justifiable PDF* support that is very large, using the support as input for the interval analysis will generally result in an extremely wide output interval. While it is theoretically correct to state that the final result will range over this output interval, disregarding the probability of the PDF* tails in this case clearly strongly devaluates the interval analysis.

When the upper and lower bounds of a non-deterministic property are well-defined but information on the type of the distribution is missing, it belongs to the class of uncertain variabilities. In this case, the interval model represents perfectly the available information. However, especially for variabilities with a very large PDF* support, the determination of the corresponding interval bounds is not always unambiguous, since the probability of the values that are located in the tails of the commonly applied PDFs with large support is typically very low. If these tails cannot be justified adequately with experimental data, there is no reason to unconditionally use the PDF* support for the interval analysis. In this case, the analyst should implement the bounds which he considers realistic with respect to the available experimental data. Often, the $3\sigma$-bounds are assumed to be realistic interval bounds. This conversion does not necessarily reduce the truthfulness of the uncertainty representation when there is little information on the actual tails of the PDF*. Still, if the tails of the PDF* are expected to have little probability, the impact of the subjective interval bounds on the interval analysis result is much larger than the impact of subjective PDF* support limits on the probabilistic analysis result. Therefore, variabilities with unknown PDF* support but a well-known normal-like behaviour near the center of the PDF* are best modelled probabilistically.

For invariable uncertainties, generally a subjective interval is required. In this case, care should be taken not to interpret the interval quantity as the actual range in the physical product. It merely represents the values the analyst considers possible at the time the analysis is performed. Therefore, similar to the application of the probabilistic concept for invariable uncertainties, it is important to acknowledge the subjectivity in the result of the analysis. However, since the interval concept requires less subjective information to be added to the problem description, there is less room for misinterpretation of the results.

To conclude, we can state that the probabilistic concept remains the most valuable for the representation of certain variabilities and uncertain variabilities with unknown support but known normal-like behaviour. The omission of a known PDF* through the interval concept can only be justified when probabilistic information is not required, or the computational cost of the interval analysis is significantly lower. The interval concept is most valuable when dealing with uncertain variabilities with known support but unknown distribution, or invariable uncertainties.

 

Basic properties of the fuzzy concept

The theory of fuzzy logic was introduced by Zadeh [21] in 1965, and has gained an increasing popularity during the last two decades. Its most important property is that it is capable of describing linguistic and, therefore, incomplete information in a non-probabilistic manner.

A fuzzy set can be interpreted as an extension of a classical set. Where a classical set clearly distinguishes between members and non-members of the set, the fuzzy set introduces a degree of membership, represented by the membership function. This membership function describes the grade of membership to the fuzzy set for each element in the domain. Figure 3 shows the membership functions of some typical normal fuzzy numbers.

Some typical membership functions that describe linguistic variables

Figure 3: Some typical membership functions that describe linguistic variables

While the concept of fuzzy logic was invented in 1967, it resulted mainly in practical applications during the last two decades. The works of Dubois and Prade [22, 23] contributed to a large extent to this evolution. The concept has been most successful in the application to controller design, known as fuzzy control [24].

Zadeh [25] extended the theory of fuzzy sets to a basis for reasoning with possibility. In this interpretation, the membership function is considered as a possibility distribution function, providing information on the values that the described quantity can adopt. More generally, the possibility is defined as a subjective measure that expresses the degree to which the analyst considers that an event can occur. It provides in a system of defining intermediate possibilities between strictly impossible and strictly possible events. Through this interpretation, the fuzzy concept has become a tool to model subjective knowledge numerically in a non-probabilistic concept. This has drawn the attention of the numerical community, since knowledge of uncertainties in a numerical model is commonly based on expert opinion. This has lead to the first attempts to use the fuzzy concept in a non-deterministic framework, resulting in some applications in structural optimisation under uncertainty [26,27]. Also, this has initiated the development of the Fuzzy Finite Element Method (FFEM) for numerical analysis of non-deterministic models [28].

However, the application of the fuzzy concept for non-deterministic numerical modelling is not straightforward. The main problem of the representation of a model property through a fuzzy set, is that the membership function does not relate to an objective measurable quantity. The level of membership that is assigned to different members of a fuzzy set is completely based on the subjective beliefs of the analyst. Therefore, also the fuzzy results obtained from the analysis will be biased with the subjective input. Hence, these results may only be interpreted in reference to the assumed fuzzy input. This poses an important restriction on the use of the fuzzy approach for numerical design validation purposes.

For a fuzzy representation of certain variabilities, the known PDF* has to be converted to a compatible membership function. A number of methods have been developed for this purpose [23,29]. The basic law for the conversion follows from the consistency principle, which states that the degree of possibility of an event is greater than or equal to its degree of probability. These conversion techniques always rely on some sort of subjective judgement. It is the authors' opinion that forcing the application of fuzzy sets into the domain of certain variabilities through a conversion of PDFs as described above is rather irrational. Available objective probabilistic data is replaced by a subjective description, resulting in the loss of very valuable information. This loss is generally unjustifiable. Therefore, the conversion of a PDF* to a membership function should not be done.

For uncertain variabilities, the fuzzy concept can be used for a hybrid uncertainty model. It stems from an alternative interpretation of a possibility distribution introduced by Dubois and Prade [30] based on the Evidence Theory. In this approach, a fuzzy number is used to represent a class of probability random quantities that have a cumulative distribution function (CDF*) in between boundaries derived directly from the possibility distribution. The left boundary on the compatible CDFs coincides with the increasing branch of the fuzzy number. The right boundary coincides with the complement of the decreasing branch of the fuzzy number. Figure 4 clarifies this approach. In this concept, the possibilistic approach becomes a tool to simultaneously examine the effect of a set of different PDFs in a single analysis. While the ability of this concept to model classes of probabilistic data seems extremely powerful, it has only been applied very rarely in uncertainty analysis.

Possibility distribution of a fuzzy number and corresponding lower and upper boundaries for CDF compatible with the fuzzy number

Figure 4: Possibility distribution of a fuzzy number and corresponding lower and upper boundaries for CDF* compatible with the fuzzy number

Finally, an invariable uncertainty requires a fuzzy set that represents the subjective expectation of the analyst. When the invariable uncertainty represents an open design decision subject to optimisation, the analyst can express his preference of the quantity through the possibility distribution. Still, when interpreting the results, reference to the chosen input membership functions is imperative.

Considering the explicit subjective nature of a fuzzy set, it is concluded that it is most useful to describe uncertainties. The more objective information becomes available on a non-deterministic model property, the less the fuzzy concept is appropriate to describe it.

 

Hybrid non-deterministic modelling concepts

If for an uncertain variability there is information about the range but no information on the probability of occurrence, every PDF* over this range becomes equally plausible and should be taken into consideration. However, the information of the likelihood inside the range is not needed for the interval object, which makes it perfectly suited to model this kind of uncertainty. An interval property can consequently be interpreted as a collective description of all possible probability density functions over the considered interval. For uncertain variabilities without objective information on the actual range, a subjective interval has to be chosen in order to apply the interval concept. For special cases, however, it might be possible to introduce a parametrical representation of the PDF* of an uncertain variability, using the unknown PDF* properties as parameters. For instance, it might be known that the quantity is normally distributed, but the mean value and variance of the distribution is not exactly known. A parametrical PDF* is introduced:

$f_X(x,m,\sigma) = \frac{1}{\sqrt{2 \pi} \sigma} \exp( - \frac{(x-m)^2}{2\sigma^2})$

(1)

with $m$ and $\sigma$ two parameters representing the mean value and standard deviation of the distribution. The interval or convex concept can then be applied to represent the uncertainty on the parameters of the probabilistic description. The resulting numerical description of the quantity takes into account all possible PDFs that can be generated using the limited probabilistic knowledge and added non-probabilistic uncertainty description. Therefore, it is called a hybrid model of uncertainty. Elishakoff et al. [51] introduced this hybrid concept for the uncertainty description by applying the concept of convex modelling on parameters in a probabilistic description of seismic excitation. He obtained the worst and best case response taking into consideration a large set of plausible probabilistic excitation functions through a single analysis.

The key idea of an alternative hybrid approach is to search for an upper and lower bound on the probabilistic reliability using the theory of Interval Probabilities. The method is based on the work of Ditlevsen [52] and has been successfully applied to describe bounds on the reliability of a system for which the interdependencies between the random quantities are unknown [53].

Finally, also the concept of Fuzzy Randomness is clearly a hybrid approach. It adopts principles from both probabilitstic and fuzzy theory. This concept allows impreciseness in the values that are assumed by a random variable. The fuzzy theory is used to represent this imreciseness. The translation of the classical properties of random numbers (e.g. mean value, variance) to this hybrid approach has been established in an early paper by Kwakernaak [49]. Recently, the application of this theory in civil engineering applications is gaining interest in literature [50].

 

Assessment of non-deterministic parameters in the engineering process

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.

 

Overview of stages in the engineering process

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.

 

Discussion of introduction of non-determinism 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.

 

Summary of decisions on non-determinism in successive steps of engineering analysis

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":

  • the specification of design loads or the specification of a probability of failure may be based on a statistical interpretation
  • product quality and control may lead to the establishment of quality distribution over the full range of nominally identical products
  • the decision to replace a component during its service life may be based on statistical information gathered previously with similar products

 

Application of non-probabilistic approaches in a design process

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.

 

Numerical non-determinism in a design process

The general objective of the application of numerical tools in a design process is to increase the design quality of an initial prototype by simulation of its realistic physical behaviour. While numerical analyses have proven to be very valuable in increasing design quality, an exact quantification of the design quality reached through the analysis is not always straightforward. This is mainly due to the fact that in any step of the design, there remains non-determinism in the numerical analysis. There generally is an evolution of the type of non-determinism encountered during a typical design process. Or as formulated by Ross [43]: As more information about a problem becomes available, the mathematical description of non-determinism can transform from one theory to the next in the characterization of the uncertainty as the uncertainty diminishes or, alternatively, as the information granularity increases and becomes specific.

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.

Typical occurrence of non-determinism in the product quality predictions during a design process

Figure 5: Typical occurrence of non-determinism in the product quality predictions during a design process

Individual model properties are modelled with fuzzy numbers in the initial design stages. After some design decisions are taken, the property is described by an interval with fixed bounds. As soon as production has started, actual values of that property exhibit a distribution.

 

Numerical reliability analysis

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.

According to its definition, reliability belongs clearly to the probabilistic framework in the frequentist context. On the one hand, this complicates probabilistic analysis of designs intended for limited production, since the fact that the product is only produced in limited quantity strongly complicates a decent a posteriori verification of the non-deterministic numerical predictions. Furthermore, for most designs intended for limited production, an unverifiably high reliability is requested (e.g. spacecraft). But an even bigger problem lies in probabilistic reliability analysis in the absence of trustworthy objective information. As discussed in section "Numerical concepts for non-deterministic numerical modelling", while applying the probabilistic concept for the representation of subjective information is possible, results from such an analysis should definitely not be interpreted as indication for an absolute frequency of occurrence. The subjectiveness devaluates the use of the probabilistic results in a reliability context. This subjectiveness of (parts of) the information is not always detected. For instance, neglecting unknown correlation between properties by assuming them as independent is a common simplification that is sometimes implicitly made, but that can have important consequences. This implicit assumption of independence between probabilistic quantities was one of the important errors that were the source of the Challenger space shuttle disaster [44]. In this case, the impact of different extreme weather conditions on the launch was analysed for each condition individually beforehand. The impact of a combination of more than one of these events, however, was never checked. Although each of the events had a very low probability of occurrence, the probability of their combination proved to be not simply a multiplication of the probability of the single events. The correlation between the conditions was clearly misjudged, leading to a plausible but unaccounted for weather situation with disastrous consequences.

The lack of credibility of numerical predictions of reliability is generally compensated by safety factors. However, one could argue that using these safety factors after applying sophisticated and computationally expensive numerical procedures is not a really economical situation. Much effort is spent on a numerical prediction, which, in the end, still has to be corrected based on practical experience. In this context, the non-probabilistic approaches could prove their value.

The application of the interval concept in numerical reliability studies is often referred to as anti-optimisation. This name stems from the fact that from all numerical models within the interval input boundaries, the one with the least favourable analysis result is the most interesting from reliability point of view. Finding this least favourable result is mathematically equivalent to performing a numerical optimisation aimed at the worst case result with respect to the input intervals.

The concept of anti-optimisation has been introduced as the basis for a non-probabilistic reliability framework [45]. This requires an evolution from a reliability concept as probability of failure towards range of acceptable behaviour. This means that the design must assure that the performance remains within an acceptable domain, without specifying a likelihood of failure. Reliability then becomes a crisp criterion distinguishing between either acceptable or unacceptable designs. The most important benefit of the anti-optimisation concept is that it broadens the objectivity of reliability studies to uncertain variabilities with known range, because the interval model perfectly represents these uncertainties without the need for subjective input. For instance, this enables a fast assessment of dimension tolerances on a design, without knowing the actual distribution of the dimension within the bounds of the prescribed tolerance. For some cases, it can be shown that the anti-optimisation procedure results in the same choice of design parameters as a probabilistic analysis if the required reliability tends to one [46]. The anti-optimisation in this case proves to be far less expensive in computation time.

The numerical implementation of the anti-optimisation approach is subject to an important requirement. Since the result of the analysis is the source of a crisp decision between acceptable and unacceptable designs, approximate results should always be kept on the safe side of the exact result. This means that if approximate solution procedures are used in the numerical implementation, they should guarantee conservatism in their result. On the other hand, this conservatism should not be excessively high in order for the result to be of any practical value.

Also the fuzzy concept can be usefully applied in a reliability framework to perform a possibilistic reliability analysis. In the interpretation of the membership function as a degree of possibility, the fuzzy outcome of an analysis could be used to define a possibility of failure. This possibility is clearly influenced by the subjectiveness that is implicitly incorporated in the fuzzy input of the analysis. This means that for the same problem, different analysts can and generally will end up with different possibilities of failure. This could be compensated by defining a personal threshold value for the allowed possibility of failure in the final decision on acceptable or unacceptable designs. However, due to the necessary amount of personal interpretation of the analyst, possibility of failure only has a relative value. Therefore, this approach is extremely difficult to standardise in a general reliability framework.

A different application of the fuzzy concept in reliability analysis is based on the use of the membership function as limit CDFs as explained in section "Numerical concepts for non-deterministic numerical modelling". It was shown by Ferrari et al. [47] that if the input membership functions represent boundaries on the CDFs of the input parameters, the membership function resulting from fuzzy analysis on this input forms reliable boundaries on the actual CDF* of the result. Therefore, the fuzzy result of a FFE analysis can be used to derive bounds on the probability of failure. A simple example illustrates this. Suppose that a FFE analysis results in a membership function $\mu_{bar \lambda}(\lambda)$ representing a crucial eigenfrequency of a design as illustrated in figure 6. Suppose furthermore that a crisp criterion states that the design is acceptable if this eigenfrequency is kept below the value $\lambda^{**}$. The fuzzy result envelopes the exact CDF* of the eigenfrequency. This means that the bounds on the probability that the eigenfrequency of the design lies below $\lambda^{**}$ can be derived from the fuzzy result. The probability interval is obtained from taking the value of the envelope curves at $\lambda^{**}$ as indicated in the figure by $ul P'_f$ and $bar P'_f$. The most conservative statement resulting from the analysis is that the probability of failure equals $(1 - ul P'_f)$ in the worst case.

Example of the application of the fuzzy outcome of a FFE analysis to predict bounds on the probability of failure

Figure 6: Example of the application of the fuzzy outcome of a FFE analysis to predict bounds on the probability of failure

It is clear that also the above non-probabilistic reliability methods are subject to the limitation that whenever there is subjective information involved in the problem definition, the results can not be interpreted as absolute measures of design quality. In an absolute reliability context, the amount of expert knowledge required in the distinction between a good or bad design is proportional to the amount of subjectiveness incorporated in the description of the non-determinism. Still, subjective analysis can be of great value when used in a relative framework, as for instance a design optimisation procedure.

 

Conclusion

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.

Next, it is shown that in the framework of numerical design analysis, there generally is an evolution of the type of the non-determinism from uncertainty towards variability. Correspondingly, the non-probabilistic approaches tend to be most valuable in early design stages, whereas the probabilistic approach remains indispensable in later stages. This leads to the conclusion that the non-probabilistic approaches should be regarded as complementary rather than competitive to the probabilistic approach. However, not only the class of the non-deterministic properties encountered in the problem definition, but also the intended output determines to what extend the different non-deterministic approaches are appropriate numerical modelling tools for the treated problem. It is discussed how the non-probabilistic approaches can be of value in a typical design process. From the discussion, it has become clear that the value of the non-probabilistic approaches in an absolute reliability analysis is rather limited. It is concluded that non-probabilistic approaches will fail to convince in areas where absolute reliability measures are primordial. However, the application of subjective probability in this context has the same limitations. Absolute reliability analysis should always be performed in a frequentist interpretation, based on objectively available data. A small assumption in the probabilistic description of the input can lead to large misjudgement of the actual reliability of the design. This should always be kept in mind when applying numerical methods for absolute design reliability predictions.

 

Acknowledgements

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".

 

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