Modelling Process for Engineering Simulation J M Smith Volume 2

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ďƒ“ Compusis 2013

Modelling Process for Engineering Simulation

CONTENTS 1. Engineering simulation 1.1 Introduction 1.2 Simulation project phases 1.2.1 Preparatory phase 1.2.2 Modelling phase 1.2.3 Assessment phase 1.3 Simulation planning 1.3.1 Problem description 1.3.2 Modelling plan 1.3.3 Assessment criteria 1.3.4 Category of risk 1.3.5 Verification and validation plan 2. Simulation process 2.1 Object domain 2.1.1 Object idealisation 2.2 Mathematical domain 2.2.1 Mathematical model 2.3 Computational domain 2.3.1 Computational model 2.4 Behavioural domain 3. Simulation error 3.1 Sources of error 3.1.1 Procedural error 3.1.2 Modelling error 3.1.3 Computation error 3.1.4 Prediction error 3.2 Error estimation 3.2.1 Exterior error estimation 3.2.2 Interior error estimation 3.2.3 Uncertainty estimation 4. Model verification and Validation 4.1 Model validation 4.1.1 Treatment of idealisation and data error 4.1.2 Consistency checking 4.2 Model verification 4.2.1 Treatment of approximation and calculation error 4.2.2 Software verification 4.3 Simulation model improvement cycle Bibliography

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1. ENGINEERING SIMULATION 1.1 Introduction The purpose of an engineering simulation is usually to represent the behavioural characteristics of an engineered object (or system) in order to predict its response to design loadings. Engineering simulation is an application of scientific simulation, which combines mathematical modelling, numerical methods and scientific computing (see Fig.1.1). Scientific simulation supports the scientific method (where scientific hypotheses are conjectured and tested for validity), by providing a virtual simulated environment for conjecture and experimentation.

Figure 1.1 Scientific simulation

Scientific simulation applications are prevalent in a wide range of fields, and engineering simulation is one such application, where: a) mathematical modelling - facilitates a mathematical description of the object of interest; b) numerical approximation - provide an algorithmic device for establishing and solving the system of equations that constitutes the model; c) scientific computing - provides the computational resources required to process the system of equations. The overall simulation process combines technical activities such as scientific, mathematical and computational tasks (necessary to simulate the engineered object) together with managerial activities such as planning, resourcing, and quality control (necessary to produce simulations of consistent quality). Within the simulation process however, opportunities for error inevitably arise which can lead to inaccurate predictions, erroneous assessments and potential in-use failure of the engineered object. However, systematic management can be employed to control the simulation process and minimise the opportunity for error. A prerequisite for achieving this is to first describe the simulation process and its sources of error, thereby enabling effective modelling, verification and validation procedures to be systematically developed and applied.

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1.2 Simulation project phases In this Primer it is assumed that the engineering simulation contributes to an assessment of an engineered object, that simulation errors and uncertainties are identified and treated, that the simulation is validated, and that confidence limits are established for critical results. In order to develop a quality management system for engineering simulation, it is first necessary to identify the simulation process and its sources of error. Engineering simulations are typically processed as projects which require the following inputs: a) clear and accurate definition of the problem; b) anticipated behaviour of the problem; c) criteria against which the results are to be assessed (the assessment criteria). The engineering simulation process involves several phases. First, the problem needs to be clearly defined and identified as amenable to simulation by computational methods. Then, assumptions need to be made to enable the problem to be idealised and then developed into a computational model, the ensuing set of equations solved, and the accuracy of the solution confirmed. Finally, model predictions need to be interpreted and evaluated in the context of the problem assessment criteria, conclusions need to be drawn, and the findings reported (see Fig. 1.2).

Input problem and assessment criteria

Engineering assumptions and conceptual mathematical idealisation

Problem diagnosis and simulation planning

Preparatory phase

yes

Are the results valid?

no

Approximate numerical model

Modelling phase

Assessment phase Assess results and report conclusions

Error and uncertainty assessment

Results processing

Computational solution

Figure 1.2 Phases of an engineering simulation project

This process can be formalised as a logical progression of three phases (see Fig. 1.2), which can be summarised as follows: a) Preparatory phase: i. the scope and objectives of the simulation are discussed and agreed with the customer; ii. the problem is described and diagnosed and a simulation approach planned; b) Modelling phase: iii. engineering assumptions are made and the problem is idealised as a conceptual mathematical model; iv. a discrete approximation of the mathematical model is created in software and solved computationally to an acceptable degree of accuracy;

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c) Assessment phase: v. the results are interpreted, and the validity of the simulation is assessed and established (unsuccessful validation may necessitate a return to an earlier phase): vi. the results are analysed and assessed in the context of the problem assessment criteria, conclusions are drawn, and a simulation report is produced.

1.2.1 Preparatory phase In the preparatory phase, the analyst and the customer discuss what the customer would like to achieve, how the organisation can assist, and agree on a problem description. The preparatory phase is the initial phase of the simulation process which needs to be undertaken before any detailed modelling begins. It involves defining and diagnosing the engineering problem, making initial engineering assumptions, and planning the simulation. A systematic approach to problem diagnosis is required to identify all realistic models and solution approaches in order to decide on the most suitable type of model and solution to be adopted. The overall nature of the systemâ€™s behaviour, domain, boundary conditions, loading and model parameters need to be considered, and potentially critical components and failure modes identified. The criteria by which the results will be assessed (the problem assessment criteria), how the simulation might be verified and validated, and the level of risk associated with the simulation, all need to be considered at this stage. A simulation plan and contract proposal (which includes a specification of the simulation) should be included in the output from the preparatory phase.

1.2.2 Modelling phase In the first stage of the modelling phase, an idealised model capable of serving as a basis for the subsequent computational model is developed from the problem description by making engineering assumptions. Sources of idealisation error introduced during the mathematical modelling phase will include the equations chosen to govern the behaviour of the model (e.g. the element and solution types), the domain of the model, the boundary conditions, and other model parameters. Idealisation error will be estimated and treated in the assessment phase. The second stage of the modelling phase focuses on transforming the idealised mathematical model into a discretised computational model that can be solved numerically, and it is at this stage that the model is actually constructed in software. The computational modelling process introduces numerical error through discretisation, truncation, and number round-off. Computational error will be estimated and treated in the assessment phase.

1.2.3 Assessment phase The assessment phase of the simulation process involves several activities: the model is verified and validated, the results are assessed in the context of the problem assessment criteria, conclusions are drawn, and a simulation report is prepared.

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In the first stage of the assessment phase, it must be demonstrated that both the idealised mathematical model and the discretised computational model are capable of producing sufficiently accurate and valid results to substantiate the problem assessment conclusions. Hence, model verification and validation must be carried out to confirm that the numerical approximation and calculation, and the idealisation and data assumptions, are valid with respect to their associated bounds and limits of applicability. Following successful verification and validation, results from the simulation can be compared with the allowable values stated in the problem assessment criteria. If the results fall within the allowable range, the assessment criteria can be deemed to have been satisfied, and the assessment conclusions can be drawn and published. Simulation reports must detail all critical results together with a discussion of their accuracy, and the verification and validation activities undertaken. Any queries regarding the validity of significant results must be answered by a credible physical explanation and no critical result should be accepted without assessment to ensure its reasonableness. Where validation is unsuccessful, it may be necessary to return to an earlier stage of the simulation process; for example, to revise the idealisation assumptions. Once the assessment phase and other activities have been satisfactorily completed, the simulation project can be concluded and contractual commitments settled. Finally, a comprehensive set of records from each stage of the process needs to be produced and archived to provide an audit trail and to demonstrate conformance with the applicable quality and regulatory standards.

1.3 Simulation planning The success of a simulation will be strongly influenced by the quality of its planning. Most problems allow more than one possible modelling approach and all feasible modelling alternatives should be considered. In practice, the range of modelling alternatives will be constrained by the type of results required, the problem assessment criteria, industrial norms, time and budget constraints, etc.

1.3.1 Problem description A clear definition of the scope of work is required to ensure that the simulation meets its technical and contractual objectives. The problem description defines the scope of the simulation and the engineered object to be represented, the criteria against which it will be assessed (the assessment criteria), the main problem variables (e.g. displacement, time, etc. - which will largely determine the type of simulation, e.g. structural, fluid, etc.), and initial planning of how the model will be validated. This process of describing the problem involves consideration of the end-use of the engineered object, how its predicted behaviour will be assessed, and the level of confidence required in the assessment conclusions. The scope of the simulation, assessment and validation need to be proportional to the time, resources and information available. Sources of error and uncertainty in the simulation need to be identified together with suitable methods of treatment, and documented in the problem description.

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1.3.2 Modelling plan A modelling plan should, as a minimum, specify the problemâ€™s governing equations (e.g. element types), main variables, boundary conditions, parameters, the solution type (e.g. direct, iterative, superposition), and an estimate of the resources required to process it. All possible loadings, modes of behaviour, and potentially critical components need to be identified, and all likely failure modes noted. The assessment criteria will have a strong influence on the type of load model, and hence on the type of simulation. For example, where bending moments are to be assessed, the use of centreline loadings on beam and shell elements is implied. Additional parameters to be computed (e.g. stresses, displacements, etc.) also need to be determined. The engineered object of the simulation can typically be decomposed into a hierarchy of components and assemblies - a feature that can be exploited to provide a systematic, hierarchical approach to simulation modelling. Individual components and assemblies can often be modelled, verified and validated concurrently as the model assembly proceeds, enabling these entities to be incorporated into the emerging complex model with greater confidence (see Fig. 1.3a). This hierarchical approach to modelling also has the advantage of isolating the component interfaces, which can then be verified and validated independently of the components themselves (e.g. see Fig. 1.3b). Complete model domain (system)

model sub-domain (sub-system)

model sub-domain (sub-system)

components

model component (part)

model component (part)

model component (part)

model component (part)

fabricated joints

Figure 1.3 a) Hierarchical modelling assembly, b) Component domains of a fabricated sub-assembly

In order to gain a better prior understanding of problem behaviour, the analyst should try to anticipate the principal results of the simulation. This can often be achieved by carrying out elementary calculations, applying formulae from handbooks, referencing experimental results, or by using alternative models to predict the anticipated response of the simulation model.

1.3.3 Assessment criteria Assessment criteria need to be identified and referenced to the relevant documents such as assessment standards and codes of practice. A category of risk must be defined for the assessment (see Appendix A). The assessment criteria are typically established by industrial practice and the potential consequences of failure, and will strongly influence how the simulation model is designed and developed. The assessment criteria can often suggest particular failure modes, the required level of modelling detail and simulation

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accuracy, the method of validation, and the level of confidence required in the results. Uncertainty is often allowed for in assessment criteria, for example through codes of practice and design standards (e.g. uncertainty in wind loading could be treated by reference to the relevant standards on wind loading).

1.3.4 Category of risk The organisation needs to agree with the customer, considering any safety and legal requirements, the degree of risk appropriate to the simulation, and the stringency of procedures appropriate to that degree of risk. NAFEMS QSS utilises a â€˜category of riskâ€™ concept whereby three categories of risk are established, enabling simulations to be classified as either Vital, Important or Advisory (see Appendix A). The stringency and qualification of the procedures, and the scope of applicability for each procedure used in the simulation, needs to support the simulation category of risk applied. These risk categories refer not only to the significance of the simulation results in the assessment of the engineered object, but also to the consequences of failure of the object with regard to risk to human life or the cost of malfunction. For example, the simulation of the flight behaviour of a commercial airliner implies a model of high accuracy and correspondingly high risk categorisation. The required level of training, qualification, and relevant experience of personnel involved in the simulation must also be appropriate to the assigned category of risk.

1.3.5 Verification and validation plan A major source of concern in simulation is the extent to which a model is valid and its predictions accurate. Any description will necessarily be approximate and modelling error will inevitably arise. Therefore, before any real confidence can be placed in a model, it needs to be shown to be fit for the purpose of describing the engineering problem of interest. The objective of validation is to confirm (preferably be some independent means) that the model, the solution and the results, are adequate for supporting the assessment conclusions. Therefore, it is essential, early in the modelling process, to consider how the simulation will be validated. In practice, the validation approach will largely be dictated by the assessment criteria, though it will also be strongly influenced by the availability and reliability of corroborative sources of data, such as that from experimental models, experience data, and alternative mathematical models. It is important to carefully consider and note (as far as possible) all the idealisation assumptions made, so that the consequences of those assumptions on the accuracy and validity of the results can be determined. It is necessary to be constantly vigilant for implicit assumptions which might inadvertently be made. Ultimately, the validity of the model will largely be determined by the quality of the underlying idealisation assumptions. The accuracy and extent of the validation may be influenced by the margin by which the engineered object is deemed to be fit for its intended use. For example, if critical results fall within the allowable range by a wide margin, then the validation requirements may be relaxed in comparison to a simulation that predicts critical results close to the allowable values.

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A strategy of falsification can be applied to assess a modelâ€™s validity, which attempts to refute a modelâ€™s validity rather than to confirm it. The advantage of a falsification approach is that only a single falsifying observation is required for refutation, no matter how many confirming observations have been made. Should a refuting observation be discovered, the cause of the refutation needs to be identified, and the model either repaired or discarded. At the conclusion of the model planning stage, a clear idea of the type of mathematical model required, a specification of the problem assessment criteria, and a plan for model validation should have emerged.

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2. SIMULATION PROCESS In the previous chapter, the simulation project was described in terms of a sequence of operational phases. In this chapter, the simulation will be described as a detailed process, suitable for systematic management in controlling the process and minimising the opportunity for error. Within the simulation modelling process, four distinct domains can be identified (see Fig. 2.1): a) object domain – comprising the engineered object and the parameters which determine its behaviour; b) mathematical domain – comprising the idealised model and data in mathematical form; c) computational domain – comprising the computational model and solution. d) behavioural domain – comprising the behavioural interpretation of the solution.

object

parameters

behaviour

model validation

idealisation

data

solution

model verification

computation approximation

calculation

converged?

discrete solution

computational domain mathematical domain object and behavioural domain

Figure 2.1 Flow diagram showing the simulation process and its domains

2.1 Object domain The object domain contains the engineered object of the simulation with its parameters, which together will determine the behaviour of the object. The engineered object might exist as a physical object in the real-world, or simply as a design concept. For the purposes of engineering simulation, the object and its parameters need

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to be described in a form amenable to engineering simulation, i.e. they need to be transformed into an idealised mathematical model with data.

2.1.1. Object idealisation There is no unique way to idealise an engineered object, and several different idealisations can potentially be developed for describing the same object or system (see Fig. 2.2). However, while one idealisation might be ‘better’ than another in some sense - for example, it might be more accurate - that advantage can easily be offset if inordinate amounts of human and computer resources are required to process it. In general, the success of an idealisation depends not only on the validity and accuracy of its predictions, but also on its demand on resources. Therefore, an idealisation should always be as simple as possible, without compromising its fitness for purpose.

Figure 2.2 Possible real-world and idealised representations of a structural joint

During the idealisation process, there are many opportunities for simplification. For example, in the case of symmetry in geometry and loading, only one section of the symmetric region needs to be idealised provided that appropriate symmetric boundary conditions and loadings are applied (see Fig. 2.3). Dimensional reduction is another potential source of domain simplification – for example, it may be possible to reduce a threedimensional domain to a two-dimensional domain without compromising it’s validity.

model domain

model domain

problem domain

Figure 2.3 Tensioned plate with hole, showing quarter symmetry domain reduction

Idealisation should be carried out in such a way that the opportunity for idealisation error is minimised, and it is good practice to begin by representing the idealised object in the simplest possible way and improving it gradually in stages. This iterative approach to idealisation can provide insights into the behaviour of each successive idealisation, and the interim findings can be used to contribute to the next improved idealisation level in the hierarchy (see Fig. 1.3). For example, a three-dimensional problem might initially be idealised as a one- or two-dimensional model (see Fig. 2.4), a time varying problem as non-time varying, and a non-linear problem as linear.

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A domain sub-division approach can be employed in the idealisation process to minimise uncertainty and error. This approach involves dividing the domain into a set of sub-domains with well-defined characteristics. For example, if there is some uncertainty about the joint properties of a fabricated object, the model could be assembled from a set of well-defined regions representing the fabricated components (e.g. see Fig. 1.3b). This modelling approach isolates the joints, enabling the uncertain joint properties to be systematically investigated using sensitivity analysis.

Figure 2.4 Plate and solid idealisation hierarchies of a channel section

2.2 Mathematical domain The mathematical domain consists of the mathematical model, its data, and its solution. These components are idealisations of the engineered object, its parameters, and its behaviour, stated in mathematical terms. The relationship between the model, data and solution can be expressed as: mathematical model Ë„ data â†’ solution which emphasises that the mathematical model, acted on by its data, produces the solution. It is instructive to state the model in this format, emphasising the specialising effect of data on the model. For models describing engineering behaviour, the analyst can usually draw on the body of equations belonging to classical mechanics (such as the equation of motion). Such models typically employ differential equations that describe the main variables in terms of rates of change. Due to the relative complexity of most engineering problems, the equations describing the mathematical model cannot usually be solved analytically (i.e. exactly). Therefore a computational model must be constructed to approximately represent the mathematical model, which can then be solved numerically (i.e. approximately). A mathematical model will have a definite purpose and scope, which needs to be defined at the outset of the modelling process. The model will also have a limited range of applicability and validity, and care must be taken to ensure that the model is not applied outside this range.

2.2.1 Mathematical model The mathematical modelling process begins by identifying the essential attributes and variables of the problem to be idealised. The behavioural relationships between those variables need to be identified so that, for the

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purposes of mathematical modelling, appropriate equations can be chosen (e.g. element and solution types) to describe those relationships. Model attributes are described in terms of: a) constants - attributes that have fixed values for all models, such as gravitational acceleration; b) parameters - attributes that have fixed values for particular models, such as material density; c) variables - attributes that have changing values, such as time. When selecting which problem attributes to represent in the idealised model, the impact of those attributes on the system behaviour needs to be considered. If it appears that a particular attribute does not significantly affect the behaviour of the model, it might be possible to exclude it, or at least relegate its importance (and its required level of accuracy). Often, the significance of a particular attribute cannot be reliably judged at the outset of modelling, but using a simplified model for pilot study, the impact of that attribute on the behaviour of the model can be evaluated by varying its value and observing the resulting change in the modelâ€™s behaviour (sensitivity analysis). Should an important attribute be inadvertently omitted, that should become apparent at the validation stage. At this stage of the modelling process, an idealised, conceptual mathematical model will have been conceived, together with a specification of its main variables, governing equations, boundary conditions and parametric data (e.g. loading). However, it is only at the computational modelling stage that the model is explicitly formulated as a system of equations and solved.

2.3 Computational domain Mathematical models of engineered objects are often described by differential equations which represent the behaviour of variables that express rates of change, such as strains and accelerations. Such mathematical models invariably assume spatial and temporal continuity, thus the domain and range of their variables will potentially comprise an infinite number of possible values. Other than in the simplest of models, no closed-form analytical solution of those continuous mathematical model equations will exist, therefore it is necessary to employ numerical methods (such as the finite element method) to approximate the mathematical model into a form that can be processed computationally. However, if numerical methods are used to approximate the mathematical model, any idealisation error in the mathematical model will be compounded by numerical error in the computational model.

2.3.1 Computational model As computational models are based on discrete mathematics, the continuous mathematical model must be discretised before it can be processed computationally. This will involve a transformation of the model equations from the domain of the mathematical continuum to the discrete numerical domain. The transformed model will then yield an approximate numerical solution consisting of a finite number of values. A computational model is typically formed from a matrix system of differential equations, and its corresponding numerical solution will consist of a finite sequence of values at discrete points approximating the exact solution function, the difference being due to numerical error.

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The size of a model in terms of the number of equations and corresponding matrix dimensions (degrees of freedom) will motivate some consideration of the computational resources required to process it, especially if the model is large. For example, a matrix inversion algorithm will execute more efficiently when processed in core memory, but a large amount of (expensive) physical memory may be required to achieve this. Computing resource requirements are problem dependent, and a suitable computer system will need to be made available for the task of processing the model (which should have been foreseen at the planning stage). In the commonly used displacement formulation of the finite element method, the raw solution output consists of discrete nodal displacements. A continuous field of results data can subsequently be derived from the discrete solution by averaging the results across the spatial domain of the model. The averaging process involves an extrapolation of computed values (e.g. extrapolation of gauss point stresses to the nodes), and while this often results in higher values, this needs to be confirmed and the differences quantified. Given a fine enough mesh however, the difference between the discrete and continuous results sets should not be significant. Indeed, evaluating this difference through an assessment of the adequacy of mesh discretisation is one of the core activities of the model verification process.

2.4 Behavioural domain In the behavioural domain, the computed results are interpreted as the predicted behaviour of the engineered object. It is shown in Fig. 2.1 that the behavioural and the object domains are related by the fact that the object is the input of the simulation, and its predicted behaviour is its output. However, it must be remembered that the idealised object and its predicted behaviour are only approximations of the actual object and its true behaviour. Indeed, how closely these are related depends on the quality of the simulation, which will be estimated by the verification and validation processes. If the predicted behaviour of the object exactly matches the true behaviour, then the domains will be coincident. In reality however, an exact match of the domains is impossible, the difference being due to simulation error (see Fig. 2.5).

actual object

valid object idealisation

valid object idealisation

solution

actual behavior

valid predicted behavior

invalid predicted behavior

Figure 2.5 Object and behavioural domains of the simulation

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3. SIMULATION ERROR 3.1 Sources of error Differences in behaviour between the object as it actually (or conceptually) exists, and the model’s predictions, are inevitable. This difference is herein referred to as error, and contributions to it can arise from several sources, which can broadly be categorised as: a) procedural error – mistakes made in simulation processing; b) modelling error – occurs in the mathematical idealisation and data (see Table 3.1); c) computation error – occurs in the solution approximation and calculation (see Table 3.2). d) prediction error – occurs in the interpretation and application of the solution. Error in the simulation process must always be evaluated, and this can be done in various ways. For example, through the application of experience, comparison with experimental test results, error estimation algorithms, simple calculations, sensitivity analysis, etc. In practice, exact measures of error will be unobtainable, though it is often possible to obtain estimates of error that can be used to guide error treatment and reduce the estimated error to acceptably low levels. Some error estimation techniques can be applied before modelling begins (such as simple calculations or the application of experience), whereas others require a results set to operate on and are therefore only applicable after a solution has been obtained (such as discretisation error estimators and sensitivity studies). Error assessment and treatment should be carried out in two ordered stages; numerical error treatment first, followed by idealisation error treatment. This is because in practice, the validity of the idealisation will often be evaluated on the basis of the computed solution, as it is typically the only results set available. Therefore it is necessary to ensure that the numerical solution is as accurate as possible before embarking on idealisation error assessment. Uncertainty Uncertainty arises where it is not possible to define a parameter’s value accurately, such as in environmental conditions, the quality of fabrication, or the magnitude of natural forces. For example, wind loads are highly variable, and the magnitude of earthquakes cannot be reliably predicted. Methods of treating uncertainty include the application of safety factors, experience, probabilistic methods to bound the uncertainty in statistical terms, and sensitivity analysis to determine the effect of an uncertain parameter on the overall results. Many uncertainties can be bounded through reference to codes of practice, for example, some building codes address uncertainty in wind loading through conservative estimates of maximum wind speeds. However, even where it is possible to identify and bound uncertainty, simulation error will still be present. The previously mentioned error sources: procedure, modelling, computation and prediction, will now be discussed.

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3.1.1 Procedural error To reduce procedural error (mistakes made at any stage of the simulation process), the simulation needs to be carried out in a controlled manner - such as within the constraints of a formalised quality management system, which employs a detailed set of procedures covering all the important processes and activities that contribute to the quality of the delivered simulation product. Indeed, the focus of this Primer is on the development, implementation and maintenance of such a system, and is discussed in detail in the latter part of this book.

3.1.2 Modelling error The simulation modelling process necessarily involves making assumptions, which will inevitably introduce idealisation and data error.

3.1.2.1 Idealisation error In the process of simulation modelling, idealisation error arises in assumptions about the behaviour, domain and boundary conditions of the engineered object, and in its subsequent description as an idealised mathematical model. Idealisation error can originate from several sources which can be categorised as follows: a) Governing equations The principle ingredient determining the fundamental behaviour of the model is its governing equations - those mathematical functions which describe the fundamental relationships between the main variables of the problem (e.g. forces and displacements). The governing equations embody the key behavioural assumptions of the model and its behaviour will both be determined and constrained by them. It is therefore, imperative that an appropriate choice of governing equations is made. For example, if time variation is considered significant, the problem might be modelled as a propagation problem employing the equation of motion as one of its governing equations. Using the finite element method, the temporal governing equations are introduced in the solution type, while the spatial governing equations are introduced in the element formulation (shape functions). Some behavioural assumptions involve dimensional reduction; for example, the assumption of plate theory reduces a three-dimensional problem to a two-dimensional problem by linearizing its through-thickness behaviour. This type of reduction is often used in engineering simulation where bending behaviour is of interest, and represents a common source of idealisation error. Several different behavioural assumptions can be applied over disparate regions of a domain, for example, plate theory might be used to model the system in some regions, whereas three-dimensional solid theory might be more appropriate in others. The interface at such behaviourally dissimilar regions is a potential source of error which needs to be considered. b) Domain space

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The domain space represents the spatial and temporal extent of the system being modelled. In the majority of models, the idealised domain represents a truncated portion of the actual domain, therefore conditions at the boundaries of the truncated domain need to be carefully considered as they can contribute a significant source of idealisation error. Where the behaviour at a boundary is not well-defined, or where boundary conditions have to be assumed, the potential for idealisation error can be significant. In other cases, for example where the eliminated portion of a domain is symmetric to the modelled portion, appropriate symmetry boundary conditions can be applied and domain truncation error avoided. Clearly, where a model domain is complete, there will be no domain truncation error. Another type of domain idealisation that can lead to significant error is the elimination, or simplification, of domain details. For example, a geometrical feature might be ignored, simplified, or 'smeared' across a region; or a smoothed intersection might be approximated by a sharp-edged intersection which could introduce a discontinuity and possibly a singularity. The approximation of a smooth curved surface by a faceted representation can introduce similarly spurious results. c) Boundary conditions The type of governing equation, together with its associated domain extent, will limit the range of boundary conditions that can be applied. For example, an idealisation based on plate theory will require boundary conditions having degrees of freedom compatible with plate theory, i.e. mid-plane rotations â€“ and a structural dynamics propagation model can support dashpot boundary conditions, whereas a steady-state model cannot. Where a domain is reduced on the basis of symmetry, symmetric boundary conditions will need to be applied. In some cases, the domain of the model may need to be extended in order for its boundary conditions to become sufficiently definable. d) Constitutive relations This refers to fundamental relationships within the model such as the stress-strain law, which can be linear or non-linear. While actual problem behaviour may be distinctly non-linear, many problems can be assessed using a linear solution provided that the assessment criteria contains sufficient conservatism to justify this approach.

3.1.2.2 Data error Having determined which variables to include in the model, and which governing equations best describe their relationships, values for the associated constants and parameters need to be decided. In engineering problems, model parameters typically describe material properties such as elasticity or density. These values are normally acquired from experimental data, the variability of which can be a source of error. Such data is typically acquired from measurements and observations associated with units of measure, which must be identified and consistently adhered to throughout the simulation. In general, engineering materials are reasonably well-defined, but the idealisation of advanced composite materials can be challenging. When modelling natural materials there will always be some uncertainty, and industrial codes of practice often provide guidance on their idealisation.

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The required accuracy of the data will depend on the required accuracy of the simulation, which will be indicated by the applied category of risk. The sensitivity of the model to specific input data must be considered: data which has little impact on the results can be of lower accuracy than data to which the model is more sensitive - though all data must be sufficiently accurate to support the problem assessment criteria and the assigned category of risk. Where data is unobtainable, estimates may be made, but their accuracy must fall within the bounds determined by the sensitivity of the model to that data. Finally, it must be ensured that any supplementary operations on data are meaningful in the context of the model, and that they are controlled and verified. Prior to computing the numerical solution, the model data needs to be checked for completeness and to verify that its accuracy is consistent with the accuracy required for the engineering assessment. Model data in the softwareâ€™s database should be checked by retrieving it from the database and comparing it with the original source data.

3.1.3 Computation error The process of approximating an infinite continuous mathematical model by a finite numerical model leads to the introduction of numerical error (as distinct from idealisation and parametric data error which arises in the modelling assumptions). Computation error arises from two main sources â€“ discretisation (and/or truncation) error, and calculation error â€“ both of which can accumulate to an appreciable amount, and which will compound any idealisation error that is already present.

3.1.3.1 Approximation error Approximation error arises from domain discretisation and series truncation. a) Spatial discretisation error Discretising the spatial domain of the mathematical continuum introduces discretisation error which, in the commonly used displacement formulation of the finite element method, leads to an underestimate of the predicted values. In the limit however, the finite element solution should converge to the analytical solution of the idealised mathematical model as the number of elements in the spatial domain is increased (see Fig. 3.1).

Stress concentration factor = 2.77

Stress concentration factor = 2.97

Figure 3.1 Effect of spatial discretisation on the accuracy of a stress concentration

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Many commercial mathematics and simulation programs have built-in algorithms which can be invoked to estimate and adaptively minimise spatial discretisation error. However, it should be noted that these adaptive methods are essentially self-referencing techniques and that their potential for improving the validity of the overall model is limited. b) Temporal discretisation error In time-dependent problems, numerical integration of the equation of motion can contribute a significant source of numerical error. Where a time-stepping solution is used, the temporal domain is discretised into a finite number of time-steps and numerical integration is carried out at each time-step to establish an approximate transient solution. This process introduces temporal discretisation error (see Fig. 3.2). c) Series truncation error Series truncation error can arise, for example, where an eigenseries is truncated and a reduced set of modes is used in a mode-superposition solution. This can adversely affect the accuracy of the solution since the truncation will result in the spatial distribution of the loading being approximated, and its frequency content curtailed.

Stress (N/m2)

transient stress at a point

Time (sec) Full solution Truncated modal solution

Figure 3.2 Effect of temporal discretisation and modal truncation on the accuracy of a time-history response

3.1.3.2 Calculation error Other significant sources of numerical error can arise in the calculation of the solution, arising from iteration, number round-off, and bugs in software. Iteration error arises, for example, in the computation of a nonlinear solution, where an unstable solution may exhibit error growth and divergence. Round-off error inevitably arises in finite-precision computation. In modern, long word-length computers however, round-off error is not usually a significant source of error, though care needs to be taken to avoid illconditioned equations which can magnify small round-off errors to significant proportions.

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Modelling Process for Engineering Simulation

Arithmetical errors can occur from unsound mathematical operations or bugs in software. In the latter case, any software errors detected should be reported to the software supplier for repair, during which time the software should be quarantined and an alternative means of computation found. In practice, â€˜work-aroundsâ€™ (which need to be verified) are usually found by the analyst or provided by the software vendor. Software suppliers often classify software errors on the basis of their obviousness and potential consequences.

Table 3.1 Modelling error sources

Table 3.2 Computation error sources

idealisation

approximation

governing equations (e.g. element or solution type)

discretisation (e.g. spatial, temporal)

domain space (e.g. geometric extent)

truncation (e.g. eigenseries)

boundary conditions (e.g. fixed/free) constitutive relations (e.g. material model)

data

calculation

loads (e.g. magnitude, direction, distribution)

arithmetical (e.g. mistakes, bugs)

coefficients (e.g. material or geometric values)

numerical (e.g. round-off, iteration)

mathematical modelling error sources

computational modelling error sources

idealisation

data

approximation

calculation

governing equations

loads

discretisation

arithmetical

domain

coefficients

truncation

numerical

boundary conditiions

constitutive relations

Sources of computation error

Sources of modelling error

3.1.4 Prediction error Finally, error in the interpretation the computed results for the purpose of predicting the behaviour of the object, and in the application of those predictions, is considered.

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It is imperative that the results generated by the simulation are as reliable as possible, and it needs to be demonstrated that a model is fit for its intended purpose of supporting the assessment conclusions, i.e. the validity of the model and the accuracy of its predictions needs to be established (e.g. by comparing the simulation predictions with observations of real-world behaviour). For a model to be deemed valid, it must, with respect to accepted bounds, accurately replicate the relevant behavioural characteristics of the engineered object. The validated predictions then need to assessed in the context of the problem assessment criteria and assessment conclusions on the fitness for purpose of the object drawn. Most results will need to be post-processed to convert them into a form suitable for verification, validation and assessment, and these operations can constitute a source of error. Therefore, assessments which cannot be carried out on the raw results data directly (i.e. prior to graphical interpretation or mathematical manipulation), need to be checked to ensure that the underlying data operations are accurate and that the manipulated data is meaningful in terms of its end-use. In any event, no significant result should be accepted without a critical assessment to ensure its reasonableness. If possible, confidence bounds should be placed on all critical results. It must be appreciated that the predicted behaviour of the object, based on the simulation results, will be limited in terms of its applicability, and care must be taken to ensure that predictions are not inadvertently applied outside the range for which they were intended and/or validated. Applying a prediction beyond the range of its validation is inference, and will require additional justification. Prediction and application techniques can, and should, be proceduralised within the framework of a quality management system (see later chapters).

3.2 Error estimation The application and effectiveness of any technique used for reducing error in the simulation is problem dependent, thus it is not possible to present a common approach covering all cases. Error estimation and treatment needs to be customised and the analyst must decide the most appropriate techniques to use in each particular case. In general however, error estimation methods can be grouped into two categories: exterior or interior methods.

3.2.1 Exterior error estimation Exterior methods such as those employing experience, experimental data and alternative calculations, exploit information extracted from the real-world and can provide an objective estimation of error. Three such exterior error estimation methods are: a) Experimental data â€“ provides a direct and truly independent estimate of idealisation error and is intuitively attractive. Results from experimental models are often used for comparison with computational results and an experimental test can often overcome the independence problems of comparative alternative simulation models and the difficulties of comparison with experience data. Reliable test data provides independent assurance that the idealisation assumptions are appropriate, and can also indicate idealisation error bounds.

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Modelling Process for Engineering Simulation

However, experimental testing is likely to be the most expensive means of testing as experimental models are often purpose built, and may have the longest lead time or simply may not be practicable. Experimental tests also introduce experimental error, so test results must also be validated with respect to the actual behaviour of the real-world system being tested. b) Experience data – which is typically available in the form of historical data, previous assessments, design codes, the analyst’s own experience, etc., and is the most widely used basis from which to validate an idealised model. Clearly, the main difficulty with this type of validation data is matching it to the present simulation context. Therefore, when appealing to experience, extreme care is required in judging its closeness and relevance to the present situation, and whether the experience data is accurate. Furthermore, the reliability of the data may be in doubt, or it may be lacking in detail. Analysts usually rely on their intuition in deciding whether one problem is sufficiently close to the other, and past experience in results assessment is invaluable for developing this intuition. c) Alternative calculations – provide insights into the system’s behaviour and are often used to augment experience-based judgement. They are often used in practice and are developed independently by analysts who repeat the simulation using alternative models and simulation software. Some advantages of this approach are its convenience and ease of results comparison. A disadvantage is the questionable independence of the software and analysts; ostensibly different software may be based on the same source code and may share common errors. Also, as analyst training becomes standardised in educational establishments, many analysts, due to their similar training, may tend to follow the same modelling approach. Finally, it should be noted that alternative simulation models are generally considered a weaker basis for validation than independent experimental models.

3.2.2 Interior error estimation Interior methods exploit information contained within the model itself and thus cannot be considered entirely objective. Nevertheless they have an important role to play. Interior methods can be used to extract information from a particular model or from a sequence of models. Two interior methods are described below, presented in the logical sequence in which the analyst might apply them. a) Hierarchical modelling – develops a sequence of models, each of which increases the analyst’s understanding of the problem. It involves representing a system at one level of idealisation and then representing the same system again at a hierarchically improved level, for example, analysing in twodimensions first, and then reanalysing in three-dimensions (e.g. see Fig. 2.6). Hence, starting from a simple model, an adequately improved model should eventually emerge from the modelling sequence, each model guiding development to the next. This adaptive approach accumulates confidence in the final model and invariably requires less time overall than if a fully detailed model was constructed at the outset, only to discover that it is erroneous, due to some aspect of its behaviour not being foreseen.

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Modelling Process for Engineering Simulation

b) Sensitivity analysis â€“ can be used to indicate the influence of a model parameter on the accuracy of the results. This information can then be applied to improve the model. For example, sensitivity studies might be used to investigate the effect of changing a boundary condition. Formal sensitivity analysis capabilities are intrinsic to some simulation programs, and will automatically calculate the effect of a small change to an input parameter on a resulting output parameter. Sensitivity analysis should be performed after the model improvement process has reduced the numerical error to an insignificant level, as the technique assumes that the model is of sufficient accuracy that first-order derivative information (the rate of change of the parameter) can be used reliably. In summary, interior error estimation techniques rely on data which is self-generated by the model, but nevertheless, provided that the underlying modelling assumptions are valid, a sequence of improved models based on interior error estimates should improve its accuracy. Conversely, exterior error estimation methods, such as the application of experience data, experimental testing and scoping calculations, exploit independent and objective information extracted directly from the real-world.

3.2.3 Uncertainty estimation Sources of uncertainty in the simulation need to be identified together with suitable methods of treatment, for example: a) use of experience, expert judgement, peer review, existing test data or direct testing to bound the uncertainty; b) use of probability methods to quantify the uncertainty in statistical terms; c) use of sensitivity studies to analyse the uncertainty using numerical or experimental methods; d) modelling to a lower level of detail whereby the source of the uncertainty can be eliminated.

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Modelling Process for Engineering Simulation

4. MODEL VERIFICATION AND VALIDATION In the previous two chapters, sources of simulation error were introduced and categorised, and a range of techniques for estimating and reducing the simulation error were discussed. It will now be shown, within the formal processes of model verification and validation, how these error estimation methods can be used to evaluate both the mathematical model, where the sources of error are categorised in terms of idealisation, data and solution (i.e. the verified computational solution), and the computational model, where the sources of error are categorised in terms of approximation and calculation. Model validation is characterised by the treatment of mathematical modelling error and can be defined as: the process of determining the degree to which the simulation model is an accurate representation of the object from the perspective of its intended use. The process of validation involves assessing and treating errors arising in the mathematical model (idealisation and data sources) which can be estimated from the degree of correspondence between the predictions of the mathematical model and an external corroborative system (e.g. a real-world or conceptual counterpart). Hence, model validation challenges the mathematical model. Model verification is characterised by the treatment of computational error and can be defined as: the process of determining the degree to which the computational model is an accurate representation of the underlying mathematical model and itâ€™s solution. The process of verification involves assessing and treating the error arising within the computational model (approximation and calculation sources) which can be estimated by determining the degree of internal coherence in the computation and by comparison with an analytical solution. Hence, model verification challenges the computational model.

4.1 Model validation It must be demonstrated that the simulation model is capable of producing results that are valid for the purpose of supporting the problem assessment conclusions. Model validity can be assessed through a process of estimating the error and uncertainty in the model using techniques such as comparison with experimental test or referenced results, application of experience, alternative calculations and sensitivity analysis. Prior to embarking upon the model validation process, it must be ensured that the computed solution does not contain significant numerical error, otherwise it will not be clear whether the overall error has arisen from the idealisation or from the numerical approximation, as the error in its totality will be a complex combination from the two sources. Hence, numerical error must first be minimised before any meaningful assessment of idealisation error can be carried out. Thus, error estimation and treatment must be applied initially to reduce

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Modelling Process for Engineering Simulation

the numerical error (the main sources being approximation and calculation) to an acceptable level before being applied to treat the idealisation error (the main sources being idealisation and parametric data). Validation will ideally involve a comparison of the modelâ€™s predicted values with those from an independent source such as experimentation and/or observation. Since the corroborative system needs to capture the relevant physics of the problem, experimental corroborative systems are preferred over experiential, analytical or computational alternatives. However, even experimental models may fail to capture all the relevant physics, for example, due to the presence of unanticipated, and therefore unmeasured, loading and failure modes. Experimental procedure errors (e.g. metrological mistakes) and scientific reduction errors tend to occur randomly but diminish as the experimentation cycle matures. In practice however, it is likely that there will be no real-world corroborative counterpart with which to compare the mathematical model. In such cases, the model validation may be limited to a consideration of the idealisation and data assumptions, together with the verified computational solution. The idealisation and data assumptions inherent in the mathematical model, comprising the governing equations, domain, boundary conditions, constitutive equations and parametric data, need to be reviewed to ensure that they can adequately represent the true behaviour of the engineered object. Assessing the degree of model validity is by no means an exact science, and often rests on judgement based largely on an assessment of modelling and computation error. Where the degree of model validity is unacceptably low, it may be necessary to revise the modelling assumptions. If model validity cannot be improved upon in this way, exploratory tests may need to be carried out to obtain more information and a better understanding of the system and/or model behaviour. In any event, model validation needs to be achieved at an acceptable cost.

4.1.1 Treatment of idealisation and data error Idealisation and data error arises within the mathematical modelling process and can be investigated as follows: a) Governing equations - limit the modelâ€™s range of possible modes of behaviour. If the equations (e.g. the element shape functions) fail to adequately capture the true behaviour of the engineered object, then the resulting model will invariably be invalid and its predictions erroneous. Analysts often ground their assessment in experience-based judgement supported by simple calculations which, for example, could give an indication of the relative importance of time-dependent behaviour and whether to include the equation of motion. To similar effect, differences in behaviour due to selected element types could be investigated using hierarchical modelling (e.g. using elements based on alternative shape functions). b) Domain - simplification can give rise to appreciable error, for example, where the spatial domain is prematurely truncated and the imposed boundary conditions artificially distort behaviour within the domain. Where domain simplification is used, sensitivity studies can help to determine the magnitude of the resulting error. c) Boundary conditions - are often difficult to abstract from a real-world situation. In addition, there is likely to be a limited range of options available in the simulation software for representing boundary conditions

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Modelling Process for Engineering Simulation

(typically, fixed or free). As a result, boundary conditions are often oversimplified in the model. Moreover, the modelâ€™s behaviour is usually sensitive to boundary conditions, and in some cases minor changes to these conditions can have a major impact on the computed results. To assess the impact of the boundary conditions on the solution, sensitivity studies can be used to gain an insight into the effects of adjusting them. Although there are some situations in which direct examination of the results can indicate the validity or otherwise of a boundary condition or loading idealisation, most situations rely on the application of experience, simple calculations, hierarchical modelling and sensitivity analysis. d) Constitutive relations â€“ refer to fundamental constituent relationships such as stress-strain, which can be linear or nonlinear. It is necessary to confirm that an appropriate and sufficiently accurate constitutive model has been defined, and this can often be achieved by reconstituting the constitutive relations from the results. e) Parametric data values - are typically obtained from the results of experimental tests and can be of varying quality. In some cases, sensitivity studies can be used to determine what effect a particular parameter has on the solution, thereby providing an estimate of the effect that any variability in the parameterâ€™s value might have on the solution accuracy. A posteriori consistency checks can also be useful.

4.1.2 Consistency checking Consistency checks are made to ensure that the solution is consistent with the idealisation and data assumptions. Such checks might include a comparison of reaction forces with those applied, and the compatibility of the results with the nature of the simulation: e.g. rotations not exceeding the limits of small deflection theory, and stresses not exceeding their elastic limit in a linear analysis.

4.2 Model verification While convergence of a sequence of approximate solutions is algorithmic in nature (e.g. mesh enrichment), convergence of calculation error (i.e. from software and process) could be described as evolutionary (i.e. as mistakes are rectified the errors reduce, but not in any predictable sense). In practice however, given experienced analysts using mature software, the solution error is more likely to be dominated by numerical approximation exhibiting algorithmic convergence.

4.2.1 Treatment of approximation and calculation error Approximation and calculation error can arise within the computational modelling process as the continuous mathematical model is transformed into a discrete computational model, and can be investigated as follows: a) Spatial discretisation error - can be estimated from field discontinuities between adjacent elements to indicate whether a discretisation is acceptable, or whether the domain needs to be enriched with more elements, thereby reducing the spatial discretisation error (see Fig. 3.1). Spatial discretisation error can be estimated in practice by manually halving the discretisation interval and recalculating the results to give an indication of the level of results accuracy and the rate of convergence. Alternatively, mathematically based numerical error estimators can be used which are often intrinsic to

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simulation software. However, even with the aid of sophisticated error estimators, it can still be difficult to objectively assess the numerical error in any given model as several error estimation techniques may be applicable, each having a distinct theoretical basis and quantifying the various sources of error in a unique way. In addition, the results convergence rates are likely to be different in disparate domains, e.g. spatial discretisation error in a particular model may converge towards zero at a different rate to say, temporal discretisation error; or the spatial discretisation error may have a modest impact on a particular result, whereas the temporal discretisation error may have a more marked effect. Thus, even with the aid of sophisticated mathematical error estimation techniques, error estimation remains something of an art. b) Temporal discretisation error - can again be estimated by manually halving the discretisation interval and recalculating the results to give an indication of the level of results accuracy and rate of convergence. Where an unconditionally stable implicit time integration scheme (such as the Newmark method) is used to solve the equation of motion, the solution will always be numerically stable whatever the time-step size, though a coarse discretisation can result in period elongation and/or amplitude decay. However, when using implicit time integration, such temporal discretisation error can be predictably controlled using a time-step size which is a small enough fraction (e.g. 1/20) of the period of the highest natural frequency of interest. c) Truncation error â€“ can be estimated for example, in a truncated-eigenseries, base-excitation response spectrum analysis, by quantifying the error in the load distribution based on the percentage of participating mass. The mode-coefficient ratio gives another measure of error due to eigenseries truncation, but this time due to curtailment of the modelâ€™s frequency range. Alternatively, manually increasing the number of modes in the superposition and re-computing the results can give an indication of results accuracy, but in this type of simulation, convergence is unlikely to be monotonic. d) Iteration error â€“ can be controlled by specifying an appropriate number of sufficiently small solution steps (thereby minimising the opportunity for solution instability and divergence) and setting acceptable convergence bounds for the solution (with due consideration to computational cost).

4.2.2 Software verification The software verification process is essentially the same as the model verification process since, within the scope of its testing, software can be deemed verified if it can accurately represent the mathematical model of interest and its solution.

4.3 Simulation model improvement cycle The sources of simulation error previously introduced and discussed can be summarised as: a) process error consists of mistakes made during the simulation process, and can be controlled using simulation procedures covering, but not limited to, modelling, verification and validation activities. b) modelling error consists of idealisation and parametric data error, which are formally treated by model validation procedures, based on a comparison between the predicted behaviour of the simulation and a corroborative counterpart.

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Modelling Process for Engineering Simulation

c) computation error consists of numerical approximation and calculation error, which are formally treated by model verification procedures, based on a comparison of the computational and mathematical models. d) prediction error consists of mistakes made during the interpretation and application of the computed results in predicting and assessing the behaviour of the engineered object. Prediction error is controlled using appropriate simulation procedures. These sources simulation error are shown schematically in Fig. 4.1 as a simulation cycle (and more generally in Fig. 4.2). From the figures it can be seen that the simulation cycle consists of: a) input domains: engineered object; mathematical model; computational model; b) output domains: discrete solution; continuous solution; predicted behaviour; c) processes: calculation; verification; validation; This schematic illustrates how the quality of the simulation can be improved in a cyclical manner as follows: 1. The simulation object of interest and its parametric data are described in mathematical terms, entailing the mathematical model. 2. The mathematical model, consisting of idealisation and data assumptions about the engineered object and its parameters, is discretised, entailing the computational model. 3. The solution of the computational model is processed and its approximation and calculation error is estimated. Where the numerical accuracy of the discrete solution is insufficient, the source of the approximation and calculation error is treated, and the improved computational model re-solved. This iterative numerical improvement process continues until the approximation and calculation error falls within accepted bounds for numerical solution convergence. 4. The discrete solution is extrapolated to produce the continuous solution for comparison against a simplified mathematical model (e.g. Roark, Blevins, etc.), which represents the solution parameters of interest (e.g. peak displacement or stress). Should the two solutions not correspond sufficiently closely, then the reasons for th lack of correlation are sought and corrected, either within the mathematical model or the computational model, or both. Steps 3 and 4 constitute the model verification process. 5. Once acceptable correlation has been achieved between the mathematical model and computational model, the solution can be interpreted in terms of the objectâ€™s behaviour. The behavioural parameters of interest are interpreted and compared against the observed real-world behaviour of the object. If the predicted behaviour corresponds sufficiently closely to the observed behaviour, then the simulation model can be deemed valid within the scope of that comparison. 6. Where no real-world corroborative counterpart exists for comparison, the validation may need to be based on an experience based assessment of the verified computational solution supplemented by consistency and other checking of the governing equations, domain, boundary conditions, constitutive equations and data, to ensure that they represent the behaviour of the object as intended. Steps 5 and 6 constitute the model validation process.

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Modelling Process for Engineering Simulation

engineered object

mathematical model

computational model

model input approximation

idealisation & data

verification (computation error)

validation (modelling error)

extrapolation

interpretation

predicted behaviour

calculation

continuous solution

discrete solution

Object domain: Object: Parameters: Behaviour:

The object of the simulation (real-world or conceptual) Parametric data which affects the object behaviour (e.g. loading) The behaviour of the object with parametric data

Mathematical domain: Idealisation: Data:

An idealised mathematical description of the object Parametric data values

Computational domain: Approximation: Calculation:

The discretised mathematical model The solution process

Discrete solution domain: Displacements: Stresses:

Nodal displacements Gauss point stresses

Continuous solution domain: Displacements: Interpolated displacements Stresses: Averaged extrapolated stresses Behavioural domain: Assessment:

Prediction and assessment of the object behaviour

Figure 4.1 Simulation process flow diagram schematising a model improvement cycle through verification and validation activities

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