Assessing Errors in Analysis Models

This article discusses errors in analysis and methods to reduce or quantify them. The approach described in the SAFESA series of documents, published by NAFEMS (Ref. R0039 , R0040 , R0041 ) attempts to formalise the measurement and treatment of error in analysis. This article gives an interpretation of some of the ideas behind this. While quantifying error is a laudable concept, it may not always be viable. But a consideration (if not a quantification) of errors should be a minimum requirement of any analysis quality system. Errors are inevitable in numerical analysis and experienced analysts have a ‘feel’ for the accuracy of a particular result.

Many analysts might argue that producing a result that is meaningful and sufficiently accurate does not require a formal quantification or error, just the supervision of an analyst with suitable experience. Attempting to quantify all errors is akin to quantifying professional experience, and can be difficult.

Nonetheless, even an informal consideration of the different types of errors that arise in numerical analysis can be a useful way of building confidence in results.

Uncertainty and Error

Many of the differences in the actual behaviour of a structure and that predicted by analysis are due to uncertainty in the physical description of the structure - due to natural variability in the loading, in environmental conditions, methods of manufacture or the operating regime. Uncertainties are often dealt with by specifying conservative loadings or material properties in codes of practice or standards. The analyst in these cases does not have to make further allowance for these uncertainties in the model.

A good example of uncertainty is wind loading on a building or structure. Codes of practice address the uncertainty in wind loading by prescribing a design wind speed based on conservative probabilistic considerations, so that the loads will account, with a high degree of confidence, for all wind conditions likely to be encountered by the structure during its life.

In this context uncertainties can be distinguished from modelling errors. In simple terms, this is a distinction between variability in the definition of the problem (uncertainty) and variability in the definition of the model created to represent it (error). (The analyst normally has full responsibility for the latter and perhaps some of the former.)

Modelling Error

Different categories of modelling approximation are listed below. These are grouped according to the SAFESA documents; but are not definitive. Some examples of sources of error are given for each category.

Mathematical Model

• How well the material formulation in the FE software represents the actual material.
  
• Approximations by representing a 3D structure with a 2D model.

Domain

• Geometric simplifications incorporated into the model e.g. the omission of small details.
  
• The extent of the model surrounding the area of interest.

Boundary Conditions

• Representing a reaction load as a displacement restraint.

Applying point loads or bolt loads

• Primitives (Elements)
  
• Elements have limitations on the behaviour that they can represent. This may not just be limited to the accuracy of the approximation of displacement or stress (for example) across an element but can also include an inability to represent some types of behaviour entirely. Examples in structural analysis include shear representation in certain types of shell elements and more obviously, beam elements not representing local stress concentrations for example, where a bracket might be attached or two beams are connected together.

The aim of the SAFESA approach is to put a figure on the maximum possible analysis error (i.e. bound the error). This can be achieved by first bounding individual errors such as those identified above, and then combining them to give a total maximum error value. It is usual in any analysis project to carry out several runs of improving accuracy and provide validating calculations. The method of quantifying individual errors in the SAFESA approach is achieved by extending and refining this process.

Analyses typically solve for a maximum value of a result such as stress, deflection or energy absorption, and compare the maximum value with an ‘allowable’ value of the same parameter. Safety factors on the input load or allowable value can be specified which account for the uncertainties in some or all of the data. Strictly speaking, however, the purpose of safety factors should not be to allow for errors in the analysis process. Instead, a total maximum error value obtained using the SAFESA or a similar approach, can be used to factor the results before comparing them with the allowable values.