A model approach - mathematical modelling

Materials World magazine
5 Nov 2011
Geometry of the bench supporting a coordinate measuring machine

The National Physical Laboratory’s Louise Wright explains how mathematical modelling can simulate the way products will respond to different environments before they are built, helping manufacturers improve design efficiency and avoid mistakes.

Competition from countries that produce cheaper products has meant many Western countries are increasingly moving towards more advanced manufacturing projects. The UK government is committed to expanding advanced manufacturing capabilities, which means smaller, more efficient or bespoke products, and those designed for extreme conditions or hard-to-access places.

When investing large amounts of money in developing increasingly complex products, we need to know they will work. Building a product, testing it, then rebuilding it is at best inefficient and at worst impossible. And if you’re making a product that operates deep within the ocean, each profile can hardly be tested in-situ.

Answering the question

In response to these challenges, and to make things easier for the ever-advancing manufacturing industry, a large body of expertise has developed on mathematical modelling, thanks partly to advances in computer programmes. Models can use information on materials and design plans to model how a product will behave under different conditions and identify potential pitfalls before it is built.

Mathematical modelling involves simulating the manufacturing process or the conditions the product will see during use. This allows manufacturers to identify at the design stage key issues that need to be considered, such as what materials to use and the correct geometries. This reduces the need to repeatedly design, build, test and redesign.

A second benefit of modelling is that it provides an easy way to explore the effects of uncertainty on a design. Every manufacturing process has uncertainties associated with it, for example materials exhibit variability in their properties and parts are machined to comply with a specified tolerance. These sources of uncertainty mean the final product has variability. A model allows the designer to check the product will still be able to perform as required, in spite of the inherent variability of the manufacturing process.

The National Physical Laboratory (NPL) has become a world leader in techniques for mathematical modelling. Thanks to a combination of measurement expertise and government-funded maths and software programmes, NPL’s scientists have developed understanding of uncertainties, and the requisite techniques for uncertainty analysis. Thanks to these government programmes, this research and expertise is available to the advanced manufacturing industry through the laboratory.

Materials modelling employs a variety of techniques, the most widely used of which is Finite Element Analysis (FEA). This numerical technique finds approximate solutions of partial differential equations and integral equations. It is particularly useful in the design of new products and to investigate failures in existing components.

The perfect combination

FE models can simulate the deformation behaviour of materials and stress/strain concentrations and can allow the user to assess the effect of geometric or material changes. A key factor in obtaining accurate results is the definition of the material behaviour. The material properties must be derived from precise measurement techniques, and the choice of material model must be based on a sound understanding of the material of interest. NPL’s combination of materials scientists, metrologists and mathematicians means it can obtain suitable data, select an appropriate model, and validate FE models’ results against experiments.

By using a range of modelling techniques, supported by a clear description of the material’s properties, different conditions can be simulated and modelled. Examples of different situations that can be modelled include –

  • Heat flow, both transient and static, which includes nonlinear materials, conduction, cavity and farfield radiation.
  • Stress analysis, both static and dynamic, for nonlinear materials, contact problems, and damage simulation.
  • Vibration analysis, including determination of natural frequencies and behaviour under driven vibrational load.

NPL also works in other areas, including acoustics, electrical charge transport and mass transport, and is developing new ways to simulate challenging problems that involve nonlinearities, discontinuities and singularities.

Examples of modelling work NPL has carried out include applying inverse methods to obtain the thermal conductivity of the components of a layered sample, for example a piece of a turbine blade with a thin coating. The properties of individual layers of such a material cannot always be measured directly, since suitable samples of a single material cannot always be obtained – usually because they are too fragile or they are oxide layers that form in-situ. However, it is important to measure thermal conductivity of these layers to understand temperature distribution when the product is in use, since thermal effects can determine the efficiency and lifetime of a product, or even whether it works correctly.

Inverse methods involve systematically altering different variables within a model to reach an ideal result. A model was constructed of the laser flash thermal diffusity experiment, which requires materials to be homogeneous for direct determination of thermal conductivity. The model simulated heat transport in a layered sample, and an optimisation algorithm was used to vary the thermal conductivity of one layer and find the value that gave the best fit of model results to measured values. The value obtained agreed well with values for similar materials given in published literature.

Another project involved FE modelling to help with the design of a bench to support a coordinate measuring machine. The accuracy of the measurement results depends on the metrologist knowing the location of the machine when the measurement is made, and this can be affected by the deflection of the bench under load. The model enabled the metrologist to ensure that the proposed bench design would have minimal deflection prior to commissioning its manufacture, which saved time and money that would have been spent on repeated prototyping.

A third example concerns the monitoring of a footbridge. As part of ongoing work on structural health monitoring, NPL is making long-term measurements, using a variety of sensors, on a footbridge under various loading scenarios. An FE model has been used to predict the deformation of the bridge under load (static and dynamic), enabling the metrologist to place the measurement sensors in positions that will generate the most useful data.

Other projects have involved FE simulation of a calibration experiment for a high-temperature thermometer to reduce measurement uncertainty; boundary element propagation of acoustic waves to understand hydrophone (a device which picks up acoustic energy underwater) performance beyond the range at which measurement is straightforward; and FE models for design of calorimeters for measurement of radiation dosimetry for medical applications, such as radiotherapy for cancer treatment.

The modelling process offers considerable advantages to manufacturers who require a clear understanding of the product they are designing. The usefulness of a model is only limited by how well the material’s properties can be described and how well the problem can be defined.

NPL’s experience in measurement of materials and their practical limits, such as cracking, allows us to develop these models based on manufacturers’ designs. This provides the manufacturer with insight into how the product will respond to likely operating conditions and what might go wrong, and ultimately how it should be designed, before they embark on the time and cost intensive process of actually building and testing it.

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