# Research

### Introduction

A mathematical model is used to describe the behaviour of a physical system in terms of equations involving variables and parameters representing aspects of the system. For example, Newton's law of motion *F* = *ma* relates the acceleration *a* that an object of mass *m* experiences when subjected to a force *F*. Models are important in metrology as they allow a value of one variable to be estimated from measurements of other variables. For example, in a mass spectrometer, the mass of a particle is estimated using the above law from measurements of *F* and *a*.

### Science

A mathematical model describes the relationships between the quantities that affect the behaviour of a system. The model equation constrains the possible combinations of variable values that are thought to be possible. For example, the equation *L(t) = L*_{0}* + c(t-t*_{0}*)* models the length *L(t)* of a metal rod as a function of temperature *t* in terms of its length *L*_{0} at temperature *t*_{0} and its coefficient *c* of thermal expansion. Once *L*_{0} and *c* are prescribed, the model assigns a unique *L(t)* to each *t*.

A model allows us to determine the values of one set of variables once the values of others are known. These values are usually determined by solving sets of mathematical equations derived from those used to build the model. Depending on the type of model, these equations may represent a system of linear or nonlinear algebraic equations or differential equations. More often than not, some of the variable values are derived from measurements and have associated uncertainties. This means that the solutions of the equations that give the values of the remaining variables will also have associated uncertainties that need to be evaluated in order to assess their quality. The method of determining the solution values will also need to take into account the fact that some of the input information is not exact.

Modelling involves:

- Model building: physical, empirical, algebraic and continuous models
- Solution of linear and nonlinear algebraic equations: numerical linear algebra, least-squares techniques
- Solution of linear and nonlinear differential equations: finite element method, boundary element method
- Numerical analysis: efficient and numerically accurate solution algorithms
- Data approximation: fitting a model to data
- Optimisation: determining best estimates of model parameters, either to explain data or to achieve optimal functionality of the modelled system
- Uncertainty evaluation: propagation of uncertainties in model inputs to model outputs
- Model validation: deciding whether a model explains satisfactorily system behaviour on the basis of measurement results and other knowledge
- Scientific software development: encoding algorithms in library level software, validation and verification of software

### Services

NPL has a highly experienced team of technical experts in the area of mathematical and statistical modelling. We are part of a world-class physical metrology organization, with experience and expertise in this field that is unique in the UK. We offer a range of services including advice, consultancy and training in the areas of modelling, experimental data analysis, and software development.

## Modelling research

- Data analysis combines the model of the physical system and the measurement data to provide estimates of system parameters for use in mathematical modelling.
- A mathematical model is the key to understanding the behaviour of a measurement system and interpreting the data it produces. Models are developed on the basis of physical theory and from the analysis of data.
- Conformity assessment is defined as any activity undertaken to determine, directly or indirectly, whether a product, process, system, person or body meets relevant standards and fulfils specified requirements.
- Model solving is the process of determining estimates of the model parameters from the measured data by solving the mathematical equations constructed as part of the model. In general, this involves developing an algorithm that will determine the values for the parameters that best explain the data. These algorithms are often referred to as estimators.
- The aim of multiscale modelling is to predict the behaviour of complex materials, including biomaterials such as proteins, across a range of length and time scales.