Service overview
Mathematical modelling, numerical methods, multiphysics, uncertainty analysis and digital twins. This service is part of that division and is scoped individually around what you need to learn, decide or publish.
A mathematical model is an explicit statement of what you believe about a system. Writing that belief as equations forces precision: which quantities matter, how they interact and what is assumed constant. The best models are as simple as the question allows. They explain behaviour, expose scaling and parameters that matter, and make predictions that can fail, which is what makes them scientific.
Modelling is as much craft as calculation. Choosing state variables, deciding which effects to include, nondimensionalising to reveal governing parameters and testing against limiting cases are skills that distinguish a trustworthy model from an elaborate one. This service supports those steps in process engineering, transport and reaction systems, dynamics and theoretical sections of research papers.
If you plan to publish the outcome, see manuscript & publication support, for example computational & simulation manuscripts or original research article writing.
What this service covers
Problems this service can address
- Models that are either too complex to identify or too simple to be useful
- Parameters that cannot be estimated from available data
- Unstated assumptions that undermine conclusions
Methods and analytical approaches
- Conservation-law and constitutive modelling
- Non-dimensionalisation and scaling analysis
- Parameter estimation and identifiability checks
- Analytical and asymptotic approximations
Applications
- Process and system dynamics
- Transport and reaction models
- Theoretical sections of research papers
Relevant engineering disciplines
Scope, inputs and deliverables
Scope of work
- Definition of the physics, geometry and quantities of interest
- Model set-up with verification and, where data allow, validation
- Parametric or sensitivity studies agreed at scoping
- Results and documentation suitable for reports or publication
- Feasibility check on: system description and objectives
Typical inputs required from you
- System description and objectives
- Available measurements
- Intended use of the model
Deliverables
- Model formulation document
- Assumption and limitation register
- Parameter estimates with uncertainty where data allow
Highly specialised requests are first checked for feasibility from this information. Please do not send confidential data through the public form; describe it first and agree secure transfer afterwards.
Project workflow and method
- 01Enquiry and requirement assessment
- 02Technical scoping and proposal
- 03Execution
- 04Quality review and delivery, with an agreed round of revisions
Read the full project workflow.
In depth: how the work is done
From physical principles to equations
Most engineering models start from conservation laws for mass, momentum, energy or charge, closed by constitutive relations that describe material or system behaviour. We identify the control volume or system boundary, the state variables, the fluxes and sources, and the constitutive relations, noting which are well established and which are empirical. The resulting equations are checked for dimensional consistency, which catches many errors at an early stage.
Scaling and nondimensionalisation
Nondimensionalisation recasts equations in terms of dimensionless groups, reducing the number of parameters and revealing which effects dominate. Characteristic scales for length, time and magnitude are chosen from the physics. Dimensionless numbers identify regimes in which terms can be neglected, guiding simplification in a principled way, and they allow results to be transferred between scales or systems. The scaled model also tends to be better conditioned numerically.
Parameters, identifiability and estimation
Models contain parameters that must be taken from literature, measured or fitted to data. Identifiability analysis asks whether the available data can determine the parameters uniquely; if several combinations fit equally well, estimates are not meaningful individually. Sensitivity analysis shows which parameters most affect outputs, guiding which measurements to prioritise. Estimation uses appropriate statistical criteria, and uncertainty in parameters is propagated to predictions.
Approximation, limiting cases and testing
Analytical and asymptotic approximations, such as perturbation methods and quasi-steady assumptions, provide insight and independent checks on numerical solutions. Limiting cases, for which the answer is known or obvious, test whether the model behaves sensibly at extremes. Predictions are then compared with data not used for fitting. A model that cannot be falsified in any way is of little use, so we also state what behaviour would indicate model failure.
Presenting a model in a paper
The theory section of a paper needs to be both rigorous and readable. We help present assumptions before equations, define every symbol once, explain the physical meaning of key terms, and place long derivations in an appendix. Comparisons with limiting cases or prior models are highlighted, since they show that the model is consistent with established results. Clear presentation of modelling choices allows reviewers to judge the model's merits and its limits fairly.
Quality control and good practice
Quality-control approach
- Verification: mesh, time-step and solver sensitivity are examined before results are reported.
- Validation: results are compared with measurements or reference solutions where these exist, and the absence of such data is stated.
- Boundary conditions and material or fluid properties are recorded with their sources.
- Sensitivity to the least certain inputs is shown or explained.
Common pitfalls this service helps avoid
- Adding detail without evidence. Extra terms and parameters that data cannot constrain make a model less predictive, not more.
- Fitting many parameters to little data. Non-identifiable parameters give a good fit and bad predictions.
- Skipping dimensional checks. Inconsistent units are among the most common and avoidable modelling errors.
- No limiting-case tests. Models that fail simple known cases should not be trusted in complex ones.
What good reporting looks like
A modelling report states the system and purpose, assumptions, governing equations with derivations or references, scaling analysis, parameter sources and estimation methods, solution approach, comparison with limiting cases and data, sensitivity analysis and the model's limitations. Notation is defined and consistent, and the report is written so that a reader could reconstruct and re-implement the model. Predictions are accompanied by uncertainty where parameters are uncertain.
An illustrative scenario
Hypothetical example written for explanation. It is not a client case, a result or a testimonial.
Consider a researcher modelling the transient temperature of a battery pack under load. A lumped-capacitance model with heat generation and convective loss is formulated, nondimensionalised and found to be governed by a ratio of internal conduction to surface heat transfer. That number shows that a lumped model is adequate for small cells but not for large ones. Parameters are fitted to a measured temperature curve, identifiability checks show that two parameters are correlated, and the report recommends an additional experiment to separate them.
Questions to settle before you begin
- What question must the model answer?
- Which state variables and physical laws are essential?
- Which dimensionless groups might govern behaviour?
- Can my data determine all parameters uniquely?
- What would show the model is wrong?
Key terms
- Constitutive relation
- An equation describing the response of a material or process, such as Fourier's law.
- Nondimensionalisation
- Rewriting equations using dimensionless variables.
- Identifiability
- Whether parameter values can be uniquely determined from data.
- Asymptotic approximation
- A simplified solution valid when a parameter is very large or small.
- Lumped model
- A model that treats a spatially varying system as a single uniform entity.
Working together on Computational Engineering & Modelling projects
Computational engagements begin by agreeing the question, the quantities of interest and the accuracy that matters, so that modelling effort goes where it changes the answer. We ask for geometry or system descriptions, operating conditions, material or property data and any measurements for validation. Early on, a small pilot case often checks feasibility and cost. Models are verified and, where data allow, validated before parametric results are produced. Software, versions and settings are recorded so results can be reproduced. Where inputs are missing, assumptions are listed and agreed rather than quietly filled in.
Frequently asked questions
Do you provide analytical derivations?
Where an analytical treatment is feasible and appropriate, yes. We document each step so it can be checked.
Do I need to have data before building a model?
Not always. A model can be developed from first principles and later tested. Data are needed to validate and to estimate parameters that cannot be derived.
Can you build a model for a process I describe?
Yes, provided the description and available knowledge are sufficient. We will state assumptions clearly and list what must be measured.
What is the difference between a mathematical model and a simulation?
The model is the set of equations and assumptions; simulation is the numerical solution of it. The two services are related.
Can you derive analytical solutions?
Where tractable, yes. Otherwise, approximations or numerical solutions are suggested.
How is this service scoped and priced?
After your enquiry we review what you need, ask for missing information and send a written scope and quotation. Price depends on complexity, deliverables and timeline. See the project workflow.