**Lecturer: Dr Stephen Millmore**

The prerequisite course for Multiphysics Modelling of Four States of Matter consider how gas, liquid, solid and plasma can be modelled individually. In this course, the techniques required to simulate multiple materials in a single computational domain, i.e. multiphysics methods, are covered. The challenges in obtaining thermodynamically consistent behaviour between materials are identified, and then two important approaches in overcoming these challenges are detailed. Ghost fluid methods enforce sharp interfaces between materials, and maintain individual evolution equations, whilst diffuse interface methods combine evolution equations, and introduce a mixture region between materials. The advantages and disadvantages of each technique is considered, and implementation is covered in detail. Practicals in this course will allow students to implement their own multiphysics codes, between materials with very different properties. This course combines physics, mathematics and computational methods, and is suitable for students with backgrounds in any of these areas, including engineering.

Students should leave the course knowing:

- How ghost fluid methods can be used to provide dynamic communication of information between multiple materials in a single computational domain
- How the governing equations of compressible fluid flow for different materials can be incorporated into a single system for a diffuse interface representation
- How these techniques can be used to model surface tension, cavitation and material fracture

**Lectures:**

- Introduction to computational multiphysics

Definition of multiphysics problems; representation of material interfaces; the challenges of multiphysics modelling; overview of multiphysics methods - Level set methods

Implicit boundary methods and an introduction to the uses of level set methods; mathematical properties of level set methods applied to interfaces; numerical methods for the level set function; reinitialisation techniques - Ghost fluid methods (Part 1)

Dynamic boundary conditions; derivation of the original ghost fluid method; techniques to improve the ghost fluid method; numerical requirements for implementing a ghost fluid method - Diffuse interface methods

Mixture quantities for a diffuse interface; the Baer-Nunziato system of equations for diffuse interfaces; velocity and pressure equilibrium limits; speed of sound in a mixture; numerical methods for diffuse interface approaches - Ghost fluid methods (Part 2)

Improving accuracy through Riemann problem-based ghost fluid methods; mixed material Riemann problems; interfaces between very different materials; ghost fluid methods in multiple dimensions; - Cavitation and surface tension

Cavitation from liquids in tension; a diffuse interface approach to cavitation; surface forces in a discretised domain; a sharp interface approach to surface tension

**Practicals:**

All practicals require students to write their own code, typically in C++, in which demonstrators will be familiar, though other languages may be used.

- Mixed-material exact Riemann solvers

Expanding the exact Riemann solver developed at the end of CCM1 to allow for two materials either side of the interface. Preparatory work for later practicals - Level set methods

Incorporating a level set method into existing Euler equation codes as a passive scalar. Demonstrating that this can track a contact discontinuity - The original ghost fluid method

Incorporating a true material interface between two ideal gases, using level set methods and the original ghost fluid method of Fedkiw et al. - The original ghost fluid method for multiple interfaces

Increasing the applicability of the ghost fluid method code such that it can deal with the complex case of a multi-interface simulation - The Riemann ghost fluid method

Implementing a Riemann problem-based ghost fluid method, and using this to solve scenarios the basic method could not cope with - Surface tension

Implementing the techniques to deal with surface tension in a toy one-dimensional problem

**Prerequisites:**

- Scientific Programming in C++ (or other C++ courses)
- Numerical Methods for Compressible Fluid Dynamics
- Simulation of Matter Under Extreme Conditions

**Recommended reading:**

- Osher, Stanley, Ronald Fedkiw, and K. Piechor. “Level set methods and dynamic implicit surfaces.”
*Appl. Mech. Rev.*57.3 (2004): B15-B15. - Fedkiw, Ronald P., et al. “A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method).”
*Journal of computational physics*152.2 (1999): 457-492. - Allaire, Grégoire, Sébastien Clerc, and Samuel Kokh. “A five-equation model for the simulation of interfaces between compressible fluids.”
*Journal of Computational Physics*181.2 (2002): 577-616.

**Picture credit:**

- Tim Wallis
- Stephen Millmore