Course Description 

Topic: Non-matching coupling meets geometric dimensionality reduction: finite element methods for mixed-dimensional problems 

Many problems in computational mechanics involve thin structures embedded in a surrounding medium, such as fibres reinforcing an elastic material or vascular networks interacting with biological tissue. Resolving every geometric detail with a conforming mesh can be prohibitively expensive. Geometric dimensionality reduction replaces thin structures with lower-dimensional models, while non-matching methods allow their discretisation to remain independent of the surrounding mesh.

This lecture introduces the mathematical and computational ideas behind combining these approaches. Starting from immersed finite element formulations, we will discuss weak coupling through Lagrange multipliers and show how reduced multiplier spaces connect bulk fields to lower-dimensional models while retaining selected features of the unresolved interface. We will address stability, approximation and practical assembly on independent meshes. Examples from fibre-reinforced materials and vascularised tissues will illustrate the methods, with implementation examples drawn from ImmersX, an open-source framework built on deal.II.

Course Description 

Topic: Computational Magneto-Mechanics of Soft Materials: From Fundamentals to Multiscale Modelling  

Soft magneto-responsive materials exhibit a rich variety of coupled physical phenomena, including large deformations, field-dependent stiffness, viscoelastic effects and programmable mechanical anisotropy. Understanding and predicting these responses requires computational frameworks capable of connecting magnetic interactions at the microscale with the nonlinear mechanical behaviour observed at the macroscale.

This lecture will provide an introduction to computational magneto-mechanics of soft materials, with particular emphasis on magnetorheological elastomers. Starting from the fundamentals of continuum magneto-mechanics and coupled constitutive modelling, we will explore computational strategies to capture finite deformations, magnetic interactions and microstructure-dependent material responses. Special attention will be given to multiscale modelling approaches and their role in identifying emerging mechanisms, including magnetically induced viscoelasticity and remanence-driven anisotropy.

Selected examples will illustrate how these computational tools can be combined with experimental observations to understand material behaviour, guide the design of multifunctional soft systems and enable applications in soft robotics and mechanobiology. 

Course Description 

Topic: An Introduction to Shape and Topology Optimisation 

Shape and topology optimisation look for the best distribution of material inside a design domain. They are now widely used in engineering design and additive manufacturing. This lecture is a self-contained introduction aimed at PhD students and early-career researchers, and assumes no prior knowledge of optimisation. Starting from the classical problem of minimising the compliance of a structure for a given amount of material, we will see how the design problem is formulated, why it is ill-posed and how sensitivities are computed with the adjoint method. We will then present three complementary ways to describe and update the design. Density-based methods (SIMP) treat the design as a material density. The shape derivative tells how the objective changes when the boundary moves, and leads to level-set methods. The topological derivative measures the effect of nucleating a small hole. The lecture highlights the ideas that connect these approaches and the practical questions in choosing one, and closes with a brief look at current research directions.