The course is thought for students attending an MSc in Engineering who want to deepen the numerical treatment of coupled problems arising in different engineering applications. The course is divided into two parts: in the first one (5 CFU), numerical methods to address general coupled problems arising in practical scenarios will be introduced and analyzed; in the second part (3 CFU), the general methods introduced in the first part will be applied to a specific context, namely the numerical modeling of the cardiovascular system, which is characterized by several coupling of different nature. This will allow to introduce the students towards projects inspired by the clinical needs. Thus, this course presents a modular structure, so that an MSc could possibly introduce only one of these two moduli among its courses.
The students will be addressed during the course towards new and advanced numerical strategies to handle complex coupled problems that could arise in different engineering contexts. 
In order to have practical answers of the new numerical strategies introduced in this course, several laboratories with the computer will be considered, where the students will implement some of the new numerical technique in a Finite Elements library.

The learning process will involve lectures on the theory and computer laboratories about the application of examples of
numerical methods.

The course could be performed also in a short version by 5 CFU called "053464 - ADVANCED NUMERICAL METHODS FOR COUPLED PROBLEMS". The present description accounts for both the courses (the extended and the short one).

For both the version of the course (extended and short), the lectures and the laboratories will provide students with the knowledge and understanding of
* Numerical techniques for heterogeneous problems based on domain decomposition;
* Numerical approximation of the coupling between a fluid and a structure;
* Numerical treatment of the coupling between geometrically heterogeneous models;

Limited to the extended version of the course, the lectures and the laboratories will provide students also with the knowledge and understanding of

* Application to the cardio-vascular system

Moreover, both for the extended and the short courses, the ability to apply the previous knowledge to some examples on the calculator will be stimulated. In particular the student is required to
* be able to run executables introduced during laboratories;
* critically discuss the results of the numerical experiments above in prespective of the theory;

The instructor expects a broad comprehension of the subjects which should not be limited to the statement of theoretical results.
Instead, the acquired knowledge should enable students to express critical judgment.
The students are expected to express their answers in a mathematically rigorous and clear way.

In what follow, the topics of the courses will be presented. Part I is common (extended and short version of the course), whereas part II refers only to the extended course

PART I (5 CFU) (Extended and short course)

I Numerical techniques for heterogeneous problems based on domain decomposition
I.1 Brief review on Domain Decomposition methods and formulation of the problem
I.2 Numerical strategies
I.2.1 Loosely coupled algorithms
I.2.2 Preconditioned monolithic systems
I.3 Convergence analysis

II Numerical approximation of the coupling between a fluid and a structure
II.1 A brief review of the numerical approximation of the Navier-Stokes equations
II.2 A brief review of the numerical approximation of the elasto-dynamics equations
II.3 The case of an enclosed fluid
II.3.1 Position of the coupled problem
II.3.2 Numerical algorithms based on monolithic strategies
II.3.3 Numerical algorithm based on partitioned schemes
II.4 The case of an immersed structure
II.4.1 The Immersed Boundary method
II.4.2 The Fictitious Domain method
II.4.3 The Extended-Finite Element method

III Numerical treatment of the coupling between geometrically heterogeneous models
III.1 Coupling between 3D and 1D models
III.2 Coupling between 3D and 0D models
III.3 Numerical treatment of defective boundary conditions
III.3.1 Lagrange multipliers approach
III.3.2 Penalization techniques
III.3.3 Methods based on optimal control

PART 2 (3 CFU) (Extended course)

IV Application to the circulatory system
IV.1 Brief introduction to the basis of physiology
IV.2 Mathematical and numerical modeling of blood and vessel wall
IV.3 Numerical issues in the fluid-structure interaction problem: the added mass effect
IV.4 The reduced one-dimensional model
IV.5 Lumped parameters models and outflow boundary conditions

V Application to valve dynamics

VI Application to the heart function
VI.1 The electrical propagation
VI.1.1 The Bidomain equations
VI.1.2 Reduced models: Monodomain and Eikonal equations
VI.1.3 Numerical strategies
VI.2 The mechanical contraction and electro-mechanical coupled problem
VI.2.1 Constitutive laws for the myocardium
VI.2.2 Electro-mechanical coupling mechanisms
VI.2.3 Numerical strategies
VI.3 Towards an integrated heart functioning modeling

The topics that the students should have previously attended refer to the basic concepts of numerical analysis, in particular to:

* solution of linear and non-linear systems;

* approximation of ordinary differential equations;

* approximation of data and integrals;

The students should also have a consolidated knowledge about

* continuous formulation of fluid-dynamics and solid mechanics;

* numerical discretization of partial differential equations;

* Finite Element methods for diffusion-advection and Stokes/Navier-Stokes equations.

The exam consists only in an oral discussion and is designed to test the student on the knowledge and understanding of:

For both the extended and the short course:
* Numerical techniques for heterogeneous problems based on domain decomposition;
* Numerical approximation of the coupling between a fluid and a structure;
* Numerical treatment of the coupling between geometrically heterogeneous models;

For the extended course:

* Application to the cardio-vascular system 

The instructor expects a broad comprehension of the subjects which should not be limited to the statement of theoretical results.
Instead, the acquired knowledge should enable students to express critical judgment.
The students are expected to express their answers in a mathematically rigorous and clear way.