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Multi-scale Modelling: Macroscopic Analysis with Finite Elements

Code: 45738
Credits: 3
2026/2027
Degree programme Type Course
Applied Nanoscience: From Materials to Devices OP 1

Contact lecturer

Name :
Albert Beardo Ricol
Email :
albert.beardo@uab.cat

Teaching staff

Carles Navau Ros
F. Xavier Alvarez Calafell

Group languages

You can consult this information at the end of the document.

Prerequisites

Basic understanding of calculus and partial differential equations is required to follow the course. It is also necessary to have experience in mathematical modeling of macroscopic phenomena in the context of physics.

Objectives

Understand the mathematical foundation underlying the Finite Element method for solving partial differential equations. Students will learn to obtain the weak formulation of generic differential equations to implement physical models with complex boundary conditions and couplings between different phenomena. From a more applied perspective, students will see in detail examples of multiscale and multiphysics modeling, including magnetic systems, diffusion, fluid mechanics and elasticity, as well as couplings such as thermoelasticity and thermoelectricity.

Learning outcomes

  • CA18 (Evaluate complex systems with commercial finite element simulators.) Evaluate complex systems with commercial finite element simulators.
  • KA17 (Identify different scales of simulation, from atomistic models to continuous models.) Identify different scales of simulation, from atomistic models to continuous models.
  • SA22 (Use multiscale simulation or multiphysics in complex systems, incorporating and analysing their coupling.) Use multiscale simulation or multiphysics in complex systems, incorporating and analysing their coupling.
  • SA23 (Analyse systems, in physics or engineering, by developing differential equations with finite elements.) Analyse systems, in physics or engineering, by developing differential equations with finite elements.

Contents

PART 1: Fundamentals of the finite element method.

Weak formulation of partial differential equations.

Boundary conditions.

Types of elements and meshing.

Variational principles.

Galerkin method.

Lagrange multipliers.

Stabilization, convergence and error estimation.


PART 2: Multiscale modeling.

Thermal, electrical and molecular diffusion.

Elasticity.

Couplings: Thermoelasticity, thermoelectricity.

Magnetic systems.

Fluid mechanics: Laminar and turbulent fluids.

Emerging phenomena at the nanoscale: Non-diffusive transport.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Modeling 21 0.84 SA23
Microscopic to macroscopic modeling 10 0.4 KA17
COMSOL Multiphysics 32 1.28 CA18, SA22

At the beginning of the course, all students will receive an official COMSOL Multiphysics license. Students will use the software throughout the course to experiment with the various examples discussed. Therefore, the sessions will not be purely lecture-based, but will combine explanations and demonstrations with hands-on in-class work. At the end of the course, students will share different models they have chosen and developed with their classmates. Thus, the teaching will not be entirely teacher-student, but will also incorporate a collaborative learning approach among students.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Written report and individual interview 35 4 0.16 KA17, SA23
Oral defense of the final modeling project 50 6 0.24 CA18, SA22
COMSOL Multiphysics: Implementation Example 15 2 0.08 CA18

Written report and individual interview (35%). It will consist of an example of numerical implementation of a differential equation from scratch. The preparation of this report will be guided during the first part of the course.


Implementation example (15%). Demonstration of the implementation of differential equations with boundary conditions using COMSOL Multiphysics.


Oral defense of the final modeling project (50%). It will consist of developing an example of complex modeling of a physical phenomenon in one of the areas that will be discussed in class. The project will be defended orally in front of the rest of the students at the end of the course.

Bibliography

The Finite Element Method: Its basis and fundamentals. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu. Springer, DOI: 10.1016/C2009-0-24909-9)


Introduction to the Finite Element Method. H. Ottosen and N.S. Petersson.


Multiscale Model Reduction: Multiscale Finite Element Methods and their Generalizations (Applied Mathematical Sciences, 212). Eric Chung, Yalchin Efendiev, Thomas Y. Hou.

Software

COMSOL Multiphysics (an official license will be provided to students).


Programming languages ​​to be chosen by the student: Python, C++, or similar.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TEm) Theory (master) 1 English first semester morning-mixed