Numerical Modelling of Geodynamic Processes
A.Y. 2026/2027
Learning objectives
The course unit aims to provide the students with the basic tools for numerical modeling of simple geological problems, using, in particular, the finite element method.
Expected learning outcomes
Ability to critically use sophisticated numerical algorithms already implemented.
Ability to independently develop simple numerical algorithms for solving complex geophysical problems.
Ability to independently develop simple numerical algorithms for solving complex geophysical problems.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Introduction to Numerical Modelling in Geodynamics
- The role of numerical modelling in the Geosciences.
- Introduction to the Finite Difference Method (FDM) and the Finite Element Method (FEM).
2. Introduction to Scientific Programming in Fortran
- Structure of a Fortran program.
- Variables, operators, and control structures.
- Arrays and memory management.
- Data input and output.
- Implementation of simple numerical algorithms.
3. Governing Equations of Geodynamic Processes
- Conservation equations for mass, momentum, and energy.
- Initial and boundary conditions.
4. From Continuous to Numerical Formulation
- Introduction to the weak formulation.
- Derivation of the weak form through integration by parts.
- Spatial discretization and construction of the algebraic system.
5. Fundamentals of the Finite Element Method
- Domain discretization and mesh generation.
- Types of finite elements (in 1D and 2D).
- Node numbering and element connectivity.
6. Shape Functions and Numerical Integration
- Definition and properties of shape functions.
- Interpolation within finite elements.
- Local coordinates and isoparametric mapping.
- Jacobian transformation.
- Numerical quadrature and Gauss integration.
7. Application of the Finite Element Method to the Energy Equation
- Derivation of the finite element formulation.
- Assembly of element and global matrices.
- Numerical implementation of the energy equation.
- Treatment of boundary conditions.
8. Marker Transport and Advection
- Introduction to marker-in-cell methods.
- First-order Runge-Kutta integration.
- Second-order Runge-Kutta integration.
- Stability and accuracy of time integration schemes.
9. Verification, Validation, and Error Analysis
- Spatial and temporal discretization errors.
- Numerical convergence.
- Benchmark problems.
- Model verification and validation.
10. Analysis and Visualization of Numerical Results
- Processing of numerical data.
- Scientific visualization techniques.
- Interpretation of numerical results.
- Critical discussion of simulations and model limitations.
- The role of numerical modelling in the Geosciences.
- Introduction to the Finite Difference Method (FDM) and the Finite Element Method (FEM).
2. Introduction to Scientific Programming in Fortran
- Structure of a Fortran program.
- Variables, operators, and control structures.
- Arrays and memory management.
- Data input and output.
- Implementation of simple numerical algorithms.
3. Governing Equations of Geodynamic Processes
- Conservation equations for mass, momentum, and energy.
- Initial and boundary conditions.
4. From Continuous to Numerical Formulation
- Introduction to the weak formulation.
- Derivation of the weak form through integration by parts.
- Spatial discretization and construction of the algebraic system.
5. Fundamentals of the Finite Element Method
- Domain discretization and mesh generation.
- Types of finite elements (in 1D and 2D).
- Node numbering and element connectivity.
6. Shape Functions and Numerical Integration
- Definition and properties of shape functions.
- Interpolation within finite elements.
- Local coordinates and isoparametric mapping.
- Jacobian transformation.
- Numerical quadrature and Gauss integration.
7. Application of the Finite Element Method to the Energy Equation
- Derivation of the finite element formulation.
- Assembly of element and global matrices.
- Numerical implementation of the energy equation.
- Treatment of boundary conditions.
8. Marker Transport and Advection
- Introduction to marker-in-cell methods.
- First-order Runge-Kutta integration.
- Second-order Runge-Kutta integration.
- Stability and accuracy of time integration schemes.
9. Verification, Validation, and Error Analysis
- Spatial and temporal discretization errors.
- Numerical convergence.
- Benchmark problems.
- Model verification and validation.
10. Analysis and Visualization of Numerical Results
- Processing of numerical data.
- Scientific visualization techniques.
- Interpretation of numerical results.
- Critical discussion of simulations and model limitations.
Prerequisites for admission
Students are expected to have a basic knowledge of differential and integral calculus, linear algebra, and ordinary differential equations. Familiarity with fundamental concepts of numerical methods is recommended but not strictly required.
Basic programming skills and prior experience with any scientific programming language (e.g., Fortran, Python, C, or MATLAB) are considered an advantage but are not mandatory.
Basic programming skills and prior experience with any scientific programming language (e.g., Fortran, Python, C, or MATLAB) are considered an advantage but are not mandatory.
Teaching methods
Teaching activities will take place in the Numerical Modelling of Geodynamic Processes laboratory and will include both lectures and hands-on laboratory sessions.
Theoretical topics will be presented through PowerPoint lectures. Laboratory sessions will focus on the implementation and analysis of numerical algorithms for the simulation of simplified geodynamic processes.
During the practical sessions, each student will have access to a computer workstation and will develop, under the supervision of the instructor, a numerical algorithm aimed at simulating a physical process of geodynamic interest. The laboratory activities are also designed to foster the understanding of the numerical methods introduced during the course and their application to practical problems.
Theoretical topics will be presented through PowerPoint lectures. Laboratory sessions will focus on the implementation and analysis of numerical algorithms for the simulation of simplified geodynamic processes.
During the practical sessions, each student will have access to a computer workstation and will develop, under the supervision of the instructor, a numerical algorithm aimed at simulating a physical process of geodynamic interest. The laboratory activities are also designed to foster the understanding of the numerical methods introduced during the course and their application to practical problems.
Teaching Resources
- Computational Methods for Geodynamics (2010). Ismail-Zadeh, Alik and Tackley, Paul. Cambridge University Press.
- 101 geodynamic modelling: how to design, interpret, and communicate numerical studies of the solid Earth (2022). van Zelst, I. and Crameri, F. and Pusok, A. E. and Glerum, A. and Dannberg, J. and Thieulot, C.. Solid Earth
Additional teaching materials and lecture slides will be made available during the course.
- 101 geodynamic modelling: how to design, interpret, and communicate numerical studies of the solid Earth (2022). van Zelst, I. and Crameri, F. and Pusok, A. E. and Glerum, A. and Dannberg, J. and Thieulot, C.. Solid Earth
Additional teaching materials and lecture slides will be made available during the course.
Assessment methods and Criteria
Assessment is based on a practical laboratory examination and an oral discussion.
During the practical laboratory examination, students will be required to modify the numerical algorithm implemented during the course. Subsequently, each student must prepare and submit a written report detailing the implemented modifications, presenting the obtained results, and providing a critical analysis and discussion of their significance. The submitted report will be evaluated through an oral discussion.
The oral discussion will take place according to the official examination schedule published on the UNIMI website. The practical laboratory examination must be completed approximately ten days prior to the scheduled oral discussion; the exact date and arrangements for the practical examination will be agreed upon with the course instructor.
During the practical laboratory examination, students will be required to modify the numerical algorithm implemented during the course. Subsequently, each student must prepare and submit a written report detailing the implemented modifications, presenting the obtained results, and providing a critical analysis and discussion of their significance. The submitted report will be evaluated through an oral discussion.
The oral discussion will take place according to the official examination schedule published on the UNIMI website. The practical laboratory examination must be completed approximately ten days prior to the scheduled oral discussion; the exact date and arrangements for the practical examination will be agreed upon with the course instructor.
GEOS-04/A - Solid Earth Geophysics - University credits: 6
Practical exercises with elements of theory: 36 hours
Lessons: 24 hours
Lessons: 24 hours
Professor:
Regorda Alessandro
Shifts:
Turno
Professor:
Regorda AlessandroProfessor(s)