ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
Numerical simulation of mathematical models in different areas of Engineering, Medicine, and Applied Sciences, in general, formulated as ordinary or partial differential equations.
The choice of problems and methods for their resolution will cover the following topics:
- Steady and dynamic problems, in one or several dimensions, with finite differences and finite elements.
- Simulation of non-linear and / or coupled phenomena, with scalar, vector and / or tensor unknowns.
- Examples in mechanics of solids, fluids, thermal, electrostatic, acoustics, and fluid structure interaction, and in the field of biomedicine and engineering.
- Use of software packages (MATLAB and COMSOL).
For each one of the considered applications, a brief description of the real problem will be made, concise writing of the corresponding mathematical model, the available and relevant data will be identified, a practical description will be made of the numerical method to be used, its programming and implementation in one-dimensional cases, its computer resolution using calculation packages in dimensions 2 and 3, the analysis and criticism of the computed results, the validation of the model, the manipulation of the results to carry out post-process calculations, and the drafting and presentation of conclusions.
1. Introduction. Relevance of numerical simulation. The process of simulation of industrial problems. Mathematical technologies involved. Lectures: 0.5.
2. 1D contour problems with Dirichlet boundary conditions. Existence of solution. Review of their resolution through finite differences. Convergence of the method. Calculation of the exact solution, discrete solution, numerical order of convergence, graphic representations, and post-processing. Application to the stationary heat equation. Problem solving with spherical symmetry Lectures: 1.5; interactive laboratory classes 3.5.
3. 1D contour problems with general boundary conditions. Existence of solution. Deduction and implementation of discretization by finite differences. Programming of the method. Design of academic tests. Application in disease propagation. Application to the calculation of the mass balance of a reactor in a stationary state; calculation of the analytical and numerical solution; analysis of numerical instabilities. Reproduction of tables and graphs of results in examples published in the bibliography. Lectures: 1.5; interactive laboratory classes 2.5.
4. Non-linear 1D contour problems. Implementation of functional iteration and Newton's algorithms to solve nonlinearity. Design of non-linear academic tests. Application of the implemented algorithms to the calculation of the mass balance of a reactor with non-linear decay of the solute. Lectures: 1; interactive laboratory classes 3.
5. Dynamic 1D contour problems. Discretization of evolutive problems. Implementation of the obtained algorithms. Stability of discretization. Design of linear and non-linear evolutive academic tests. Application of the implemented algorithms to the calculation of the mass balance of a dynamic reactor with linear and non-linear decay; numerical verification of mass conservation. Reproduction of tables and graphs of results in examples published in the bibliography. Lectures: 1; interactive laboratory classes 3.
6. Resolution of initial value problems associated with systems of ordinary differential equations. Post-processing of the solution and validation with models published in the literature. Resolution of a deterministic model of pandemic evolution. Lectures: 1.5; interactive laboratory classes 0.5.
7. Finite element method to solve 1D contour problems: variational formulation, existence and uniqueness of weak solution, order of convergence. Calculation of the matrices and elementary vectors corresponding to Lagrange elements of order 1 and 2. Implementation of the method for different boundary conditions. Verification of the algorithm for academic tests designed in 3. Application to a bar elongation model, and to a heat transfer model with one or more materials. Finite element approach of the applications already introduced in 3. Lectures: 3.5; interactive laboratory classes 4.5.
8. Finite element method to solve non-linear and evolutive 1D boundary problems. Incorporation of time dependence. Application to the calculation of the mass balance of a reactor in the linear and non-linear, stationary and dynamic assumptions. Interactive laboratory classes 3.
9. Generalization of the finite element method to the 2D case: variational formulation, existence and uniqueness of weak solution, order of convergence. Introduction to MATLAB's PDE Modeler. Analysis of included physical models. Introduction to the concept of CAD. Realization of CADs for elementary and non-elemental geometries. Analysis of the 2D simulation methodology. Resolution of problems with known academic solution. Identification of the main pointers generated, and their manipulation to carry out post-process programs on the results. Calculation of the error in L2 and H1 norms. Solving the 2D heat equation. Physical interpretation of the results and post-processing of the results. Lectures: 1, interactive laboratory classes 4.
10. MATLAB simulation of 2D models in solids mechanics, electrostatics and heat transfer. Application to multi-material geometries. Incorporation of time dependence. Adaptive mesh. Simplification of 3D problems using cylindrical symmetry hypothesis, or using reduced models. Physical interpretation of the results. Programming of post-process calculations. Lectures 0.5, interactive laboratory classes 3.5.
11. Simulation with COMSOL of 2D models. Resolution of 2D models with non-regular data, with special emphasis on the concept of distribution and the weak solution. Analysis of results when the weak solution is known. Application to the simulation of heat exchangers under the hypothesis of cylindrical symmetry. Lectures 0.5, interactive laboratory classes 3.
12. Simulation with COMSOL of 3D models in solids mechanics and heat transfer. Comparison of the results with those obtained with the simplified 2D models. Lectures 0.5, interactive laboratory classes 2.5.
13. 3D simulation of the acoustics of a room. Deduction of the Helmholtz equation. Eigenvalues, frequency, amplitude. Effects of different factors: location of furniture, materials, location of the sound source. Comparison with Benchmark examples from the bibliography. Lectures 0.5, interactive laboratory classes 3.5.
14. Introduction to the modelling of fluid-structure coupled problems. Application in Medicine: 3D simulation of blood flow in an artery and its interaction with walls. Lectures 0.5, interactive laboratory classes 2.5.
• Basic bibliography.
- CALDWELL J., DOUGLAS K.S. Mathematical Modelling. Case Studies and Projects. Kluwer texts in the Mathematical Sciences. Kluwer Academic Publishers. Vol. 28, 2004. Available online through the agreement between Springer and USC.
- PENA, F. and QUINTELAP. Numerical Simulation Workshop Course. Notes and software codes available in the Virtual
Course. 2024-25.
- QUINTELA P. Matemáticas en Ingeniería con MATLAB. Serv. Publicaciones Universidad de Santiago de Compostela. 2000.
- QUINTELA P. Métodos Numéricos en Ingeniería. Tórculo Edicións. Santiago de Compostela. 2001.
- VIAÑO, J.M – FIGUEIREDO, J., Implementação do Método de Elementos Finitos. Notas. 2000.
- Software tutorials: Software User Guides of MATLAB and COMSOL.
• Complementary Bibliography
- AHMED I., MODU G.U., YUSUF A., KUMAM P., YUSUF, I. A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes. Results in Physics 21 (2021).
- BERMÚDEZ A., Continuous Thermomechanics. Birkhäuser Verlag. 2005.
- GURTIN M.E., An Introduction to Continuum Mechanics. Academic Press. New York, 1981.
- JOHNSON C. Numerical Solution of Partial Differential Equations by Finite Element Method. Cambridge Univ. Press, 1987.
- QUARTERONI A., SALERI F., Scientific Computing with MATLAB. Springer. 2003.
- RAVIART P.A. - THOMAS J.M. Introduction à l’Ánalyse Numérique des Équations aux Dérivées Partielles. Masson. 1983.
- SINGIRESU S.R., Applied Numerical Methods for Engineers and Scientists. Prentice Hall, 2002.
- TIAN Y., ZHANG T., YAO H., TADÉ M.O. Computation of Mathematical Models for Complex Industrial processes. Advances in Process Systems Engineering. Vol. 4. World Scientific, 2014.
General
Understand the concepts, methods, and main results of different branches of mathematics, along with some historical perspective of their development.
Apply both theoretical and practical knowledge acquired, as well as the capacity of analysis and abstraction in the definition and approach to new problems and finding solutions in both academic and professional contexts.
Communicate, both in writing and orally, knowledge, procedures, results and ideas in mathematics to an audience both specialized and non-specialized.
Study and learn independently, to organize time and resources, new knowledge and techniques in any scientific or technological field.
Specific
Understand and use mathematical language.
Realize rigorous proofs of some classical theorems in different areas of mathematics.
Develop demonstrations of mathematical results, formulate conjectures, and envision strategies to confirm or deny them.
Identify faulty reasoning errors, proposing new proofs or counterexamples.
Assimilating the definition of new mathematical concepts, connected with others already known, and be able to use it in different contexts.
Learn to abstract the properties and essential facts of a problem, as distinct from those occasional or purely circumstantial.
Propose, analyse, validate, and interpret simple models of real situations, using the mathematical tools most appropriate to the aims pursued.
Plan and implement algorithms and mathematical methods to solve problems in academic, technical, financial, or social.
Using computer applications of symbolic and numerical calculus, optimization, graphic visualization, and scientific software, in general, to experiment in mathematics and solve problems.
Transverse
Using literature and research tools general and specific resources of Mathematics libraries, including Internet access.
Optimally management of work time and organize the available resources, setting priorities, alternative routes and identifying logical errors in decision making.
Reasonably prove or disprove the arguments of others.
Working in interdisciplinary teams, providing order, abstraction capacity, and logical reasoning.
Read scientific texts not only in official languages but also in others relevant in science, especially the English.
4 hours a week in which are expository (1 hr per week), interactive laboratory classes (3 hours per week) and tutorials in very small groups in the computer lab (2 hours). Overall, the student will receive 14 hours of expository class, 40 hours of interactive laboratory classes, 3 hours of control of the practices, of the acquired knowledge, and of the degree of progress of the learning process.
The student will have a virtual Web, in which will be available different material on the subject, plus it can be used as a meeting point with the teacher and other students of the subject. Recent articles published in scientific journals of applied sciences, medicine and engineering will be used in the classes as examples of the problems to be studied and to reproduce their results.
During the course, interaction with the students will be facilitated through the teachers' e-mail and the virtual tools available at the Universidade de Santiago.
A number of practical exercises will be proposed for personal resolution.
Throughout the course, the students personal work will be verified by carrying out 3 controls to check their level of results in the different practices carried out.
Exam (10 points): The final exam will be theory and practice, in which will raise theoretical issues, practices and the usage of software packages for the design and implementation of what was studied during the course. The exam will be carried out in a computer classroom.
Personal Work (10 points): it includes the evaluation of the student's work throughout the course. There will be 3 controls, which combine a virtual test of the theoretical knowledge studied with the realization of practices to evaluate their level in implementing and using the theoretical knowledge, the algorithms studied and the practical material of the subject. Likewise, the theoretical knowledge shown in their virtual tests will be taken into account. The continuous assessment tests are the same and on the same date for all subject groups. The grade of the personal work will be the average of the grades obtained in the three controls of the subject. The dates of the continuous evaluation tests will be announced on the first day of the course and will be held during the timetable foreseen for the subject.
Students who obtain more than a 7 in their personal work may choose to keep a 7 as their final mark for the subject without the need to take the exam.
The exam will be the same for all students in the subject. For the students who take the exam, the final grade of the subject will be the maximum between the grade of the exam and the arithmetic mean between the grade of the personal work and the grade of the exam.
In the second opportunity of the evaluation, the student keeps the mark obtained with his/her personal work throughout the course. The final grade of the second opportunity will be governed by the same criteria as in the first opportunity.
The grade will be considered as Not Presented if the student did not participate in any control, and did not take the exam.
The set of basic, general, specific and transversal skills previously described are evaluated at 100% with the evaluation system proposed.
In the case of fraudulent exercises or tests, the provisions of the Regulations for the Evaluation of Students' Academic Performance and for the Revision of Qualifications will apply.
WORK IN THE CLASSROOM Hours
Expository classes 14
Interactive laboratory classes 39
Interactive control classes 3
Tutorial class 2
Total working hours in the classroom 58
PERSONAL WORK STUDENT Hours
Individual self-study or group: 25
Writing exercises, conclusions or other work: 20
Programming / testing or other work at computer / lab 35
Recommended reading, library activities or similar 12
Total PERSONAL WORK STUDENT 92
Have taken courses in differential equations, numerical methods and mathematical modelling is recommended.
Francisco Jose Pena Brage
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813194
- fran.pena [at] usc.es
- Category
- Professor: Temporary PhD professor
Peregrina Quintela Estevez
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813223
- peregrina.quintela [at] usc.es
- Category
- Professor: University Professor
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLIL_01 | Galician, Spanish | Computer room 4 |
| 11:00-12:00 | Grupo /CLIL_01 | Spanish, Galician | Computer room 4 |
| 05.23.2025 10:00-14:00 | Grupo /CLE_01 | Computer room 2 |
| 07.01.2025 16:00-20:00 | Grupo /CLE_01 | Computer room 2 |