NUMERICAL MODELS FOR ENGINEERING
The course teaches how to implement and use modern computer simulation techniques for the solution of engineering problems described by differential equations. Students are introduced to differential equations by applying it to the description of typical engineering problems, by solving it with numerical methods, and by analyzing the obtained results. The first part of the course refers to one-dimensional problems solved by proprietary codes based on finite differences and finite elements methods. In the second part of the course, the discussion is extend to the problems of multiple dimensions, unsteady, solved with the help of commercial numerical codes.
Knowledge and understanding:
At the end of the course, students understand differential equations describing typical engineering problems and their physical meaning, numerical methods commonly used for their solution, and know how to write simple codes for solving mono-dimensional problems. In addition, students are able to understand the basic principles behind commercial codes for the solution of boundary value problems. Finally, students understand the approximations introduced in computer simulations, both from the mathematical models used to describe real problems, and from the numerical models for the solution of differential equations.
Ability to apply knowledge and understanding:
The student must demonstrate that he / she is able to:
- develop models based on differential equations that describe several engineering problems;
- implement numerical codes for the solution of one-dimensional problems;
- use commercial codes for the solution of differential problems that describe engineering applications;
- evaluate the approximations introduced in computer simulations of engineering problems;
- analyze and present the numerical results.
Autonomy of judgment:
Students must demonstrate that they have developed the ability to use critically and autonomously simulation models for engineering problems described by differential equations, and to evaluate the results in a sound scientific manner, in terms of adequacy of both the model used and the obtained results.
Students should be able to explain in a simple way, even to people who are not experts in the field, with a simple, clear and scientifically rigorous manner, the model used and the results obtained in the solution of engineering problems described by differential equations.
Students do so by presenting their results in reports and presentations.
Students should be able to update their knowledge through the consultation of texts and publications related to the numerical modeling of differential problems, and use commercial codes, from the knowledge and method of analysis acquired during the course.
The course is intended for students of Engineering Master courses, who are familiar with the fundamentals of engineering acquired in Batchelor courses.
Students who have not taken an engineering course should have acquired the fundamentals of engineering and the following knowledge, provided in Engineering Calculus courses:
- Concepts of limits, integration and derivation of functions of a single variable;
- Functions of multiple variables, partial derivatives, and superficial integrals;
- Differential and series of functions.
One dimensional steady problems (4.0 CFU): Introduction to modeling engineering problems described by differential equations. Ordinary Differential Equations (ODEs). Numerical solution using the finite difference method of mono-dimensional problems with outline conditions (BVP). Implementation of finite difference methods methods for BVP solution. Estimates of the approximation in numerical solution with finite differences.
Finite Element Method (FEM): weak formulations for boundary value problems, one-dimensional approximation functions. Higher order finite elements. Implementation of finite element method for one-dimensional problems with boundary conditions. Verification and validation of numerical models.
Steady two dimensional applications (3.0 CFU): Introduction to Partial Differential Equations (PDEs); two-dimensional steady problems with boundary conditions: finite differences and finite elements. Physical formulation of the finite difference method for two-dimensional problems. Weak formulation of two-dimensional problems, two-dimensional interpolating functions, and properties of two-dimensional finite elements. Three-dimensional finite elements.
Using commercial codes for modeling engineering problems: planning a FEM model; implementation of boundary conditions; generation of computational grids; analysis of results: post-processing of results, verification and validation of numerical solutions.
Unsteady problems and vectorial problems (2.0 CFU): Formulation of unsteady problems: Initial Value Problems (IVP); finite differences for IVP solution; stability and accuracy of numerical solutions. Implementation of models for numerical solution of IVP. Time discretization of unsteady problems spatially discretized with finite differences and finite elements.
Introduction to vector problems described by PDE. Application of numerical models to the solution of engineering problem chosen by students.
Lecture notes available from the course website.
J. Fish, T. Belytschko, A First Course in Finite Elements, Wiley, 2007.
R. W. Lewis, P. Nithiarasu, K. Seetharamu, Fundamentals of the Finite Element Method for Heat and Fluid Flow, Wiley, 2004.
FURTHER SUGGESTED READINGS:
O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method, 7th Edition, Elsevier, 2007.
Nam-Ho Kim, B. V. Sankar, Introduction to Finite Element Analysis and Design, John Wiley & Sons, 2009.
R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
Two intermediate exercises are available, which can be evaluated upon request of students.
During the course, students carry out two reports, the first one related to a one-dimensional problem solved by a proprietary code, and the second one on a engineering problem chosen by students and solved by commercial code.
The exam is divided into two parts, which take place on the same day: a written test (1 hour duration) consisting of an exercise related to the numerical modeling of a real problem. Students must introduce the model describing a practical problem, the numerical discretization of the related differential model, and an estimate of the approximations introduced by the numerical model.
In the second part of the oral exam, students discuss the theoretical aspects of the course, starting with the problem solved in the first phase of the exam, and presents the results contained in the reports carried out during the course, with the help of a presentation (optional). During the oral examination students are evaluated based on the understanding of the topics discussed, the ability to apply the concepts learned for the solution of practical engineering problems and to present the results obtained in a simple and rigorous manner.
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