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##### Module Notes

Faculty Member (Members):

Postgraduate, Spring Semester

*Module Type:*Specialization Courses

*Teaching Language:*English/Greek

*Course Code:*GCHM_C741

*ECTS Credits:*8

*Module Availability on Erasmus Students:*No

##### Module Details

At the end of this course, the students should:

- Comprehend the basic principles of the Finite Element and Pseudospectral Methods, and of grid generation techniques.
- Be able to solve problems in the fields of computational transport phenomena and computational fluid dynamics.
- Be able to use FEM codes in their personal research activities.

The student will develop the following skills:

- High level programming.
- Understanding the limitations and advantages of each numerical method.
- Choosing the most suitable numerical scheme for a physical problem.
- Using FEM software.

- The finite difference method for solving ordinary and partial differential equations. The upwinding scheme for convection - diffusion problems. Band solvers and the Thomas solver.
- Orthogonal polynomials: Chebyshev, Jacobi, Fourier. Principles of weighted Galerkin residual, least squares and collocation methods. Solution of one-dimensional problems with periodic conditions or conditions Dirichlet, Robin through Pseudospectral methods.
- The weak form of a Differential Equation. Examples of derivation of the weak form of an ODE. Essential and natural boundary conditions and their introduction in the weak form.
- The Galerkin finite element method in 1D problems. All steps for discretizing a linear differential equation of a boundary value problem according to FE Method. Application of boundary conditions. Construction of local basis functions in physical space. The linear and the quadratic polynomials Lagrange.
- Formation of the linear algebraic problem. Solution of the linear system of equations. Band and sparse solvers.
- Solution of a nonlinear problem of boundary conditions in 1D via Newton-Raphson method.
- Parent element in 1D-3D: linear, square, triangular, tetrahedral and hexahedral elements. Lagrangian and Hermitian basis functions.
- Calculation of spatial integrals via the Gauss method in 1D-3D prototype geometries.
- Practical applications of the above. Demonstration of a FEM code in 1D. Solution of a nonlinear reaction - diffusion problem.
- Comparison of Lagrangian and Hermitian cubic polynomials. Problem in solving differential equations with third or fourth order derivatives or systems of differential equations. Accuracy and convergence results.
- The Galerkin finite element method in 2D & 3D.
- Practical applications of the above. Demonstration of a FEM code in 2D & 3D.
- Solution of parabolic problems via FE method. Time discretization and application of initial conditions. Solution of a nonlinear time-dependent problem.
- Calculation of eigenvalues of elliptic problems by the method of finite element Galerkin. Applications on fluid mechanics.
- Structured and unstructured meshes. Techniques for the construction of grid: Elliptic and algebraic methods.
- Quality indices of a grid. Dynamic adaptive and boundary fitted grid generation techniques.

**Course textbook**

Burnett D.S., *Finite Element Analysis: From Concepts **to Applications*, Addison Wesley, 1987 (ISBN-10:

0201108062).

**Additional studying**:

- Zienkiewicz, O.C., Taylor, R.L., & Zhu, J.Z.,
*The Finite Element Method: Its Basis and Fundamentals, Seventh Edition*, Butterworth-Heinemann, 2013 (ISBN-10: 1856176339). - Zienkiewicz, O.C., Taylor, R.L., & Nithiarasu, P.,
*The Finite Element Method For Fluid Dynamics, Sixth Edition*, Butterworth-Heinemann, 2005 (ISBN-10: 0750663227). - Chung, T.J.,
*Computational Fluid Dynamics*, Cambridge University Press, 2010 (ISBN-10: 0521769698). - Liseikin, V.D.,
*Grid Generation, Scentific Computation, Second Edition*, Springer, 2009 (ISBN-10: 9048129117). - Canuto, C., Hussaini, M.Y., Quarteroni, A., & Zang, T.A.,
*Spectral Methods: Fundamentals in Single Domains, Fourth Edition*, Springer, 2011 (ISBN-10: 3540307257).

- Lectures are presented via Powerpoint, while the related problems are solved on the board. Slides are given to students in digital format.
- Six (6) sets of exercises are given during the semester. The students are asked to solve them within a week’s time during which they can ask for clarifications.
- Basic FEM codes of 1D-3D are given to the students.