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.
Required courses have not been set. The students however, must have good knowledge of Differential and Integral calculus, solving methods of Differential equations, Programming, Differential Geometry and undergraduate level Numerical Methods.
- 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.
Burnett D.S., Finite Element Analysis: From Concepts
to Applications, Addison Wesley, 1987 (ISBN-10:
- 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.
The course grade is determined by the exercises (45%), and a research project (55%) based on the recent scientific literature (e.g. articles from Journal of Computational Physics).