Graduate School

Meshless and other advanced numerical methods

This course is part of the programme:
Physics (Third Level)

Objectives and competences

The course systematically explores and establishes the theory, principles, and procedures that lead to mesh-free methods. Course in Mesh Free and Other Advanced Numerical Methods gives basic understanding and application of a class of emerging numerical methods. Students prepare a seminary that elaborates one of the topics.

Prerequisites

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Content (Syllabus outline)

The purpose of this course is to introduce the state-of-the-art computational meshless methods based on the radial basis functions. These newly developed techniques allow us to numerically solving a large class of partial differential equations without mesh generation. In contras to the traditional computational methods such as finite element and finite difference methods, the meshless methods introduced in this course are especially effective for solving problems with complicate shape and higher dimension.

The major topics of the course are as follows:

1. Introduction of radial basis functions

2. Function reconstruction

3. Kansa’s method

4. The method of particular solutions

5. The method of fundamental solutions

6. The localized method of particular solutions

7. Meshless collocation methods for time-dependent problems

Intended learning outcomes

Knowledge and understanding:

  • Different meshless methods.
  • Their practical application in an engineering or scientific problem.
  • Readings

    • Wen Chen, Z.J. Fu, C.S. Chen, Recent Advances in Radial Basis Function Collocation Methods, Springer, 2014.
    • M.A. Golberg and C.S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, Southampton, Boston, 1997
    • C.S. Chen, A. Karageorghis, Y.S. Smyrlis, The Method of Fundamental Solutions – A Meshless Method, Dynamic Publications, Inc., 2008.
    • C.S. Chen, Y.C. Hon, R.S. Schaback, Scientific Computing with Radial Basis Functions, Department of Mathematics, University of Southern Mississippi, USA.
    • M.A. Golberg and C.S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods – Numerical and Mathematical Aspects, ed. M.A. Golberg, Computational Mechanics Publications, 1998, pp.103-176.

    Assessment

  • seminar work
  • oral exam
  • Lecturer's references

    Full professor of Mathematics at the University of Nova Gorica.

    1. Ji Lin, Wen Chen, C.S. Chen, Numerical treatment of acoustic problems with boundary singularities by the singular boundary method, Journal of Sound and Vibration, 333, 3177 (2014)

    2. Ming Li, C.S. Chen, C.C. Chu, D.L. Young, Transient 3D heat conduction in functionally graded materials by the method of fundamental solutions, Engineering Analysis with Boundary Elements, 45, 62 (2014)

    3. C.Y. Lin, M.H. Gu, D.L. Young, C.S. Chen, Localized method of approximate particular solutions with Cole–Hopf transformation for multi-dimensional Burgers equations, 40, 78 (2014)

    4. Ming Li, C.S. Chen, A. Karageorghis, The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions, Computers & Mathematics with Applications, 66, 2400 (2013)

    5. Ming Li, Wen Chen, C.S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Engineering Analysis with Boundary Elements, 37, 1300 (2013)

    University course code: 3FIi04

    Year of study: 1

    Lecturer:

    ECTS: 6

    Workload:

    • Lectures: 45 hours
    • Individual work: 135 hours

    Course type: elective

    Languages: english

    Learning and teaching methods:
    lectures, seminars, practicals, individual hours with the lecturer