• Biomedical Engineering, Semester 3, position 4

The scope of this course is to provide a thorough theoretical and practical foundation for Finite Element Method (FEM). In the first part of the course the students will become familiar with the application of FEM to the solution of boundary value problems in elasticity. The formulation of weak form of the solution will be in the main focus, regarding problems of statics and dynamics of simpler structures. The main goal of this part of the course is to give a full and detailed view of the development of linearized formulation for diverse structural problems, by employing displacement based FEM. The matrix formulation of the problem will be presented in details. While the main focus is on methodology applied through coding in various computer codes, the students will also learn the basics of use of diverse commercial software.

Upon fulfilling the course the students will be able to: 1. Write their own computer codes for building stiffness matrix by employing finite elements of truss type, beam type as well as continuum finite elements in 2D and 3D. 2. Write their own computer codes in order to perform static and dynamic finite element analysis of simpler structures. 3. Write their own computer codes for post-processing of resulting displacement values in nodal points, in order to recover strains and stresses. 4. Understand the foundation on which commercial FEM software are build.

Introduction to numerical modeling. Virtual work principle and its application to the formulation of weak form of the solution in boundary value problems. Shape functions for the displacement field interpolation. Building of the stiffness matrix for one- two- and three-dimensional finite elements. Building of the stiffness matrix for complex structures and solution strategies based on reduced stiffness matrix and Lagrange multipliers. Post-processing of results, recovery of strain and stress field on the basis of known displacement field. Dynamic problems and time integration. Implicit and explicit integration schemes.

Coding in MATLAB package apt for the evaluation of stiffness matrix for finite elements of different types, i.e. truss elements, beam elements and continuum elements in 2D and 3D. Assembling of global stiffness matrix for structural problems. Application of boundary conditions. Linearization of the problem and different methods for obtaining solution for the system of linear algebraic equations. Reduced stiffness matrix and Lagrange multipliers method. Development of codes for recovery of strains and stress field. Static and dynamic analysis of structures.

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“The finite element method: static and dynamic finite element analysis”, Thomas Hughes, Prentice-Hall 2010. ISBN: 0-13-317025-X “The finite element method: a practical course”, G.R. Liu and S.S. Quek, Butterworth-Heinemann, 2003. ISBN: 0-7506-5866-5

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 25
Laboratory exercises: 0
Calculation tasks: 3
Seminar paper: 0
Project: 2
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 2
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 3
Test: 0
Test: 0
Final exam: 10

Activity during lectures: 0
Test/test: 0
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 50
Final exam: 50
Requirement for taking the exam (required number of points): 0

“The finite element method: static and dynamic finite element analysis”, Thomas Hughes, Prentice-Hall 2010. ISBN: 0-13-317025-X; “The finite element method: a practical course”, G.R. Liu and S.S. Quek, Butterworth-Heinemann, 2003. ISBN: 0-7506-5866-5