# 330-0903/01 – Finite Element Method in Mechanics (MKPME)

 Gurantor department Department of Applied Mechanics Credits 10 Subject guarantor doc. Ing. Zdeněk Poruba, Ph.D. Subject version guarantor doc. Ing. Jiří Podešva, Ph.D. Study level postgraduate Study language Czech Year of introduction 2015/2016 Year of cancellation Intended for the faculties USP, FS Intended for study types Doctoral
Instruction secured by
HOR80 prof. Ing. Petr Horyl, CSc., dr.h.c.
MAW007 Ing. Pavel Maršálek, Ph.D.
POD10 doc. Ing. Jiří Podešva, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Examination 2+2
Combined Examination 18+10

### Subject aims expressed by acquired skills and competences

Students will extend and make deeper their theoretical knowledge of the background of FEM and the numerical procedures that lead to the practical use of the method. Especially the problematics of solving nonlinear tasks will be deepen.

### Teaching methods

Lectures
Individual consultations
Tutorials
Project work

### Summary

Variational Methods. Principle of stationary potential energy. Problems having many degrees of freedom (DOF). Potential energy of an elastic body. The Rayleigh-Ritz method. Galerkin and other weighted residual methods (MWR). Examples: Uniform bar, Beam dynamics. Galerkin FEM in two dimensions. Bar and Beam Elements. Displacement-based elements. Shape functions. Stiffness matrix. Properties of stiffness matrices. Timoshenko beam element. Boundary conditions. Applied mechanical loads. Equilibrium equations. Stresses. FEM Concepts. Elements of arbitrary orientation – local and global matrices. Assembly of elements ( assembly and structure node numbers ). Exploiting sparsity, numbering and sparsity. Solution of equations. Structural symmetry. Connecting dissimilar elements. Eccentric stiffeners. Rigid elements. Basic Elements. Preliminaries: Strain-displacement relations, Stress-strain relations. Interpolation and shape functions. Formulas for element matrices. Linear triangle ( constant-strain triangle CST ). Quadratic triangle ( LST ). Bilinear rectangle ( Q4 ). Quadratic rectangle ( Q8, Q9 ). Rectangular solid elements. Choice of interpolation functions. Nature of a finite element solution. Isoparametric Elements. Example- bar element. Bilinear quadrilateral ( Q4 ). Transformation. [B] matrix and stiffness matrix. Numerical integration and Gauss quadrature. One, two and three dimensions. Stiffness matrix integration. Static condensation. Stress calculation. Analysis of axisymmetric solids. Elasticity relations. Axisymmetric solid elements. Loads without axial symmetry. FEM in Structural Dynamics. Dynamic equation. Mass and damping matrices. Consistent and lumped (diagonal) mass matrix. Proportional damping, Eigenfrequencies (natural frequencies), eigenmodes (mode shapes) and solutions method. Reduction of the number of DOF. Response History. Modal methods. Harmonic response. Direct integration methods-explicit or implicit. Central differences-stability conditions. Newmark family of methods. Heat Transfer and Selected Fluid Problems. Heat transfer: introduction. Finite element formulation. Transient thermal analysis – Modal method and direct integration. Acoustics and FE formulation. Boundary absorption. Fluid - structure interaction. Buckling. Geometric nonlinearity-Green strain. Energy considerations. Initial stress stiffness matrix (geometric stiffness matrix ). Linear buckling. Imperfection. Nonlinear buckling. Nonlinearity. Newton-Raphson method. Arc-length method. Convergence criteria. Problems of gaps and contact.

### Compulsory literature:

Cook R. D., Malkus D.S., Plesha M.E., Witt R.J. CONCEPTS AND APPLICATIONS OF FINITE ELEMENT ANALYSIS. 4th edition. J. Wiley & Sons, Inc. NY, 2002, p. 719, ISBN 0-471-35605-0 REDDY, J.N., An Introduction Nonlinear Finite Element Analysis, Oxford University Press, 2004, p. 463, ISBN 0-19-852529-X BHATTI,M.A., Advanced Topics in Finite Element Analysis of Structures: with Mathematica and Matlab Computations, Wiley, 2006, p.590, ISBN-13 978-0-471- 64807-9

### Recommended literature:

Examples for ANSYS solutions: http://www.mece.ualberta.ca/tutorials/ansys/ Zhi-Hua Zhong. Finite Element Procedures for Contact-Impact Problems. Oxford University Press, 1993, p. 371, ISBN 0-19 856383-3 WRIGGERS, P., Nichtlineare Finite-Element Metoden, Springer, 2005, p. 495, ISBN 3-540-67747

### Way of continuous check of knowledge in the course of semester

The workout of the individual project on the given theme.

### Další požadavky na studenta

Adequate appropriate knowledge in the area of finite element method.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Variational Methods. Principle of stationary potential energy. Problems having many degrees of freedom (DOF). Potential energy of an elastic body. The Rayleigh-Ritz method. Galerkin and other weighted residual methods (MWR). Examples: Uniform bar, Beam dynamics. Galerkin FEM in two dimensions. Bar and Beam Elements. Displacement-based elements. Shape functions. Stiffness matrix. Properties of stiffness matrices. Timoshenko beam element. Boundary conditions. Applied mechanical loads. Equilibrium equations. Stresses. FEM Concepts. Elements of arbitrary orientation – local and global matrices. Assembly of elements ( assembly and structure node numbers ). Exploiting sparsity, numbering and sparsity. Solution of equations. Structural symmetry. Connecting dissimilar elements. Eccentric stiffeners. Rigid elements. Basic Elements. Preliminaries: Strain-displacement relations, Stress-strain relations. Interpolation and shape functions. Formulas for element matrices. Linear triangle ( constant-strain triangle CST ). Quadratic triangle ( LST ). Bilinear rectangle ( Q4 ). Quadratic rectangle ( Q8, Q9 ). Rectangular solid elements. Choice of interpolation functions. Nature of a finite element solution. Isoparametric Elements. Example- bar element. Bilinear quadrilateral ( Q4 ). Transformation. [B] matrix and stiffness matrix. Numerical integration and Gauss quadrature. One, two and three dimensions. Stiffness matrix integration. Static condensation. Stress calculation. Analysis of axisymmetric solids. Elasticity relations. Axisymmetric solid elements. Loads without axial symmetry. FEM in Structural Dynamics. Dynamic equation. Mass and damping matrices. Consistent and lumped (diagonal) mass matrix. Proportional damping, Eigenfrequencies (natural frequencies), eigenmodes (mode shapes) and solutions method. Reduction of the number of DOF. Response History. Modal methods. Harmonic response. Direct integration methods-explicit or implicit. Central differences-stability conditions. Newmark family of methods. Heat Transfer and Selected Fluid Problems. Heat transfer: introduction. Finite element formulation. Transient thermal analysis – Modal method and direct integration. Acoustics and FE formulation. Boundary absorption. Fluid - structure interaction. Buckling. Geometric nonlinearity-Green strain. Energy considerations. Initial stress stiffness matrix ( geometric stiffness matrix ). Linear buckling. Imperfection. Nonlinear buckling. Nonlinearity. Newton-Raphson method. Arc-length method. Convergence criteria. Problems of gaps and contact.

### Conditions for subject completion

Full-time form (validity from: 2015/2016 Winter semester)
Task nameType of taskMax. number of points
Min. number of points
Examination Examination
Mandatory attendence parzicipation:

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Combined form (validity from: 2015/2016 Winter semester)
Task nameType of taskMax. number of points
Min. number of points
Examination Examination
Mandatory attendence parzicipation:

Show history

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### Occurrence in special blocks

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