Southern Federal University
Faculty of mathematics, mechanics and computer science
Milchakova str. 8a, Rostov-on-Don, 344090
Phone.: (863) 2975 111;
Fax: (863) 2975 113;
SYLLABUS
3-semester course
Finite Element Modeling in Nonlinear Problems
for Masters Program Computational Mechanics and Biomechanics.
5 ECTS Credits
Course objectives
The course «Finite Element Modeling in Nonlinear Problems» is aimed at
studying various nonlinear mathematical and physico-technical models of practical
importance, gaining knowledge of finite element technologies to solve nonlinear problems
of mathematical physics, and making use of modern computational software with
advanced capabilities for nonlinear analysis of real-world problems.
The relevance of the course is due to the importance of nonlinear problems of
mathematical physics and computational mechanics, their great practical importance and
nontrivial implementations in modern computational software. Indeed, the majority of
practically important problems of structural analysis are nonlinear, and this nonlinearity
can show up in different ways. The concept of nonlinearity includes geometric
nonlinearities (large displacements, large deformations), physical nonlinearities in
2
materials behavior, problems of plasticity,
stability, fracture mechanics, contact problems, and nonlinearities in coupled physical and
mechanical problems. Dynamic problems of interaction between solid and liquid media
can represent coupled highly nonlinear problems with fast processes.
Techniques, Skills, etc.
After completing the course, the students are expected to be able to:

set different nonlinear problems for structure analysis;

know characteristics of different nonlinear solvers;

choose and implement a suitable numerical method for given nonlinear
problem;

choose material models for specific nonlinear problems;

set contact problems for structure analysis;

use finite element packages (ANSYS, FlexPDE etc.) for solution of nonlinear
structure problems;

present coherent arguments to answer questions both orally and in writing.
Teaching
The following methods and forms of study are used in the course:

Lectures

Lectures - disputes, lectures – discussion

Presentation for lectures

Labs, practical work/training

Problem sets

Finite element packages ANSYS and FlexPDE, technical documentation for
ANSYS and FlexPDE

Methodic manuals and school-books
3


Self-study
Use of different reference books and Internet resources
At the end of the course including practical work, the students are supposed to do
problem sets and write a report, make an oral presentation and participate in discussion.
Upon the successful completion, the students will gain 5 credits.
Course content
N
Subject
NO
Form of Duration
Lesson
1.
Introduction.
(hrs)
Lecture
0,5
Buckling analysis of Lecture
0,5
Course organization, its
aims and structure.
Information on main and
additional readings.
2.
elastic system. Linear or
Laboratory
2
Lecture
1
Lecture
2
eigenvalue buckling
theory. Euler buckling
behavior of a column
3.
Finite element (FE)
approximations for linear
buckling analysis.
Geometric, incremental or
initial stress stiffness
matrix
4.
Snap-through
analysis of flat thin-walled
elastic structures.
Date
4
Buckling failure of von
Mises truss
5.
Nonlinear buckling
Lecture
2
problems and FE
Laboratory
4
approximations ANSYS
Self-study
2
Lecture
3
Lecture
1
of nonlinear systems of
Laboratory
2
equations in ANSYS and
Self-study
technique for linear
buckling analysis.
6.
Solution of
nonlinear systems of
equations. NewtonRaphson method.
Modified Newton scheme.
Quasi-Newton method.
Convergence of nonlinear
FE solvers.
7.
Solution technique
FlexPDE
8.
Arc-Length
Lecture
2
Laboratory
2
Lecture
2
analysis. Geometric and
Laboratory
2
material nonlinearities in
Self-study
2
Lecture
2
Method. ANSYS
technique for nonlinear
buckling analysis.
9.
Nonlinear structural
structural analysis.
Nonlinear constitutive
equations.
10.
1D plasticity
5
models. Basis of plasticity
Laboratory
2
Lecture
1
Laboratory
2
Self-study
2
Lecture
1
mechanics, fracture
Laboratory
2
criteria. FEM in fracture
Self-study
4
Lecture
2
Self-study
2
theory. Invariants of stress
and strain tensors. Yield
criterions (Tresca, Hubervon Mises/Hill), isotropic
hardening, kinematic
hardening, Drucker-Prager
model. Rate-dependent
plasticity. General
statements of plasticity
problems. Flow rules. FE
approximations for
plasticity problems and FE
algorithms for solution of
plasticity problems.
11.
ANSYS technique
for plasticity analysis.
12.
Basis of fracture
mechanics. Stress singular
finite elements. J-Integral
calculation in FEM.
13.
Basis types of
highly nonlinear problems
for fast processes. Explicit
dynamic. Material models.
14.
Solution methods of Lecture
2
6
highly nonlinear problems
Self-study
2
Lecture
2
nonlinear problems for
Laboratory
2
fast processes in ANSYS
Self-study
4
Lecture
2
Self-study
4
Lecture
2
nonlinear problems in
Laboratory
8
ANSYS, different solvers
Self-study
2
Lecture
2
Lecture
2
solution methods for
Laboratory
2
contact problems.
Self-study
2
for fast processes.
15.
Solution of highly
LS-DYNA.
16.
Solution of highly
nonlinear problems for
fast processes in ANSYS
AUTODYN.
17.
Solution of coupled
for coupled problems in
ANSYS. Interface
between structure analysis
and fluid dynamics in the
finite element packages.
18.
Statement details of
contact problems.
Boundary conditions in
contact problems. Contact
type boundary conditions.
Constitutive equations at
the contact interface
19.
Algorithms of
Augmented Lagrangian
method, Lagrange
7
multiplier method, penalty
function method.
20.
FE approximations.
Lecture
2
Lecture
2
and dynamic contact
Laboratory
6
problems in ANSYS.
Self-study
4
Contact and target finite
elements.
21.
Solutions of static
Requirements
During the session students are required to

attend class lectures;

be prepared for laboratory hours;

attend the laboratory;

write a report;

represent the report results in oral presentation at the colloquium;

be prepared to participate in final course discussion.
Grade determination

Class participation - 40%

Laboratory work - 40%

Written report and its presentation – 10%

Participation in discussion – 10%
8
Literature
Main
ANSYS. Theory Reference for ANSYS and ANSYS Workbench. Rel. 11.0. Ed. P.
Kothnke / SAS IP Inc. Canonsburg, 2007.
ANSYS. Basic Analysis Procedures Guide. Rel. 11.0. / SAS IP Inc. Canonsburg,
2007
ABAQUS. Theory Manual. Ver. 6.6. ABAQUS Inc., 2006.
BatheK.-J. Finite element procedure. Prentice-Hall, 1996.
Hughes T.J.R. The Finite element method. Linear static and dynamic finite element
analysis. Prentic-Hall: New Jersey, 1987.
Madenci E., Guven I. The finite element method and applications in engineering
using ANSYS. Springer, 2006.
Wriggers P. Computational contact mechanics. Springer, 2006.
Wriggers P. Nonlinear finite element method. Springer, 2008.
Zienkiewicz O.C. The Finite Element Method in Engineering Science. 1971.
McGraw Hill.
Zienkiewicz O.C., Morgan K. Finite Elements and Approximation. 1983. John
Wiley & Sons.
Zienkiewicz O.C., Taylor R.L. The finite element method. V. 1. The Basic. 2000.
Zienkiewicz O.C., Taylor R.L. The finite element method. V. 2. Solid Mechanics.
2000.
Additional
Bathe K -J., Wilson C. L. Numerical Methods in Finite Element Analysis. Prentice
Hall , Inc., Englewood Cliffs, NJ, 1976.
Belytschko T., Liu W.K., Moran B. Nonlinear finite elements for continua and
structures. Wiley J. & Sons, Inc., 2000.
Gallagher R.H. Finite Element Analysis. Fundamentals. 1975. Prentice Hall.
9
Nakasone
N.,
Stolarski
T.A.,
Yoshimoto S. Engineering analysis with ANSYS software. Elsevier, 2006.
Oden J. T. Finite Elements of Nonlinear Continua. McGraw-Hill, Book Company,
New York, 1972.
Internet Resources
ANSYS
11.0,
www.ansys.com,
www.pdesolutions.com,
www.ansys.msk.ru, www.ansyssolutions.ru, www.cae.ru
www.emt.ru,