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
1-semester course
Computer algebra methods and their applications to nonlinear
biomechanics
Master Program Computational Mechanics and Biomechanics.
3 ECTS Credits
Course description
This course studies material nonlinearity as an important property of
biological tissue models. The method of gathering material properties of the
models is the main object of this course, namely, Saint-Venant semi-inverse
method. It allows to determine and test various material parameters in set of
simple experiments such as extension, torsion, disclination, and so on, of elastic
canonic-shaped bodies. Along with that students get a deeper understanding of
constitutive equation theory and approaches to solid media modeling. They also
get a fully theoretical task of finding a square root of Cauchy deformation measure
tensor, being left tensor of deformations.
Secondary aim of this course is to get use of Maple software and its freeware
and open-source analogs as Maxima along with solvers Octave and SciLab. All
tasks are intended to be realized in two different ways using different software.
Recommended previous knowledge
Linear algebra and differential equations courses are obligatory. Strong
background in theoretical and/or continuum mechanics is also mandatory (tensor
concepts and notations, stress-strain relations, theory of constitutive equations).
Knowledge in computational algorithms is optional, as necessary algorithms such
as shooting method will be discussed during the course.
Techniques, Skills, etc.
After completing the course, the students are expected to be able to:
·
Get Lame coefficients of any orthogonal coordinate system.
·
Develop and solve boundary value problem (BVP) from semi-inverse
deformation of canonic-shaped elastic bodies and unit strain energy function.
·
Become an experienced user of advanced engineering environments
such as Maple, Maxima, SciLab, Octave.
Teaching
The course will be taught mainly in lecture format along with class
examples and demonstrations. Biological issues will be presented using
demonstrations. Readings will be drawn from text sources indicated below.
Problems (Individual Tasks) will be assigned as follows: first and second one on
second and third weeks respectively, and main third upon completion of first two
(usually it takes three weeks). At the end of the term the qualification will take
place. Upon the successful completion, the students will gain 3 credits.
Course content
№ Subject (theory)
№
3 and definition. Software:
14
history, capabilities,
5
licensing,
comparison.
2
Assignments
(practice)
21
Computer
algebra methods
6
Subject
Nonlinear
Duration
(in hrs)
Maxima
4
computer algebra
system
Maple
biomechanics.
Nonlinear
7
computer algebra
theory
8 of elasticity in
system
4
biomechanics. Theory of
Saint-Venant semi-inverse
method.
Curvilinear
coordinates. Covariant
Scilab
4
package
derivative. Christoffel
3
symbols of second kind.
Lame coefficients and
gathering of them.
Automatization of
4
gathering Lame coefficients
problem #1: «The
both in Maple and Maxima.
orthogonality check
Gathering a Christoffel
and gathering of
symbols of second kind.
Lame coefficients for
Semi-inverse SaintVenant method and its
application to the problem
5
Individual
of inflating of left heart
ventricle
a curvilinear
coordinate system»
in Maple and
Maxima.
4
4
Deformation gradient
6
Individual
and two methods of getting
problem #2:
it from semi-inverse
«Computing of left
representation. Algorithms.
Distortion and
distortion tensor
Individual
using pre-defined
problem #3:
Couchy deformation
«Computer
measure coefficient
realization of semimatrix».
inverse method for
rotation tensors. Cauchy
7
8
deformation measure and
its computer realization.
Unit strain energy
Gathering
function
8
for isotropic and
deformation
transversal isotropic media.
gradient
Governing equation for
4
4
8
solving the problems
of nonlinear
biomechanics»
Piolla tensor. Gathering of
Piolla tensor.
9
Equilibrium equation
Gathering
4
9 of solid media using Piolla Piolla stress
stress tensor. Algorithms.
tensor and
Typical boundary
comparing it with
conditions.
known analytical
one
1
Biomechanical
Creating a
10 applications of semi-inverse program for
method. Stretching and
getting divergence
torsion of an artery with
of second-rank
residual stresses.
tensor and getting
Blastosphere with inner
equilibrium
pressure. Stretching and
equations and
torsion of muscular tissue.
BCs
4
Shooting method in
Maple
1
11
Creating a
8
program for
numeric
experiment in
CAS Maple
Scilab, Octave and
Using
4
Maxima
1
packets integration SciLab for getting
12
a numeric
solution of BVP
gathered.
Coupling it with
Supersoft tissue
CAS Maxima
Getting the
modeling. An approach to
resulting charts.
13 in-vitro experiments with
Completing the
real tissue.
The equilibrium
report.
Final
stability of nonlinear elastic lesson. Defending
bodies. The method of
14
applying small deformation
to a finite one.
4
reports.
4
Requirements During the session students are required to
·
attend class lectures;
·
attend and conduct a research in the laboratory;
·
write reports;
·
present the report results in oral presentation at the end of course;
·
be prepared to participate in discussions.
Grade determination
·
Class participation - 30%
·
Laboratory work - 45%
·
Written report and its presentation – 30
Literature
Core
1. Biomechanics : principles and applications / edited by D. Schneck and J. D.
Bronzino. CRC Press, 2003. – 300 p.
2. Fung Y.C. Biomechanics: Mechanical Properties of Living Tissues. – Springer,
New York. 1993.
3. Sachse F. B. Computational Cardiology: Modeling of Anatomy,
Electrophysiology, and Mechanics. - Springer-Verlag, Berlin-Heidelberg, 2004. –
327 p.
4. Taber L.A. Nonlinear theory of elasticity. Applications in Biomechanics. – Word
Scientific Publishing, 2004. – 399 p.
Internet Resources
1. Maple User Contributed Applications http://maplesoft.com/applications/
2. Maxima documentation http://maxima.sourceforge.net/documentation.html
3. Scilab Online documentation
http://www.scilab.org/support/documentation/manuals
Contact
Assistant Dmitry Sukhov, devitor@mail.ru