Thu, 3 May 01: HW12 due.
Sat, 28 Apr 01:
No-crib-sheet final exam:
The
final exam
for on-campus students will be in
NEB 101, on Sat, 28 Apr 01, 2pm-5pm.
Closed book, closed notes,
no crib sheet.
IMPORTANT:
Sat, 28 Apr 01, is both part of Week 16 and part of the Exam week.
Wed, 25 Apr 2001
EXAM WEEK
Wed, 25 Apr 01:
Lecture 43:
Magnetodynamics:
More examples governed by equations similar to the transient heat
transfer problems;
Elastodynamics:
Governing equations of motion, boundary conditions (displacement or
Dirichlet, force or Neumann),
derivation of weak form (principle of virtual work).
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Sat, 21 Apr 2001
Did you know:
A recent report on
research directions in computational mechanics.
Sat, 21 Apr 2001
Mon, 23 Apr 01:
Lecture 42:
Isoparametric mapping:
Explanation on routine
shape.f,
elements with variable number of nodes (4 to 9);
Programming techniques:
Element matrices;
arrays ID, IEN, LM;
use of LM array to assemble element matrices into global matrices,
cases where there are several dofs at each node (elastodynamics).
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Sat, 21 Apr 2001
WEEK 16
Fri, 20 Apr 01: HW11 due.
Lecture 41:
Isoparametric mapping:
Overall review of transformation of coordinates (integration variables)
from physical coordinates to parametric (natural) coordinates,
element node numbering,
construction of
shape functions for bilinear quadrilateral elements and
for 9-node Lagrangian elements,
computation of conductance matrix and capacitance matrix
for transient heat transfer problem; similarly for the stiffness matrix
and mass matrix for the dynamic vibration of a 2-D membrane.
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Sat, 21 Apr 2001
Did you know:
You may be interested in
a talk
on a new method of teaching the FEM at the
undergraduate level
(see
abstract)
at the
6th US National Congress on Computational Mechanics, Dearborn, MI, 1-4 Aug 2001.
Note that this undergraduate course contains a once graduate-level
material (substructuring analysis, a form of domain decomposition) that
is not taught in the present FEM1 course.
Thu, 19 Apr 2001
Wed, 18 Apr 01:
Remarks on
HW8
(element matrices of identical elements, assembly into global
matrices); Fortran code
dlearn to go with Hughes [2000].
Lecture 40:
Isoparametric mapping:
Computation of gradient of basis functions, Jacobian matrix and
its determinant for the
isoparametric mapping,
interpolation of coordinates,
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Fri, 20 Apr 2001
Mon, 16 Apr 01:
Remarks on the etymology of
quadrature
and
HW12.
Lecture 39:
Isoparametric mapping:
2/3-D transient heat transfer problem,
integration of capacitance matrix C and conductance matrix
K (similarly for structural dynamics);
node (dof) numbering,
ID array to account for homogeneous Dirichlet boundary condition in the
global stiffness matrix (programming concepts);
transformation of coordinates from physical space to parametric space,
isoparametric mapping, bi-unit square and bi-unit cube as parent
elements, computation of gradient of basis functions.
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Sun, 15 Apr 2001
WEEK 15
Fri, 13 Apr 01: HW10 due.
Lecture 38:
Gauss-Legendre
quadrature
:
Optimality in 1-D;
in 2-D on bi-unit square;
in 3-D on bi-unit cube, lack of optimality, other integration rules.
2/3-D heat transfer:
Preliminary setup for programming of isoparametric elements.
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Sun, 15 Apr 2001
Did you know
The meaning of the word
quadrature,
which was often used to designate numerical integration? Don't feel
bad if you did not. On Friday, 13 Apr
01, I asked a prominent Stanford University mathematician, who
did not know the meaning of this word either, and he used this word
extensively in his talk.
(There was also another well-known professor of Greek origin
participating in this conversation; he did not know the answer either.)
I knew that the word had to do with the number "4", since
"quad" means four, as in quadrilateral elements, but could not connect
the number "4" to numerical integration per se.
So I decided to hit the encyclopedia to
find it out myself:
To measure (integrate) areas,
you can divide that area into rectangles, which
have four edges, and are special forms of quadrilateral elements, and
thus the meaning of the word
quadrature.
Similarly, the word
cubature
(subdivide a volumes into cubes)
is used to designate
a method to measure
(integrate)
volumes, and the word
rectification
(subdividing a line into straight segments)
for measuring the length of possibly curved lines.
Read more about
quadrature and cubature
in Greek antiquity, and their relations to modern analysis.
The
first use
of
"quadrature"
in English
appears to be in 1596, and of "quadrilateral" in 1650.
Sun, 15 Apr 2001
Wed, 11 Apr 01:
Lecture 37:
Gauss-Legendre
quadrature
:
In 1-D,
derivation of lower-order rules (1 point, 2 points, 3 points),
comparison with trapezoidal and Simpson rules;
Isoparametric mapping in 1-D, parent element in parametric space
(bi-unit interval), child elements in physical space;
in 2-D on bi-unit square (parent element).
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Mon, 9 Apr 01:
Lecture 36:
2/3-D transient heat transfer problem:
Discretization,
properties of energy inner product
(importance of homogeneous essential boundary condition,
superposition of solution, selection of particular function
satisfying essential boundary condition in 2-D, convenient function in
case of constant essential boundary condition),
relation to definition of positive-definite matrices,
positive definiteness of conductance matrix
K
.
Gauss
quadrature
:
Transformation of coordinate in 1-D.
Reading assignment: Hughes [2000],
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
WEEK 14
Fri, 6 Apr 01: HW8 due.
Lecture 35:
Finite-element basis functions:
Lagrangian elements:
General formula, element shape, global shape.
Hermitian elements:
Element shape, global shape.
2/3-D transient heat transfer problem:
Divergence theorem,
weak form,
discretization.
Reading assignment: Hughes [2000],
Chap 1,
pp.48-51
(Hermite [cubic] interpolation [FE basis] functions, Euler-Bernoulli
beam);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Wed, 4 Apr 01:
Lecture 34:
Finite-element basis functions:
Hermitian elements:
Convenient numbering of nodal dofs,
Euler-Bernoulli beam dynamics,
(consistent) mass matrix;
transient heat problem,
(consistent) capacitance matrix;
more general higher-order Hermitian elements.
Lagrangian elements:
2nd-order PDEs,
higher-order basis functions,
transformation of
coefficients in polynomial series
to nodal dofs.
Reading assignment: Hughes [2000],
Chap 1,
pp.48-51
(Hermite [cubic] interpolation [FE basis] functions, Euler-Bernoulli
beam);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Mon, 2 Apr 01:
Lecture 33:
Remark:
Connection between definition of positive definite matrices and
of definition of inner product.
Finite-element basis functions:
4th-order PDE,
weak form, requirements of continuity,
higher-order basis functions,
transformation of
coefficients in polynomial series
to nodal dofs,
Hermitian elements (cubic, quintic),
use of Hermitian elements for 2nd-order PDE.
Reading assignment: Hughes [2000],
Chap 1,
pp.48-51
(Hermite [cubic] interpolation [FE basis] functions, Euler-Bernoulli
beam);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
WEEK 13
Fri, 30 Mar 01:
Lecture 32:
Finite-element basis functions:
Norm induced by inner product,
energy norm.
Properties of FE basis functions (cont'd):
non-satisfaction of natural boundary conditions;
FE basis functions:
Nodal property,
transformation of
coefficients in polynomial series
to nodal dofs,
identification of nodal basis functions,
the linear case.
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem),
pp.48-51
(Hermite [cubic] interpolation [FE basis] functions, Euler-Bernoulli
beam);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
HW12:
Please download.
Wed, 28 Mar 01:
Lecture 31:
Transient problems:
Elastic bar, heat transfer: similarities and differences;
Properties of
mass
M
or
capacitance
C
,
and
stiffness
or
conductance
K
before accounting for boundary conditions.
Non-singularity
of K:
(2) via definition of positive-definite matrices;
importance of homogeneous essential boundary condition;
Cases where singularity of
K
is desired, role of non-singular
M
:
Flexible structures under free flight;
finite-element discretization and data structure in 1-D
(Hughes [2000], Chap 1),
properties of FE basis functions (compact
support, bandedness, weaker differentiability requirement,
non-satisfaction of natural boundary conditions).
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem),
pp.48-51
(Hermite [cubic] interpolation [FE basis] functions, Euler-Bernoulli
beam);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Mon, 26 Mar 01:
Lecture 30:
Dynamic problem, mass matrix, stiffness matrix:
Discretization of equation of motion of elastic bar (varying cross
section and Young's modulus, distributed load, concentrated load at free end,
prescribed motion at fixed end),
inertia and stiffness operators in WF2,
superposition,
mass matrix
M
as Gram matrix with (generalized) L^2 inner
product, stiffness matrix
K
as Gram matrix with (generalized) a_2(.,.) inner product;
Strain energy of elastic bar,
physical meaning of a_2(.,.) inner product;
Energy functional (potential energy),
principle of minimum potential energy,
directional derivative,
recovery of weak form
(principle of virtual work).
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
WEEK 12
Fri, 23 Mar 01: HW7 due.
Lecture 29:
Stiffness matrix (cont'd):
Elastic bar with varying cross section and varying Young's modulus,
under distributed axial load, with
prescribed boundary displacement
and
prescribed concentrated load:
Derivation of equation of motion,
boundary conditions,
initial conditions.
Rigid-body (zero-energy) mode,
function space in which a_2(.,.) is an inner product,
importance of homogeneous essential boundary condition.
Non-singularity
of K: (1) via
Gram
matrix of basis functions wrt a_2(.,.) inner product.
Remark on Gram matrices
constructed with
different operators that are not necessarily inner products.
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem);
Chap 2 and 3
(2/3-D problems, isoparametric elements, programming concepts);
Chap 7, pp.418-422
(3-D
transient heat transfer problem).
HW11:
Please download.
Wed, 21 Mar 01:
Lecture 28:
Stiffness matrix (cont'd):
Energy inner product a_2(.,.) of WF2
and homogeneous essential b.c.,
rigid (zero-energy) mode and properties of inner product,
global basis functions (Fourier, polynomials) and
finite-element hat functions,
stiffness matrix
K
elastic bar problem.
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem); Chap 7, pp.418-422
(3-D
transient heat transfer problem).
Mon, 19 Mar 01:
Lecture 27:
Stiffness matrix (cont'd):
Properties of determinant,
linear dependence of family of functions implies zero
Gram
determinant using L^2 inner product, proof;
stiffness operator a_2(.,.) of WF2
and homogeneous essential b.c.,
stiffness matrix K as
Gram
matrix.
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem); Chap 7, pp.418-422
(3-D
transient heat transfer problem).
WEEK 11
Fri, 16 Mar 01:
HW6 due.
Handout: Download excerpts
from Strang [1980] Linear Algebra and Its Applications
(properties
of determinant).
Lecture 26:
Stiffness matrix (cont'd):
Expansion of a function in terms of basis functions,
examples
(Fourier series, series of polynomials),
Gram
matrix using L^2 inner product, test of linear independence, zero Gram
determinant implies linear dependence, proof (null space, zero eigenvalue,
properties of inner product).
Reading assignment: Hughes [2000], Chap
1, pp.1-23, pp.37-46
(Galerkin method on 1-D
model problem); Chap 7, pp.418-422
(3-D
transient heat transfer problem).
HW10:
Please download.
Wed, 14 Mar 01:
Lecture 25:Basis functions:
Brief review of discretization
of WF2 (1-D model problem), selection of basis functions, linear independence,
definition, example with vectors/functions, expansion of a vector/function
in terms of basis vectors/functions, computation of components (coefficients)
in the expansion, inner product (dot product, L^2 inner product),
Gram
matrix.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
Mon, 12 Mar 01: Return of HW5; discussion, symmetry of stiffness
matrix in method of weighted residual and weak form, related questions.
Remark on measure of a set, countability of a set, sets of measure
zero, e.g., set of rational numbers in [a,b], Dirichlet function and its
Lebesgue integration.
Handout: Download excerpts
from Chae [1995] Lebesgue Integration
.
Lecture 24:Discretization of weak forms:
1-D model problem:
discretization of WF1 and WF2, Petrov-Galerkin,
Bubnov-Galerkin,
symmetry and non-symmetry of a(.,.) and K.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
WEEK 10
WEEK 9 : Spring break week; no class.
Fri, 2 Mar 01: HW6 due date extended.
Lecture 23:Weak form (cont'd): 1-D model problem: spaces
of trial solutions and of weighting functions, abstract weak forms WF1
and WF2 with function spaces, derivative in the generalized sense, definition
of Sobolev space H^1, examples, reason for the need of H^1 smoothness;
discretization of WF1 and WF2.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
HW8:
Please download.
Wed, 28 Feb 01:
Lecture 22:Weak form (cont'd): 1-D model problem: Equivalence
of WF2 with SF, FLCV3 (homogeneous Dirichlet b.c.'s on weighting functions),
application of FLCV3, "essential" and "natural" boundary conditions, relationship
between FLCV1, FLCV2, FLCV3; superposition, discretization.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
Mon, 26 Feb 01:
Lecture 21:Weak form (cont'd): 1-D model problem: Equivalence
of WF1 with SF, FLCV1 and FLCV2 (boundary conditions of weighting functions),
application of FLCV2, relationship of FLCV2 and FLCV1, meaning of "essential"
and "natural" boundary conditions.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
WEEK 8
Fri, 23 Feb 01: HW5
due.
Lecture 20:Weak form (cont'd): Additional remark on
Lebesgue
integral and Riemann
integral, and additional example; 1-D model problem (cont'd): abstract
forms of WF1 and WF2, equivalence of WF1 and WF2 with SF, FLCV.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
HW7:
Please download.
Wed, 21 Feb 01:
Lecture 19:Weak form (cont'd): Remark on
Lebesgue
integral and Riemann
integral), example; Finlayson & Scriven [1966] (Crandall's remark
on self-adjoint problems and symmetry); 1-D model problem (cont'd): weak
form WF1 and weak form WF2, symmetry and lack of symmetry (depending on
choice of boundary conditions of weighting functions, homogeneous essential
boundary condition), road map for equivalence with SF.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
Mon, 19 Feb 01:
Lecture 18:Weak form (Principle of virtual work): Review
of road map, Finlayson & Scriven [1966] (historical perspective, open
problems in the past, boundary conditions, Galerkin method), L^2 inner
product between two functions ("L" for Lebesgue
integral, Riemann
integral, completeness of basis functions, Pascal triangle; 1-D model
problem: Choice of weighting functions, integration by parts, two weak
forms, meaning of "weak" and "strong", equivalence with SF.
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
WEEK 7
Fri, 16 Feb 01: HW4
due; HW5 due date
extended.
Remark on
HW5; construction of basis functions
for trial solution.
Lecture 17:Method of weighted residual (cont'd): Transient
heat transfer problem (3-D), capacitance matrix C, conductance matrix
K, ODEs, (apparent) non-symmetry, computation of initial conditions for ODEs;
Finlayson & Scriven [1966].
Reading assignment: Hughes [2000], Chap
1, pp.1-8 (Galerkin method on 1-D model problem); Chap 7, pp.418-422 (3-D
transient heat transfer problem).
HW6:
Please download.
Wed, 14 Feb 01: The
explosion of Ariane Flight 501 (firm
schedule; room 303 Aero, 10:40am. Please
note
that this lecture is for another class, not in your usual lecture room).
HW3 returned; comment;
you can resubmit this HW report.
Lecture 16: Method of weighted residual: Finlayson &
Scriven [1966], discretization of time-dependent problems (trial solution,
boundary condition, initial condition, weighting function), transient heat
transfer problem (parabolic equation) with Dirichlet boundary condition,
system of ODEs, matrix form.
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
Did you know: That there is a person
who would die of hunger every 3.6 seconds, and that 3/4
of those are children under 5 years old? You can help to donate
food to these hungry people without
costing you a cent, just by clicking at the Hunger
Site of the UN
World Food Program
once a day.
To remind me to click at the Hunger Site every day, I created the Hunger-Site
stickers (in MS Word doc format) for all computers in my office and
in my lab. You are welcomed to use these stickers for your computers.
(The best place to put a sticker is just below
the screen of your monitor.) Better yet, since I am using Linux
,
I have set up my system to automatically display the Hunger
Site every morning at 8am, so that I can ``donate'' food
every morning before I start my day. If you are interested in knowing how
I did this setup, just ask me.
Mon, 12 Feb 01:
Lecture 15:
Method of weighted residual: Properties of
weighting function and trial solution, (apparent) non-symmetry. Remark on Finlayson
& Scriven [1966], time-dependent problem, properties of trial solution,
discretization, system of ODEs, initial conditions.
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
WEEK 6
Fri, 9 Feb 01: HW4 due date extended.
Remark on HW4, The
explosion of Ariane Flight 501
Lecture 14:
1-D model problem (cont'd): Method of weighted
residual, superposition, discretization of trial solution and of weighting
function, discrete weighted residual form, "discrete FLCV", linear system
Kd=F,
stiffness matrix K, solution.
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
HW5:Please
download.
Wed, 7 Feb 01:
Lecture 13:
1-D model problem (cont'd):
Method of weighted
residual, properties of trial solution and weighting functions, discretization,
superposition, selection of prescribed function, problem with homogeneous
b.c.'s.
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
Mon, 5 Feb 01: Remarks:
HW4
(Reynolds Transport Theorem), Fourier's
work.
Lecture 12:
1-D model problem:
Trial solution, residue,
test (weighting) functions, weighted-residual form, equivalence with strong
form, fundamental lemma of calculus of variations (FLCV), proof.
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
WEEK 5
Fri, 2 Feb 01: HW3
due.
Lecture 11:
Galerkin
method (cont'd): Derivative of an integral with variable bounds (cont'd),
1-D model problem, general road map (strong form, weighted residual method,
equivalence, discretization, weak forms, equivalence).
Reading assignment: Finlayson & Scriven
[1966] on weighted residual method; Hughes [2000], Chap 1, pp.1-8 (Galerkin
method on 1-D model problem).
HW4:Please
download.
Wed, 31 Jan 01:
Lecture 10: Galerkin
method (cont'd): Adiabatic boundary condition in heat transfer
problems. Remark on classifications of matrices in terms of its diagonalizability
(simple, semi-simple, defective, Jordan
canonical form), eigenvalue problem.
Reading assignment: Eriksson
et al., Sec 10.3.1 and Sec 10.3.2, pp.252-253 (diagonalizability, and
non-diagonalizability of a matrix, integrating linear ODEs).
Mon, 29 Jan 01: Remark on Galerkin
method and boundary integral equations
Lecture 9: Galerkin
method (cont'd): Heat transfer problem: Boundary conditions (Dirichlet,
Neumann, Fourier's
law of heat conduction, and Newton's
law of heat cooling, Robin
); 1-D model problem, exact solution, verification, derivative of an integral
with variable bounds.
Reading assignment: Eriksson
et al., Sec 6.2, pp.113-117 (1-D heat problem, Galerkin method), Sec
13.5, pp.313-316 (heat equation in higher dimension, derivation).
WEEK 4
Fri, 26 Jan 01: HW2
due.
Lecture 8: Galerkin
method (cont'd): Poisson equation, boundary conditions, applications,
derivation of governing equation for heat transfer problem (balance of
heat in control volume, Fourier's law of heat conduction, localization
of integral form of balance of heat equation, PDE for transient heat transfer,
steady state).
Reading assignment: Finlayson & Scriven
[1966], Eriksson
et al., Chap 6, pp.104-117 (Galerkin method).
HW3:Please
download.
Wed, 24 Jan 01:
Lecture 7: Complete Hadamard's
example of ill-posed PDE. Galerkin
method:
Model problem in 2-D (Poisson
equation), applications in various engineering areas, tensor form, indicial
notation (div, grad, and all that), component form; classification, positive
definiteness of conductivity matrix, elliptic PDE. (Portrait
of Galerkin.)
Reading assignment: Finlayson & Scriven
[1966], Tables from Reddy [1986], Eriksson
et al., Chap 6, pp.104-117 (Galerkin method).
Mon, 22 Jan 01: Remark on downloading
handouts (Finlayson & Scriven [1966] and tables from Reddy [1986]),
team organization.
Lecture 6: PDEs (cont'd): Boundary conditions.
Well-posedness, behavior of solution under perturbation of data; chaos;
Hadamard's
example of an ill-posed PDE.
Reading assignment: Finlayson & Scriven
[1966], Tables from Reddy [1986], Eriksson
et al., Chap 6, pp.104-117 (Galerkin method).
WEEK 3
Fri, 19 Jan 01: HW1
due.
Lecture 5: PDEs (cont'd): Conics (cont'd), normal form,
matrix form, eigenvalue problem, rotation and translation of coordinate
system. Linear operators, nonlinear operators, affine operators, examples.
Reading assignment: Eriksson
et al., Chap 6, pp.104-117 (Galerkin method).
HW2:Please
download.
GiD:
a free academic version of a mesh generator and post-processor; please
install it in your computer, and produce some nice meshes (in the HW3;
invent your own meshes; be creative); I will ask some of you to present
your meshes in a future lecture (please print out a hardcopy).
Wed, 17 Jan 01:
Lecture 4: Classification of PDEs (cont'd): Proof of
conservation property of classification of PDEs under change of coordinates;
normal form of conics, matrix form, diagonalization of real symmetric matrices.
Reading assignment: Eriksson
et al., Chap 4, pp.61-70 (review of linear algebra; in particular,
eigenvalue problems).
Mon, 15 Jan 01: Martin Luther King, Jr. holiday. No class.
WEEK 2
Fri, 12 Jan 01:
Lecture 3:Classification of PDEs (cont'd):
Change
of coordinates; conservation of classification of PDE under change of coordinates,
statement; conics, general equation in the plane, normal form (eigenvalue
problem, change of coordinates), classification of conics. Examples of
PDEs and their classification in the plane; local feature of their classification.
Reading assignment: Eriksson
et al., Chap 1 (Introduction) and Chap 2 (History).
HW1
Did you know: Around this time last year, I read in a Saturday Gainesville Sun recently that some researchers have ranked the 100 most influential speeches in US history, and that the "I have a dream" speech of Martin Luther King, Jr., was ranked number one, far above the second runner up, based not only on its content, but also on its masterful and emotional delivery, fittingly on the steps of the Lincoln Memorial in Washington, D.C., in Aug 1963. One of the famous lines in that speech: "I have a dream that my little four children will one day live in a nation where they will not be judged by the color of their skin, but by the content of their character."
Wed, 10 Jan 01: Remarks on course web site,
syllabus, Matlab, etc.
Lecture 2:Classification of PDEs:
Examples of simple second-order linear PDEs in engineering. Lecture plan
on this topic. General form of PDEs; notation; definition of order of PDEs,
quasi-linearity, linearity; classification of 2nd-order quasi-linear PDEs;
conics, geometric definition.
Reading assignment: Matlab
Primer by the late UF math professor K. Sigmon; Cooperative
learning techniques.
You should practice
using Matlab in parallel to your reading
of the Matlab Primer starting from today.
HW1:Please
download. Reproduce all examples
in the
Matlab
Primer. HW sets are numbered using the Week numbers.
Thus,
in Week 1, the HW set is designatedby HW1.
Mon, 8 Jan 01: Course organization, cooperative learning, video presentation of the use of FEM in industry and other examples, history of the development of the FEM.
WEEK 1