mm-273
===
Subject: Re: Hexagonal and pentagonal finite element> I
am trying to write the shape function (seredenpity) of two
linear > elements: the hexagonal prism and the pentagonal
prism (a prism with > respectively 12 vertices and 10
vertices).Sorry never used these before. But you might be able
to constructthem via a triangular prism (6 vertices). The
triangular prism iseasy, take the tensor product of a linear
triangle and a linear 1Delement (edge).> BTW, does anyone know
the shape functions for a linear pyramid (5 vertex) ?You can
use so-called rational basis functions. We have not testhem
extensively, however...phi0 = .25*(zeta + xi - 1.)*(zeta + eta
- 1.)/((1. - zeta) + eps);phi1 = .25*(zeta - xi - 1.)*(zeta +
eta - 1.)/((1. - zeta) + eps);phi2 = .25*(zeta - xi -
1.)*(zeta - eta - 1.)/((1. - zeta) + eps); phi3 = .25*(zeta +
xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);phi4 =
zeta;where (xi, eta, zeta) are points in the reference pyramid
with basexi = [-1,1] (cross) eta = [-1,1] and zeta = [0,1]. eps
is a smallnumber to prevent dividing by zero when zeta=1.Hope
===
element> Sorry never used these before. But you might be able
to construct> them via a triangular prism (6 vertices). The
triangular prism is> easy, take the tensor product of a linear
triangle and a linear 1D> element (edge).I tried decomposing it
into 3 linear wedges. But I had to add two new points (center
of both hexahedron basis), thus creating some fake edges. And
some algorithm like contouring (which rely on these
interpolations functions) failed to produce the right result
(the fake edges introduced fake points).>>BTW, does anyone
know the shape functions for a linear pyramid (5 vertex) ?>
You can use so-called rational basis functions. We have not
tes> them extensively, however...> phi0 = .25*(zeta + xi -
1.)*(zeta + eta - 1.)/((1. - zeta) + eps);> phi1 = .25*(zeta -
xi - 1.)*(zeta + eta - 1.)/((1. - zeta) + eps);> phi2 =
.25*(zeta - xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps); >
phi3 = .25*(zeta + xi - 1.)*(zeta - eta - 1.)/((1. - zeta) +
eps);> phi4 = zeta;> where (xi, eta, zeta) are points in the
reference pyramid with base> xi = [-1,1] (cross) eta = [-1,1]
and zeta = [0,1]. eps is a small> number to prevent dividing
by zero when zeta=1.Ok, my pyramid base is [0,1]x[0,1] and I
came with:phi0 = (1. - xi)*(1. - eta)*(1. - zeta);phi1 = xi
*(1. - eta)*(1. - zeta);phi2 = xi * eta *(1. - zeta);phi3 =
(1. - xi)* eta *(1. - zeta);phi4 = zeta;But I was told there
could be some problems for vertex zeta=1, which I guess is
===
Hexagonal and pentagonal finite element> Sorry never used
these before. But you might be able to construct> them via a
triangular prism (6 vertices). The triangular prism is> easy,
take the tensor product of a linear triangle and a linear 1D>
element (edge).> I tried decomposing it into 3 linear wedges.
But I had to add two new > points (center of both hexahedron
basis), thus creating some fake edges. > And some algorithm
like contouring (which rely on these interpolations >
functions) failed to produce the right result (the fake edges
introduced > fake points).>>BTW, does anyone know the shape
functions for a linear pyramid (5 vertex) ?You can use
so-called rational basis functions. We have not tes> them
extensively, however...> phi0 = .25*(zeta + xi - 1.)*(zeta +
eta - 1.)/((1. - zeta) + eps);> phi1 = .25*(zeta - xi -
1.)*(zeta + eta - 1.)/((1. - zeta) + eps);> phi2 = .25*(zeta -
xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps); > phi3 =
.25*(zeta + xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);>
phi4 = zeta;where (xi, eta, zeta) are points in the reference
pyramid with base> xi = [-1,1] (cross) eta = [-1,1] and zeta =
[0,1]. eps is a small> number to prevent dividing by zero when
zeta=1.> Ok, my pyramid base is [0,1]x[0,1] and I came with:>
phi0 = (1. - xi)*(1. - eta)*(1. - zeta);> phi1 = xi *(1. -
eta)*(1. - zeta);> phi2 = xi * eta *(1. - zeta);> phi3 = (1. -
xi)* eta *(1. - zeta);> phi4 = zeta;> But I was told there
could be some problems for vertex zeta=1, which I > guess is
rela to your eps.> What do you think ?> MathieuCorrect. If a
conforming displacement model is used, ata pyramid vertex the
Jacobian matrix is singular and its ordinaryinverse does not
exist. This is true for any polyhedra corner at whichmore than
3 faces meet. Gradients and strains are undefined therebecause
they involve the Jacobian inverse.For the shape functions of
the 5-node pyramid (which arewell known) see Sec 17.8 of the
class Notes
athttp://caswww.colorado.edu/courses.d/AFEM.d/AFEM.Ch17.d/
===
element> Correct. If a conforming displacement model is used,
at> a pyramid vertex the Jacobian matrix is singular and its
ordinary> inverse does not exist. This is true for any
polyhedra corner at which> more than 3 faces meet. Gradients
and strains are undefined there> because they involve the
Jacobian inverse.> For the shape functions of the 5-node
pyramid (which are> well known) see Sec 17.8 of the class
Notes at>
http://caswww.colorado.edu/courses.d/AFEM.d/AFEM.Ch17.d/
===
the process of writing code to model force fields within
thecrushing chamber of a rock crusher. I have a method that
shows promise. Theprocess requires input functions describing
pressure fields on the 3dcrushing faces. Once the functions
are input they are integra over theaffec area. The integrals
tend to be rather straightforward but toocalculation intensive
to do manually on a large scale basis. I have testhe method
using Mathcad and in cases where I can calculate a
theoreticallyexact solution that does not require numeric
methods, the Mathcadinvariably give the exact answer to high
degree of precision.My problem is that in order to move on to
Mathcad method works fine for manual input of aparticular
problem but I need to automate the process. I can write all
thecode, but the integration routines that I have tried
(gleened from mynumeric-methods class in college) never seem
to be as precise as the Mathcadmethod. I am getting bogged
down in the search for a best method forintegrating these
functions instead of working on the rock crusher stuffwhich is
my primary project.Are there public domain (not necessarily C++
code) methods that will improvemy precision? What kind of
methods are used in programs like Mathcad, Matlabetc that
allow them to take any function at random and kick out a
===
C++ integration codeX-NewsReader: GRn 3.2n February 9, 1999> I
am in the process of writing code to model force fields within
the> crushing chamber of a rock crusher. I have a method that
shows promise. The> process requires input functions
describing pressure fields on the 3d> crushing faces. Once the
functions are input they are integra over the> affec area. The
integrals tend to be rather straightforward but too>
calculation intensive to do manually on a large scale basis. I
have tes> the method using Mathcad and in cases where I can
calculate a theoreticallyexact solution that does not require
numeric methods, the Mathcad> invariably give the exact answer
to high degree of precision.> My problem is that in order to
move on to the next step, I need to write> some standalone
code. The Mathcad method works fine for manual input of a>
particular problem but I need to automate the process. I can
write all the> code, but the integration routines that I have
tried (gleened from my> numeric-methods class in college)
never seem to be as precise as the Mathcad> method. I am
getting bogged down in the search for a best method for>
integrating these functions instead of working on the rock
crusher stuff> which is my primary project.> Are there public
domain (not necessarily C++ code) methods that will improve>
my precision? What kind of methods are used in programs like
Mathcad, Matlab> etc that allow them to take any function at
random and kick out a precise> answer most of the time?May I
ask a question? Why are you concerned with precisionof
computations when dealing with what would seem to be arather
random actions? At any instant in time, the crusher/may/ be
exerting a force on a point and in the next instantit will be
doing nothing (the rock broke and the mechanismhasn't move far
===
C++ integration code> Are there public domain (not necessarily
C++ code) methods that will improve> my precision? What kind of
methods are used in programs like Mathcad, Matlab> etc that
allow them to take any function at random and kick out a
precise> answer most of the time?> Jon JuhlinThe basic method
is to compute an integral with N points and 2*Npoints, and to
stop doubling the number of points when the estimatesagree to
within a specified tolerance.To point you to specific
algorithms and software, you should tell uswhether(1) the
integrals are 1-D or multidimensional(2) the bounds of
integration are finite or infinite (is the
integralimproper?)(3) there exist integrable singularities in
your functionThe best-known software for numerical integration
(quadrature) is theFortran package QUADPACK. The original
Fortran 77 version is
athttp://www.netlib.org/quadpack/index.html , and a
translation to moremodern Fortran 90 by John Burkardt is
athttp://www.csit.fsu.edu/~burkardt/f_src/quadpack/
quadpack.html .Alan Miller has several F90 quadrature codes
athttp://users.bigpond.net.au/amiller/ -- search for
integration.The Numerical Recipes library, which you can buy
online at www.nr.comin C++ and Fortran versions, has
quadrature code that may be adequatefor your application.Alan
Genz has much Matlab and Fortran quadrature code
athttp://www.math.wsu.edu/faculty/genz/software/software.html
.If you want a stand-alone solution, I recommend using C++ or
===
code> I am in the process of writing code to model force
fields within the> crushing chamber of a rock crusher. I have
a method that shows promise. The> process requires input
functions describing pressure fields on the 3d> crushing
faces. Once the functions are input they are integra over the>
affec area. The integrals tend to be rather straightforward but
too> calculation intensive to do manually on a large scale
basis. I have tes> the method using Mathcad and in cases where
I can calculate a theoreticallyexact solution that does not
require numeric methods, the Mathcad> invariably give the
exact answer to high degree of precision.> My problem is that
in order to move on to the next step, I need to write> some
standalone code. The Mathcad method works fine for manual
input of a> particular problem but I need to automate the
process. I can write all the> code, but the integration
routines that I have tried (gleened from my> numeric-methods
class in college) never seem to be as precise as the Mathcad>
method. I am getting bogged down in the search for a best
method for> integrating these functions instead of working on
the rock crusher stuff> which is my primary project.> Are
there public domain (not necessarily C++ code) methods that
will improve> my precision? What kind of methods are used in
programs like Mathcad, Matlab> etc that allow them to take any
function at random and kick out a precise> answer most of the
time?> Jon JuhlinIt would be helpful if you would provide a
little more detail. If you aretalking about 'symbolic'
integration, i.e. finding exact, closed form solutionsfor
integrals, you would be on a totally different path than if
you simply wantto evaluate integrals numerically. For the
latter case, you might do a searchfor QUADPACK or search for
Prof. Ooura's website which has some very robustcodes for
integration.--There are two things you must never attempt to
prove: the unprovable -- and
===
methods for solving systems of linear equationsI need some
info for those methods (for sparse matrices): relaxation
method,block iteration method, ADI method, cg Method, Buneman
Algorithm fordiscrete Poisson equation, Multigrid MethodI was
looking for it, but i didnt found any simple
theoreticalexplanations with solved examples... so im looking
for simple theoreticalexplanations with solved examples...
please, help me and name someweb-pages or pdf/doc documents...
===
of linear equations> I need some info for those methods (for
sparse matrices): relaxation method,> block iteration method,
ADI method, cg Method, Buneman Algorithm for> discrete Poisson
equation, Multigrid Method> I was looking for it, but i didnt
found any simple theoretical> explanations with solved
examples... so im looking for simple theoretical> explanations
with solved examples... please, help me and name some>
Recipes provides an introduction to some of thesetopics. You
can download the pdf files of the relevant sections
===
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am a student on biometrical genetics in plants and I would
like toknow more about maximum likelihood ratio and maximum
look forward youas soon as possible.sincerely
===
&estimation> I am a student on biometrical genetics in plants
and I would like to> know more about maximum likelihood ratio
attention and I look forwardThis topic should be covered in
===
Re: Eigenvectors of the laplacian> I am looking for
information on the properties of the Laplacian on> a bounded
domain of R^n.> In particular, if one consider a bounded open
set O with a regular> boundary, I would like to have a
complete study of the properties of> the eigenvectors
{v1,v2,...}> of the laplacian with Neumann boundary
conditions.> In fact, I am looking forward some approximation
results.> For example, given a C^1 regular function f on O,>
which is the convergence speed of the approximation>
sum{i<N}{<f,vi> vi} when N-->infinity ?> This is in fact the
study of the spectrum {<f,vi>}> of a regular function f.> At
last (but not least), I would like to have some insight> about
discrete approximation results.> For example, when I have a
grid that approximates the domain> O, and the laplacian is
Gabrielasymptotic behavior of the spectrum, extending the
famous workby (again IIRC) the famous paper of Herman Weyl.--
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God
is not willing to do everything and thereby take away our free
will and that share of glory that rightfully belongs to us. --
===
laplacian> Some books on functional analysis discuss this. I
think the bookFunctional Analysis in Normed Spaces by
Kantorovich and Akilov> (Oxford, New York, 1964) discusses it,
but my copy is elsewhere> at the moment.Does this book deals
with approximation in the (Hilbert)basis of the eigenvectors
===
Kalman filter helpYou might also look at
http://www.taygeta.com/kalman.html>Try this URL>
http://ourworld.compuserve.com/homepages/PDJoseph/kalman.htm>I
bought the lessons from Peter Joseph and the seemed to be quite
clear.>> Can you suggest a book that is not too heavy on the
math? I am mostly>> looking for implementation examples. I
have looked at enough>> derivations and would rather look at
real world examples.>> Try R.M. RogeApplied Mathematics in
Integra Navigation Systems, >> published by AIAA. It is all
about using Kalman filters to combine >> navigation solutions
from multiple sensors.>> Lars