Subject: numerical integration
From: charron@grbb.polymtl.ca (Guy Charron)
Date: 17 Dec 1996 19:22:48 GMT
Salut everyone,
I have a problem I'm sure I am not the first one to encounter, so I would appreciate a little feedback on this. To solve a differential equation in an electromagnetic
problem, I am using Green functions. The solution is known, but expressed in the form
of a complicated multidimensional integral. I have to perform an integral from
zero to infinity of a product of two bessel functions and a rational fonction. As the
variable tend toward infinity, the integrand tend toward zero.
For some values of parameters, the function may need to be integrated to around 10
before the integral converges, some other cases require that it has to
be integrated up until 100 000 or so. What would be the best way to minimise
computation time in such a case.
There is also the problem of how many points should be used to perform the
integral. The main problem is that the integrand oscillate, and that there is
a slight difference between the positive part and the negative part of the function
that slowly add up to give me the result. What would be a good way to tackle this
problem?
Currently, I'm working with Matlab. I am using a Gauss rule, ( Can anyone point
me some reference on this) where basically the algorithm chooses points and calculate
a weigth for them so that the integral is exact for a polynom of order 2*n-1 if I
use n points. To tackle both problem, I do the integral on 0 to 100, then on 100 to 200 etc, until the new contribution becomes sufficently small. I have to check the
integrande before though, because in some cases, to evaluate it I multiply large number
by small number, and the programm doesn't know what to do exactly with that ( but I
am sure that the small number is way smaller than 1/(large number)).
So basically, that is where I am. Any comments anyone?
--
Guy Charron
-pensee de la semaine-
La vie n'a de sens que celui qu'on lui donne
-Croenberg, sur le film "Crash"-
lundi le 16 dec.
Subject: please discard the previous article
From: clement@cs.sc.edu (George Clement)
Date: 17 Dec 1996 20:29:29 GMT
clement@cs.sc.edu (George Clement) writes:
>kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) writes:
>>In article <32B607FB.211A3107@asu.edu>,
>>Hans D Mittelmann wrote:
>>:George Clement wrote:
>>:>
>>:> Hi everybody,
>>:>
>>:> I am new to this group.
>>:>
>>:> I have a question.
>>:>
>>:> Is there a formula using which one can calculate
>>:> 1 + x + x^2 + x^3 + ...... + x^n
>>:>
>>:> Given values for x and n, I would like to calculate the
>>:> progressive expression using a formula (instead of calculating
>>:> it iteratively).
>>:>
>>:> Thanks for all the help,
>>:>
>>:> George Clement
>>:> gclement@imonics.com
>>:Hi,
>>:are you sure you didn't have that in high-school? It's a finite
>>:geometric series with value
>>: (1-x^(n+1))/(1-x)
>>:
>>(unless x=1, and in this case the sum is (n+1))
>> Also, if we call the sum s_n then: s_0 = 1, and recursively
>> s_(n+1) = 1 + x * s_n
>> this time with no restrictions on x.
>>(Some high schools, I heard, are so busy teaching self-esteem that they
>>have no time for a finite geometric series :-)
>>Cheers, ZVK (Slavek).
>I just noticed that the formula works only if x is a whole number > 1.
>Is iteration the only way of calculating this series if x is fractional? like x=2.08
>Thanks,
>George Clement
Sorry, please discard the previous mail. I was trying the
formula for values x=0.08.
Thanks,
George Clement
Subject: Re: help required in finding a formula
From: clement@cs.sc.edu (George Clement)
Date: 17 Dec 1996 20:00:35 GMT
kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) writes:
>In article <32B607FB.211A3107@asu.edu>,
>Hans D Mittelmann wrote:
>:George Clement wrote:
>:>
>:> Hi everybody,
>:>
>:> I am new to this group.
>:>
>:> I have a question.
>:>
>:> Is there a formula using which one can calculate
>:> 1 + x + x^2 + x^3 + ...... + x^n
>:>
>:> Given values for x and n, I would like to calculate the
>:> progressive expression using a formula (instead of calculating
>:> it iteratively).
>:>
>:> Thanks for all the help,
>:>
>:> George Clement
>:> gclement@imonics.com
>:Hi,
>:are you sure you didn't have that in high-school? It's a finite
>:geometric series with value
>: (1-x^(n+1))/(1-x)
>:
>(unless x=1, and in this case the sum is (n+1))
> Also, if we call the sum s_n then: s_0 = 1, and recursively
> s_(n+1) = 1 + x * s_n
> this time with no restrictions on x.
>(Some high schools, I heard, are so busy teaching self-esteem that they
>have no time for a finite geometric series :-)
>Cheers, ZVK (Slavek).
I just noticed that the formula works only if x is a whole number > 1.
Is iteration the only way of calculating this series if x is fractional? like x=2.08
Thanks,
George Clement
Subject: Re: Problem: Gravity simulator
From: scott@eviews.com (Scott Ellsworth)
Date: Tue, 17 Dec 96 23:08:59 GMT
In article <593rm7$3nh$1@o.online.no>, larsmg@online.no (Lars Marius Garshol) wrote:
>In article
> ,
>ghh@mpq.mpg.de says...
>>
>>The problem you describe is usually solved by "adaptive stepsize control"...
>>If you have trouble finding references for these, I can tell you some
>>literature pointers.
>...
> Do you know of
>any online sites that carry this kind of information?
You will likely have to just hit the books on this one. Before you do,
though, visit Amara Graps' Web site, on which she has put together a very
complete study of the N-body problem, the various ways to solve it with
differing numbers of bodies, and the people working on the problem. She may
be found at The nbody stuff may be found in
papers/nbody.html, last time I checked.
Also, she has good taste in pictures and illustrations on the Web site.
Scott
Scott Ellsworth scott@eviews.com
"When a great many people are unable to find work, unemployment
results" - Calvin Coolidge, (Stanley Walker, City Editor, p. 131 (1934))
"The barbarian is thwarted at the moat." - Scott Adams
Subject: Re: Problem: Gravity simulator
From: Gerhard Heinzel
Date: Wed, 18 Dec 1996 10:40:46 +0100
On 16 Dec 1996, Lars Marius Garshol wrote:
> In article ,
> ghh@mpq.mpg.de says...
> >
> >The problem you describe is usually solved by "adaptive stepsize control".
> >A good method to solve this kind of problem are so-called "embedded
> >Runge-Kutta Formulae" (some well known variants are called
> >Runge-Kutta-Fehlberg.) Another variety which may be useful here is called
> >Runge-Kutta-Nystrom (for second order diff. equn's without first
> >derivatives).
> >
> >If you have trouble finding references for these, I can tell you some
> >literature pointers.
>
> I do have trobule, so yes, please, I'd like to have as many pointers as possible. Do you know of
> any online sites that carry this kind of information?
>
> Thanks,
> --Lars M.
>
>
If you can read german, there is a long article in the german computer
magazine "c't" 8/92 including sample code. Here are some references:
I do have some C-programs as well, but they are not well documented, or
only in german.
\begin{thebibliography}{99}
\bibitem{Hairer1}
Hairer,E.; N{\o}rsett,S.P.; Wanner,G.: {\em Solving Ordinary Differential
Equations I : Nonstiff Problems}: Springer Series in Comp. Math. 8 (1987)
\bibitem{Fe69}
Fehlberg, E.: {\em Klassische Runge--Kutta--Formeln f"unfter und siebenter
Ordnung mit Schrittweiten--Kontrolle}: Computing 4 (1969) pp. 93--106 mit
Berichtigung in Band 5 (1970), p.184
\bibitem{Fe85}
Fehlberg, E.: {\em Some Old and New Runge--Kutta Formulas with Stepsize Control
and Their Error Coefficients}: Computing 34 (1985) pp. 265--270
\bibitem{DP80}
Dormand, J.R., Prince, P.J.: {\em A family of embedded Runge--Kutta
formulae}: Journal of Computat. and Appl. Math. 6 (1980) pp. 19--26
\bibitem{DP81}
Dormand, J.R., Prince, P.J.: {\em High order embedded Runge--Kutta
formulae}: Journal of Computat. and Appl. Math. 7 (1981) pp. 67--75
\bibitem{DP86}
Dormand, J.R., Prince, P.J.: {\em Runge--Kutta Triples}: Comp. \& Maths.
with Appls. Vol. 12A (1986) pp.1007--1017
\bibitem{DP86a}
Dormand, J.R., Prince, P.J.: {\em A reconsideration of some embedded Runge--
Kutta formulae}: Journal of Computat. and Appl. Math. 15 (1986) pp. 203--211
\bibitem{DP89}
Dormand, J.R., Prince, P.J.: {\em Practical Runge--Kutta Processes}:
SIAM J.Sci.Stat.Comput. 10 (1989) pp. 977--989
\bibitem{Dormand}
Mitgeteilt von Dr. John Dormand, Teesside Polytechnic,
Middlesbrough, Cleveland, Gro"sbritannien.
\bibitem{art}
Shampine, L.F., Watts, H.A.: {\em The Art of Writing a Runge--Kutta Code
II.}: Applied Math. and Computation 5 (1979) pp. 93--121
\bibitem{Cal90}
Calvo, M., Montijano, J.I., R\'{a}ndez, L.: {\em A new embedded pair of Runge--
Kutta--Formulas of orders 5 and 6}: Computers Math. Applic. 20 (1990)
pp.15--24
\bibitem{Randez}
Mitgeteilt von Dr. Luis R\'{a}ndez, Universidad de Zaragoza, Spanien.
\bibitem{Verner}
Verner, J.H.: {\em Some Runge--Kutta Formula Pairs}: SIAM J. Numer. Anal. 28
(1991) pp.496--511
\bibitem{DPN87a}
Dormand, J.R., El--Mikkawy, M.E.A., Prince, P.J.: {\em Families of
Runge--Kutta--Nystrom Formulae}: IMA J.Numer.Anal. 7 (1987), pp.235--250
\bibitem{DPN87b}
Dormand, J.R., El--Mikkawy, M.E.A., Prince, P.J.: {\em High--Order
Embedded Runge--Kutta--Nystrom Formulae}: IMA J.Numer.Anal. 7 (1987),
pp.423--430; {\em Korrigendum} Band 11 (1991), p.297
\bibitem{DPN87c}
Dormand, J.R., Prince, P.J.: {\em Runge--Kutta--Nystrom Triples}: Comp. \&
Maths. with Appls. 13 (1987) pp.937--949
\bibitem{FeNy72}
Fehlberg, E.: {\em Klassische Runge--Kutta--Nystr"om--Formeln mit
Schrittweiten--
Kontrolle f"ur Differentialgleichungen $\ddot{x}=f(t,x)$}, Computing 10
(1972), pp. 305--315
\bibitem{FiNy85}
Filippi, S., Gr"af, J.: {\em Ein Runge--Kutta--Nystr"om--Formelpaar der Ordnung
11(12) f"ur Differentialgleichungen der Form $y''=f(x,y)$}, Computing 34
(1985), pp. 271--282
\bibitem{Hairer2}
Hairer,E.; Wanner,G.: {\em Solving Ordinary Differential
Equations : Stiff and Differential-Algebraic Problems}: Springer Series in
Comp. Math. 14 (1991)
\bibitem{Fatunla}
Fatunla,S.O.: {\em Numerical Methods for Initial Value Problems in Ordinary
Differential Equations}: Academic Press (1988)
\bibitem{Bloch}
Bloch, R.: {\em A Practical Method to Integrate Some Stiff Systems}:
Computers and Biomedical Research 24 (1991) pp. 420--428
\end{thebibliography}
Good Luck,
Gerhard Heinzel
=====================================================================
Gerhard Heinzel E-mail: ghh@mpq.mpg.de
Max-Planck-Institut fuer Quantenoptik
Hans-Kopfermann-Str. 1 Phone: +49(89)32905-268
D-85748 Garching -252
Germany Fax: +49(89)32905-200
=====================================================================
Subject: FE shell element not using rotations
From: damianm@eram.esi.com.au (Damian McGuckin)
Date: 18 Dec 1996 23:11:21 +1100
A long while ago, Thomas Hughes developed a (quadrilateral) shell element
for use in stress analysis problems which had a finite thickness and had
no rotations associated with it.
Basically, it was a degenerated brick element which started off in life
in side view as
Z
| 8*-------------------------*7
| /: /|
o---X / : / |
/ / : / |
/ / 4#...................../...#3
Y / . / /
/ . / /
5*-------------------------*6 /
| . | /
| . | /
|. |/
1#-------------------------#2
where any node has displacements only associated with it, namely
(u , v , w )
i i i
However the degrees of freedom associated with the nodes, i.e. those
variables which appear in the equations which we solve ( e.g. K * u = F),
are
(u , v , w ) for i = 1, 2, 3, 4
i i i
and
(u', v', w') for i = 5, 6, 7, 8
i i i
where
u' = u - u if i > 3, i.e. are relative displacements.
i i i-4
This apparently worked but has the disadvantage that the shell is not
symmetric about the X-Y central plane.
You could just as easily draw the above a 12 noded element and have the
degrees of freedom about the central plane as the actual displacements
and the degrees of freedom on the top and bottom of the element as the
relative displacements. However, whether this made a viable finite
element is guess on my part.
Has anybody seen any work done on something like this variation, did
it fail, did it work, or have people just lost interest?
As an aside, do you submit this FE stuff to groups like sci.engr and
sci.engr.{civil, mech} or don't they appreciate it?
Thanks - Damian
--
Damian McGuckin / Pacific Eng Sys Int'l | d a m i a n m @ e s i . c o m . a u
Ph:+61-2-99063377 ... Fx:+61-2-99063468 | unsolicited email not wanted here !
Views and opinions here are mine and not those of any past or present employer.
