Can you " Approximate " what you called as Random ?
Nimo
For the Last 1 month or so
I'm extensively investigating about
" Analysis & Synthesis" concepts;
I'm going to close this chapter
by this very last question.
Lets see...,
L u(.) = f___(1)
L is linear or Integral operator,
u(.) is un-known function,
f is known function;
well, we would restrict our problem only
to Numerical analysis only
{ I mean boundary value problems }
For simplicity we will take u in one variable only
=> i.e. u(x)
now, what I'm saying is
lets take the ' f ' values as Random;
we will take Pi digits for first 1000 places.
that eq(1) can be solved in variety of ways
we will go with method of weighted residuals[ Galerkin appraoch ]
for the mesh analysis.
lets take all the basis functions as piecewise st.lines
u =~ U = Sig j=1_N aj *Bj
where
=> aj are co-efficients,
=> Bj are basis functions {St.lines}
Doubt:-
At the end we will get an equation as a Result
Can that equation would interpolate all the Pi digits correctly or
at least approximatley.
If not then, why there is so much hype that
FEM is better for " complex " Geometries
those who said that, it is good for " Complex " Geometries
How will they define the word " Complex " ?
Thnx a lot for the answers
____
Why should I refuse a good dinner
simply because I don't understand
the digestive processes involved?
Oliver Heaviside
Peter Spellucci
In article <1adaf238-667b-40fe...@e24g2000vbe.googlegroups.com>,
in R^1 an interval is hardly a complex geometry
hence on the very entry you are on the false track
and yes, if you you extract finitely many random numbers,
and built a linear model with sufficient parameters and
a stable scheme (means: the matrix is at most mildly illconditioned)
then the result will reproduce this, but the sense of all this
will be questionable at least.
what concerns FEM via FD on complex geometries:
you have a car , I assume. open the engine and extract a piston.
look at it: this is an elastic body (yes) and engineers might want
to compute its deformation under different loads.
well , we are in R^3, the equations of elasticity are known,
material properties are known, and for small loads the linearity
of the model can be assumed, hence we are at (1).
you can well discretize this by FD or FEM, but think how much
trouble you will get with the boundaries:
for FD it will end up in constructing a special formula for every
boundary point separately. happy discretizing.
hth
peter
Nimo
Spellucci) wrote:
> In article <1adaf238-667b-40fe-ae03-309c86dfc...@e24g2000vbe.googlegroups.com>,
>
>
>
> Nimo <azeez...@gmail.com> writes:
> >hi....,
>
> >For the Last 1 month or so
> >I'm extensively investigating about
> >" Analysis & Synthesis" concepts;
>
> >I'm going to close this chapter
> >by this very last question.
>
> >Lets see...,
>
> >L u(.) = f___(1)
>
> >L is linear or Integral operator,
while writing, small mistake occurred here
sorry for that. It should be like,
L is Linear differential or Integral operator.
Exactly; peter you are 100% correct;
thnx a lot for the clarifications;
In this regard, I want to say two statements here,
stat:1) For computer "simulation" numerical PDE's are opted
stat:2) but not, "computers" to solve PDE's numerically
the difference is very obvious in those 2 statements.
______
=> L u(.) = f____(1)
{ for the sake of discussion, we will put Linear operators
aside and will directly deal with functions only }
then (1) becomes
u(.) = f___(2)
u(.) is unknown function
f is known.
As there is no differential operator here;
I think we don't need of any " weak formulation " anymore
so in general we can write u(.) as
u=~ U = Sig j=1_N aj Bj(.) + error
where
aj's are co-efficients
Bj(.) are basis functions
up to here, I think everything is clear
for discretization,
we will opt the method of weighted residuals[Galerkin approach]
at the end will get a resulting function.
_____
In the above procedure a small modification, lets see
=> u(x) = f(x)___(1)
=> u(x)=~ U(x) = Sig j=1_N aj Bj(x) + small error___(2)
aj are co-effs
Bj(x) are St.lines
=> Residual(R) = U(x)-f(x) = aj Bj(x) - f(x)___(3)
now, for discretization we will invoke
method of weighted residuals
=> Integral w(x) * residual = 0______(4)
substituting (3) in (4) we get
=> Integral w(x) * { U(x)-f(x) } = 0
=> Integral wi(x) * { aj Bj(x)-f(x) } = 0 ; i=1,2,3,..n
what my point is I'll use ' f ' as random; I mean
I'll use ' Pi ' digits for first 500 or 1000 places
Q1) can the resulting function(St.line) can interpolate all
the Pi values correctly ? if no, why?
then what's the use with Element analysis stuff ?
ans)
__________
Q2) well, in the unknown function u(.)
I'll restrict u(.) to u(x)
my u(x) has certain specification;
let me tell you all those things
so that you can help me very clearly
a) it doesn't has any discontinuities;
I mean, it is smooth.
b) u(x) is "Always" ' increasing '
c) I've to discretize it for millions of points,
{ think like fluid flow problem }
users suggested me to opt
Spectral element method to solve the problem
I think both FEM and SEM are same, but SEM is
extension to FEM by choosing basis functions as
"higher" degree polynomials
I'm very confusing,
How to discretize it and solve the problem?
ans)
______
so long
nimo
Book reference or
any web links especially for ' SEM '
thnx a lot