manual-full.tex

FreeFem++可以生成高质量的有限元网格。可以用于流体力学

\itemtt[false]  means ``false'' in  \Bool\ value.
\itemtt[pi]   is the \Real ~approximation value of $\pi$.
\end{description}

\medskip

Here is how to show all the types, and all the operator and functions.
\bFF
 @dumptable(@cout);
\eFF \index{dumptable}
To execute a system command in the string (not implemented on Carbon
MacOs)
\bFF
  @exec("shell command");
\eFF
On MS-Windows, we need the full path. For example, if there is the command
``ls.exe'' in the subdirectory ``\verb|c:\cygwin\bin\|'', then we must write
\bFF
  @exec("c:\\cygwin\\bin\\ls.exe");
\eFF
\index{exec}


\subsection{Arithmetic}
In integers, $+,\, -,\, *$ express the usual arithmetic summation (plus),
subtraction (minus) and multiplication (times), respectively.
The operators $/$ and $\%$ yield the quotient and the remainder from the division of the first expression by the second.
If the second number of $/$ or $\%$ is zero the behavior is undefined.
The \key{maximum} or \key{minimum} of two integers $a,\, b$ are obtained
by \texttt{max($a$,$b$)} of
\texttt{min($a$,$b$)}.
The power $a^b$ of two integers $a,\, b$ is calculated by writing \verb|a^b|.
\begin{example} Calculations with the integers
\label{exm:int}
\bFF
@int a = 12, b = 5;
@cout <<"plus, minus of "<<a<<" and "<<b<<" are "<<a+b<<", "<<a-b<<@endl;
@cout <<"multiplication, quotient of them are "<<a*b<<", "<<a/b<<@endl;
@cout <<"remainder from division of "<<a<<" by "<<b<<" is "<<a%b<<@endl;
@cout <<"the minus of "<<a<<" is "<< -a << @endl;
@cout <<a<<" plus -"<<b<<" need bracket:"<<a<<"+(-"<<b<<")="<<a+(-b)<<@endl;
@cout <<"max and min of "<<a<<" and "<<b<<" is "<<@max(a,b)<<","<<@min(a,b)<< @endl;
@cout <<b<<"th power of "<<a<<" is "<<a^b<< @endl;
b=0;
@cout <<a<<"/0"<<" is "<< a/b << @endl;
@cout <<a<<"\%0"<<" is "<< a\%b << @endl;
\eFF
produce the following result:
\bFF
plus, minus of 12 and 5 are 17, 7
multiplication, quotient of them are 60, 2
remainder from division of 12 by 5 is 2
the minus of 12 is -12
12 plus -5 need bracket :12+(-5)=7
max and min of 12 and 5 is 12,5
5th power of 12 is 248832
12/0 : long long long
Fatal error : ExecError  Div by 0 at exec line  9
Exec error : exit
\eFF
\end{example}

By the relation $integer\subset real$, the operators
``$+,\, -,\, *,\, /,\, \%$'' and ``\ttCC{@max,\, @min,\, \^}''
are also applicable in real-type. However,  $\%$
calculates the remainder of the integral parts of two real numbers.

The following example similar to Example \ref{exm:int}
\bT
@real a=sqrt(2.), b = pi;
@cout <<"plus, minus of "<<a<<" and "<<pi<<" are "<< a+b <<", "<< a-b << @endl;
@cout <<"multiplication, quotient of them are "<<a*b<<", "<<a/b<< @endl;
@cout <<"remainder from division of "<<a<<" by "<<b<<" is "<< a%b << @endl;
@cout <<"the minus of "<<a<<" is "<< -a << @endl;
@cout <<a<<" plus -"<<b<<" need bracket :"<<a<<"+(-"<<b<<")="<<a + (-b) << @endl;
\eT
gives the following output:
\bT
plus, minus of 1.41421 and 3.14159 are 4.55581, -1.72738
multiplication, quotient of them are 4.44288, 0.450158
remainder from division of 1.41421 by 3.14159 is 1
the minus of 1.41421 is -1.41421
1.41421 plus -3.14159 need bracket :1.41421+(-3.14159)=-1.72738
\eT

By the relation
$$
bool\subset int \subset real\subset complex,
$$
the operators
``$+,\, -,\, *,\, /$'' and ``\ttCC{\^}'' are also applicable in complex-type,
but ``\%,\, max, min'' fall into disuse.
Complex number such as \texttt{5+9i},\, i$=\sqrt{-1}$, can be a little tricky.
For real variables \texttt{a=2.45, b=5.33}, we must write the complex numbers
\texttt{a+b*i} and \ttCC{a+@sqrt(2.0)*i} as
\bT
@complex z1 = a+b*1i, z2=a+@sqrt(2.0)*1i;
\eT
The imaginary and real parts of complex number \texttt{z} is obtained by
\ttCC{@imag} and \ttCC{@real}.
The conjugate of $a+bi$ ($a,b$ are real) is defined by $a-bi$, which
is denoted by \ttCC{@conj(a+b*1i)} in \freefempp.

The complex number $z=a+ib$ is considered as
the pair $(a,b)$ of real numbers $a,\, b$.
Now we draw the point $(a,b)$ in the plane (Cartesian rectangular system
of axes) and mark on the $x$-axis the real numbers in the usual way, on the
$y$-axis the imaginary numbers with $i$ as unit.
By changing Cartesian coordinate $(a,b)$ to the polar coordinate $(r,\phi)$,
the complex number $z$ has another expression $z=r(\cos \phi+i\sin\phi )$,
$r=\sqrt{a^2+b^2}$ and $\phi=\tan^{-1}(b/a)$;
$r$ is called the \key{absolute value} and $\phi$ the \key{argument} of $z$.
In the following example, we shall show them using \freefempp programming,
and \key{de Moivre's formula} $z^n=r^n(\cos n\phi+i\sin n\phi)$.

\begin{example}~
\label{exm:complex}
\bFF
@real a=2.45, b=5.33;
@complex  z1=a+b*1i, z2 = a+sqrt(2.)*1i;
@func @string pc(@complex z) // printout complex to (real)+i(imaginary)
{
   @string r = "("+real(z);
   if (@imag(z)>=0) r = r+"+";
   @return r+@imag(z)+"i)";
}
// printout complex to |z|*(cos(arg(z))+i*sin(arg(z)))
@func @string toPolar(@complex z)
{
   @return @abs(z)+"*(cos("+@arg(z)+")+i*sin("+@arg(z)+"))";
}
cout <<"Standard output of the complex "<<pc(z1)<<" is the pair "
     <<z1<<endl;
cout <<"Plus, minus of "<<pc(z1)<<" and "<<pc(z2)<<" are "<< pc(z1+z2)
     <<", "<< pc(z1-z2) << endl;
cout <<"Multiplication, quotient of them are "<<pc(z1*z2)<<", "
     <<pc(z1/z2)<< endl;
cout <<"Real/imaginary part of "<<pc(z1)<<" is "<<@real(z1)<<", "
     <<@imag(z1)<<endl;
cout <<"Absolute of "<<pc(z1)<<" is "<<@abs(z1)<<endl;
cout <<pc(z2)<<" = "<<toPolar(z2)<<endl;
cout <<"  and polar("<<@abs(z2)<<","<<@arg(z2)<<") = "
     << pc(@polar(abs(z2),arg(z2)))<<endl;
cout <<"de Moivre's formula: "<<pc(z2)<<"^3 = "<<toPolar(z2^3)<<endl;
cout <<"conjugate of "<<pc(z2)<<" is "<<pc(@conj(z2))<<endl;
cout <<pc(z1)<<"^"<<pc(z2)<<" is "<< pc(z1^z2) << endl;
\eFF
Here's the output from Example \ref{exm:complex}
\bT
Standard output of the complex (2.45+5.33i) is the pair (2.45,5.33)
Plus, minus of (2.45+5.33i) and (2.45+1.41421i) are (4.9+6.74421i), (0+3.91579i)
Multiplication, quotient of them are (-1.53526+16.5233i), (1.692+1.19883i)
Real/imaginary part of (2.45+5.33i) is 2.45, 5.33
Absolute of (2.45+5.33i) is 5.86612
(2.45+1.41421i) = 2.82887*(cos(0.523509)+i*sin(0.523509))
  and polar(2.82887,0.523509) = (2.45+1.41421i)
de Moivre's formula: (2.45+1.41421i)^3
                         = 22.638*(cos(1.57053)+i*sin(1.57053))
conjugate of (2.45+1.41421i) is (2.45-1.41421i)
(2.45+5.33i)^(2.45+1.41421i) is (8.37072-12.7078i)
\eT
\end{example}

\subsection{\setS{One Variable Functions}}
\index{functions}
\begin{description}
  \item[Fundamental functions] are built into \freefempp.
The \emph{power function} \ttCC{x\^}$\alpha (=x^\alpha)$;
the \emph{exponent function} \ttCC{@exp(x)} ($=e^x$);
the \emph{logarithmic function} \ttCC{@log(x)}($=\ln x$) or
\ttCC{@log10(x)} ($=\log_{10}x$);
the \emph{trigonometric functions} \ttCC{@sin(x), @cos(x), @tan(x)}
depending on angles measured by \emph{radian};
the inverse of $\sin x,\, \cos x,\, \tan x$ called \emph{circular function} or \emph{inverse trigonometric function} \ttCC{@asin(x)}(=$\arcsin x$), \ttCC{@acos(x)}(=$\arccos x$), \ttCC{@atan(x)}(=$\arctan x$);
the \emph{hyperbolic function},
\[
\sinh x=\left( e^x-e^{-x}\right)/2,\qquad
\cosh x=\left( e^x+e^{-x}\right)/2.
\]
and $\tanh x=\sinh x/\cosh x$ written
by \ttCC{@sinh(x)}, \ttCC{@cosh(x)}, \ttCC{@asinh(x)} and \ttCC{@acosh(x)}.
\[
\sinh^{-1}x=\ln \left[x+\sqrt{x^2+1}\right],\qquad
\cosh^{-1}x=\ln \left[x+\sqrt{x^2-1}\right].
\]
\itemtt[Elementary Functions]
is the class of functions consisting of the functions in this section
(polynomials, exponential, logarithmic, trigonometric, circular) and
the functions obtained from those listed by the four arithmetic operations
\[
f(x)+g(x),\, f(x)-g(x),\, f(x)g(x),\, f(x)/g(x)
\]
and by superposition $f(g(x))$, in which four arithmetic operarions and superpositions are permitted finitely many times.
In \freefempp, we can create all elementary functions.
The derivative of an elementary function is also elementary.
However, the indefinite integral of an elementary function cannot always be expressed in terms of elementary functions.
\begin{example}
The following give the example
to make the boundary using elementary functions.
\emph{Cardioid}
\bT
@real b = 1.;
@real a = b;
@func @real phix(@real t)
{
   @return (a+b)*cos(t)-b*cos(t*(a+b)/b);
}
@func @real phiy(@real t)
{
   @return (a+b)*sin(t)-b*sin(t*(a+b)/b);
}
@border C(t=0,2*pi) { x=phix(t); y=phiy(t); }
@mesh Th = @buildmesh(C(50));
\eT
\end{example}
Taking the principal value, we can define $\log z$ for $z\neq 0$ by
\[
\ln z = \ln |z|+\arg z.
\]
Using \freefempp, we calculated
\ttCC{@exp(1+4i)}, \ttCC{@sin(pi+1i)}, \ttCC{@cos(pi/2-1i)} and \ttCC{@log(1+2i)}, we then have
\begin{eqnarray*}
-1.77679-2.0572i,& 1.88967 10^{-16}-1.1752i,\\
9.44833 10^{-17}+1.1752i, & 0.804719+1.10715i.
\end{eqnarray*}
\end{description}

\subsection{Two Variable Functions}
\label{sec:TwoVarFunctions}
\subsubsection{\setS{Formula}}
The general form of real functions with two independent variables $x,\, y$ is
usually written as $z=f(x,y)$. In \freefempp, \ttCC{x} and \ttCC{y} are
reserved word in Section \ref{sec:Global}.
When two independent variables are \ttCC{x} and \ttCC{y},
we can define a function without argument, for example
\bT
@func f=@cos(x)+@sin(y) ;
\eT
Remark the function's type is given by the expression's type.
The power of functions are given in \freefempp such as
\ttCC{x\^{}1}, \ttCC{y\^{}0.23}.
In \ttCC{func}, we can write an elementary function as follows
\bT
@func f = @sin(x)*@cos(y);
@func g = (x^2+3*y^2)*@exp(1-x^2-y^2);
@func h = @max(-0.5,0.1*@log(f^2+g^2));
\eT

Complex valued function create functions with 2 variables \ttCC{x, y} as follows,
\bT
@mesh Th=square(20,20,[-pi+2*pi*x,-pi+2*pi*y]); // $]-\pi,\pi[^2$
@fespace Vh(Th,P2);
@func z=x+y*1i;  // $z=x+iy$
@func f=@imag(sqrt(z));  // $f=\Im\sqrt{z}$
@func g=@abs( sin(z/10)*exp(z^2/10) ); // $g=|\sin z/10\exp z^2/10|$
Vh fh = f; plot(fh);  // contour lines of $f$
Vh gh = g; plot(gh);  // contour lines of $g$
\eT
We call also by \emph{two variable elementary function} functions
obtained from elementary functions $f(x)$ or $g(y)$
by the four arithmetic operations
and by superposition in finite times.

\subsubsection{\setS{FE-function}}\index{FE-function}
Arithmetic built-in functions are able to construct a new function by
the four arithmetic operations and superposition of them
(see \emph{elementary functions}), which are called
\key{formulas} to distinguish from
FE-functions.
We can add \emph{new formulas} easily, if we want.
Here, FE-function is an element of
finite element space (real or complex) (see Section \refSec{Finite Elements}).
Or to put it another way: \emph{formulas} are the
mathematical expressions combining its numerical analogs, but
it is independent of meshes (triangulations).

Also, in \texttt{freefem++}, we can give an arbitrary symbol to FE-function
combining numerical calculation by FEM.
The projection of a formula $f$ to FE-space is done as in
\bT
func f=x^2*(1+y)^3+y^2;
mesh Th = square(20,20,[-2+2*x,-2+2*y]); // square $]-2,2[^2$
fespace Vh(Th,P1);
Vh fh=f;  // fh is the  projection of f to Vh (real value)
func zf=(x^2*(1+y)^3+y^2)*exp(x+1i*y);
Vh<complex> zh = zf; // zh is the projection of zf
// to complex value Vh space   \index{FE function!complex}
\eT
The construction of \ttCC{fh} (=$f_h$) is explained in
\refSec{Finite Elements}.

\begin{note}
The command \ttCC{@plot} is valid only for real  FE-functions.
\end{note}
Complex valued function create functions with 2 variables \ttCC{x, y} as follows,
\bT
@mesh Th=square(20,20,[-pi+2*pi*x,-pi+2*pi*y]); // $]-\pi,\pi[^2$
@fespace Vh(Th,P2);
@func z=x+y*1i;  // $z=x+iy$
@func f=@imag(sqrt(z));  // $f=\Im\sqrt{z}$
@func g=@abs( sin(z/10)*exp(z^2/10) ); // $g=|\sin z/10\exp z^2/10|$
Vh fh = f; plot(fh);  // Fig. \ref{cfunc1} isovalue of $f$
Vh gh = g; plot(gh);  // Fig. \ref{cfunc2} isovalue of $g$
\eT
\twoplot[height=5cm]{cfunc1}{cfunc2}{$\Im\sqrt{z}$ has branch}
{$|\sin (z/10)\exp (z^2/10)|$}

\subsection{Array}
\index{array}
An \emph{array} stores multiple objects, and
there are 2 kinds of arrays:
The first is the \emph{vector} that is arrays with \emph{integer indices}
and
arrays with \emph{string indices}.

In the first case, the size of this array
must be know at the execution time, and the implementation is done
with the \ttCC{KN<>} class so all the vector operator of
 \ttCC{KN<>} are implemented. The sample
\bT
@real [int] tab(10), tab1(10); // 2 array of 10 real
@real [int] tab2;    //  bug array with no size
tab = 1.03;                //  set all the array to 1.03
tab[1]=2.15;
@cout << tab[1] << " " << tab[9] << " size of tab = "
     << tab.n << " min: " << tab.min << "  max:" << tab.max
     << " sum : "   << tab.sum <<   endl; //
tab.resize(12); //  change the size of array tab
  // to 12 with preserving first value
tab(10:11)=3.14; //  set unset value
cout <<" resize tab: " <<  tab << endl;
\eT
\index{array!resize}\index{resize}\index{max}\index{array!max} \index{min}\index{array!min}\index{sum}\index{array!sum}
produce the output
\bT
2.15 1.03 size of tab = 10 min: 1.03  max:2.15 sum : 11.42
 resize tab: 12
        1.03    2.15    1.03    1.03    1.03
        1.03    1.03    1.03    1.03    1.03
        3.14    3.14
\eT

It is also possible to make an array of FE function, with the same syntax,
and we can treat them as \emph{vector valued function} if we need them.
\index{array}\index{array!FE function}
\begin{example}
In the following example, Poisson's equation is solved under 3 different given
functions $f=1,\, \sin(\pi x)\cos(\pi y),\, |x-1||y-1|$, whose solutions are