# Prototypes of FORTRAN functions in the static library libaem.a

## FORTRAN subroutine conformsystem_( )

The purpose of this function/subroutine is to compute the Jacobian matrices of real potential \Phi = \mathrm{Re}\!\left\{\Omega\right\}, stream function \Psi = \mathrm{Im}\!\left\{\Omega\right\}, and velocity components v_x = \mathrm{Re}\!\left\{-\overline{\frac{d\Omega}{dz}}\right\} and v_y = \mathrm{Re}\!\left\{-\overline{\frac{d\Omega}{dz}}\right\} at control points on the slit elements, with respect to the degrees of freedom of the elements:

void conformsystem_ ( 	double *, int *,
double *, double *, double *, double *, int *,
double *, double *, int *, double *, double *, double *, double *,
int *, double *, double *,
double *, double *, double *, double *, double *, double *, double *, double *,
double *, double *, double *, double *,
double *, int *);
/*
SUBROUTINE CONFORMSYSTEM ( THETA,ntheta,
&                             ReZmin,ImZmin,ReZmax,ImZmax,nslit,
&                             ReCn,ImCn,ncoef,Q,ReV0,ImV0,Phi0,
&                             varout,ReZ,ImZ,
&                             PHI,PSI,Vx,Vy,J_PHI,J_PSI,J_Vx,J_Vy,
! work arrays:
&                             J_PHI1,J_PSI1,J_Vx1,J_Vy1,
&                             Xdof,ndof)
*/

Following the approach of Steward (2015), we consider a collection of slit elements, each element j creating a complex potential \Omega_j(z) at location z=x+iy in the \mathbb{C}-plane, with the form: \displaystyle \Omega_j(z)= Q_j\times\left(\frac{\ln Z_j(z)}{2\pi}-1\right) + \sum_{k=1}^{n_\mathrm{coef}} c_{j,k}Z_j^{-k}(z) Q_j is a sink term, and the c_{j,k} are the complex coefficients of a Laurent series that can be adjusted to match various boundary conditions. The mapping function Z_j(z) for each slit element is given by:

\displaystyle \begin{cases} \zeta_j(z;a_j,b_j)= \displaystyle\frac{z-\frac{1}{2}(a_j+b_j)}{b_j-a_j} & \\ & \\ Z_j(z) = \zeta_j(z)+\sqrt{\zeta_j(z)+1}\sqrt{\zeta_j(z)-1} & \\ \end{cases}

where a_j is the complex affix of the start point of slit element j, and b_j the affix of its end point. The function \zeta\mapsto \zeta + \sqrt{\zeta+1}\sqrt{\zeta-1} is the Joukowsky transformation (Joukowsky, 1910).

By superposition, the total (complex) potential at location z in the complex plane is simply the sum of the contributions of all slit elements in the domain: \displaystyle \Omega_\mathrm{tot}(z) = \Phi_\mathrm{tot}(z) + i\,\Psi_\mathrm{tot}(z) = \Phi_0 - \overline{v_0}\,z + \sum_{j=1}^{n_\textrm{slit}} \Omega_j(z) where v_0 = v_{0,x}+i\,v_{0,y} is a complex number giving the ‘background’ uniform flow, \overline{v_0} its complex conjugate, and \Phi_0 an offset value for the real potential.

Denoting n_\textrm{slit} the number of slit elements in the system and n_\textrm{coef} the number of complex-valued coefficients in the Laurent series, there are (2 n_\textrm{coef} +1)n_\textrm{slit} real degrees of freedom. We define n_\theta control points along each slit element, such that:

\displaystyle \begin{cases} \theta_m = \pi\frac{m-\frac{1}{2}}{n_\theta}\qquad 1\leq m\leq n_\theta & \\ & \\ z(\theta_m) = \frac{1}{2}(b_j-a_j)\cos(\theta_m) + \frac{1}{2}(a_j+b_j)& \\ \end{cases}

Then the matrices J_\Phi, J_\Psi, J_{v_x}, and J_{v_y} are of the form:

\displaystyle \small\begin{array}{r|c|} & \overbrace{ \mathrm{Re}\!\left\{c_1\right\}\ \mathrm{Im}\!\left\{c_1\right\} \ \cdots\ \mathrm{Re}\!\left\{c_{n_\textrm{coef}}\right\}\ \mathrm{Im}\!\left\{c_{n_\textrm{coef}}\right\}\ Q }^{\small\textrm{slit } 1} \quad\overbrace{ \mathrm{Re}\!\left\{c_1\right\}\ \mathrm{Im}\!\left\{c_1\right\} \ \cdots\ \mathrm{Re}\!\left\{c_{n_\textrm{coef}}\right\}\ \mathrm{Im}\!\left\{c_{n_\textrm{coef}}\right\}\ Q }^{\small\textrm{slit } 2} \qquad\cdots\qquad \overbrace{ \mathrm{Re}\!\left\{c_1\right\}\ \mathrm{Im}\!\left\{c_1\right\} \ \cdots\ \mathrm{Re}\!\left\{c_{n_\textrm{coef}}\right\}\ \mathrm{Im}\!\left\{c_{n_\textrm{coef}}\right\}\ Q }^{\small\textrm{slit } n_\textrm{slit}} \\ & \\ \hline \textrm{slit }1 \left\{\begin{array}{c} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_{n_\theta} \end{array}\right. & \\ \textrm{slit }2 \left\{\begin{array}{c} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_{n_\theta} \end{array}\right. & \Large\texttt{J(nslit\,*\,ntheta,(2\,*\,ncoef+1)*\,nslit)}\\ \vdots\quad\left\{ \begin{array}{c} ~\\ ~\\ ~\\ ~\\ \end{array} \right.\quad & \\ \textrm{slit }n_\textrm{slit} \left\{\begin{array}{c} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_{n_\theta} \end{array}\right. & \\ \hline \end{array}

General form of the potential created by a slit element. In each figure, the gray levels show the value of \Phi = \mathrm{Re}\left\{\Omega(z)\right\} while the dashed white lines are contours of the stream function \Psi = \mathrm{Im}\left\{\Omega(z)\right\}. Black arrows show the flux vector q=-\overline{\frac{d\Omega}{dz}}. (Left) Potential created by a pure sink. (Center) Potential created by a single, real-valued coefficient in the Laurent series; note that the real potential \Phi is continuous between the ‘top’ and ‘bottom’ of the slit, while the stream function \Psi is discontinuous. (Right) Potential created by a single, pure imaginary coefficient in the Laurent series; there is now a jump in the real potential \Phi between the two sides of the slit while the stream function, in turn, is continuous.

## FORTRAN subroutine slit_( )

void slit_ ( double *, double *, int *,
double *, double *, double *, double *, int *,
double *, double *, int *,double *,double *,double *,double *,
int *,
double *, double *, double *, double *,double *,double *,double *,double *);
/*
SUBROUTINE SLIT (ReZ,ImZ,nz,
&                   ReZmin,ImZmin,ReZmax,ImZmax,nslit,
&                   ReCn,ImCn,ncoef,Q,ReV0,ImV0,Phi0,
&                   varout,
&                   PHI,PSI,Vx,Vy,J_PHI,J_PSI,J_Vx,J_Vy)
*/

## FORTRAN subroutine arealsink_( )

void arealsink_ ( double *, double *, int *,
double *, double *, int *, double *,
double *, double *, double *, double *);
/*
SUBROUTINE AREALSINK ( Xvert,Yvert,nvert,
&                         Xquery,Yquery,nquery,gamma0,
! output arrays:
&                         PHI,Vx,Vy,A)
*/

## FORTRAN subroutine inpolygon_( )

void inpolygon_ ( double *, double *, int *,
double *, double *, int *,
double *);
/*
SUBROUTINE INPOLYGON( Xvert,Yvert,nvert,
&                        Xquery,Yquery,nquery,
! Output array
&                        RES)
*/

## LAPACK FORTRAN subroutine dgels_( )

void dgels_ ( char *, int *, int *, int *, double *, int *, double *, int *, double *, int *, int *);

/*
SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
\$                  INFO )
*180/span>
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*184/span>
*     .. Scalar Arguments ..
CHARACTER          TRANS
INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*/