analyticalSolutions.c

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//
// analytical functions
//
#include "analyticalSolutions.h"
 
int PlaneCouetteFlow_u(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
  Couette Flow between two parallel plates. One moving relative to the other with constant seperation (width).
  Flow is axial. u != 0, v = w = 0 [Plates are very wide(z) and very long(x)]
  vx = Velocity of moving plate in the x-dir.
  h  = width between plates in the y-dir.
  Axis of orgin: bottom plate
  xs,ys,zs = axis offset
 
  \del^2(\vecV) = 0
  
  d^2u/dy^2 = 0
  u(y) =vx/h * Y
 
  u(0) = 0
  u(h) = vx
*/
        int i;
        double vx=rwork[0], h=rwork[1];
        double ys=rwork[3];
        double Y;
 
        for (i=0; i<nPoints; i++)
        {
             Y = x[i*3+1] - ys;
                
             u[i] = vx*(Y/h);
        }
 
        return 0;
}
 
int diffusionSin1D(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
  int i;
  double omega_x = iwork[0];
 
  for (i=0; i<nPoints; i++)
    {
      u[i] = -pow(2.0*PI*omega_x,2.0)*sin(2.0*PI*omega_x*x[i*3+0]);
    }
        
    return 0;
}
int diffusionSin2D(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
        int i;
        double omega_x=iwork[0];
        double omega_y=iwork[1];
 
        for (i=0; i<nPoints; i++)
        {
             u[i] = -pow(2.0*PI*omega_x,2.0)*sin(2.0*PI*omega_x*x[i*3+0]) 
                    -pow(2.0*PI*omega_y,2.0)*sin(2.0*PI*omega_y*x[i*3+1]) ;
        }
 
        return 0;
}
int diffusionSin3D(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
        int i;
        double omega_x=2.0*PI*iwork[0];
        double omega_y=2.0*PI*iwork[1];
        double omega_z=2.0*PI*iwork[2];
 
        for (i=0; i<nPoints; i++)
        {
             u[i] = -pow(2.0*PI*omega_x,2.0)*sin(2.0*PI*omega_x*x[i*3+0]) 
                    -pow(2.0*PI*omega_y,2.0)*sin(2.0*PI*omega_y*x[i*3+1])
                    -pow(2.0*PI*omega_z,2.0)*sin(2.0*PI*omega_z*x[i*3+2]) ;
        }
 
        return 0;
}
int diffusionSin1D_r(int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
{
        int i;
        double omega_x=iwork[0];
 
        for (i=0; i<nPoints; i++)
        {
             r[i] = pow(2.0*PI*omega_x,2.0)*sin(2.0*PI*omega_x * x[i*3+0]);
        }
        
        return 0;
}
int diffusionSin2D_r(int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
{
        int i;
        double omega_x=iwork[0];
        double omega_y=iwork[1];
 
        for (i=0; i<nPoints; i++)
        {
          r[i] = pow(2.0*PI*omega_x,2.0) * sin(2.0*PI*omega_x * x[i*3+0]) +
                 pow(2.0*PI*omega_y,2.0) * sin(2.0*PI*omega_y * x[i*3+1]) ;
        }
        
        return 0;
}
int diffusionSin3D_r(int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
{
        int i;
        double omega_x=iwork[0];
        double omega_y=iwork[1];
        double omega_z=iwork[2];
 
        for (i=0; i<nPoints; i++)
        {
          r[i] = pow(2.0*PI*omega_x,2.0) * sin(2.0*PI*omega_x * x[i*3+0]) +
                 pow(2.0*PI*omega_y,2.0) * sin(2.0*PI*omega_y * x[i*3+1]) +
                 pow(2.0*PI*omega_z,2.0) * sin(2.0*PI*omega_z * x[i*3+2]) ;
        }
        
        return 0;
}
int LinearAD_DiracIC(int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
/*
    The exact solution of
    
    u_t + \deld(b u - a \grad u) = 0
    
    on an infinite domain with dirac initial data
    
    u0 = \int u0 \delta(x - x0)
 
    also returns advective, diffusive, and total flux
*/
{
        int i, j;
        double b[3]={rwork[0],rwork[1],rwork[2]};
        double n=rwork[3], a=rwork[4], tStart=rwork[5], u0=rwork[6];
        double x0[3]={rwork[7],rwork[8],rwork[9]};
        double y[3], exp_arg, yDoty, t = T + tStart;
 
    for (i=0; i<nPoints; i++)
        {
                for (yDoty=0.0,j=0; j<3; j++)
                {
                        y[j] = x[i*3+j] - x0[j] - b[j]*t;
                        yDoty += y[j]*y[j];
                }
                exp_arg = yDoty / (4.0 * a * t) ;
                
                if (exp_arg > 100)
                {
                        u[i] = 0.0;
                }
                else
                {
                        u[i] = u0*exp(-exp_arg) / pow( (4.0*a*PI*t),(n/2.0) );
                }
        }
 
    return 0;
}
int LinearAD_DiracIC_advectiveVelocity(int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
{
  double b[3]={rwork[0],rwork[1],rwork[2]};
  double *u = (double *)malloc(nPoints*sizeof(double));
 
        int i, j, iret;
        
        iret = LinearAD_DiracIC(iwork, rwork,  nPoints, T, x, u);
 
        for (i=0; i<nPoints; i++)
        {
            for (j=0; j<3; j++)
            {
                 f[i*3+j] = b[j]*u[i];
            }
        }
        free(u);
 
        return iret;
}
int LinearAD_DiracIC_diffusiveVelocity(int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
{
        int i, j, iret;
        double a=rwork[4];
        
        iret = LinearAD_DiracIC_du(iwork, rwork, nPoints, T, x, f);
 
        for (i=0; i<nPoints; i++)
        {
            for (j=0; j<3; j++)
            {
                 f[i*3+j] = -a * f[i*3+j];
            }
        }
        return iret;
}
int LinearAD_DiracIC_du(int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
{
        int i, j, iret;
        double b[3]={rwork[0],rwork[1],rwork[2]};
        double a=rwork[4], tStart=rwork[5];
        double x0[3]={rwork[7],rwork[8],rwork[9]};
        double y[3], t=T + tStart;
        double *u = (double *)malloc(nPoints*sizeof(double));
 
        iret = LinearAD_DiracIC(iwork, rwork, nPoints, T, x, u);
 
        for (i=0; i<nPoints; i++)
        {
                for (j=0; j<3; j++)
                {
                        y[j] = x[i*3+j] - x0[j] - b[j]*t;
                        f[i*3+j] = u[i]*2.0*y[j] / (4.0*a*t);
                }
        }
        free(u);
 
        return iret;
}
 
int LinearAD_DiracIC_totalVelocity(int *iwork, double *rwork, int nPoints, double T, double *x, double *f)
{
        int i, j, iret;
        double *f_adv = (double *)malloc(nPoints*3*sizeof(double));
        
        iret = LinearAD_DiracIC_advectiveVelocity(iwork, rwork, nPoints, T, x, f_adv);
        iret += LinearAD_DiracIC_diffusiveVelocity(iwork, rwork, nPoints, T, x, f);
 
        for (i=0; i<nPoints; i++)
        {
            for (j=0; j<3; j++)
            {
                 f[i*3+j] += f_adv[i*3+j];
            }
        }
        free(f_adv);
 
        return iret;
}
int LinearAD_SteadyState(int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
/*
    The exact solution for
    (bu - au_x)_x = 0
    u(0) = 1
    u(1) = 0
*/
{
  int i;
  double  b=rwork[0], a=rwork[1];
  double x, D=0.0, C;
 
  if (b != 0.0)
  {
        D = (1.0/(exp(b/a)-1.0));
  }
  C = (-D)*exp(b/a);
 
  for (i=0;i<nPoints;i++)
  {
      x = X[i*3+0];
      if (D != 0.0)
      {
        u[i] = (-D)*exp(b*x/a) - C;
      }
      else
      {
        u[i] = 1.0 - x;
      }
  } 
 
  return 0;
}
 
int LinearADR_Decay_DiracIC(int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
/*
The exact solution of
    
    u_t + \deld(bu - a \grad u) + cu= 0
    
    on an infinite domain with Dirac initial data.
    Also returns the fluxes (by inheritance).
*/
{
        int i, iret;
        double c=rwork[10], tStart=rwork[5]; 
        double t = T + tStart;
 
        iret = LinearAD_DiracIC(iwork, rwork, nPoints, T, x, u);
        for (i=0; i<nPoints; i++)
        {
             u[i] *= exp(-c*t);
        }
 
        return iret;
}
int LinearADR_Decay_DiracIC_dr(int *iwork, double *rwork, int nPoints, double T, double *x, double *u, double *dr)
{
  int i, iret;
  double c=rwork[10];
        
  iret = LinearADR_Decay_DiracIC(iwork, rwork, nPoints, T, x, u);
 
  for (i=0; i<nPoints; i++)
  {
       dr[i] = c;
  }
 
  return iret;
}
int LinearADR_Decay_DiracIC_r(int *iwork, double *rwork, int nPoints, double T, double *x, double *u, double *r)
{
  int i, iret;
  double c=rwork[10];
 
  iret = LinearADR_Decay_DiracIC(iwork, rwork, nPoints, T, x, u);
        
  for (i=0; i<nPoints; i++)
  {
       r[i] = u[i] * c;
  }
 
  return iret;
}
int LinearADR_Sine(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
/*
    An exact solution and source term for
    
    \deld (\vec b u - \ten a \grad u) + c u + d = 0
    
    where
    
    u(x) = sin[\gvec \omega \cdot \vec x + \omega_0) ] = sin(Ax - b)
    r(u,x) = - [(\ten a \gvec \omega] \cdot \gvec \omega) u
             - (\vec b \cdot \omega) cos(Ax - b)
           = c u + D cos(Ax - b)
           = cu + d
           
    also returns the advective, diffusive, and total flux at a point.
*/
{
  int i=0;
  double omega[3]={rwork[0],rwork[1],rwork[2]};
  double omega0=rwork[3];
  double Y;
 
  for (i=0;i<nPoints;i++)
  {
          Y = omega[0]*x[i*3+0] + omega[1]*x[i*3+1] + omega[2]*x[i*3+2] + omega0;
          u[i] = sin(Y);
  }
 
  return 0;
}
int LinearADR_Sine_advectiveVelocity(int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
{
        int i, j, iret;
        double b[3]={rwork[4],rwork[5],rwork[6]};
        double *u = (double *)malloc(nPoints * sizeof(double));
 
        /*for (i=0;i<nPoints;i++)
        {
          printf(" %d x = %f, %f, %f\n",i,x[i*3+0],x[i*3+1],x[i*3+2]);
        }
        */
        iret = LinearADR_Sine(iwork, rwork, nPoints, t, x, u);
 
        for (i=0; i<nPoints; i++)
        {
                for (j=0; j<3; j++)
                {
                  f[i*3+j] = u[i]*b[j];
                }
        }
        free( u );
 
        return iret;
}
int LinearADR_Sine_diffusiveVelocity(int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
{
        int i, j, iret;
        double a[3][3]={ {rwork[7],rwork[8],rwork[9]},{rwork[10],rwork[11],rwork[12]},{rwork[13],rwork[14],rwork[15]} };
        double *du = (double *)malloc(nPoints*3*sizeof(double));
 
        iret = LinearADR_Sine_du(iwork, rwork, nPoints, t, x, du);
 
// matrix multiplication of matrix a(3x3) and a vector x of nPoints.
        for (i=0; i<nPoints; i++)
        {
            for (j=0; j<3; j++)
            {
                f[i*3+j] = -( a[j][0]*du[i*3+j] + a[j][1]*du[i*3+j] + a[j][2]*du[i*3+j] );
            }
        }
        free(du);
 
        return iret;
}
int LinearADR_Sine_dr(int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *dr)
{
        int i, iret;
        double c=rwork[16];
        
        iret = LinearADR_Sine(iwork, rwork, nPoints, t, x, u);
 
        for (i=0; i<nPoints; i++)
        {
           dr[i] = c;
        }
 
        return iret;
}
int LinearADR_Sine_du(int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
{
  int i, j;
  double omega[3]={rwork[0],rwork[1],rwork[2]};
  double omega0=rwork[3];
  double Y;
 
  for (i=0;i<nPoints;i++)
  {
      Y = cos( omega[0]*x[i*3+0] + omega[1]*x[i*3+1] + omega[2]*x[i*3+2] + omega0 );
      for (j=0; j<3; j++)
      {
          f[i*3+j] = omega[j] * Y;
      }
  }
 
  return 0;
}
int LinearADR_Sine_r(int *iwork, double *rwork, int nPoints, double t, double *x, double *u, double *r)
{
  int i, j, iret;
  double omega[3]={rwork[0],rwork[1],rwork[2]};
  double omega0=rwork[3];
  double b[3]={rwork[4],rwork[5],rwork[6]};
  double a[3][3]={ {rwork[7],rwork[8],rwork[9]},{rwork[10],rwork[11],rwork[12]},{rwork[13],rwork[14],rwork[15]}};
  double c=rwork[16]; 
  double Y, D, E;
  double MM[3];
 
  iret = LinearADR_Sine(iwork, rwork, nPoints, t, x, u);
 
// matrix multiplication of 3x3 matrix and 3x1 vector
  for (j=0; j<3; j++)
  {
          MM[j] = a[j][0]*omega[0] + a[j][1]*omega[1] + a[j][2]*omega[2];
  }
  E = - ( MM[0]*omega[0] + MM[1]*omega[1] + MM[2]*omega[2] ) - c;
 
// dot product of vector b and vector omega
  D = -( b[0]*omega[0] + b[1]*omega[1] + b[2]*omega[2] );
 
  for (i=0; i<nPoints; i++)
  {
          Y = omega[0]*x[i*3+0] + omega[1]*x[i*3+1] + omega[2]*x[i*3+2] + omega0;
          r[i] = c*u[i] + D*cos(Y) + E*sin(Y);
  }
 
  return iret;
}
int LinearADR_Sine_totalVelocity(int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
{
        int i, j, iret;
        double *f_adv = (double *)malloc(nPoints*3*sizeof(double));
 
        iret = LinearADR_Sine_advectiveVelocity(iwork, rwork, nPoints, t, x, f_adv);
        iret += LinearADR_Sine_diffusiveVelocity(iwork, rwork, nPoints, t, x, f);
 
        for (i=0;i<nPoints;i++)
        {
                for (j=0; j<3; j++)
                {
                        f[i*3+j] += f_adv[i*3+j];
                }
        }
        free(f_adv);
 
        return iret;
}
/********************************************************************/
int NonlinearAD_SteadyState(int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
/*
    (bu^q - a(u^r)_x)_x = 0
    u(0) = 1
    u(1) = 0
*/
{
        int i, iret=0;
        int q=iwork[0], r=iwork[1];
        double b=rwork[0], a=rwork[1];
        double x, lu, f0, ff, fC;
        double C, rtmC=0.0, Ctmp, dC, nlC=0.0, nlD=0.0;
 
    if (q==2 && r==1)
        {
                if (b != 0.0)
                {
                        printf( "Solving for sqrt(-C) for q=2, r=1\n");
                        rtmC = sqrt(1.5);
                        while ( fabs( f(rtmC,b,a,q,r) ) > 1.0E-8 )
                        {
                                rtmC -= ( f(rtmC,b,a,q,r) / df(rtmC,b,a,q,r) );
                        }
                        printf( "sqrt(-C)= %lf \n",rtmC);
                }
        }
    else if (q==1 && r==2)
        {
                printf( "\nSolving for C in q=1,r=2\n" );
                C = 1.0 + 1.0E-10;
                f0 = f(C,b,a,q,r);
                printf("f0 = %lf\n", f0);
                while ( fabs(f(C,b,a,q,r)) > (1.0E-7*fabs(f0) + 1.0E-7))
                {
                        dC = -f(C,b,a,q,r)/df(C,b,a,q,r);
                  printf("dc = %lf\n",dC);
                  Ctmp = C + dC;
                  while ( (fabs(ff=f(Ctmp,b,a,q,r)) > 0.99*fabs(fC=f(C,b,a,q,r))) || (Ctmp <= 1.0) )
                        {
                                printf("f(%lf) = %lf\n", Ctmp, ff);
                                printf("f(%lf) = %lf\n", C, fC);
                                printf( "ls\n" );
                                dC *= 0.9 ;
                                Ctmp = C + dC ;
                        }
                  printf( "out\n" );
                  printf( "%lf \n",Ctmp);
// are we printing the new values from the functions f & df?
                  printf(" f(%lf) = %lf\n",Ctmp, f(Ctmp,b,a,q,r));
                  printf("df(%lf) = %lf\n",Ctmp, df(Ctmp,b,a,q,r));
                  C=Ctmp;
                }
                printf("C = %lf\n",C);
                nlC = C;
                nlD = 0.5*(2.0*C*log(C*(C-1)) - 4.0*C + 2.0 - b/a);
                printf( "D = %lf\n",nlD);
        }
     else
        {
                printf("q,r not implemented\n");
                return 0;
        }
// Main Loop for uOfX
 
     if(q==1 && r==2)
     {
       iret = LinearAD_SteadyState(iwork, rwork, nPoints, t, X, u) ;
     }
 
     for (i=0; i<nPoints; i++)
        {
             x = X[i*3+0];
                
             if (q==2 && r==1)
                {
                    if (b != 0.0)
                    {
                        u[i] = rtmC * tanh( (-b*rtmC/a)*(x - 1.0) );
                    }
                    else
                    {
                        u[i] = 1.0 - x;
                    }
                }
             else if (q==1 && r==2)
                {
                    lu = u[i];
                    f0 = uOfX_f(a, b, nlC, nlD, x, lu) ;
                    while ( fabs(uOfX_f(a, b, nlC, nlD, x, lu)) > (1.0E-6*fabs(f0) + 1.0E-6) )
                    {
                        lu -= uOfX_f(a, b, nlC, nlD, x, lu) / uOfX_df(nlC, lu) ;
                    }
                    u[i] = lu;
                }
              else
                {       
                        printf("q,r not implemented");  
                }
         }
      return iret;
}
 
int NonlinearADR_Decay_DiracIC(int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
/*
The approximate analytical solution of
    
    u_t + \deld(bu - a \grad u) + cu^d= 0
    
    on an infinite domain with Dirac initial data.
    Also returns the fluxes (by inheritance).
*/
{
    int i, iret;
    double c=rwork[10], d=rwork[11], tStart=rwork[5];
    double t = T + tStart;
 
    iret = LinearAD_DiracIC(iwork, rwork, nPoints, T, x, u);
 
    for (i=0; i<nPoints; i++)
    {
         if (u[i] > 0.0)
         {
             u[i] *= exp( -(2.0*c*t*pow(u[i],(d-1.0)))/(d+1.0) );
         }
    }
 
    return iret;
}
int NonlinearADR_Decay_DiracIC_dr(int *iwork, double *rwork, int nPoints, double T, double *x, double *u, double* dr)
{
    int i, iret;
    double c=rwork[10], d=rwork[11];
        
    iret = NonlinearADR_Decay_DiracIC(iwork, rwork, nPoints, T, x, u);
 
    for (i=0; i<nPoints; i++)
    {
         dr[i] = d*c*pow(u[i],(d-1.0));
    }
 
    return iret;
}
 
int NonlinearADR_Decay_DiracIC_r(int *iwork, double *rwork, int nPoints, double T, double *x, double *u, double* r)
{
    int i, iret;
    double c=rwork[10], d=rwork[11];
 
    iret = NonlinearADR_Decay_DiracIC(iwork, rwork, nPoints, T, x, u);
        
    for (i=0; i<nPoints; i++)
    {
         r[i] = c*pow(u[i],d);
    }
 
    return iret;
}
 
int NonlinearDAE(int *iwork, double *rwork, int nPoints, double T, double *x, double *u)
/*
    The exact solution of
    
    u_t = - a max(u,0)^p
    u(0) = 1
*/
{
  int i;
  double q, t, a=rwork[0], p=rwork[1];
 
  for (i=0;i<nPoints;i++)
    {
      t = x[i*3+0];
      
      if (p == 1.0)
        {
                u[i] =  exp( -a*t );
        }
      else
        {
                q = 1.0/(1.0 - p);
                u[i] = pow( max( (1.0 - (1.0 - p)*a*t), 0.0 ), q ) ;
        }
    }
 
  return 0;
}
int NonlinearDAE_f(int *iwork, double *rwork, int nPoints, double t, double *x, double *f)
/*
    The exact solution of
    
    u_t = - a max(u,0)^p
    u(0) = 1
*/
{
        int i, j;
        double a=rwork[0], p=rwork[1];
 
        for (i=0;i<nPoints;i++)
        {
            for (j=0; j<3; j++)
            {
                  f[i*3+j] = -a*pow(max(f[i*3+j],0.0),p) ;
            }
        }
 
        return 0;
}
   
int PlanePoiseuilleFlow_u(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
  Poiseuille Flow between two parallel fixed plates with constant seperation (width)
 
  h  = width between plates
  GradP = Pressure gradient (neg.)
  mu = viscosity
  q = rate of flow per unit width 
  xs,ys,zs = axis offset
  Axis of origin: bottom plate
  Vmax = maximum velocity at the centerline
 
  -\delp + mu \del^2(\u_x) = 0
 
  given either -GradP: u_{max} = -GradP / (2 * mu) * (h/2)^2
         or     q    : u_{max} = 3/2 * (q / h)
 
  u(y) = 4 * u_{max} * Y * (h-Y) / h^2
 
  u(0) = u(h) = 0
  u(h/2) = u_{max}  
  u_{max} = -GradP / (2 * mu) * (h/2)<sup>2</sup>\n
  or \n
  u_{max} = 3/2 * (q / h) 
*/
        int i;
        double h=rwork[0], mu=rwork[1];
        double GradP = rwork[2], q = rwork[3];
        double ys=rwork[5];
        double umax;
        double Y;
 
        if(GradP != 0.0)
          {
             umax = -GradP *(1.0/(2.0*mu)) * (h*h/4.0);
          }
        else if(q != 0.0)
          {
             umax = (3.0/2.0)*q/h;
          }
        else
          {
             printf("Please enter input values for either q(Flow Rate per unit depth) or GradP(Pressure Gradient)\n");
             exit (0);
          }
 
        for (i=0; i<nPoints; i++)
        {
             Y = x[i*3+1] - ys;
               
             u[i] = (4.0*umax*Y*(h-Y))/(h*h);
        }
 
        return 0;
}
int PoiseuillePipeFlow(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
  Poiseuille Flow through a circular pipe
  Velocity in the direction of flow (z-dir)
 
  R  = radius of pipe
  GradP = Pressure gradient (neg.)
  mu = viscosity   
  Q = Rate of Flow 
  Axis of origin: center of pipe
  xs,ys,zs = axis offset
 
  x-momentum equation in cylindrical coordinates
  Vr = Vtheta = 0
  -\\delp + mu \\del^2(Vx) = 0
 
  No transformation of coordinates is required for x.
  
  Vmax = -GradP * R^2/(4 * mu)\n
         or  \n  
  Vmax = 2 * Q / Pipe Area\n
 
  Vx(r) = Vmax *  (1 - r^2/R^2)
 
  Vx(0) = Vmax
  Vx(R) = 0
  Vmax = -GradP * R<sup>2</sup>/(4 * mu)\n
  or\n
  Vmax = 2 * Q / Pipe Area\n
 
*/
        int i;
        double r=rwork[0], mu = rwork[1];
        double GradP=rwork[2], Q=rwork[3];
        double ys=rwork[5];
        double umax;
        double Y;
 
        if(GradP != 0.0)
          {
             umax = -GradP *(r*r/(4.0*mu));
          }
        else if(Q != 0.0)
          {
             umax = 2.0*Q/(PI * r*r);
          }
        else
          {
             printf("Please enter input values for either Q(Flow Rate) or GradP(Pressure Gradient)\n");
             exit (0);
          }
 
        for (i=0; i<nPoints; i++)
        {
             Y = x[i*3+1] - ys;
                
             u[i] = umax*(1.0 - ((Y*Y)/(r*r)));
        }
 
        return 0;
}
int PoiseuillePipeFlow_P(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
  Poiseuille Flow through a circular pipe
  Velocity in the direction of flow (z-dir)
 
  R  = radius of pipe
  GradP = Pressure gradient (neg.)
  mu = viscosity   
  Q = Rate of Flow 
  Axis of origin: center of pipe
  xs,ys,zs = axis offset
  P = pressure
 
  x-momentum equation in cylindrical coordinates
  Vr = Vtheta = 0
  -GradP + mu \del^2(Vx) = 0
 
  For Q given:\n
  GradP = (8 * Q * mu) / (R<sup>2</sup> * Pipe Area)
  assumuption: P1 = atmospheric pressure = 0.0
*/
        int i;
        double r=rwork[0], mu=rwork[1];
        double GradP=rwork[2], Q=rwork[3], L=rwork[4];
        double xs=rwork[5];
        double X;
 
 
        if(GradP == 0.0 && Q != 0.0)
          {
             GradP = - (8.0*Q*mu) / (PI*pow(r,4.0));
          }
        else if(GradP == 0.0)
          {
             printf("Please enter input values for either Q(Flow Rate) OR GradP(Pressure Gradient)\n");
             exit(0);
          }
 
        for (i=0; i<nPoints; i++)
        {
             X = x[i*3+0] - xs;
                
             u[i] = -GradP*L*(1.0-X);
        }
 
        return 0;
}
int poissonsEquationExp1D(int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
{
/*
    -u_{xx} - f = 0
    u = K x(1-x)y(1-y)z(1-z)e^{x^2 + y^2 + z^2}
    f = -K {> [y(1-y)z(1-z)][4x^3 - 4x^2 + 6x - 2]+
           [x(1-x)z(1-z)][4y^3 - 4y^2 + 6y - 2]+
           [x(1-x)y(1-y)][4z^3 - 4z^2 + 6z - 2]}e^{x^2 + y^2 + z^2} 
*/
 
  int i=0;
  double K=rwork[0];
  double x;
 
  for (i=0;i<nPoints;i++)
    {
      x = X[i*3+0];      
      u[i] = K*x*(1.0-x)*exp(x*x);
    }
 
  return 0;
}
int poissonsEquationExp2D(int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
{
  int i=0;
  double K=rwork[0];
  double x, y;
 
  for (i=0;i<nPoints;i++)
    {
      x = X[i*3+0];      
      y = X[i*3+1];      
      u[i] = K*x*(1.0-x)*y*(1.0-y)*exp(x*x + y*y);
    }
  return 0;
}
int poissonsEquationExp3D(int *iwork, double *rwork, int nPoints, double t, double *X, double *u)
{
  int i=0;
  double K=rwork[0];
  double x, y, z;
 
  for (i=0;i<nPoints;i++)
    {
      x = X[i*3+0];      
      y = X[i*3+1];      
      z = X[i*3+2];
      u[i] = K*x*(1.0-x)*y*(1.0-y)*z*(1.0-z)*exp(x*x + y*y + z*z);
    }
 
  return 0;
}
int poissonsEquationExp3D_dr(int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *dr)
{
        int i;
        
        for (i=0; i<nPoints; i++)
        {
           dr[i] = 0.0;
        }
 
        return 0;
}
 
int poissonsEquationExp1D_r(int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
{
  int i=0;
  double K=rwork[0];
  double x;
 
  for (i=0;i<nPoints;i++)
  {
      x = X[i*3+0];  
    
      r[i] = K*(4.0*(1.0-x)*pow(x,3.0) - 4.0*x*x + 6.0*(1.0-x)*x - 2.0)*exp(x*x) ;
  }
 
  return 0;
}
int poissonsEquationExp2D_r(int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
{
  int i=0;
  double K=rwork[0];
  double x, y;
 
  for (i=0;i<nPoints;i++)
  {
      x = X[i*3+0];      
      y = X[i*3+1]; 
     
      r[i] = K*(y*(1.0-y)*(4.0*(1.0-x)*pow(x,3) - 4.0*x*x + 6.0*x*(1.0-x) - 2.0) + 
                x*(1.0-x)*(4.0*(1.0-y)*pow(y,3) - 4.0*y*y + 6.0*y*(1.0-y) - 2.0))*exp(x*x + y*y);
  }
 
  return 0;
}
int poissonsEquationExp3D_r(int *iwork, double *rwork, int nPoints, double t, double *X, double *u, double *r)
{
  int i=0;
  double K=rwork[0];
  double x,y,z;
 
  for (i=0; i<nPoints; i++)
  {
      x = X[i*3+0];      
      y = X[i*3+1];      
      z = X[i*3+2];
 
      r[i] = K*(y*(1.0-y)*z*(1.0-z)*(4.0*(1.0-x)*pow(x,3) - 4.0*x*x + 6.0*x*(1.0-x) - 2.0) +
                x*(1.0-x)*z*(1.0-z)*(4.0*(1.0-y)*pow(y,3) - 4.0*y*y + 6.0*y*(1.0-y) - 2.0) +
                x*(1.0-x)*y*(1.0-y)*(4.0*(1.0-z)*pow(z,3) - 4.0*z*z + 6.0*z*(1.0-z) - 2.0))*exp(x*x + y*y + z*z);
  }
 
  return 0;
}
 
int STflowSphere_P(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
 Stokes Flow around Sphere of radius r^s and center x^s, y^s, z^s.
 Velocity is u,v,w and viscosity is mu.
 Array rwork contains all the above variables.
 Input  : t, x
                : x[npoints*3] = {x,y,z} for each point
                : rwork[] = {uu,v,w,rS,xS,yS,zS,mu}
 Output : P
*/
  int i;
  double vx=rwork[0], vy=rwork[1], vz=rwork[2];
  double rS=rwork[3], xS=rwork[4], yS=rwork[5], zS=rwork[6];
  double mu=rwork[7];
  double eR[3],eTHETA[3];
  double norm_v,theta,r;
 
  for (i=0; i<nPoints; i++)
    {
      coords(vx,vy,vz,
             xS,yS,zS,
             &x[i*3],
             &r,
             &theta,
             &norm_v,
             eR,
             eTHETA);
      if (r >= rS)
        u[i] = (-3.0*rS*mu*norm_v*cos(theta))/(2.0*r*r);
      else
        u[i]=0.0;
    }
 
  return 0;
}
 
int STflowSphere_Vx(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
 Stokes Flow around Sphere of radius r^s and center x^s, y^s, z^s.
 Velocity is u,v,w and viscosity is mu..
 Input  : t, x
                : x[npoints*3] = {x,y,z} for each point
                : rwork[] = {uu,v,w,rS,xS,yS,zS,mu}
 Output : Vx
*/
  int i;
  double vx=rwork[0], vy=rwork[1], vz=rwork[2];
  double rS=rwork[3], xS=rwork[4], yS=rwork[5], zS=rwork[6];
  double eR[3],eTHETA[3];
  double norm_v,theta,r;
  double vR,vTHETA;
 
  for (i=0; i<nPoints; i++)
    {
      coords(vx,vy,vz,
             xS,yS,zS,
             &x[i*3],
             &r,
             &theta,
             &norm_v,
             eR,
             eTHETA);
      vel(rS,norm_v,r,theta,&vR,&vTHETA);
      u[i] = vR * eR[0]+ vTHETA * eTHETA[0];
    }
 
  return 0;
}
 
int STflowSphere_Vy(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
 Stokes Flow around Sphere of radius r^s and center x^s, y^s, z^s.
 Velocity is u,v,w and viscosity is mu.
 Array rwork contains all the above variables.
 Input  : t, x
                : x[npoints*3] = {x,y,z} for each point
                : rwork[] = {uu,v,w,rS,xS,yS,zS,mu}
 Output : Vy
*/
  int i;
  double vx=rwork[0], vy=rwork[1], vz=rwork[2];
  double rS=rwork[3], xS=rwork[4], yS=rwork[5], zS=rwork[6];
  double eR[3],eTHETA[3];
  double norm_v,theta,r;
  double vR,vTHETA;
 
  for (i=0; i<nPoints; i++)
    {
      coords(vx,vy,vz,
             xS,yS,zS,
             &x[i*3],
             &r,
             &theta,
             &norm_v,
             eR,
             eTHETA);
      vel(rS,norm_v,r,theta,&vR,&vTHETA);
      u[i] = vR * eR[1] + vTHETA * eTHETA[1];
    }
 
  return 0;
}
int STflowSphere_Vz(int *iwork, double *rwork, int nPoints, double t, double *x, double *u)
{
/*
 Stokes Flow around Sphere of radius r^s and center x^s, y^s, z^s.
 Velocity is u,v,w and viscosity is mu.
 Array rwork contains all the above variables.
 Input  : t, x
                : x[npoints*3] = {x,y,z} for each point
                : rwork[] = {uu,v,w,rS,xS,yS,zS,mu}
 Output : Vz
*/
  int i;
  double vx=rwork[0], vy=rwork[1], vz=rwork[2];
  double rS=rwork[3], xS=rwork[4], yS=rwork[5], zS=rwork[6];
  double eR[3],eTHETA[3];
  double norm_v,theta,r;
  double vR,vTHETA;
 
  for (i=0; i<nPoints; i++)
    {
      coords(vx,vy,vz,
             xS,yS,zS,
             &x[i*3],
             &r,
             &theta,
             &norm_v,
             eR,
             eTHETA);
      vel(rS,norm_v,r,theta,&vR,&vTHETA);
      u[i] = vR * eR[2] + vTHETA* eTHETA[2];
    }
 
  return 0;
}
 
/********************************************************************/
void coords(double vx, double vy, double vz,
            double xS, double yS, double zS, 
            double* x, 
            double* r, 
            double* theta, 
            double* norm_v,
            double* eR, 
            double* eTHETA)
{
  double xBar[3],
    norm_xBar,
    xBar_dot_v,
    eTHETA_dot_eR,
    norm_eTHETA;
  
  *norm_v = sqrt(vx*vx+vy*vy+vz*vz);
  
  xBar[0]=x[0]-xS;
  xBar[1]=x[1]-yS;
  xBar[2]=x[2]-zS;
  
  norm_xBar = sqrt(xBar[0]*xBar[0] + xBar[1]*xBar[1] + xBar[2]*xBar[2]);
  
  *r = norm_xBar;
 
  if (norm_xBar != 0.0)
    {
      eR[0]=xBar[0]/norm_xBar;
      eR[1]=xBar[1]/norm_xBar;
      eR[2]=xBar[2]/norm_xBar;
    }
  else
    {
      eR[0]=0.0;
      eR[1]=0.0;
      eR[2]=0.0;
    }
  xBar_dot_v = xBar[0]*vx + xBar[1]*vy + xBar[2]*vz;
  
  if (norm_xBar != 0.0)
    *theta = acos(xBar_dot_v/(norm_xBar*(*norm_v)));
  else
    *theta = 0.0;
 
  eTHETA[0] = eR[0] - vx/(*norm_v);
  eTHETA[1] = eR[1] - vy/(*norm_v);
  eTHETA[2] = eR[2] - vz/(*norm_v);
  
  eTHETA_dot_eR = eTHETA[0]*eR[0] + eTHETA[1]*eR[1] + eTHETA[2]*eR[2];
  
  eTHETA[0] -= eTHETA_dot_eR*eR[0];
  eTHETA[1] -= eTHETA_dot_eR*eR[1];
  eTHETA[2] -= eTHETA_dot_eR*eR[2];
  
  norm_eTHETA = sqrt(eTHETA[0]*eTHETA[0] + eTHETA[1]*eTHETA[1] + eTHETA[2]*eTHETA[2]);
  
  if (norm_eTHETA > 0.0)
    {
      eTHETA[0]/= norm_eTHETA;
      eTHETA[1]/= norm_eTHETA;
      eTHETA[2]/= norm_eTHETA;
    }
  else
    {
      eTHETA[0]=0.0;
      eTHETA[1]=0.0;
      eTHETA[2]=0.0;
    }
}
 
void vel(double rS, double norm_v, double r, double theta, double* vR, double* vTHETA)
{
  if(r > rS)
    {
      *vR = norm_v * cos(theta) * ( 1.0 - (3.0*rS/(2.0*r)) + (pow(rS,3.0)/(2.0*pow(r,3.0))) );
      *vTHETA = -norm_v * sin(theta) * ( 1.0 - (3.0*rS/(4.0*r)) - (pow(rS,3.0)/(4.0*pow(r,3.0))) );
    }
  else
    {
      *vR = 0.0;
      *vTHETA = 0.0;
    }
}
/********************************************************************/
double uOfX_df(double nlC, double lu)
{
        return ( (2.0*nlC/(lu-nlC)+2.0) );
}
double uOfX_f(double a, double b, double nlC, double nlD, double x, double lu)
{
         return ( (2.0 * (nlC * log(nlC-lu) - (nlC-lu)) - nlD) - b*x/a );
}
double f(double C, double b, double a, int q, int r)
{
        if (q==2 && r==1)
        {
                if (b != 0.0)
                {
                        return ( C*tanh(b*C/a) - 1.0 );
                }
                else
                {
                        printf("function f(q=2,r=1,b=0) returns -99999.99\n");
                        return -99999.99;
                }
        }
        else if (q==1 && r==2)
        {
                return ( 2.0*C*(log(C-1.0) - log(C)) + 2.0 + b/a );
        }
        else 
        {
                printf("q,r not implemented: function f returns -99999.99\n");
                return -99999.99;
        }
}
double df(double C, double b, double a, int q, int r)
{
        if (q==2 && r==1)
        {
                if (b != 0.0)
                {
                        return ( C*(1.0/pow(cosh(b*C/a),2))*(b/a) + tanh(b*C/a) );
                }
                else
                {
                        printf("function df(q=2,r=1,b=0) returns -99999.99\n");
                        return -99999.99;
                }
        }       
        else if (q==1 && r==2)
        {
                return ( 2.0*(log(C-1.0) - log(C)) + 2.0*C*(1.0/(C-1.0) - 1.0/C) ) ;
 
        }
        else 
        {
                printf("q,r not implemented: function df returns -99999.99\n");
                return -99999.99;
        }
 
}
/********************************************************************/