fe_monomial_shape_3d.c

一个用来实现偏微分方程中网格的计算库

#if DIM == 3
  
  libmesh_assert (j<6);
  
  libmesh_assert (i < (static_cast<unsigned int>(order)+1)*
              (static_cast<unsigned int>(order)+2)*
              (static_cast<unsigned int>(order)+3)/6);

  const Real xi   = p(0);
  const Real eta  = p(1);
  const Real zeta = p(2);

    // monomials. since they are hierarchic we only need one case block.
  switch (j)
    {
      // d^2()/dxi^2
    case 0:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	  return 2.;
  	  
  	case 5:
  	case 6:
  	case 7:
  	case 8:
  	case 9:
  	  return 0.;

  	  // cubic
  	case 10:
  	  return 6.*xi;

  	case 11:
  	  return 2.*eta;

  	case 12:
  	case 13:
  	  return 0.;

  	case 14:
  	  return 2.*zeta;

  	case 15:
  	case 16:
  	case 17:
  	case 18:
  	case 19:
  	  return 0.;

  	  // quartics
  	case 20:
  	  return 12.*xi*xi;

  	case 21:
  	  return 6.*xi*eta;

  	case 22:
  	  return 2.*eta*eta;

  	case 23:
  	case 24:
  	  return 0.;

  	case 25:
  	  return 6.*xi*zeta;

  	case 26:
  	  return 2.*eta*zeta;

  	case 27:
  	case 28:
  	  return 0.;

  	case 29:
  	  return 2.*zeta*zeta;

  	case 30:
  	case 31:
  	case 32:
  	case 33:
  	case 34:
  	  return 0.;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = nx * (nx - 1);
          for (unsigned int index=2; index < nx; index++)
            val *= xi;
          for (unsigned int index=0; index != ny; index++)
            val *= eta;
          for (unsigned int index=0; index != nz; index++)
            val *= zeta;
          return val;
  	}
      }


      // d^2()/dxideta
    case 1:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	  return 0.;

  	case 5:
  	  return 1.;

  	case 6:
  	case 7:
  	case 8:
  	case 9:
  	  return 0.;

  	  // cubic
  	case 10:
  	  return 0.;

  	case 11:
  	  return 2.*xi;

  	case 12:
  	  return 2.*eta;

  	case 13:
  	case 14:
  	  return 0.;

  	case 15:
  	  return zeta;

  	case 16:
  	case 17:
  	case 18:
  	case 19:
  	  return 0.;

  	  // quartics
  	case 20:
  	  return 0.;

  	case 21:
  	  return 3.*xi*xi;

  	case 22:
  	  return 4.*xi*eta;

  	case 23:
  	  return 3.*eta*eta;

  	case 24:
  	case 25:
  	  return 0.;

  	case 26:
  	  return 2.*xi*zeta;

  	case 27:
  	  return 2.*eta*zeta;

  	case 28:
  	case 29:
  	  return 0.;

  	case 30:
  	  return zeta*zeta;

  	case 31:
  	case 32:
  	case 33:
  	case 34:
  	  return 0.;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = nx * ny;
          for (unsigned int index=1; index < nx; index++)
            val *= xi;
          for (unsigned int index=1; index < ny; index++)
            val *= eta;
          for (unsigned int index=0; index != nz; index++)
            val *= zeta;
          return val;
  	}
      }

      
      // d^2()/deta^2
    case 2:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	case 5:
  	  return 0.;
  	  
  	case 6:
  	  return 2.;

  	case 7:
  	case 8:
  	case 9:
  	  return 0.;

  	  // cubic
  	case 10:
  	case 11:
  	  return 0.;

  	case 12:
  	  return 2.*xi;
  	case 13:
  	  return 6.*eta;

  	case 14:
  	case 15:
  	  return 0.;

  	case 16:
  	  return 2.*zeta;

  	case 17:
  	case 18:
  	case 19:
  	  return 0.;

  	  // quartics
  	case 20:
  	case 21:
  	  return 0.;

  	case 22:
  	  return 2.*xi*xi;

  	case 23:
  	  return 6.*xi*eta;

  	case 24:
  	  return 12.*eta*eta;

  	case 25:
  	case 26:
  	  return 0.;

  	case 27:
  	  return 2.*xi*zeta;

  	case 28:
  	  return 6.*eta*zeta;

  	case 29:
  	case 30:
  	  return 0.;

  	case 31:
  	  return 2.*zeta*zeta;

  	case 32:
  	case 33:
  	case 34:
  	  return 0.;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = ny * (ny - 1);
          for (unsigned int index=0; index != nx; index++)
            val *= xi;
          for (unsigned int index=2; index < ny; index++)
            val *= eta;
          for (unsigned int index=0; index != nz; index++)
            val *= zeta;
          return val;
  	}
      }

      
      // d^2()/dxidzeta
    case 3:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	case 5:
  	case 6:
  	  return 0.;

  	case 7:
  	  return 1.;
  	  
  	case 8:
  	case 9:
  	  return 0.;

  	  // cubic
  	case 10:
  	case 11:
  	case 12:
  	case 13:
  	  return 0.;

  	case 14:
  	  return 2.*xi;

  	case 15:
  	  return eta;

  	case 16:
  	  return 0.;

  	case 17:
  	  return 2.*zeta;

  	case 18:
  	case 19:
  	  return 0.;

  	  // quartics
  	case 20:
  	case 21:
  	case 22:
  	case 23:
  	case 24:
  	  return 0.;

  	case 25:
  	  return 3.*xi*xi;

  	case 26:
  	  return 2.*xi*eta;

  	case 27:
  	  return eta*eta;

  	case 28:
  	  return 0.;

  	case 29:
  	  return 4.*xi*zeta;

  	case 30:
  	  return 2.*eta*zeta;

  	case 31:
  	  return 0.;

  	case 32:
  	  return 3.*zeta*zeta;

  	case 33:
  	case 34:
  	  return 0.;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = nx * nz;
          for (unsigned int index=1; index < nx; index++)
            val *= xi;
          for (unsigned int index=0; index != ny; index++)
            val *= eta;
          for (unsigned int index=1; index < nz; index++)
            val *= zeta;
          return val;
  	}
      }

      // d^2()/detadzeta
    case 4:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	case 5:
  	case 6:
  	case 7:
  	  return 0.;

  	case 8:
  	  return 1.;
  	  
  	case 9:
  	  return 0.;

  	  // cubic
  	case 10:
  	case 11:
  	case 12:
  	case 13:
  	case 14:
  	  return 0.;

  	case 15:
  	  return xi;

  	case 16:
  	  return 2.*eta;

  	case 17:
  	  return 0.;

  	case 18:
  	  return 2.*zeta;

  	case 19:
  	  return 0.;

  	  // quartics
  	case 20:
  	case 21:
  	case 22:
  	case 23:
  	case 24:
  	case 25:
  	  return 0.;

  	case 26:
  	  return xi*xi;

  	case 27:
  	  return 2.*xi*eta;

  	case 28:
  	  return 3.*eta*eta;

  	case 29:
  	  return 0.;

  	case 30:
  	  return 2.*xi*zeta;

  	case 31:
  	  return 4.*eta*zeta;

  	case 32:
  	  return 0.;

  	case 33:
  	  return 3.*zeta*zeta;

  	case 34:
  	  return 0.;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = ny * nz;
          for (unsigned int index=0; index != nx; index++)
            val *= xi;
          for (unsigned int index=1; index < ny; index++)
            val *= eta;
          for (unsigned int index=1; index < nz; index++)
            val *= zeta;
          return val;
  	}
      }


      // d^2()/dzeta^2
    case 5:
      {
        switch (i)
  	{
  	  // constant
  	case 0:
  	  
  	  // linear
  	case 1:
  	case 2:
  	case 3:
  	  return 0.;

  	  // quadratic
  	case 4:
  	case 5:
  	case 6:
  	case 7:
  	case 8:
  	  return 0.;

  	case 9:
  	  return 2.;

  	  // cubic
  	case 10:
  	case 11:
  	case 12:
  	case 13:
  	case 14:
  	case 15:
  	case 16:
  	  return 0.;

  	case 17:
  	  return 2.*xi;

  	case 18:
  	  return 2.*eta;

  	case 19:
  	  return 6.*zeta;

  	  // quartics
  	case 20:
  	case 21:
  	case 22:
  	case 23:
  	case 24:
  	case 25:
  	case 26:
  	case 27:
  	case 28:
  	  return 0.;

  	case 29:
  	  return 2.*xi*xi;

  	case 30:
  	  return 2.*xi*eta;

  	case 31:
  	  return 2.*eta*eta;

  	case 32:
  	  return 6.*xi*zeta;

  	case 33:
  	  return 6.*eta*zeta;

  	case 34:
  	  return 12.*zeta*zeta;
  	  
  	default:
          unsigned int o = 0;
          for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
          unsigned int i2 = i - (o*(o+1)*(o+2)/6);
          unsigned int block=o, nz = 0;
          for (; block < i2; block += (o-nz+1)) { nz++; }
          const unsigned int nx = block - i2;
          const unsigned int ny = o - nx - nz;
          Real val = nz * (nz - 1);
          for (unsigned int index=0; index != nx; index++)
            val *= xi;
          for (unsigned int index=0; index != ny; index++)
            val *= eta;
          for (unsigned int index=2; index < nz; index++)
            val *= zeta;
          return val;
  	}
      }

      
    default:
      libmesh_error();
    }

#endif
  
  libmesh_error();
  return 0.;  
}



template <>
Real FE<3,MONOMIAL>::shape_second_deriv(const Elem* elem,
				        const Order order,
				        const unsigned int i,
				        const unsigned int j,
				        const Point& p)
{
  libmesh_assert (elem != NULL);
      
  // call the orientation-independent shape function derivatives
  return FE<3,MONOMIAL>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}