fe_xyz_shape_3d.c

一个用来实现偏微分方程中网格的计算库

    default:
      libmesh_error();
    }

#endif
  
  libmesh_error();
  return 0.;  
}



template <>
Real FE<3,XYZ>::shape_second_deriv(const ElemType,
			           const Order,
			           const unsigned int,
			           const unsigned int,
			           const Point&)
{
  std::cerr << "XYZ polynomials require the element\n"
            << "because the centroid is needed."
            << std::endl;
  
  libmesh_error();
  return 0.;
}



template <>
Real FE<3,XYZ>::shape_second_deriv(const Elem* elem,
			           const Order order,
			           const unsigned int i,
			           const unsigned int j,
			           const Point& p)
{
#if DIM == 3

  libmesh_assert (elem != NULL);
  libmesh_assert (j<3);
  
  // Only recompute the centroid if the element
  // has changed from the last one we computed.
  // This avoids repeated centroid calculations
  // when called in succession with the same element.
  if (elem->id() != old_elem_id)
    {
      centroid = elem->centroid();
      old_elem_id = elem->id();
    }  
  
  const Real x  = p(0);
  const Real y  = p(1);
  const Real z  = p(2);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real zc = centroid(2);
  const Real dx = x - xc;
  const Real dy = y - yc;
  const Real dz = z - zc;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = static_cast<Order>(order + elem->p_level());
#endif
  libmesh_assert (i < (static_cast<unsigned int>(totalorder)+1)*
              (static_cast<unsigned int>(totalorder)+2)*
              (static_cast<unsigned int>(totalorder)+2)/6);

    // monomials. since they are hierarchic we only need one case block.
  switch (j)
    {
      // d^2()/dx^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.*dx;

  	case 11:
  	  return 2.*dy;

  	case 12:
  	case 13:
  	  return 0.;

  	case 14:
  	  return 2.*dz;

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

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

  	case 21:
  	  return 6.*dx*dy;

  	case 22:
  	  return 2.*dy*dy;

  	case 23:
  	case 24:
  	  return 0.;

  	case 25:
  	  return 6.*dx*dz;

  	case 26:
  	  return 2.*dy*dz;

  	case 27:
  	case 28:
  	  return 0.;

  	case 29:
  	  return 2.*dz*dz;

  	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 *= dx;
          for (unsigned int index=0; index != ny; index++)
            val *= dy;
          for (unsigned int index=0; index != nz; index++)
            val *= dz;
          return val;
  	}
      }


      // d^2()/dxdy
    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.*dx;

  	case 12:
  	  return 2.*dy;

  	case 13:
  	case 14:
  	  return 0.;

  	case 15:
  	  return dz;

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

  	  // quartics
  	case 20:
  	  return 0.;

  	case 21:
  	  return 3.*dx*dx;

  	case 22:
  	  return 4.*dx*dy;

  	case 23:
  	  return 3.*dy*dy;

  	case 24:
  	case 25:
  	  return 0.;

  	case 26:
  	  return 2.*dx*dz;

  	case 27:
  	  return 2.*dy*dz;

  	case 28:
  	case 29:
  	  return 0.;

  	case 30:
  	  return dz*dz;

  	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 *= dx;
          for (unsigned int index=1; index < ny; index++)
            val *= dy;
          for (unsigned int index=0; index != nz; index++)
            val *= dz;
          return val;
  	}
      }

      
      // d^2()/dy^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.*dx;
  	case 13:
  	  return 6.*dy;

  	case 14:
  	case 15:
  	  return 0.;

  	case 16:
  	  return 2.*dz;

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

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

  	case 22:
  	  return 2.*dx*dx;

  	case 23:
  	  return 6.*dx*dy;

  	case 24:
  	  return 12.*dy*dy;

  	case 25:
  	case 26:
  	  return 0.;

  	case 27:
  	  return 2.*dx*dz;

  	case 28:
  	  return 6.*dy*dz;

  	case 29:
  	case 30:
  	  return 0.;

  	case 31:
  	  return 2.*dz*dz;

  	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 *= dx;
          for (unsigned int index=2; index < ny; index++)
            val *= dy;
          for (unsigned int index=0; index != nz; index++)
            val *= dz;
          return val;
  	}
      }

      
      // d^2()/dxdz
    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.*dx;

  	case 15:
  	  return dy;

  	case 16:
  	  return 0.;

  	case 17:
  	  return 2.*dz;

  	case 18:
  	case 19:
  	  return 0.;

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

  	case 25:
  	  return 3.*dx*dx;

  	case 26:
  	  return 2.*dx*dy;

  	case 27:
  	  return dy*dy;

  	case 28:
  	  return 0.;

  	case 29:
  	  return 4.*dx*dz;

  	case 30:
  	  return 2.*dy*dz;

  	case 31:
  	  return 0.;

  	case 32:
  	  return 3.*dz*dz;

  	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 *= dx;
          for (unsigned int index=0; index != ny; index++)
            val *= dy;
          for (unsigned int index=1; index < nz; index++)
            val *= dz;
          return val;
  	}
      }

      // d^2()/dydz
    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 dx;

  	case 16:
  	  return 2.*dy;

  	case 17:
  	  return 0.;

  	case 18:
  	  return 2.*dz;

  	case 19:
  	  return 0.;

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

  	case 26:
  	  return dx*dx;

  	case 27:
  	  return 2.*dx*dy;

  	case 28:
  	  return 3.*dy*dy;

  	case 29:
  	  return 0.;

  	case 30:
  	  return 2.*dx*dz;

  	case 31:
  	  return 4.*dy*dz;

  	case 32:
  	  return 0.;

  	case 33:
  	  return 3.*dz*dz;

  	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 *= dx;
          for (unsigned int index=1; index < ny; index++)
            val *= dy;
          for (unsigned int index=1; index < nz; index++)
            val *= dz;
          return val;
  	}
      }


      // d^2()/dz^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.*dx;

  	case 18:
  	  return 2.*dy;

  	case 19:
  	  return 6.*dz;

  	  // 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.*dx*dx;

  	case 30:
  	  return 2.*dx*dy;

  	case 31:
  	  return 2.*dy*dy;

  	case 32:
  	  return 6.*dx*dz;

  	case 33:
  	  return 6.*dy*dz;

  	case 34:
  	  return 12.*dz*dz;
  	  
  	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 *= dx;
          for (unsigned int index=0; index != ny; index++)
            val *= dy;
          for (unsigned int index=2; index < nz; index++)
            val *= dz;
          return val;
  	}
      }

      
    default:
      libmesh_error();
    }

#endif
  
  libmesh_error();
  return 0.;  
}