sigma_cpu.f90

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module sigma_cpu
  use num_types, only : rp
  use field_list, only : field_list_t
  use scratch_registry, only : neko_scratch_registry
  use field_registry, only : neko_field_registry
  use field, only : field_t
  use operators, only : dudxyz
  use coefs, only : coef_t
  use gs_ops, only : gs_op_add
  use math, only : neko_eps, col2
  implicit none
  private
 
  public :: sigma_compute_cpu
 
contains
 
  subroutine sigma_compute_cpu(if_ext, t, tstep, coef, nut, delta, c)
    logical, intent(in) :: if_ext
    real(kind=rp), intent(in) :: t
    integer, intent(in) :: tstep
    type(coef_t), intent(in) :: coef
    type(field_t), intent(inout) :: nut
    type(field_t), intent(in) :: delta
    real(kind=rp), intent(in) :: c
    ! This is the velocity gradient tensor
    type(field_t), pointer :: g11, g12, g13, g21, g22, g23, g31, g32, g33
    type(field_t), pointer :: u, v, w
 
    real(kind=rp) :: sigg11, sigg12, sigg13, sigg22, sigg23, sigg33
    real(kind=rp) :: sigma1, sigma2, sigma3
    real(kind=rp) :: invariant1, invariant2, invariant3
    real(kind=rp) :: alpha1, alpha2, alpha3
    real(kind=rp) :: dsigma
    real(kind=rp) :: pi_3 = 4.0_rp/3.0_rp*atan(1.0_rp)
    real(kind=rp) :: tmp1
    real(kind=rp) :: eps
 
    integer :: temp_indices(9)
    integer :: e, i
 
    ! some constant
    eps = neko_eps
 
 
    ! get fields from registry
    if (if_ext .eqv. .true.) then
       u => neko_field_registry%get_field_by_name("u_e")
       v => neko_field_registry%get_field_by_name("v_e")
       w => neko_field_registry%get_field_by_name("w_e")
    else
       u => neko_field_registry%get_field_by_name("u")
       v => neko_field_registry%get_field_by_name("v")
       w => neko_field_registry%get_field_by_name("w")
    end if
 
    call neko_scratch_registry%request_field(g11, temp_indices(1))
    call neko_scratch_registry%request_field(g12, temp_indices(2))
    call neko_scratch_registry%request_field(g13, temp_indices(3))
    call neko_scratch_registry%request_field(g21, temp_indices(4))
    call neko_scratch_registry%request_field(g22, temp_indices(5))
    call neko_scratch_registry%request_field(g23, temp_indices(6))
    call neko_scratch_registry%request_field(g31, temp_indices(7))
    call neko_scratch_registry%request_field(g32, temp_indices(8))
    call neko_scratch_registry%request_field(g33, temp_indices(9))
 
 
    ! Compute the derivatives of the velocity (the components of the g tensor)
    call dudxyz (g11%x, u%x, coef%drdx, coef%dsdx, coef%dtdx, coef)
    call dudxyz (g12%x, u%x, coef%drdy, coef%dsdy, coef%dtdy, coef)
    call dudxyz (g13%x, u%x, coef%drdz, coef%dsdz, coef%dtdz, coef)
 
    call dudxyz (g21%x, v%x, coef%drdx, coef%dsdx, coef%dtdx, coef)
    call dudxyz (g22%x, v%x, coef%drdy, coef%dsdy, coef%dtdy, coef)
    call dudxyz (g23%x, v%x, coef%drdz, coef%dsdz, coef%dtdz, coef)
 
    call dudxyz (g31%x, w%x, coef%drdx, coef%dsdx, coef%dtdx, coef)
    call dudxyz (g32%x, w%x, coef%drdy, coef%dsdy, coef%dtdy, coef)
    call dudxyz (g33%x, w%x, coef%drdz, coef%dsdz, coef%dtdz, coef)
 
    call coef%gs_h%op(g11, gs_op_add)
    call coef%gs_h%op(g12, gs_op_add)
    call coef%gs_h%op(g13, gs_op_add)
    call coef%gs_h%op(g21, gs_op_add)
    call coef%gs_h%op(g22, gs_op_add)
    call coef%gs_h%op(g23, gs_op_add)
    call coef%gs_h%op(g31, gs_op_add)
    call coef%gs_h%op(g32, gs_op_add)
    call coef%gs_h%op(g33, gs_op_add)
 
    do concurrent(e = 1:coef%msh%nelv)
       do concurrent(i = 1:coef%Xh%lxyz)
          ! G_ij = g^t g = g_mi g_mj
          sigg11 = g11%x(i,1,1,e)**2 + g21%x(i,1,1,e)**2 + g31%x(i,1,1,e)**2
          sigg22 = g12%x(i,1,1,e)**2 + g22%x(i,1,1,e)**2 + g32%x(i,1,1,e)**2
          sigg33 = g13%x(i,1,1,e)**2 + g23%x(i,1,1,e)**2 + g33%x(i,1,1,e)**2
          sigg12 = g11%x(i,1,1,e)*g12%x(i,1,1,e) + &
               g21%x(i,1,1,e)*g22%x(i,1,1,e) + &
               g31%x(i,1,1,e)*g32%x(i,1,1,e)
          sigg13 = g11%x(i,1,1,e)*g13%x(i,1,1,e) + &
               g21%x(i,1,1,e)*g23%x(i,1,1,e) + &
               g31%x(i,1,1,e)*g33%x(i,1,1,e)
          sigg23 = g12%x(i,1,1,e)*g13%x(i,1,1,e) + &
               g22%x(i,1,1,e)*g23%x(i,1,1,e) + &
               g32%x(i,1,1,e)*g33%x(i,1,1,e)
 
          ! If LAPACK compute eigenvalues of the semi-definite positive matrix G
          ! ..........to be done later on......
          ! ELSE use the analytical method as done in the following
 
          ! eigenvalues with the analytical method of Hasan et al. (2001)
          ! doi:10.1006/jmre.2001.2400
          if (abs(sigg11) .lt. eps) then
             sigg11 = 0.0_rp
          end if
          if (abs(sigg12) .lt. eps) then
             sigg12 = 0.0_rp
          end if
          if (abs(sigg13) .lt. eps) then
             sigg13 = 0.0_rp
          end if
          if (abs(sigg22) .lt. eps) then
             sigg22 = 0.0_rp
          end if
          if (abs(sigg23) .lt. eps) then
             sigg23 = 0.0_rp
          end if
          if (abs(sigg33) .lt. eps) then
             sigg33 = 0.0_rp
          end if
 
          if (abs(sigg12*sigg12 + &
               sigg13*sigg13 + sigg23*sigg23) .lt. eps) then
             !             G is diagonal
             ! estimate the singular values according to:
             sigma1 = sqrt(max(max(max(sigg11, sigg22), sigg33), 0.0_rp))
             sigma3 = sqrt(max(min(min(sigg11, sigg22), sigg33), 0.0_rp))
             invariant1 = sigg11 + sigg22 + sigg33
             sigma2 = sqrt(abs(invariant1 - sigma1*sigma1 - sigma3*sigma3))
          else
 
             !  estimation of invariants
             invariant1 = sigg11 + sigg22 + sigg33
             invariant2 = sigg11*sigg22 + sigg11*sigg33 + sigg22*sigg33 - &
                  (sigg12*sigg12 + sigg13*sigg13 + sigg23*sigg23)
             invariant3 = sigg11*sigg22*sigg33 + &
                  2.0_rp*sigg12*sigg13*sigg23 - &
                  (sigg11*sigg23*sigg23 + sigg22*sigg13*sigg13 + &
                  sigg33*sigg12*sigg12)
 
             ! G is symmetric semi-definite positive matrix:
             ! the invariants have to be larger-equal zero
             ! which is obtained via forcing
             invariant1 = max(invariant1, 0.0_rp)
             invariant2 = max(invariant2, 0.0_rp)
             invariant3 = max(invariant3, 0.0_rp)
 
             ! compute the following angles from the invariants
             alpha1 = invariant1*invariant1/9.0_rp - invariant2/3.0_rp
 
             ! since alpha1 is always positive (see Hasan et al. (2001))
             ! forcing is applied
             alpha1 = max(alpha1, 0.0_rp)
 
             alpha2 = invariant1*invariant1*invariant1/27.0_rp - &
                  invariant1*invariant2/6.0_rp + invariant3/2.0_rp
 
             ! since acos(alpha2/(alpha1^(3/2)))/3.0_rp only valid for
             ! alpha2^2 < alpha1^3.0_rp and arccos(x) only valid for -1<=x<=1
             !  alpha3 is between 0 and pi/3
             tmp1 = alpha2/sqrt(alpha1 * alpha1 * alpha1)
 
             if (tmp1 .le. -1.0_rp) then
                ! alpha3=pi/3 -> cos(alpha3)=0.5
                ! compute the singular values
                sigma1 = sqrt(max(invariant1/3.0_rp + sqrt(alpha1), 0.0_rp))
                sigma2 = sigma1
                sigma3 = sqrt(invariant1/3.0_rp - 2.0_rp*sqrt(alpha1))
 
             elseif (tmp1 .ge. 1.0_rp) then
                ! alpha3=0.0_rp -> cos(alpha3)=1.0
                sigma1 = sqrt(max(invariant1/3.0_rp + 2.0_rp*sqrt(alpha1), &
                     0.0_rp))
                sigma2 = sqrt(invariant1/3.0_rp - sqrt(alpha1))
                sigma3 = sigma2
             else
                alpha3 = acos(tmp1)/3.0_rp
 
                if (abs(invariant3) .lt. eps) then
                   ! In case of Invariant3=0, one or more eigenvalues are equal
                   ! to zero. Therefore force sigma3 to 0 and compute sigma1 and
                   ! sigma2
                   sigma1 = sqrt(max(invariant1/3.0_rp + &
                        2.0_rp*sqrt(alpha1)*cos(alpha3), 0.0_rp))
                   sigma2 = sqrt(abs(invariant1 - sigma1*sigma1))
                   sigma3 = 0.0_rp
                else
                   sigma1 = sqrt(max(invariant1/3.0_rp + &
                        2.0_rp*sqrt(alpha1)*cos(alpha3), 0.0_rp))
                   sigma2 = sqrt(invariant1/3.0_rp - &
                        2.0_rp*sqrt(alpha1)*cos(pi_3 + alpha3))
                   sigma3 = sqrt(abs(invariant1 - &
                        sigma1*sigma1-sigma2*sigma2))
                end if ! Invariant3=0 ?
             end if ! tmp1
          end if ! G diagonal ?
 
          ! Estimate Dsigma
          if (sigma1 .gt. 0.0_rp) then
             dsigma = &
                  sigma3*(sigma1 - sigma2)*(sigma2 - sigma3)/(sigma1*sigma1)
          else
             dsigma = 0.0_rp
          end if
 
          !clipping to avoid negative values
          dsigma = max(dsigma, 0.0_rp)
 
          ! estimate turbulent viscosity
 
          nut%x(i,1,1,e) = (c*delta%x(i,1,1,e))**2 * dsigma &
               * coef%mult(i,1,1,1)
 
       end do
    end do
 
    call coef%gs_h%op(nut, gs_op_add)
    call col2(nut%x, coef%mult, nut%dof%size())
 
    call neko_scratch_registry%relinquish_field(temp_indices)
  end subroutine sigma_compute_cpu
 
end module sigma_cpu