ArtificialDamping.f90

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subroutine artificialdamping(numcstet, numnp, &
     rho, cd_fastest, detfold, dt, f, fdot, vx6, strssvisco)

  implicit none

  integer :: numcstet
  integer :: numnp

  REAL*8 :: rho
  real*8 :: cd_fastest
  real*8 :: detfold ! change to more general
  real*8 :: dt
  real*8 :: vx6

  
  REAL*8,DIMENSION(1:3,1:3) :: id = reshape( &
       (/1.0,0.0, 0.0, &
         0.0,1.0, 0.0, &
         0.0,0.0, 1.0 /),(/3,3/) )


  REAL*8 :: f(3,3),detf,finv(3,3), fdot(3,3), t(3,3), symt(3,3), devt(3,3)
  REAL*8 :: strsscauchy(3,3), strssvisco(3,3)

  REAL*8 :: eta, c1, cl, h, du

  c1 = 1.d0
  cl = 0.1d0 

  finv(:,:) = f(:,:)

! returns the                           -1
! inverse of the deformation gradient, F  
! and determanate of F

  CALL invert3x3(finv,detf)

!   .    -1
!   F * F
!

  t = matmul(fdot,finv)
!
!        .    -1                         T
! symm(  F * F  ) = symm(T) = 0.5*( T + T )

  symt = 0.5d0*( t + transpose(t))

!
! dev( symm(T) ) = symmT - 1/3 * ( tr(symm(T)) * I
!

  devt = symt - 1.d0/3.d0*(symt(1,1)+symt(2,2)+symt(3,3))*id

!
!  h = ( J* d! * |Vol| ) ^ (1/d)
!  
!       where
!             d is the dimension fo space
!            Vol is the volume of the element
!            J is the Jacobian
!
!

  h = ( detf * vx6 )**(1./3.)

!                     n+1        n
!   DelU = h * ( log(J    - log J  ) ) / Delt
!

  du = h * ( log(detf) - log(detfold) )/dt



!
!   artifical viscosity,  max(0, -3/4 * h * rho*(c1*DelU - cL*a)

!  IF(Du.GE.0.d0)THEN
!     eta = 0.d0
!  ELSE

  eta = max(0.d0,-0.75d0*h*rho/detf*(c1*du - cl*cd_fastest))

!  ENDIF

! Cauchy Stress

  strsscauchy = 2.0*eta*devt

! PK2 stress
     
  strssvisco = detf*finv*strsscauchy*transpose(finv)

! 
! store Jacobian for next time step

  detfold = detf

end subroutine artificialdamping