v3d4_nl.f90

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SUBROUTINE v3d4_nl(coor,matcstet,lmcstet,R_in,d,ci, &
     s11,s22,s33,s12,s23,s13,numnp,nstart,nend,numcstet,numat_vol,xmu,xlambda)

!________________________________________________________________________
!
!  V3D4 - Performs displacement based computations for Volumetric 3D 
!         4-node tetrahedra large deformation (i.e. nonlinear) elastic 
!         elements with linear interpolation functions. (constant strain 
!         tetrahedra). Returns the internal force vector R_in.
!
!  DATE: 04.2000                  AUTHOR: SCOT BREITENFELD
!________________________________________________________________________

  IMPLICIT NONE
!---- Global variables
  INTEGER :: numnp          ! number of nodal points
  INTEGER :: numcstet       ! number of CSTet elements
  INTEGER :: numat_vol      ! number of volumetric materials
!--   coordinate array
  REAL*8, DIMENSION(1:3,1:numnp) :: coor,x
!--   elastic stiffness consts
  REAL*8, DIMENSION(1:9,1:numat_vol) :: ci
!--   internal force
  REAL*8, DIMENSION(1:3*numnp) :: r_in
!--  displacement vector
  REAL*8, DIMENSION(1:3*numnp) :: d
!--   CSTet stress
  REAL*8, DIMENSION(1:numcstet) :: s11, s22, s33, s12, s23, s13
!--  x, y and z displacements of nodes
  REAL*8 :: u1,u2,u3,u4,v1,v2,v3,v4,w1,w2,w3,w4
!--   coordinate holding variable
  REAL*8 :: x1,x2,x3,x4,y1,y2,y3,y4,z1,z2,z3,z4
!--  Coordinate subtractions
  REAL*8 :: x14, x24, x34, y14, y24, y34, z14, z24, z34
!--  Coordinate subtractions: These are to speed up B calculation
  REAL*8 :: x12, x13, y12, y13, z12, z13
!--  
!  REAL*8 :: c11, c21, c31
!--   6*volume, inverse(6*volume),  and the volume      
  REAL*8 :: vx6, vx6inv, vol
!--   spacial derivatives
  REAL*8 :: b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12
!--   partial derivatives of the displacement 
  REAL*8 :: dudx,dvdy,dwdz,dudy,dvdx,dvdz,dwdy,dudz,dwdx
!--   strains
  REAL*8 :: e11,e22,e33,e12,e23,e13
! -- connectivity table for CSTet
  INTEGER, DIMENSION(1:4,1:numcstet) :: lmcstet
! -- mat number for CST element
  INTEGER, DIMENSION(1:numcstet) :: matcstet
! -- node numbers
  INTEGER :: n1,n2,n3,n4
! -- dummy and counters
  INTEGER :: i,j,nstart,nend
  INTEGER :: k1n1,k1n2,k1n3,k1n4,k2n1,k2n2,k2n3,k2n4
  INTEGER :: k3n1,k3n2,k3n3,k3n4! -- 
  REAL*8 :: f11, f12, f13, f21, f22, f23, f31, f32, f33
  REAL*8 :: c11, c12, c13, c21, c22, c23, c31, c32, c33

! Cauchy Variables
  REAL*8, DIMENSION(1:numat_vol) :: xmu, xlambda

  DO i = nstart, nend
     j = matcstet(i)
     
     n1 = lmcstet(1,i)
     n2 = lmcstet(2,i)
     n3 = lmcstet(3,i)
     n4 = lmcstet(4,i)
     
     k3n1 = 3*n1
     k3n2 = 3*n2
     k3n3 = 3*n3
     k3n4 = 3*n4
     
     k2n1 = k3n1 - 1
     k2n2 = k3n2 - 1
     k2n3 = k3n3 - 1
     k2n4 = k3n4 - 1
     
     k1n1 = k3n1 - 2
     k1n2 = k3n2 - 2
     k1n3 = k3n3 - 2
     k1n4 = k3n4 - 2     
     ! k#n# dummy variables replaces:
     u1 = d(k1n1)           ! (3*n1-2)
     u2 = d(k1n2)           ! (3*n2-2)
     u3 = d(k1n3)           ! (3*n3-2)
     u4 = d(k1n4)           ! (3*n4-2)
     v1 = d(k2n1)           ! (3*n1-1)
     v2 = d(k2n2)           ! (3*n2-1)
     v3 = d(k2n3)           ! (3*n3-1)
     v4 = d(k2n4)           ! (3*n4-1)
     w1 = d(k3n1)           ! (3*n1)
     w2 = d(k3n2)           ! (3*n2)
     w3 = d(k3n3)           ! (3*n3)
     w4 = d(k3n4)           ! (3*n4)
     
     x1 = coor(1,n1)
     x2 = coor(1,n2)
     x3 = coor(1,n3)
     x4 = coor(1,n4)
     y1 = coor(2,n1)
     y2 = coor(2,n2)
     y3 = coor(2,n3)
     y4 = coor(2,n4)
     z1 = coor(3,n1)
     z2 = coor(3,n2)
     z3 = coor(3,n3)
     z4 = coor(3,n4)

     x12 = x1 - x2          ! not used in vol. calc
     x13 = x1 - x3          ! not used in vol. calc
     x14 = x1 - x4
     x24 = x2 - x4
     x34 = x3 - x4
     y12 = y1 - y2          ! not used in vol. calc
     y13 = y1 - y3          ! not used in vol. calc
     y14 = y1 - y4
     y24 = y2 - y4
     y34 = y3 - y4
     z12 = z1 - z2          ! not used in vol. calc
     z13 = z1 - z3          ! not used in vol. calc
     z14 = z1 - z4
     z24 = z2 - z4
     z34 = z3 - z4
     
     c11 =    y24*z34 - z24*y34
     c21 = -( x24*z34 - z24*x34 )
     c31 =    x24*y34 - y24*x34
     
     vx6 = -( x14*c11 + y14*c21 + z14*c31 )
     
     vx6inv = 1.d0 / vx6

! See the maple worksheet 'V3D4.mws' for the derivation of [B]
! NOTE: Factored for a more equivalent/compact form then maple's

     b1  = (y34*z24 - y24*z34) * vx6inv
     b2  = (z34*x24 - z24*x34) * vx6inv
     b3  = (x34*y24 - x24*y34) * vx6inv
     b4  = (y13*z14 - y14*z13) * vx6inv
     b5  = (z13*x14 - z14*x13) * vx6inv
     b6  = (x13*y14 - x14*y13) * vx6inv
     b7  = (y14*z12 - y12*z14) * vx6inv
     b8  = (z14*x12 - z12*x14) * vx6inv
     b9  = (x14*y12 - x12*y14) * vx6inv
     b10 = (y12*z13 - y13*z12) * vx6inv
     b11 = (z12*x13 - z13*x12) * vx6inv
     b12 = (x12*y13 - x13*y12) * vx6inv

!     print*,'B',B1,B2,B3,B4,B5,B6,B7,B8,B9,B10,B11,B12
!
! Calculate displacement gradient (H)
!
     dudx = b1*u1 + b4*u2 + b7*u3 + b10*u4
     dvdy = b2*v1 + b5*v2 + b8*v3 + b11*v4
     dwdz = b3*w1 + b6*w2 + b9*w3 + b12*w4
     dudy = b2*u1 + b5*u2 + b8*u3 + b11*u4
     dvdx = b1*v1 + b4*v2 + b7*v3 + b10*v4
     dvdz = b3*v1 + b6*v2 + b9*v3 + b12*v4
     dwdy = b2*w1 + b5*w2 + b8*w3 + b11*w4
     dudz = b3*u1 + b6*u2 + b9*u3 + b12*u4
     dwdx = b1*w1 + b4*w2 + b7*w3 + b10*w4
! 
! deformation gradients F
!
     f11 = 1.d0 + dudx
     f12 = dudy
     f13 = dudz
     f21 = dvdx
     f22 = 1.d0 + dvdy
     f23 = dvdz
     f31 = dwdx
     f32 = dwdy
     f33 = 1.d0 + dwdz

!
! Right Cauchy-Green deformation tensor
!
     c11 = f11*f11+f21*f21+f31*f31
     c12 = f11*f12+f21*f22+f31*f32
     c13 = f11*f13+f21*f23+f31*f33
     c21 = c12
     c22 = f12*f12+f22*f22+f32*f32
     c23 = f12*f13+f22*f23+f32*f33
     c31 = c13
     c32 = c23
     c33 = f13*f13+f23*f23+f33*f33
!
! Lagrangian or Green's Strain tensor E = 1/2 * (C - I)
!
     e11 = 0.5d0*(c11 - 1.d0)
     e22 = 0.5d0*(c22 - 1.d0)
     e33 = 0.5d0*(c33 - 1.d0)
     e12 = 0.5d0*c12
     e23 = 0.5d0*c23
     e13 = 0.5d0*c13

! second Piola-Kirchhoff stress tensor
!
! S = lambda * tr(E) * I + 2 * mu * E

     s11(i) = xlambda(j)*(e11+e22+e33) + 2.d0*xmu(j)*e11
     s12(i) =  2.d0*xmu(j)*e12
     s13(i) =  2.d0*xmu(j)*e13

     s22(i) = xlambda(j)*(e11+e22+e33) + 2.d0*xmu(j)*e22
     s23(i) =  2.d0*xmu(j)*e23

     s33(i) = xlambda(j)*(e11+e22+e33) + 2.d0*xmu(j)*e33

! calculate the volume

     vol = vx6 / 6.d0

! ASSEMBLE THE INTERNAL FORCE VECTOR
!
! local node 1
     r_in(k1n1) = r_in(k1n1) - vol* &
          ( s11(i)*b1*f11 + s22(i)*b2*f12 + s33(i)*b3*f13 &
          + s12(i)*( b2*f11 + b1*f12 ) &
          + s23(i)*( b3*f12 + b2*f13 ) &
          + s13(i)*( b3*f11 + b1*f13 ) )
     r_in(k2n1) = r_in(k2n1) - vol* &
          ( s11(i)*b1*f21 + s22(i)*b2*f22 + s33(i)*b3*f23 &
          + s12(i)*( b1*f22 + b2*f21 ) &
          + s23(i)*( b3*f22 + b2*f23 ) &
          + s13(i)*( b3*f21 + b1*f23 ) )
     r_in(k3n1)   = r_in(k3n1)   - vol* &
          ( s11(i)*b1*f31 + s22(i)*b2*f32 + s33(i)*b3*f33 &
          + s12(i)*( b2*f31 + b1*f32 ) &
          + s23(i)*( b3*f32 + b2*f33 ) &
          + s13(i)*( b3*f31 + b1*f33 ) )
! local node 2 
     r_in(k1n2) = r_in(k1n2) - vol* &
          ( s11(i)*b4*f11 + s22(i)*b5*f12 + s33(i)*b6*f13 &
          + s12(i)*( b5*f11 + b4*f12 ) &
          + s23(i)*( b6*f12 + b5*f13 ) &
          + s13(i)*( b6*f11 + b4*f13 ) )
     r_in(k2n2) = r_in(k2n2) - vol* &
          ( s11(i)*b4*f21 + s22(i)*b5*f22 + s33(i)*b6*f23 &
          + s12(i)*( b4*f22 + b5*f21 ) &
          + s23(i)*( b6*f22 + b5*f23 ) &
          + s13(i)*( b6*f21 + b4*f23 ) )
     r_in(k3n2)   = r_in(k3n2)   - vol* &
          ( s11(i)*b4*f31 + s22(i)*b5*f32 + s33(i)*b6*f33 &
          + s12(i)*( b5*f31 + b4*f32 ) &
          + s23(i)*( b6*f32 + b5*f33 ) &
          + s13(i)*( b6*f31 + b4*f33 ) )
! local node 3 
     r_in(k1n3) = r_in(k1n3) - vol* &
          ( s11(i)*b7*f11 + s22(i)*b8*f12 + s33(i)*b9*f13 &
          + s12(i)*( b8*f11 + b7*f12 ) &
          + s23(i)*( b9*f12 + b8*f13 ) &
          + s13(i)*( b9*f11 + b7*f13 ) )
     r_in(k2n3) = r_in(k2n3) - vol* &
          ( s11(i)*b7*f21 + s22(i)*b8*f22 + s33(i)*b9*f23 &
          + s12(i)*( b7*f22 + b8*f21 ) &
          + s23(i)*( b9*f22 + b8*f23 ) &
          + s13(i)*( b9*f21 + b7*f23 ) )
     r_in(k3n3)   = r_in(k3n3)   - vol* &
          ( s11(i)*b7*f31 + s22(i)*b8*f32 + s33(i)*b9*f33 &
          + s12(i)*( b8*f31 + b7*f32 ) &
          + s23(i)*( b9*f32 + b8*f33 ) &
          + s13(i)*( b9*f31 + b7*f33 ) )
! local node 4         
     r_in(k1n4) = r_in(k1n4) - vol* &
          ( s11(i)*b10*f11 + s22(i)*b11*f12+s33(i)*b12*f13 &
          + s12(i)*( b11*f11 + b10*f12 ) &
          + s23(i)*( b12*f12 + b11*f13 ) &
          + s13(i)*( b12*f11 + b10*f13 ) )
     r_in(k2n4) = r_in(k2n4) - vol* &
          ( s11(i)*b10*f21 + s22(i)*b11*f22+s33(i)*b12*f23 &
          + s12(i)*( b10*f22 + b11*f21 ) &
          + s23(i)*( b12*f22 + b11*f23 ) &
          + s13(i)*( b12*f21 + b10*f23 ) )
     r_in(k3n4)   = r_in(k3n4)   - vol* &
          ( s11(i)*b10*f31 + s22(i)*b11*f32+s33(i)*b12*f33 &
          + s12(i)*( b11*f31 + b10*f32 ) &
          + s23(i)*( b12*f32 + b11*f33 ) &
          + s13(i)*( b12*f31 + b10*f33 ) )

  ENDDO
  
  RETURN
END SUBROUTINE v3d4_nl