// Ureset fixes internal variables after u (displacements) have been zeroed
func (o *ElemU) Ureset(sol *Solution) (ok bool) {
for idx, _ := range o.IpsElem {
if len(o.States[idx].F) > 0 {
la.MatFill(o.States[idx].F, 0)
la.MatFill(o.StatesBkp[idx].F, 0)
}
}
return true
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o *Rod) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
// zero K matrix
la.MatFill(o.K, 0)
la.MatFill(o.M, 0) // TODO: implement mass matrix
// for each integration point
var E float64
nverts := o.Cell.Shp.Nverts
for idx, ip := range o.IpsElem {
// interpolation functions, gradients and variables @ ip
err = o.ipvars(idx, sol)
if err != nil {
return
}
// auxiliary
coef := ip[3]
Jvec := o.Cell.Shp.Jvec3d
G := o.Cell.Shp.Gvec
J := o.Cell.Shp.J
// add contribution to consistent tangent matrix
for m := 0; m < nverts; m++ {
for n := 0; n < nverts; n++ {
for i := 0; i < o.Ndim; i++ {
for j := 0; j < o.Ndim; j++ {
r := i + m*o.Ndim
c := j + n*o.Ndim
E, err = o.Model.CalcD(o.States[idx], firstIt)
if err != nil {
return
}
o.K[r][c] += coef * o.A * E * G[m] * G[n] * Jvec[i] * Jvec[j] / J
}
}
}
}
}
// add K to sparse matrix Kb
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return
}
// adds element K to global Jacobian matrix Kb
func (o Rod) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
// zero K matrix
la.MatFill(o.K, 0)
la.MatFill(o.M, 0)
// for each integration point
nverts := o.Shp.Nverts
ndim := Global.Ndim
for idx, ip := range o.IpsElem {
// interpolation functions, gradients and variables @ ip
if !o.ipvars(idx, sol) {
return
}
// auxiliary
coef := ip.W
Jvec := o.Shp.Jvec3d
G := o.Shp.Gvec
J := o.Shp.J
// add contribution to consistent tangent matrix
for m := 0; m < nverts; m++ {
for n := 0; n < nverts; n++ {
for i := 0; i < ndim; i++ {
for j := 0; j < ndim; j++ {
r := i + m*ndim
c := j + n*ndim
E, err := o.Model.CalcD(o.States[idx], firstIt)
if LogErr(err, "AddToKb") {
return
}
o.K[r][c] += coef * o.A * E * G[m] * G[n] * Jvec[i] * Jvec[j] / J
}
}
}
}
}
// add K to sparse matrix Kb
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return true
}
// CalcD computes D = dσ_new/dε_new (consistent)
func (o SmallElasticity) CalcD(D [][]float64, s *State) (err error) {
if o.Pse {
if o.Nsig != 4 {
return chk.Err("for plane-stress analyses, D must be 4x4. nsig = %d is incorrect.\n", o.Nsig)
}
if o.Kgc != nil {
return chk.Err("plane-stress analysis does not work with nonlinear K and G\n")
}
c := o.E / (1.0 - o.Nu*o.Nu)
la.MatFill(D, 0)
D[0][0] = c
D[0][1] = c * o.Nu
D[1][0] = c * o.Nu
D[1][1] = c
D[3][3] = c * (1.0 - o.Nu)
return
}
if o.Kgc != nil {
o.K, o.G = o.Kgc.Calc(s)
}
for i := 0; i < o.Nsig; i++ {
for j := 0; j < o.Nsig; j++ {
D[i][j] = o.K*tsr.Im[i]*tsr.Im[j] + 2*o.G*tsr.Psd[i][j]
}
}
return
}
// Extrapolator computes the extrapolation matrix for this Shape with a combination of integration points 'ips'
// Note: E[nverts][nip] must be pre-allocated
func (o *Shape) Extrapolator(E [][]float64, ips []*Ipoint) (err error) {
la.MatFill(E, 0)
nip := len(ips)
N := o.GetShapeMatAtIps(ips)
if nip < o.Nverts {
ξ := o.GetNodesNatCoordsMat()
ξh := o.GetIpsNatCoordsMat(ips)
ξhi := la.MatAlloc(o.Gndim+1, nip)
Ni := la.MatAlloc(o.Nverts, nip)
err = la.MatInvG(Ni, N, 1e-10)
if err != nil {
return
}
err = la.MatInvG(ξhi, ξh, 1e-10)
if err != nil {
return
}
ξhξhI := la.MatAlloc(nip, nip) // ξh * inv(ξh)
for k := 0; k < o.Gndim+1; k++ {
for j := 0; j < nip; j++ {
for i := 0; i < nip; i++ {
ξhξhI[i][j] += ξh[i][k] * ξhi[k][j]
}
for i := 0; i < o.Nverts; i++ {
E[i][j] += ξ[i][k] * ξhi[k][j] // ξ * inv(ξh)
}
}
}
for i := 0; i < o.Nverts; i++ {
for j := 0; j < nip; j++ {
for k := 0; k < nip; k++ {
I_kj := 0.0
if j == k {
I_kj = 1.0
}
E[i][j] += Ni[i][k] * (I_kj - ξhξhI[k][j])
}
}
}
} else {
err = la.MatInvG(E, N, 1e-10)
if err != nil {
return
}
}
return
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o *ElemPhi) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
// auxiliary
nverts := o.Cell.Shp.Nverts
// zero K matrix
la.MatFill(o.K, 0)
dt := sol.Dt * 2
// for each integration point
for iip, ip := range o.IpsElem {
// interpolation functions and gradients
err = o.Cell.Shp.CalcAtIp(o.X, ip, true)
if err != nil {
return
}
// auxiliary variables
coef := o.Cell.Shp.J * ip[3]
S := o.Cell.Shp.S
G := o.Cell.Shp.G
// add to right hand side vector
for m := 0; m < nverts; m++ {
for n := 0; n < nverts; n++ {
o.K[m][n] -= coef * (S[m]*S[n] + dt*dt/6.0*(o.v_0[iip][0]*G[m][0]+o.v_0[iip][1]*G[m][1])*(o.v_0[iip][0]*G[n][0]+o.v_0[iip][1]*G[n][1])) / dt
}
}
}
// add K to sparse matrix Kb
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o *ElemPhi) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
// auxiliary
β1 := Global.DynCoefs.β1
nverts := o.Shp.Nverts
// zero K matrix
la.MatFill(o.K, 0)
// for each integration point
for _, ip := range o.IpsElem {
// interpolation functions and gradients
if LogErr(o.Shp.CalcAtIp(o.X, ip, true), "InterpStarVars") {
return
}
// auxiliary variables
coef := o.Shp.J * ip.W
S := o.Shp.S
// add to right hand side vector
for m := 0; m < nverts; m++ {
for n := 0; n < nverts; n++ {
o.K[m][n] += coef * S[m] * S[n] * β1
}
}
}
// add K to sparse matrix Kb
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return true
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o *ElemU) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
// zero K matrix
la.MatFill(o.K, 0)
// for each integration point
dc := Global.DynCoefs
ndim := Global.Ndim
nverts := o.Shp.Nverts
for idx, ip := range o.IpsElem {
// interpolation functions, gradients and variables @ ip
if !o.ipvars(idx, sol) {
return
}
// check Jacobian
if o.Shp.J < 0 {
LogErrCond(true, "ElemU: eid=%d: Jacobian is negative = %g\n", o.Id(), o.Shp.J)
return
}
// auxiliary
coef := o.Shp.J * ip.W * o.Thickness
S := o.Shp.S
G := o.Shp.G
// consistent tangent model matrix
if LogErr(o.MdlSmall.CalcD(o.D, o.States[idx], firstIt), "AddToKb") {
return
}
// add contribution to consistent tangent matrix
if o.UseB {
radius := 1.0
if Global.Sim.Data.Axisym {
radius = o.Shp.AxisymGetRadius(o.X)
coef *= radius
}
IpBmatrix(o.B, ndim, nverts, G, radius, S)
la.MatTrMulAdd3(o.K, coef, o.B, o.D, o.B) // K += coef * tr(B) * D * B
} else {
IpAddToKt(o.K, nverts, ndim, coef, G, o.D)
}
// dynamic term
if !Global.Sim.Data.Steady {
for m := 0; m < nverts; m++ {
for i := 0; i < ndim; i++ {
r := i + m*ndim
for n := 0; n < nverts; n++ {
c := i + n*ndim
o.K[r][c] += coef * S[m] * S[n] * (o.Rho*dc.α1 + o.Cdam*dc.α4)
}
}
}
}
}
// add K to sparse matrix Kb
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return true
}
// adds element K to global Jacobian matrix Kb
func (o ElemUP) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
// clear matrices
ndim := Global.Ndim
u_nverts := o.U.Shp.Nverts
p_nverts := o.P.Shp.Nverts
la.MatFill(o.P.Kpp, 0)
for i := 0; i < o.U.Nu; i++ {
for j := 0; j < o.P.Np; j++ {
o.Kup[i][j] = 0
o.Kpu[j][i] = 0
}
for j := 0; j < o.U.Nu; j++ {
o.U.K[i][j] = 0
}
}
if o.P.DoExtrap {
for i := 0; i < p_nverts; i++ {
o.P.ρl_ex[i] = 0
for j := 0; j < p_nverts; j++ {
o.P.dρldpl_ex[i][j] = 0
}
for j := 0; j < o.U.Nu; j++ {
o.dρldus_ex[i][j] = 0
}
}
}
// for each integration point
dc := Global.DynCoefs
var coef, plt, klr, ρL, Cl, divvs float64
var ρl, ρ, Cpl, Cvs, dρdpl, dpdpl, dCpldpl, dCvsdpl, dklrdpl, dCpldusM, dρldusM, dρdusM float64
var r, c int
for idx, ip := range o.U.IpsElem {
// interpolation functions, gradients and variables @ ip
if !o.ipvars(idx, sol) {
return
}
coef = o.U.Shp.J * ip.W
S := o.U.Shp.S
G := o.U.Shp.G
Sb := o.P.Shp.S
Gb := o.P.Shp.G
// axisymmetric case
radius := 1.0
if Global.Sim.Data.Axisym {
radius = o.U.Shp.AxisymGetRadius(o.U.X)
coef *= radius
}
// auxiliary
divvs = dc.α4*o.divus - o.U.divχs[idx] // divergence of Eq (35a) [1]
// tpm variables
plt = dc.β1*o.P.pl - o.P.ψl[idx] // Eq (35c) [1]
klr = o.P.Mdl.Cnd.Klr(o.P.States[idx].A_sl)
ρL = o.P.States[idx].A_ρL
Cl = o.P.Mdl.Cl
if LogErr(o.P.Mdl.CalcLs(o.P.res, o.P.States[idx], o.P.pl, o.divus, true), "AddToKb") {
return
}
ρl = o.P.res.A_ρl
ρ = o.P.res.A_ρ
Cpl = o.P.res.Cpl
Cvs = o.P.res.Cvs
dρdpl = o.P.res.Dρdpl
dpdpl = o.P.res.Dpdpl
dCpldpl = o.P.res.DCpldpl
dCvsdpl = o.P.res.DCvsdpl
dklrdpl = o.P.res.Dklrdpl
dCpldusM = o.P.res.DCpldusM
dρldusM = o.P.res.DρldusM
dρdusM = o.P.res.DρdusM
// Kpu, Kup and Kpp
for n := 0; n < p_nverts; n++ {
for j := 0; j < ndim; j++ {
// Kpu := ∂Rl^n/∂us^m and Kup := ∂Rus^m/∂pl^n; see Eq (47) of [1]
for m := 0; m < u_nverts; m++ {
c = j + m*ndim
// add ∂rlb/∂us^m: Eqs (A.3) and (A.6) of [1]
o.Kpu[n][c] += coef * Sb[n] * (dCpldusM*plt + dc.α4*Cvs) * G[m][j]
// add ∂(ρl.wl)/∂us^m: Eq (A.8) of [1]
for i := 0; i < ndim; i++ {
o.Kpu[n][c] += coef * Gb[n][i] * S[m] * dc.α1 * ρL * klr * o.P.Mdl.Klsat[i][j]
}
// add ∂rl/∂pl^n and ∂p/∂pl^n: Eqs (A.9) and (A.11) of [1]
o.Kup[c][n] += coef * (S[m]*Sb[n]*dρdpl*o.bs[j] - G[m][j]*Sb[n]*dpdpl)
// for seepage face
if o.P.DoExtrap {
o.dρldus_ex[n][c] += o.P.Emat[n][idx] * dρldusM * G[m][j]
}
}
// term in brackets in Eq (A.7) of [1]
o.P.tmp[j] = Sb[n]*dklrdpl*o.hl[j] - klr*(Sb[n]*Cl*o.bs[j]+Gb[n][j])
}
// Kpp := ∂Rl^m/∂pl^n; see Eq (47) of [1]
for m := 0; m < p_nverts; m++ {
// add ∂rlb/dpl^n: Eq (A.5) of [1]
o.P.Kpp[m][n] += coef * Sb[m] * Sb[n] * (dCpldpl*plt + dCvsdpl*divvs + dc.β1*Cpl)
// add ∂(ρl.wl)/∂us^m: Eq (A.7) of [1]
for i := 0; i < ndim; i++ {
for j := 0; j < ndim; j++ {
o.P.Kpp[m][n] -= coef * Gb[m][i] * o.P.Mdl.Klsat[i][j] * o.P.tmp[j]
}
}
// inner summation term in Eq (22) of [2]
if o.P.DoExtrap {
o.P.dρldpl_ex[m][n] += o.P.Emat[m][idx] * Cpl * Sb[n]
}
}
// Eq. (19) of [2]
if o.P.DoExtrap {
o.P.ρl_ex[n] += o.P.Emat[n][idx] * ρl
}
}
// Kuu: add ∂rub^m/∂us^n; see Eqs (47) and (A.10) of [1]
for m := 0; m < u_nverts; m++ {
for i := 0; i < ndim; i++ {
r = i + m*ndim
for n := 0; n < u_nverts; n++ {
for j := 0; j < ndim; j++ {
c = j + n*ndim
o.U.K[r][c] += coef * S[m] * (S[n]*dc.α1*ρ*tsr.It[i][j] + dρdusM*o.bs[i]*G[n][j])
}
}
}
}
// consistent tangent model matrix
if LogErr(o.U.MdlSmall.CalcD(o.U.D, o.U.States[idx], firstIt), "AddToKb") {
return
}
// Kuu: add stiffness term ∂(σe・G^m)/∂us^n
if o.U.UseB {
IpBmatrix(o.U.B, ndim, u_nverts, G, radius, S)
la.MatTrMulAdd3(o.U.K, coef, o.U.B, o.U.D, o.U.B) // K += coef * tr(B) * D * B
} else {
IpAddToKt(o.U.K, u_nverts, ndim, coef, G, o.U.D)
}
}
// contribution from natural boundary conditions
if o.P.HasSeep {
if !o.P.add_natbcs_to_jac(sol) {
return
}
if !o.add_natbcs_to_jac(sol) {
return
}
}
// add K to sparse matrix Kb
// _ _
// | Kuu Kup 0 |
// | Kpu Kpp Kpf |
// |_ Kfu Kfp Kff _|
//
for i, I := range o.P.Pmap {
for j, J := range o.P.Pmap {
Kb.Put(I, J, o.P.Kpp[i][j])
}
for j, J := range o.P.Fmap {
Kb.Put(I, J, o.P.Kpf[i][j])
Kb.Put(J, I, o.P.Kfp[j][i])
}
for j, J := range o.U.Umap {
Kb.Put(I, J, o.Kpu[i][j])
Kb.Put(J, I, o.Kup[j][i])
}
}
for i, I := range o.P.Fmap {
for j, J := range o.P.Fmap {
Kb.Put(I, J, o.P.Kff[i][j])
}
}
for i, I := range o.U.Umap {
for j, J := range o.U.Umap {
Kb.Put(I, J, o.U.K[i][j])
}
}
return true
}
// add_natbcs_to_jac adds contribution from natural boundary conditions to Jacobian
func (o ElemP) add_natbcs_to_jac(sol *Solution) (ok bool) {
// clear matrices
if o.HasSeep {
for i := 0; i < o.Np; i++ {
for j := 0; j < o.Nf; j++ {
o.Kpf[i][j] = 0
o.Kfp[j][i] = 0
}
}
la.MatFill(o.Kff, 0)
}
// compute surface integral
nverts := o.Shp.Nverts
var shift float64
var ρl, pl, fl, plmax, g, rmp, rmpD float64
var drxdpl, drxdfl, drfdpl, drfdfl float64
for idx, nbc := range o.NatBcs {
// plmax shift
shift = nbc.Fcn.F(sol.T, nil)
// loop over ips of face
for jdx, ipf := range o.IpsFace {
// interpolation functions and gradients @ face
iface := nbc.IdxFace
if LogErr(o.Shp.CalcAtFaceIp(o.X, ipf, iface), "add_natbcs_to_jac") {
return
}
Sf := o.Shp.Sf
Jf := la.VecNorm(o.Shp.Fnvec)
coef := ipf.W * Jf
// select natural boundary condition type
switch nbc.Key {
case "seep":
// variables extrapolated to face
ρl, pl, fl = o.fipvars(iface, sol)
plmax = o.Plmax[idx][jdx] - shift
if plmax < 0 {
plmax = 0
}
// compute derivatives
g = pl - plmax // Eq. (24)
rmp = o.ramp(fl + o.κ*g)
rmpD = o.rampD1(fl + o.κ*g)
drxdpl = ρl * o.κ * rmpD // first term in Eq. (A.4) (without Sn)
drxdfl = ρl * rmpD // Eq. (A.5) (without Sn)
drfdpl = -o.κ * rmpD // Eq. (A.6) (corrected with κ and without Sn)
drfdfl = 1.0 - rmpD // Eq. (A.7) (without Sn)
for i, m := range o.Shp.FaceLocalV[iface] {
μ := o.Vid2seepId[m]
for j, n := range o.Shp.FaceLocalV[iface] {
ν := o.Vid2seepId[n]
o.Kpp[m][n] += coef * Sf[i] * Sf[j] * drxdpl
o.Kpf[m][ν] += coef * Sf[i] * Sf[j] * drxdfl
o.Kfp[μ][n] += coef * Sf[i] * Sf[j] * drfdpl
o.Kff[μ][ν] += coef * Sf[i] * Sf[j] * drfdfl
}
for n := 0; n < nverts; n++ { // Eqs. (18) and (22)
for l, r := range o.Shp.FaceLocalV[iface] {
o.Kpp[m][n] += coef * Sf[i] * Sf[l] * o.dρldpl_ex[r][n] * rmp
}
}
}
}
}
}
return true
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o ElemP) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
// clear matrices
la.MatFill(o.Kpp, 0)
ndim := Global.Ndim
nverts := o.Shp.Nverts
if o.DoExtrap {
for i := 0; i < nverts; i++ {
o.ρl_ex[i] = 0
for j := 0; j < nverts; j++ {
o.dρldpl_ex[i][j] = 0
}
}
}
// for each integration point
Cl := o.Mdl.Cl
β1 := Global.DynCoefs.β1
var coef, plt, klr, ρL, ρl, Cpl, dCpldpl, dklrdpl float64
for idx, ip := range o.IpsElem {
// interpolation functions, gradients and variables @ ip
if !o.ipvars(idx, sol) {
return
}
coef = o.Shp.J * ip.W
S := o.Shp.S
G := o.Shp.G
// tpm variables
plt = β1*o.pl - o.ψl[idx]
klr = o.Mdl.Cnd.Klr(o.States[idx].A_sl)
ρL = o.States[idx].A_ρL
if LogErr(o.Mdl.CalcLs(o.res, o.States[idx], o.pl, 0, true), "AddToKb") {
return
}
ρl = o.res.A_ρl
Cpl = o.res.Cpl
dCpldpl = o.res.DCpldpl
dklrdpl = o.res.Dklrdpl
// Kpp := dRpl/dpl. see Eqs. (18), (A.2) and (A.3) of [1]
for n := 0; n < nverts; n++ {
for j := 0; j < ndim; j++ {
o.tmp[j] = S[n]*dklrdpl*(ρL*o.g[j]-o.gpl[j]) + klr*(S[n]*Cl*o.g[j]-G[n][j])
}
for m := 0; m < nverts; m++ {
o.Kpp[m][n] += coef * S[m] * S[n] * (dCpldpl*plt + β1*Cpl)
for i := 0; i < ndim; i++ {
for j := 0; j < ndim; j++ {
o.Kpp[m][n] -= coef * G[m][i] * o.Mdl.Klsat[i][j] * o.tmp[j]
}
}
if o.DoExtrap { // inner summation term in Eq. (22)
o.dρldpl_ex[m][n] += o.Emat[m][idx] * Cpl * S[n]
}
}
if o.DoExtrap { // Eq. (19)
o.ρl_ex[n] += o.Emat[n][idx] * ρl
}
}
}
// add to Kb
if o.HasSeep {
// contribution from natural boundary conditions
if !o.add_natbcs_to_jac(sol) {
return
}
// add to sparse matrix Kb
for i, I := range o.Pmap {
for j, J := range o.Pmap {
Kb.Put(I, J, o.Kpp[i][j])
}
for j, J := range o.Fmap {
Kb.Put(I, J, o.Kpf[i][j])
Kb.Put(J, I, o.Kfp[j][i])
}
}
for i, I := range o.Fmap {
for j, J := range o.Fmap {
Kb.Put(I, J, o.Kff[i][j])
}
}
} else {
// add to sparse matrix Kb
for i, I := range o.Pmap {
for j, J := range o.Pmap {
Kb.Put(I, J, o.Kpp[i][j])
}
}
}
return true
}
// Update perform (tangent) update
func (o *Rjoint) Update(sol *Solution) (err error) {
// auxiliary
nsig := 2 * o.Ndim
rodH := o.Rod.Cell.Shp
rodS := rodH.S
rodNn := rodH.Nverts
sldH := o.Sld.Cell.Shp
sldNn := sldH.Nverts
// extrapolate stresses at integration points of solid element to its nodes
if o.Coulomb {
la.MatFill(o.σNo, 0)
for idx, _ := range o.Sld.IpsElem {
σ := o.Sld.States[idx].Sig
for i := 0; i < nsig; i++ {
for m := 0; m < sldNn; m++ {
o.σNo[m][i] += o.Emat[m][idx] * σ[i]
}
}
}
}
// interpolate Δu of solid to find ΔuC @ rod node; Eq (30)
var r, I int
for m := 0; m < rodNn; m++ {
for i := 0; i < o.Ndim; i++ {
o.ΔuC[m][i] = 0
for n := 0; n < sldNn; n++ {
r = i + n*o.Ndim
I = o.Sld.Umap[r]
o.ΔuC[m][i] += o.Nmat[n][m] * sol.ΔY[I] // Eq (30)
}
}
}
// loop over ips of rod
var Δwb0, Δwb1, Δwb2, σc float64
for idx, ip := range o.Rod.IpsElem {
// auxiliary
e0, e1, e2 := o.e0[idx], o.e1[idx], o.e2[idx]
// interpolation functions and gradients
err = rodH.CalcAtIp(o.Rod.X, ip, true)
if err != nil {
return
}
// interpolated relative displacements @ ip of join; Eqs (31) and (32)
for i := 0; i < o.Ndim; i++ {
o.Δw[i] = 0
for m := 0; m < rodNn; m++ {
r = i + m*o.Ndim
I = o.Rod.Umap[r]
o.Δw[i] += rodS[m] * (o.ΔuC[m][i] - sol.ΔY[I]) // Eq (31) and (32)
}
}
// relative displacents in the coratational system
Δwb0, Δwb1, Δwb2 = 0, 0, 0
for i := 0; i < o.Ndim; i++ {
Δwb0 += e0[i] * o.Δw[i]
Δwb1 += e1[i] * o.Δw[i]
Δwb2 += e2[i] * o.Δw[i]
}
// new confining stress
σc = 0.0
if o.Coulomb {
// calculate σIp
for j := 0; j < nsig; j++ {
o.σIp[j] = 0
for n := 0; n < sldNn; n++ {
o.σIp[j] += o.Pmat[n][idx] * o.σNo[n][j]
}
}
// calculate t1 and t2
for i := 0; i < o.Ndim; i++ {
o.t1[i], o.t2[i] = 0, 0
for j := 0; j < o.Ndim; j++ {
o.t1[i] += tsr.M2T(o.σIp, i, j) * e1[j]
o.t2[i] += tsr.M2T(o.σIp, i, j) * e2[j]
}
}
// calculate p1, p2 and σcNew
p1, p2 := 0.0, 0.0
for i := 0; i < o.Ndim; i++ {
p1 += o.t1[i] * e1[i]
p2 += o.t2[i] * e2[i]
}
// σcNew
σc = -(p1 + p2) / 2.0
}
// update model
err = o.Mdl.Update(o.States[idx], σc, Δwb0)
if err != nil {
return
}
o.States[idx].Phi[0] += o.k1 * Δwb1 // qn1
o.States[idx].Phi[1] += o.k2 * Δwb2 // qn2
// debugging
//if true {
if false {
o.debug_update(idx, Δwb0, Δwb1, Δwb2, σc)
}
}
return
}
// adds element K to global Jacobian matrix Kb
func (o *Rjoint) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
// auxiliary
rodH := o.Rod.Cell.Shp
rodS := rodH.S
rodNn := rodH.Nverts
sldH := o.Sld.Cell.Shp
sldNn := sldH.Nverts
// zero K matrices
for i, _ := range o.Rod.Umap {
for j, _ := range o.Rod.Umap {
o.Krr[i][j] = 0
}
for j, _ := range o.Sld.Umap {
o.Krs[i][j] = 0
o.Ksr[j][i] = 0
}
}
la.MatFill(o.Kss, 0)
// auxiliary
var coef float64
var DτDω float64
var Dwb0Du_nj, Dwb1Du_nj, Dwb2Du_nj float64
var DτDu_nj, DqbDu_nij float64
var Dwb0Dur_nj, Dwb1Dur_nj, Dwb2Dur_nj float64
var DqbDur_nij float64
// loop over rod's integration points
for idx, ip := range o.Rod.IpsElem {
// auxiliary
e0, e1, e2 := o.e0[idx], o.e1[idx], o.e2[idx]
// interpolation functions and gradients
err = rodH.CalcAtIp(o.Rod.X, ip, true)
if err != nil {
return
}
coef = ip[3] * rodH.J
// model derivatives
DτDω, err = o.Mdl.CalcD(o.States[idx], firstIt)
if err != nil {
return
}
// compute derivatives
for j := 0; j < o.Ndim; j++ {
// Krr and Ksr; derivatives with respect to ur_nj
for n := 0; n < rodNn; n++ {
// ∂wb/∂ur Eq (A.4)
Dwb0Dur_nj = -rodS[n] * e0[j]
Dwb1Dur_nj = -rodS[n] * e1[j]
Dwb2Dur_nj = -rodS[n] * e2[j]
// compute ∂■/∂ur derivatives
c := j + n*o.Ndim
for i := 0; i < o.Ndim; i++ {
// ∂qb/∂ur Eq (A.2)
DqbDur_nij = o.h*e0[i]*(DτDω*Dwb0Dur_nj) + o.k1*e1[i]*Dwb1Dur_nj + o.k2*e2[i]*Dwb2Dur_nj
// Krr := ∂fr/∂ur Eq (58)
for m := 0; m < rodNn; m++ {
r := i + m*o.Ndim
o.Krr[r][c] -= coef * rodS[m] * DqbDur_nij
}
// Ksr := ∂fs/∂ur Eq (60)
for m := 0; m < sldNn; m++ {
r := i + m*o.Ndim
for p := 0; p < rodNn; p++ {
o.Ksr[r][c] += coef * o.Nmat[m][p] * rodS[p] * DqbDur_nij
}
}
}
}
// Krs and Kss
for n := 0; n < sldNn; n++ {
// ∂wb/∂us Eq (A.5)
Dwb0Du_nj, Dwb1Du_nj, Dwb2Du_nj = 0, 0, 0
for m := 0; m < rodNn; m++ {
Dwb0Du_nj += rodS[m] * o.Nmat[n][m] * e0[j]
Dwb1Du_nj += rodS[m] * o.Nmat[n][m] * e1[j]
Dwb2Du_nj += rodS[m] * o.Nmat[n][m] * e2[j]
}
// ∂τ/∂us_nj hightlighted term in Eq (A.3)
DτDu_nj = DτDω * Dwb0Du_nj
// compute ∂■/∂us derivatives
c := j + n*o.Ndim
for i := 0; i < o.Ndim; i++ {
// ∂qb/∂us Eq (A.3)
DqbDu_nij = o.h*e0[i]*DτDu_nj + o.k1*e1[i]*Dwb1Du_nj + o.k2*e2[i]*Dwb2Du_nj
// Krs := ∂fr/∂us Eq (59)
for m := 0; m < rodNn; m++ {
r := i + m*o.Ndim
o.Krs[r][c] -= coef * rodS[m] * DqbDu_nij
}
// Kss := ∂fs/∂us Eq (61)
for m := 0; m < sldNn; m++ {
r := i + m*o.Ndim
for p := 0; p < rodNn; p++ {
o.Kss[r][c] += coef * o.Nmat[m][p] * rodS[p] * DqbDu_nij
}
}
}
}
}
}
// debug
//if true {
if false {
o.debug_print_K()
}
// add K to sparse matrix Kb
for i, I := range o.Rod.Umap {
for j, J := range o.Rod.Umap {
Kb.Put(I, J, o.Krr[i][j])
}
for j, J := range o.Sld.Umap {
Kb.Put(I, J, o.Krs[i][j])
Kb.Put(J, I, o.Ksr[j][i])
}
}
for i, I := range o.Sld.Umap {
for j, J := range o.Sld.Umap {
Kb.Put(I, J, o.Kss[i][j])
}
}
return
}
// AddToKb adds element K to global Jacobian matrix Kb
func (o *ElemU) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
// zero K matrix
la.MatFill(o.K, 0)
// for each integration point
nverts := o.Cell.Shp.Nverts
for idx, ip := range o.IpsElem {
// interpolation functions, gradients and variables @ ip
err = o.ipvars(idx, sol)
if err != nil {
return
}
// check Jacobian
if o.Cell.Shp.J < 0 {
return chk.Err("ElemU: eid=%d: Jacobian is negative = %g\n", o.Id(), o.Cell.Shp.J)
}
// auxiliary
coef := o.Cell.Shp.J * ip[3] * o.Thickness
S := o.Cell.Shp.S
G := o.Cell.Shp.G
// consistent tangent model matrix
err = o.MdlSmall.CalcD(o.D, o.States[idx], firstIt)
if err != nil {
return
}
// add contribution to consistent tangent matrix
if o.UseB {
radius := 1.0
if sol.Axisym {
radius = o.Cell.Shp.AxisymGetRadius(o.X)
coef *= radius
}
IpBmatrix(o.B, o.Ndim, nverts, G, radius, S, sol.Axisym)
la.MatTrMulAdd3(o.K, coef, o.B, o.D, o.B) // K += coef * tr(B) * D * B
} else {
IpAddToKt(o.K, nverts, o.Ndim, coef, G, o.D)
}
// dynamic term
if !sol.Steady {
α1 := sol.DynCfs.α1
α4 := sol.DynCfs.α4
for m := 0; m < nverts; m++ {
for i := 0; i < o.Ndim; i++ {
r := i + m*o.Ndim
for n := 0; n < nverts; n++ {
c := i + n*o.Ndim
o.K[r][c] += coef * S[m] * S[n] * (o.Rho*α1 + o.Cdam*α4)
}
}
}
}
}
// add Ks to sparse matrix Kb
switch {
case o.HasContact:
err = o.contact_add_to_jac(Kb, sol)
case o.Xfem:
err = o.xfem_add_to_jac(Kb, sol)
default:
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
}
return
}