// explicit Runge-Kutta step function
func erk_step(o *ODE, y []float64, x float64, args ...interface{}) (rerr float64, err error) {
for i := 0; i < o.nstg; i++ {
o.u[i] = x + o.h*o.erkdat.c[i]
la.VecCopy(o.v[i], 1, y)
for j := 0; j < i; j++ {
la.VecAdd(o.v[i], o.h*o.erkdat.a[i][j], o.f[j])
}
if i == 0 && o.erkdat.usefp && !o.first {
la.VecCopy(o.f[i], 1, o.f[o.nstg-1])
} else {
o.nfeval += 1
err = o.fcn(o.f[i], o.u[i], o.v[i], args...)
if err != nil {
return
}
}
}
var lerrm float64 // m component of local error estimate
for m := 0; m < o.ndim; m++ {
lerrm = 0.0
for i := 0; i < o.nstg; i++ {
o.w[0][m] += o.erkdat.b[i] * o.f[i][m] * o.h
lerrm += (o.erkdat.be[i] - o.erkdat.b[i]) * o.f[i][m] * o.h
}
rerr += math.Pow(lerrm/o.scal[m], 2.0)
}
rerr = max(math.Sqrt(rerr/float64(o.ndim)), 1.0e-10)
return
}
// contact_add_to_rhs adds contribution to rhs due to contact modelling
func (o *ElemU) contact_add_to_rhs(fb []float64, sol *Solution) (err error) {
// compute surface integral
var qb, db, rmp, rx, rq float64
for _, nbc := range o.NatBcs {
// loop over ips of face
for _, ipf := range o.IpsFace {
// interpolation functions and gradients @ face
iface := nbc.IdxFace
err = o.Cell.Shp.CalcAtFaceIp(o.X, ipf, iface)
if err != nil {
return
}
Sf := o.Cell.Shp.Sf
nvec := o.Cell.Shp.Fnvec
// select natural boundary condition type
switch nbc.Key {
// contact
case "contact":
// variables extrapolated to face
qb = o.fipvars(iface, sol)
xf := o.Cell.Shp.FaceIpRealCoords(o.X, ipf, iface)
la.VecAdd(xf, 1, o.us) // add displacement: x = X + u
db = o.contact_g(xf)
// compute residuals
coef := ipf[3] * o.Thickness
Jf := la.VecNorm(nvec)
rmp = o.contact_ramp(qb + o.κ*db)
rx = rmp
rq = qb - rmp
for j, m := range o.Cell.Shp.FaceLocalVerts[iface] {
μ := o.Vid2contactId[m]
fb[o.Qmap[μ]] -= coef * Sf[j] * rq * Jf // -residual
for i := 0; i < o.Ndim; i++ {
r := o.Umap[i+m*o.Ndim]
fb[r] -= coef * Sf[j] * rx * nvec[i] // -extra term
}
}
}
}
}
return
}
// contact_add_to_jac adds coupled equations due to contact modelling to Jacobian
func (o *ElemU) contact_add_to_jac(Kb *la.Triplet, sol *Solution) (err error) {
// clear matrices
for i := 0; i < o.Nq; i++ {
for j := 0; j < o.Nq; j++ {
o.Kqq[i][j] = 0
}
for j := 0; j < o.Nu; j++ {
o.Kqu[i][j] = 0
o.Kuq[j][i] = 0
}
}
// compute surface integral
var qb, db, Hb float64
dddu := make([]float64, o.Ndim)
for _, nbc := range o.NatBcs {
// loop over ips of face
for _, ipf := range o.IpsFace {
// contact
switch nbc.Key {
case "contact":
// interpolation functions and gradients @ face
iface := nbc.IdxFace
err = o.Cell.Shp.CalcAtFaceIp(o.X, ipf, iface)
if err != nil {
return
}
Sf := o.Cell.Shp.Sf
nvec := o.Cell.Shp.Fnvec
coef := ipf[3] * o.Thickness
Jf := la.VecNorm(nvec)
// variables extrapolated to face
qb = o.fipvars(iface, sol)
xf := o.Cell.Shp.FaceIpRealCoords(o.X, ipf, iface)
la.VecAdd(xf, 1, o.us) // add displacement: x = X + u
db = o.contact_g(xf)
o.contact_dgdx(dddu, xf)
// compute derivatives
Hb = o.contact_rampD1(qb + o.κ*db)
for i, m := range o.Cell.Shp.FaceLocalVerts[iface] {
μ := o.Vid2contactId[m]
for j, n := range o.Cell.Shp.FaceLocalVerts[iface] {
ν := o.Vid2contactId[n]
o.Kqq[μ][ν] += coef * Jf * Sf[i] * Sf[j] * (1.0 - Hb)
for k := 0; k < o.Ndim; k++ {
r := k + m*o.Ndim
c := k + n*o.Ndim
val := coef * Sf[i] * Sf[j] * Hb * o.κ * dddu[k] * Jf
o.Kuq[r][ν] += coef * Sf[i] * Sf[j] * Hb * nvec[k]
o.Kqu[μ][c] -= val
o.K[r][c] += val
}
}
}
}
}
}
// add Ks to sparse matrix Kb
for i, I := range o.Qmap {
for j, J := range o.Qmap {
Kb.Put(I, J, o.Kqq[i][j])
}
for j, J := range o.Umap {
Kb.Put(I, J, o.Kqu[i][j])
Kb.Put(J, I, o.Kuq[j][i])
}
}
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return
}