// run_iterations solves the nonlinear problem
func run_iterations(t, Δt float64, d *Domain, dc *DynCoefs, sum *Summary, dbgKb DebugKb_t) (diverging bool, err error) {
// zero accumulated increments
la.VecFill(d.Sol.ΔY, 0)
// calculate global starred vectors and interpolate starred variables from nodes to integration points
if !d.Sim.Data.Steady {
// compute starred vectors
for _, I := range d.T1eqs {
d.Sol.Psi[I] = dc.β1*d.Sol.Y[I] + dc.β2*d.Sol.Dydt[I]
}
for _, I := range d.T2eqs {
d.Sol.Zet[I] = dc.α1*d.Sol.Y[I] + dc.α2*d.Sol.Dydt[I] + dc.α3*d.Sol.D2ydt2[I]
d.Sol.Chi[I] = dc.α4*d.Sol.Y[I] + dc.α5*d.Sol.Dydt[I] + dc.α6*d.Sol.D2ydt2[I]
}
// set internal starred variables
for _, e := range d.Elems {
err = e.InterpStarVars(d.Sol)
if err != nil {
err = chk.Err("cannot compute starred variables:\n%v", err)
return
}
}
}
// auxiliary variables
var it int
var largFb, largFb0, Lδu float64
var prevFb, prevLδu float64
dat := d.Sim.Solver
// message
if dat.ShowR {
io.Pf("\n%13s%4s%23s%23s\n", "t", "it", "largFb", "Lδu")
defer func() {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}()
}
// iterations
for it = 0; it < dat.NmaxIt; it++ {
// assemble right-hand side vector (fb) with negative of residuals
la.VecFill(d.Fb, 0)
for _, e := range d.Elems {
err = e.AddToRhs(d.Fb, d.Sol)
if err != nil {
return
}
}
// join all fb
if d.Distr {
mpi.AllReduceSum(d.Fb, d.Wb) // this must be done here because there might be nodes sharing boundary conditions
}
// point natural boundary conditions; e.g. concentrated loads
d.PtNatBcs.AddToRhs(d.Fb, t)
// essential boundary conditioins; e.g. constraints
d.EssenBcs.AddToRhs(d.Fb, d.Sol)
// find largest absolute component of fb
largFb = la.VecLargest(d.Fb, 1)
// save residual
if d.Sim.Data.Stat {
if sum != nil {
sum.Resids.Append(it == 0, largFb)
}
}
// check largFb value
if it == 0 {
// store largest absolute component of fb
largFb0 = largFb
} else {
// check convergence on Lf0
if largFb < dat.FbTol*largFb0 { // converged on fb
break
}
// check convergence on fb_min
if largFb < dat.FbMin { // converged with smallest value of fb
break
}
}
// check divergence on fb
if it > 1 && dat.DvgCtrl {
if largFb > prevFb {
diverging = true
break
}
}
prevFb = largFb
// assemble Jacobian matrix
do_asm_fact := (it == 0 || !dat.CteTg)
if do_asm_fact {
// assemble element matrices
d.Kb.Start()
for _, e := range d.Elems {
err = e.AddToKb(d.Kb, d.Sol, it == 0)
if err != nil {
return
}
}
// debug
if dbgKb != nil {
dbgKb(d, it)
}
// join A and tr(A) matrices into Kb
if d.Proc == 0 {
d.Kb.PutMatAndMatT(&d.EssenBcs.A)
}
// initialise linear solver
if d.InitLSol {
err = d.LinSol.InitR(d.Kb, d.Sim.LinSol.Symmetric, d.Sim.LinSol.Verbose, d.Sim.LinSol.Timing)
if err != nil {
err = chk.Err("cannot initialise linear solver:\n%v", err)
return
}
d.InitLSol = false
}
// perform factorisation
err = d.LinSol.Fact()
if err != nil {
err = chk.Err("factorisation failed:\n%v", err)
return
}
}
// solve for wb := δyb
err = d.LinSol.SolveR(d.Wb, d.Fb, false)
if err != nil {
err = chk.Err("solve failed:%v\n", err)
return
}
// update primary variables (y)
for i := 0; i < d.Ny; i++ {
d.Sol.Y[i] += d.Wb[i] // y += δy
d.Sol.ΔY[i] += d.Wb[i] // ΔY += δy
}
if !d.Sim.Data.Steady {
for _, I := range d.T1eqs {
d.Sol.Dydt[I] = dc.β1*d.Sol.Y[I] - d.Sol.Psi[I]
}
for _, I := range d.T2eqs {
d.Sol.Dydt[I] = dc.α4*d.Sol.Y[I] - d.Sol.Chi[I]
d.Sol.D2ydt2[I] = dc.α1*d.Sol.Y[I] - d.Sol.Zet[I]
}
}
// update Lagrange multipliers (λ)
for i := 0; i < d.Nlam; i++ {
d.Sol.L[i] += d.Wb[d.Ny+i] // λ += δλ
}
// backup / restore
if it == 0 {
// create backup copy of all secondary variables
for _, e := range d.ElemIntvars {
e.BackupIvs(false)
}
} else {
// recover last converged state from backup copy
for _, e := range d.ElemIntvars {
e.RestoreIvs(false)
}
}
// update secondary variables
for _, e := range d.Elems {
err = e.Update(d.Sol)
if err != nil {
break
}
}
// compute RMS norm of δu and check convegence on δu
Lδu = la.VecRmsErr(d.Wb[:d.Ny], dat.Atol, dat.Rtol, d.Sol.Y[:d.Ny])
// message
if dat.ShowR {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}
// stop if converged on δu
if Lδu < dat.Itol {
break
}
// check divergence on Lδu
if it > 1 && dat.DvgCtrl {
if Lδu > prevLδu {
diverging = true
break
}
}
prevLδu = Lδu
}
// check if iterations diverged
if it == dat.NmaxIt {
err = chk.Err("max number of iterations reached: it = %d\n", it)
}
return
}
// run_iterations solves the nonlinear problem
func run_iterations(t, Δt float64, d *Domain, sum *Summary) (diverging, ok bool) {
// zero accumulated increments
la.VecFill(d.Sol.ΔY, 0)
// calculate global starred vectors and interpolate starred variables from nodes to integration points
if LogErr(d.star_vars(Δt), "cannot compute starred variables") {
return
}
// auxiliary variables
var it int
var largFb, largFb0, Lδu float64
var prevFb, prevLδu float64
// message
if Global.Sim.Data.ShowR {
io.Pf("\n%13s%4s%23s%23s\n", "t", "it", "largFb", "Lδu")
defer func() {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}()
}
// iterations
for it = 0; it < Global.Sim.Solver.NmaxIt; it++ {
// assemble right-hand side vector (fb) with negative of residuals
la.VecFill(d.Fb, 0)
for _, e := range d.Elems {
if !e.AddToRhs(d.Fb, d.Sol) {
break
}
}
if Stop() {
return
}
// join all fb
if Global.Distr {
mpi.AllReduceSum(d.Fb, d.Wb) // this must be done here because there might be nodes sharing boundary conditions
}
// point natural boundary conditions; e.g. concentrated loads
d.PtNatBcs.AddToRhs(d.Fb, t)
// essential boundary conditioins; e.g. constraints
d.EssenBcs.AddToRhs(d.Fb, d.Sol)
// debug
if Global.Debug {
//la.PrintVec("fb", d.Fb[:d.Ny], "%13.10f ", false)
//panic("stop")
}
// find largest absolute component of fb
largFb = la.VecLargest(d.Fb, 1)
// save residual
if Global.Stat {
sum.Resids.Append(it == 0, largFb)
}
// check largFb value
if it == 0 {
// store largest absolute component of fb
largFb0 = largFb
} else {
// check convergence on Lf0
if largFb < Global.Sim.Solver.FbTol*largFb0 { // converged on fb
break
}
// check convergence on fb_min
if largFb < Global.Sim.Solver.FbMin { // converged with smallest value of fb
break
}
}
// check divergence on fb
if it > 1 && Global.Sim.Solver.DvgCtrl {
if largFb > prevFb {
diverging = true
break
}
}
prevFb = largFb
// assemble Jacobian matrix
do_asm_fact := (it == 0 || !Global.Sim.Data.CteTg)
if do_asm_fact {
// assemble element matrices
d.Kb.Start()
for _, e := range d.Elems {
if !e.AddToKb(d.Kb, d.Sol, it == 0) {
break
}
}
if Stop() {
return
}
// debug
if Global.DebugKb != nil {
Global.DebugKb(d, it)
}
// join A and tr(A) matrices into Kb
if Global.Root {
d.Kb.PutMatAndMatT(&d.EssenBcs.A)
}
// initialise linear solver
if d.InitLSol {
if LogErr(d.LinSol.InitR(d.Kb, Global.Sim.LinSol.Symmetric, Global.Sim.LinSol.Verbose, Global.Sim.LinSol.Timing), "cannot initialise linear solver") {
return
}
d.InitLSol = false
}
// perform factorisation
LogErr(d.LinSol.Fact(), "factorisation")
if Stop() {
return
}
}
// debug
//KK := d.Kb.ToMatrix(nil).ToDense()
//la.PrintMat("KK", KK, "%20.10f", false)
//panic("stop")
// solve for wb := δyb
LogErr(d.LinSol.SolveR(d.Wb, d.Fb, false), "solve")
if Stop() {
return
}
// debug
if Global.Debug {
//la.PrintVec("wb", d.Wb[:d.Ny], "%13.10f ", false)
}
// update primary variables (y)
for i := 0; i < d.Ny; i++ {
d.Sol.Y[i] += d.Wb[i] // y += δy
d.Sol.ΔY[i] += d.Wb[i] // ΔY += δy
}
if !Global.Sim.Data.Steady {
for _, I := range d.T1eqs {
d.Sol.Dydt[I] = Global.DynCoefs.β1*d.Sol.Y[I] - d.Sol.Psi[I]
}
for _, I := range d.T2eqs {
d.Sol.Dydt[I] = Global.DynCoefs.α4*d.Sol.Y[I] - d.Sol.Chi[I]
d.Sol.D2ydt2[I] = Global.DynCoefs.α1*d.Sol.Y[I] - d.Sol.Zet[I]
}
}
// update Lagrange multipliers (λ)
for i := 0; i < d.Nlam; i++ {
d.Sol.L[i] += d.Wb[d.Ny+i] // λ += δλ
}
// backup / restore
if it == 0 {
// create backup copy of all secondary variables
for _, e := range d.ElemIntvars {
e.BackupIvs(false)
}
} else {
// recover last converged state from backup copy
for _, e := range d.ElemIntvars {
e.RestoreIvs(false)
}
}
// update secondary variables
for _, e := range d.Elems {
if !e.Update(d.Sol) {
break
}
}
if Stop() {
return
}
// compute RMS norm of δu and check convegence on δu
Lδu = la.VecRmsErr(d.Wb[:d.Ny], Global.Sim.Solver.Atol, Global.Sim.Solver.Rtol, d.Sol.Y[:d.Ny])
// message
if Global.Sim.Data.ShowR {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}
// stop if converged on δu
if Lδu < Global.Sim.Solver.Itol {
break
}
// check divergence on Lδu
if it > 1 && Global.Sim.Solver.DvgCtrl {
if Lδu > prevLδu {
diverging = true
break
}
}
prevLδu = Lδu
}
// check if iterations diverged
if it == Global.Sim.Solver.NmaxIt {
io.PfMag("max number of iterations reached: it = %d\n", it)
return
}
// success
ok = true
return
}
// Solve solves non-linear problem f(x) == 0
func (o *NlSolver) Solve(x []float64, silent bool) (err error) {
// compute scaling vector
la.VecScaleAbs(o.scal, o.atol, o.rtol, x) // scal := Atol + Rtol*abs(x)
// evaluate function @ x
err = o.Ffcn(o.fx, x) // fx := f(x)
o.NFeval, o.NJeval = 1, 0
if err != nil {
return
}
// show message
if !silent {
o.msg("", 0, 0, 0, true, false)
}
// iterations
var Ldx, Ldx_prev, Θ float64 // RMS norm of delta x, convergence rate
var fx_max float64
var nfv int
for o.It = 0; o.It < o.MaxIt; o.It++ {
// check convergence on f(x)
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if fx_max < o.ftol {
if !silent {
o.msg("fx_max(ini)", o.It, Ldx, fx_max, false, true)
}
break
}
// show message
if !silent {
o.msg("", o.It, Ldx, fx_max, false, false)
}
// output
if o.Out != nil {
o.Out(x)
}
// evaluate Jacobian @ x
if o.It == 0 || !o.CteJac {
if o.useDn {
err = o.JfcnDn(o.J, x)
} else {
if o.numJ {
err = Jacobian(&o.Jtri, o.Ffcn, x, o.fx, o.w, false)
o.NFeval += o.neq
} else {
err = o.JfcnSp(&o.Jtri, x)
}
}
o.NJeval += 1
if err != nil {
return
}
}
// dense solution
if o.useDn {
// invert matrix
err = la.MatInvG(o.Ji, o.J, 1e-10)
if err != nil {
return chk.Err(_nls_err1, err.Error())
}
// solve linear system (compute mdx) and compute lin-search data
o.φ = 0.0
for i := 0; i < o.neq; i++ {
o.mdx[i], o.dφdx[i] = 0.0, 0.0
for j := 0; j < o.neq; j++ {
o.mdx[i] += o.Ji[i][j] * o.fx[j] // mdx = inv(J) * fx
o.dφdx[i] += o.J[j][i] * o.fx[j] // dφdx = tra(J) * fx
}
o.φ += o.fx[i] * o.fx[i]
}
o.φ *= 0.5
// sparse solution
} else {
// init sparse solver
if o.It == 0 {
symmetric, verbose, timing := false, false, false
err := o.lis.InitR(&o.Jtri, symmetric, verbose, timing)
if err != nil {
return chk.Err(_nls_err9, err.Error())
}
}
// factorisation (must be done for all iterations)
o.lis.Fact()
// solve linear system => compute mdx
o.lis.SolveR(o.mdx, o.fx, false) // mdx = inv(J) * fx false => !sumToRoot
// compute lin-search data
if o.Lsearch {
o.φ = 0.5 * la.VecDot(o.fx, o.fx)
la.SpTriMatTrVecMul(o.dφdx, &o.Jtri, o.fx) // dφdx := transpose(J) * fx
}
}
//io.Pforan("φ = %v\n", o.φ)
//io.Pforan("dφdx = %v\n", o.dφdx)
// update x
Ldx = 0.0
for i := 0; i < o.neq; i++ {
o.x0[i] = x[i]
x[i] -= o.mdx[i]
Ldx += (o.mdx[i] / o.scal[i]) * (o.mdx[i] / o.scal[i])
}
Ldx = math.Sqrt(Ldx / float64(o.neq))
// calculate fx := f(x) @ update x
err = o.Ffcn(o.fx, x)
o.NFeval += 1
if err != nil {
return
}
// check convergence on f(x) => avoid line-search if converged already
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if fx_max < o.ftol {
if !silent {
o.msg("fx_max", o.It, Ldx, fx_max, false, true)
}
break
}
// check convergence on Ldx
if Ldx < o.fnewt {
if !silent {
o.msg("Ldx", o.It, Ldx, fx_max, false, true)
}
break
}
// call line-search => update x and fx
if o.Lsearch {
nfv, err = LineSearch(x, o.fx, o.Ffcn, o.mdx, o.x0, o.dφdx, o.φ, o.LsMaxIt, true)
o.NFeval += nfv
if err != nil {
return chk.Err(_nls_err2, err.Error())
}
Ldx = 0.0
for i := 0; i < o.neq; i++ {
Ldx += ((x[i] - o.x0[i]) / o.scal[i]) * ((x[i] - o.x0[i]) / o.scal[i])
}
Ldx = math.Sqrt(Ldx / float64(o.neq))
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if Ldx < o.fnewt {
if !silent {
o.msg("Ldx(linsrch)", o.It, Ldx, fx_max, false, true)
}
break
}
}
// check convergence rate
if o.It > 0 && o.ChkConv {
Θ = Ldx / Ldx_prev
if Θ > 0.99 {
return chk.Err(_nls_err3, Θ, Ldx, Ldx_prev)
}
}
Ldx_prev = Ldx
}
// output
if o.Out != nil {
o.Out(x)
}
// check convergence
if o.It == o.MaxIt {
err = chk.Err(_nls_err4, o.It)
}
return
}