// AddToKb adds element K to global Jacobian matrix Kb
func (o *ElastRod) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return
}
/* Jacobian
========
Calculates (with N=n-1):
df0dx0, df0dx1, df0dx2, ... df0dxN
df1dx0, df1dx1, df1dx2, ... df1dxN
. . . . . . . . . . . . .
dfNdx0, dfNdx1, dfNdx2, ... dfNdxN
INPUT:
ffcn : f(x) function
x : station where dfdx has to be calculated
fx : f @ x
w : workspace with size == n == len(x)
RETURNS:
J : dfdx @ x [must be pre-allocated] */
func Jacobian(J *la.Triplet, ffcn Cb_f, x, fx, w []float64, distr bool) (err error) {
ndim := len(x)
start, endp1 := 0, ndim
if distr {
id, sz := mpi.Rank(), mpi.Size()
start, endp1 = (id*ndim)/sz, ((id+1)*ndim)/sz
if J.Max() == 0 {
J.Init(ndim, ndim, (endp1-start)*ndim)
}
} else {
if J.Max() == 0 {
J.Init(ndim, ndim, ndim*ndim)
}
}
J.Start()
// NOTE: cannot split calculation by columns unless the f function is
// independently calculated by each MPI processor.
// Otherwise, the AllReduce in f calculation would
// join pieces of f from different processors calculated for
// different x values (δx[col] from different columns).
/*
for col := start; col < endp1; col++ {
xsafe := x[col]
delta := math.Sqrt(EPS * max(CTE1, math.Abs(xsafe)))
x[col] = xsafe + delta
ffcn(w, x) // fnew
io.Pforan("x = %v, f = %v\n", x, w)
for row := 0; row < ndim; row++ {
J.Put(row, col, (w[row]-fx[row])/delta)
}
x[col] = xsafe
}
*/
var df float64
for col := 0; col < ndim; col++ {
xsafe := x[col]
delta := math.Sqrt(EPS * max(CTE1, math.Abs(xsafe)))
x[col] = xsafe + delta
err = ffcn(w, x) // w := f(x+δx[col])
if err != nil {
return
}
for row := start; row < endp1; row++ {
df = w[row] - fx[row]
//if math.Abs(df) > EPS {
J.Put(row, col, df/delta)
//}
}
x[col] = xsafe
}
return
}
// 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
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("Test SumToRoot 01")
}
M := [][]float64{
{1000, 1000, 1000, 1011, 1021, 1000},
{1000, 1000, 1000, 1012, 1022, 1000},
{1000, 1000, 1000, 1013, 1023, 1000},
{1011, 1012, 1013, 1000, 1000, 1000},
{1021, 1022, 1023, 1000, 1000, 1000},
{1000, 1000, 1000, 1000, 1000, 1000},
}
id, sz, m := mpi.Rank(), mpi.Size(), len(M)
start, endp1 := (id*m)/sz, ((id+1)*m)/sz
if sz > 6 {
chk.Panic("this test works with at most 6 processors")
}
var J la.Triplet
J.Init(m, m, m*m)
for i := start; i < endp1; i++ {
for j := 0; j < m; j++ {
J.Put(i, j, M[i][j])
}
}
la.PrintMat(fmt.Sprintf("J @ proc # %d", id), J.ToMatrix(nil).ToDense(), "%10.1f", false)
la.SpTriSumToRoot(&J)
var tst testing.T
if mpi.Rank() == 0 {
chk.Matrix(&tst, "J @ proc 0", 1.0e-17, J.ToMatrix(nil).ToDense(), [][]float64{
{1000, 1000, 1000, 1011, 1021, 1000},
{1000, 1000, 1000, 1012, 1022, 1000},
{1000, 1000, 1000, 1013, 1023, 1000},
{1011, 1012, 1013, 1000, 1000, 1000},
{1021, 1022, 1023, 1000, 1000, 1000},
{1000, 1000, 1000, 1000, 1000, 1000},
})
}
}
// 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
}
// adds element K to global Jacobian matrix Kb
func (o Beam) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (ok bool) {
if Global.Sim.Data.Steady {
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
} else {
dc := Global.DynCoefs
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.M[i][j]*dc.α1+o.K[i][j])
}
}
}
return true
}
// adds element K to global Jacobian matrix Kb
func (o *Beam) AddToKb(Kb *la.Triplet, sol *Solution, firstIt bool) (err error) {
if sol.Steady {
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.K[i][j])
}
}
return
}
α1 := sol.DynCfs.α1
for i, I := range o.Umap {
for j, J := range o.Umap {
Kb.Put(I, J, o.M[i][j]*α1+o.K[i][j])
}
}
return
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
myrank := mpi.Rank()
if myrank == 0 {
chk.PrintTitle("Test MUMPS Sol 02")
}
ndim := 10
id, sz := mpi.Rank(), mpi.Size()
start, endp1 := (id*ndim)/sz, ((id+1)*ndim)/sz
if mpi.Size() > ndim {
chk.Panic("the number of processors must be smaller than or equal to %d", ndim)
}
b := make([]float64, ndim)
var t la.Triplet
t.Init(ndim, ndim, ndim*ndim)
for i := start; i < endp1; i++ {
j := i
if i > 0 {
j = i - 1
}
for ; j < ndim; j++ {
val := 10.0 - float64(j)
if i > j {
val -= 1.0
}
t.Put(i, j, val)
}
b[i] = float64(i + 1)
}
x_correct := []float64{-1, 8, -65, 454, -2725, 13624, -54497, 163490, -326981, 326991}
sum_b_to_root := true
la.RunMumpsTestR(&t, 1e-4, b, x_correct, sum_b_to_root)
}
// 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
}
func Assemble(K11, K12 *la.Triplet, F1 []float64, src Cb_src, g *Grid2D, e *Equations) {
K11.Start()
K12.Start()
la.VecFill(F1, 0.0)
kx, ky := 1.0, 1.0
alp, bet, gam := 2.0*(kx/g.Dxx+ky/g.Dyy), -kx/g.Dxx, -ky/g.Dyy
mol := []float64{alp, bet, bet, gam, gam}
for i, I := range e.RF1 {
col, row := I%g.Nx, I/g.Nx
nodes := []int{I, I - 1, I + 1, I - g.Nx, I + g.Nx} // I, left, right, bottom, top
if col == 0 {
nodes[1] = nodes[2]
}
if col == g.Nx-1 {
nodes[2] = nodes[1]
}
if row == 0 {
nodes[3] = nodes[4]
}
if row == g.Ny-1 {
nodes[4] = nodes[3]
}
for k, J := range nodes {
j1, j2 := e.FR1[J], e.FR2[J] // 1 or 2?
if j1 > -1 { // 11
K11.Put(i, j1, mol[k])
} else { // 12
K12.Put(i, j2, mol[k])
}
}
if src != nil {
x := float64(col) * g.Dx
y := float64(row) * g.Dy
F1[i] += src(x, y)
}
}
}
// 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
}
// Jacobian computes Jacobian (sparse) matrix
// Calculates (with N=n-1):
// df0dx0, df0dx1, df0dx2, ... df0dxN
// df1dx0, df1dx1, df1dx2, ... df1dxN
// . . . . . . . . . . . . .
// dfNdx0, dfNdx1, dfNdx2, ... dfNdxN
// INPUT:
// ffcn : f(x) function
// x : station where dfdx has to be calculated
// fx : f @ x
// w : workspace with size == n == len(x)
// RETURNS:
// J : dfdx @ x [must be pre-allocated]
func Jacobian(J *la.Triplet, ffcn Cb_f, x, fx, w []float64) (err error) {
ndim := len(x)
start, endp1 := 0, ndim
if J.Max() == 0 {
J.Init(ndim, ndim, ndim*ndim)
}
J.Start()
var df float64
for col := 0; col < ndim; col++ {
xsafe := x[col]
delta := math.Sqrt(EPS * max(CTE1, math.Abs(xsafe)))
x[col] = xsafe + delta
err = ffcn(w, x) // w := f(x+δx[col])
if err != nil {
return
}
for row := start; row < endp1; row++ {
df = w[row] - fx[row]
J.Put(row, col, df/delta)
}
x[col] = xsafe
}
return
}
func IpBmatrix_sparse(B *la.Triplet, ndim, nne int, G [][]float64, radius float64, S []float64, axisym bool) {
B.Start()
if ndim == 3 {
for i := 0; i < nne; i++ {
B.Put(0, 0+i*3, G[i][0])
B.Put(1, 1+i*3, G[i][1])
B.Put(2, 2+i*3, G[i][2])
B.Put(3, 0+i*3, G[i][1]/SQ2)
B.Put(4, 1+i*3, G[i][2]/SQ2)
B.Put(5, 2+i*3, G[i][0]/SQ2)
B.Put(3, 1+i*3, G[i][0]/SQ2)
B.Put(4, 2+i*3, G[i][1]/SQ2)
B.Put(5, 0+i*3, G[i][2]/SQ2)
}
return
}
if axisym {
for i := 0; i < nne; i++ {
B.Put(0, 0+i*2, G[i][0])
B.Put(1, 1+i*2, G[i][1])
B.Put(2, 0+i*2, S[i]/radius)
B.Put(3, 0+i*2, G[i][1]/SQ2)
B.Put(3, 1+i*2, G[i][0]/SQ2)
}
return
}
for i := 0; i < nne; i++ {
B.Put(0, 0+i*2, G[i][0])
B.Put(1, 1+i*2, G[i][1])
B.Put(3, 0+i*2, G[i][1]/SQ2)
B.Put(3, 1+i*2, G[i][0]/SQ2)
}
}
func main() {
// input matrix in Triplet format
// including repeated positions. e.g. (0,0)
var A la.Triplet
A.Init(5, 5, 13)
A.Put(0, 0, 1.0) // << repeated
A.Put(0, 0, 1.0) // << repeated
A.Put(1, 0, 3.0)
A.Put(0, 1, 3.0)
A.Put(2, 1, -1.0)
A.Put(4, 1, 4.0)
A.Put(1, 2, 4.0)
A.Put(2, 2, -3.0)
A.Put(3, 2, 1.0)
A.Put(4, 2, 2.0)
A.Put(2, 3, 2.0)
A.Put(1, 4, 6.0)
A.Put(4, 4, 1.0)
// right-hand-side
b := []float64{8.0, 45.0, -3.0, 3.0, 19.0}
// solve
x, err := la.SolveRealLinSys(&A, b)
if err != nil {
io.Pfred("solver failed:\n%v", err)
return
}
// output
la.PrintMat("a", A.ToMatrix(nil).ToDense(), "%5g", false)
la.PrintVec("b", b, "%v ", false)
la.PrintVec("x", x, "%v ", false)
}
func main() {
// input matrix in Triplet format
// including repeated positions. e.g. (0,0)
var A la.Triplet
A.Init(5, 5, 13)
A.Put(0, 0, 1.0) // << repeated
A.Put(0, 0, 1.0) // << repeated
A.Put(1, 0, 3.0)
A.Put(0, 1, 3.0)
A.Put(2, 1, -1.0)
A.Put(4, 1, 4.0)
A.Put(1, 2, 4.0)
A.Put(2, 2, -3.0)
A.Put(3, 2, 1.0)
A.Put(4, 2, 2.0)
A.Put(2, 3, 2.0)
A.Put(1, 4, 6.0)
A.Put(4, 4, 1.0)
// right-hand-side
b := []float64{8.0, 45.0, -3.0, 3.0, 19.0}
// allocate solver
lis := la.GetSolver("umfpack")
defer lis.Clean()
// info
symmetric := false
verbose := false
timing := false
// initialise solver (R)eal
err := lis.InitR(&A, symmetric, verbose, timing)
if err != nil {
io.Pfred("solver failed:\n%v", err)
return
}
// factorise
err = lis.Fact()
if err != nil {
io.Pfred("solver failed:\n%v", err)
return
}
// solve (R)eal
var dummy bool
x := make([]float64, len(b))
err = lis.SolveR(x, b, dummy) // x := inv(a) * b
if err != nil {
io.Pfred("solver failed:\n%v", err)
return
}
// output
la.PrintMat("a", A.ToMatrix(nil).ToDense(), "%5g", false)
la.PrintVec("b", b, "%v ", false)
la.PrintVec("x", x, "%v ", false)
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("Test ODE 04b (MPI)")
io.Pfcyan("Hairer-Wanner VII-p376 Transistor Amplifier (MPI)\n")
io.Pfcyan("(from E Hairer's website, not the system in the book)\n")
}
if mpi.Size() != 3 {
chk.Panic(">> error: this test requires 3 MPI processors\n")
return
}
// RIGHT-HAND SIDE OF THE AMPLIFIER PROBLEM
w := make([]float64, 8) // workspace
fcn := func(f []float64, x float64, y []float64, args ...interface{}) error {
d := args[0].(*HWtransData)
UET := d.UE * math.Sin(d.W*x)
FAC1 := d.BETA * (math.Exp((y[3]-y[2])/d.UF) - 1.0)
FAC2 := d.BETA * (math.Exp((y[6]-y[5])/d.UF) - 1.0)
la.VecFill(f, 0)
switch mpi.Rank() {
case 0:
f[0] = y[0] / d.R9
case 1:
f[1] = (y[1]-d.UB)/d.R8 + d.ALPHA*FAC1
f[2] = y[2]/d.R7 - FAC1
case 2:
f[3] = y[3]/d.R5 + (y[3]-d.UB)/d.R6 + (1.0-d.ALPHA)*FAC1
f[4] = (y[4]-d.UB)/d.R4 + d.ALPHA*FAC2
f[5] = y[5]/d.R3 - FAC2
f[6] = y[6]/d.R1 + (y[6]-d.UB)/d.R2 + (1.0-d.ALPHA)*FAC2
f[7] = (y[7] - UET) / d.R0
}
mpi.AllReduceSum(f, w)
return nil
}
// JACOBIAN OF THE AMPLIFIER PROBLEM
jac := func(dfdy *la.Triplet, x float64, y []float64, args ...interface{}) error {
d := args[0].(*HWtransData)
FAC14 := d.BETA * math.Exp((y[3]-y[2])/d.UF) / d.UF
FAC27 := d.BETA * math.Exp((y[6]-y[5])/d.UF) / d.UF
if dfdy.Max() == 0 {
dfdy.Init(8, 8, 16)
}
NU := 2
dfdy.Start()
switch mpi.Rank() {
case 0:
dfdy.Put(2+0-NU, 0, 1.0/d.R9)
dfdy.Put(2+1-NU, 1, 1.0/d.R8)
dfdy.Put(1+2-NU, 2, -d.ALPHA*FAC14)
dfdy.Put(0+3-NU, 3, d.ALPHA*FAC14)
dfdy.Put(2+2-NU, 2, 1.0/d.R7+FAC14)
case 1:
dfdy.Put(1+3-NU, 3, -FAC14)
dfdy.Put(2+3-NU, 3, 1.0/d.R5+1.0/d.R6+(1.0-d.ALPHA)*FAC14)
dfdy.Put(3+2-NU, 2, -(1.0-d.ALPHA)*FAC14)
dfdy.Put(2+4-NU, 4, 1.0/d.R4)
dfdy.Put(1+5-NU, 5, -d.ALPHA*FAC27)
case 2:
dfdy.Put(0+6-NU, 6, d.ALPHA*FAC27)
dfdy.Put(2+5-NU, 5, 1.0/d.R3+FAC27)
dfdy.Put(1+6-NU, 6, -FAC27)
dfdy.Put(2+6-NU, 6, 1.0/d.R1+1.0/d.R2+(1.0-d.ALPHA)*FAC27)
dfdy.Put(3+5-NU, 5, -(1.0-d.ALPHA)*FAC27)
dfdy.Put(2+7-NU, 7, 1.0/d.R0)
}
return nil
}
// MATRIX "M"
c1, c2, c3, c4, c5 := 1.0e-6, 2.0e-6, 3.0e-6, 4.0e-6, 5.0e-6
var M la.Triplet
M.Init(8, 8, 14)
M.Start()
NU := 1
switch mpi.Rank() {
case 0:
M.Put(1+0-NU, 0, -c5)
M.Put(0+1-NU, 1, c5)
M.Put(2+0-NU, 0, c5)
M.Put(1+1-NU, 1, -c5)
M.Put(1+2-NU, 2, -c4)
M.Put(1+3-NU, 3, -c3)
case 1:
M.Put(0+4-NU, 4, c3)
M.Put(2+3-NU, 3, c3)
M.Put(1+4-NU, 4, -c3)
case 2:
M.Put(1+5-NU, 5, -c2)
M.Put(1+6-NU, 6, -c1)
M.Put(0+7-NU, 7, c1)
M.Put(2+6-NU, 6, c1)
M.Put(1+7-NU, 7, -c1)
}
// WRITE FILE FUNCTION
idxstp := 1
var b bytes.Buffer
out := func(first bool, dx, x float64, y []float64, args ...interface{}) error {
if mpi.Rank() == 0 {
if first {
fmt.Fprintf(&b, "%6s%23s%23s%23s%23s%23s%23s%23s%23s%23s\n", "ns", "x", "y0", "y1", "y2", "y3", "y4", "y5", "y6", "y7")
}
fmt.Fprintf(&b, "%6d%23.15E", idxstp, x)
for j := 0; j < len(y); j++ {
fmt.Fprintf(&b, "%23.15E", y[j])
}
fmt.Fprintf(&b, "\n")
idxstp += 1
}
return nil
}
defer func() {
if mpi.Rank() == 0 {
io.WriteFileD("/tmp/gosl", "hwamplifierB.res", &b)
}
}()
// INITIAL DATA
D, xa, xb, ya := HWtransIni()
// SET ODE SOLVER
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
ndim := len(ya)
//numjac := true
numjac := false
var osol ode.ODE
osol.Pll = true
if numjac {
osol.Init(method, ndim, fcn, nil, &M, out, silent)
} else {
osol.Init(method, ndim, fcn, jac, &M, out, silent)
}
osol.IniH = 1.0e-6 // initial step size
// SET TOLERANCES
atol, rtol := 1e-11, 1e-5
osol.SetTol(atol, rtol)
// RUN
t0 := time.Now()
if fixstp {
osol.Solve(ya, xa, xb, 0.01, fixstp, &D)
} else {
osol.Solve(ya, xa, xb, xb-xa, fixstp, &D)
}
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
}
}
func Test_linipm02(tst *testing.T) {
//verbose()
chk.PrintTitle("linipm02")
// linear program
// min 2*x0 + x1
// s.t. -x0 + x1 ≤ 1
// x0 + x1 ≥ 2 → -x0 - x1 ≤ -2
// x0 - 2*x1 ≤ 4
// x1 ≥ 0
// standard (step 1) add slack
// s.t. -x0 + x1 + x2 = 1
// -x0 - x1 + x3 = -2
// x0 - 2*x1 + x4 = 4
// standard (step 2)
// replace x0 := x0_ - x5
// because it's unbounded
// min 2*x0_ + x1 - 2*x5
// s.t. -x0_ + x1 + x2 + x5 = 1
// -x0_ - x1 + x3 + x5 = -2
// x0_ - 2*x1 + x4 - x5 = 4
// x0_,x1,x2,x3,x4,x5 ≥ 0
var T la.Triplet
T.Init(3, 6, 12)
T.Put(0, 0, -1)
T.Put(0, 1, 1)
T.Put(0, 2, 1)
T.Put(0, 5, 1)
T.Put(1, 0, -1)
T.Put(1, 1, -1)
T.Put(1, 3, 1)
T.Put(1, 5, 1)
T.Put(2, 0, 1)
T.Put(2, 1, -2)
T.Put(2, 4, 1)
T.Put(2, 5, -1)
Am := T.ToMatrix(nil)
A := Am.ToDense()
c := []float64{2, 1, 0, 0, 0, -2}
b := []float64{1, -2, 4}
// print LP
la.PrintMat("A", A, "%6g", false)
la.PrintVec("b", b, "%6g", false)
la.PrintVec("c", c, "%6g", false)
io.Pf("\n")
// solve LP
var ipm LinIpm
defer ipm.Clean()
ipm.Init(Am, b, c, nil)
err := ipm.Solve(chk.Verbose)
if err != nil {
tst.Errorf("ipm failed:\n%v", err)
return
}
// check
io.Pf("\n")
bres := make([]float64, len(b))
la.MatVecMul(bres, 1, A, ipm.X)
io.Pforan("bres = %v\n", bres)
chk.Vector(tst, "A*x=b", 1e-8, bres, b)
// fix and check x
x := ipm.X[:2]
x[0] -= ipm.X[5]
io.Pforan("x = %v\n", x)
chk.Vector(tst, "x", 1e-8, x, []float64{0.5, 1.5})
// plot
if true && chk.Verbose {
f := func(x []float64) float64 {
return c[0]*x[0] + c[1]*x[1]
}
g := func(x []float64, i int) float64 {
return A[i][0]*x[0] + A[i][1]*x[1] - b[i]
}
np := 41
vmin, vmax := []float64{-2.0, -2.0}, []float64{2.0, 2.0}
PlotTwoVarsContour("/tmp/gosl", "test_linipm02", x, np, nil, true, vmin, vmax, f,
func(x []float64) float64 { return g(x, 0) },
func(x []float64) float64 { return g(x, 1) },
)
}
}
func Test_linipm01(tst *testing.T) {
//verbose()
chk.PrintTitle("linipm01")
// linear programming problem
// min -4*x0 - 5*x1
// s.t. 2*x0 + x1 ≤ 3
// x0 + 2*x1 ≤ 3
// x0,x1 ≥ 0
// standard:
// 2*x0 + x1 + x2 = 3
// x0 + 2*x1 + x3 = 3
// x0,x1,x2,x3 ≥ 0
var T la.Triplet
T.Init(2, 4, 6)
T.Put(0, 0, 2.0)
T.Put(0, 1, 1.0)
T.Put(0, 2, 1.0)
T.Put(1, 0, 1.0)
T.Put(1, 1, 2.0)
T.Put(1, 3, 1.0)
Am := T.ToMatrix(nil)
A := Am.ToDense()
c := []float64{-4, -5, 0, 0}
b := []float64{3, 3}
// print LP
la.PrintMat("A", A, "%6g", false)
la.PrintVec("b", b, "%6g", false)
la.PrintVec("c", c, "%6g", false)
io.Pf("\n")
// solve LP
var ipm LinIpm
defer ipm.Clean()
ipm.Init(Am, b, c, nil)
err := ipm.Solve(chk.Verbose)
if err != nil {
tst.Errorf("ipm failed:\n%v", err)
return
}
// check
io.Pf("\n")
io.Pforan("x = %v\n", ipm.X)
io.Pfcyan("λ = %v\n", ipm.L)
io.Pforan("s = %v\n", ipm.S)
x := ipm.X[:2]
bres := make([]float64, 2)
la.MatVecMul(bres, 1, A, x)
io.Pforan("bres = %v\n", bres)
chk.Vector(tst, "x", 1e-9, x, []float64{1, 1})
chk.Vector(tst, "A*x=b", 1e-8, bres, b)
// plot
if true && chk.Verbose {
f := func(x []float64) float64 {
return c[0]*x[0] + c[1]*x[1]
}
g := func(x []float64, i int) float64 {
return A[i][0]*x[0] + A[i][1]*x[1] - b[i]
}
np := 41
vmin, vmax := []float64{-2.0, -2.0}, []float64{2.0, 2.0}
PlotTwoVarsContour("/tmp/gosl", "test_linipm01", x, np, nil, true, vmin, vmax, f,
func(x []float64) float64 { return g(x, 0) },
func(x []float64) float64 { return g(x, 1) },
)
}
}
// 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
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("ode04: Hairer-Wanner VII-p376 Transistor Amplifier\n")
}
if mpi.Size() != 3 {
chk.Panic(">> error: this test requires 3 MPI processors\n")
return
}
// data
UE, UB, UF, ALPHA, BETA := 0.1, 6.0, 0.026, 0.99, 1.0e-6
R0, R1, R2, R3, R4, R5 := 1000.0, 9000.0, 9000.0, 9000.0, 9000.0, 9000.0
R6, R7, R8, R9 := 9000.0, 9000.0, 9000.0, 9000.0
W := 2.0 * 3.141592654 * 100.0
// initial values
xa := 0.0
ya := []float64{0.0,
UB,
UB / (R6/R5 + 1.0),
UB / (R6/R5 + 1.0),
UB,
UB / (R2/R1 + 1.0),
UB / (R2/R1 + 1.0),
0.0}
// endpoint of integration
xb := 0.05
//xb = 0.0123 // OK
//xb = 0.01235 // !OK
// right-hand side of the amplifier problem
w := make([]float64, 8) // workspace
fcn := func(f []float64, dx, x float64, y []float64, args ...interface{}) error {
UET := UE * math.Sin(W*x)
FAC1 := BETA * (math.Exp((y[3]-y[2])/UF) - 1.0)
FAC2 := BETA * (math.Exp((y[6]-y[5])/UF) - 1.0)
la.VecFill(f, 0)
switch mpi.Rank() {
case 0:
f[0] = y[0] / R9
case 1:
f[1] = (y[1]-UB)/R8 + ALPHA*FAC1
f[2] = y[2]/R7 - FAC1
case 2:
f[3] = y[3]/R5 + (y[3]-UB)/R6 + (1.0-ALPHA)*FAC1
f[4] = (y[4]-UB)/R4 + ALPHA*FAC2
f[5] = y[5]/R3 - FAC2
f[6] = y[6]/R1 + (y[6]-UB)/R2 + (1.0-ALPHA)*FAC2
f[7] = (y[7] - UET) / R0
}
mpi.AllReduceSum(f, w)
return nil
}
// Jacobian of the amplifier problem
jac := func(dfdy *la.Triplet, dx, x float64, y []float64, args ...interface{}) error {
FAC14 := BETA * math.Exp((y[3]-y[2])/UF) / UF
FAC27 := BETA * math.Exp((y[6]-y[5])/UF) / UF
if dfdy.Max() == 0 {
dfdy.Init(8, 8, 16)
}
NU := 2
dfdy.Start()
switch mpi.Rank() {
case 0:
dfdy.Put(2+0-NU, 0, 1.0/R9)
dfdy.Put(2+1-NU, 1, 1.0/R8)
dfdy.Put(1+2-NU, 2, -ALPHA*FAC14)
dfdy.Put(0+3-NU, 3, ALPHA*FAC14)
dfdy.Put(2+2-NU, 2, 1.0/R7+FAC14)
case 1:
dfdy.Put(1+3-NU, 3, -FAC14)
dfdy.Put(2+3-NU, 3, 1.0/R5+1.0/R6+(1.0-ALPHA)*FAC14)
dfdy.Put(3+2-NU, 2, -(1.0-ALPHA)*FAC14)
dfdy.Put(2+4-NU, 4, 1.0/R4)
dfdy.Put(1+5-NU, 5, -ALPHA*FAC27)
case 2:
dfdy.Put(0+6-NU, 6, ALPHA*FAC27)
dfdy.Put(2+5-NU, 5, 1.0/R3+FAC27)
dfdy.Put(1+6-NU, 6, -FAC27)
dfdy.Put(2+6-NU, 6, 1.0/R1+1.0/R2+(1.0-ALPHA)*FAC27)
dfdy.Put(3+5-NU, 5, -(1.0-ALPHA)*FAC27)
dfdy.Put(2+7-NU, 7, 1.0/R0)
}
return nil
}
// matrix "M"
c1, c2, c3, c4, c5 := 1.0e-6, 2.0e-6, 3.0e-6, 4.0e-6, 5.0e-6
var M la.Triplet
M.Init(8, 8, 14)
M.Start()
NU := 1
switch mpi.Rank() {
case 0:
M.Put(1+0-NU, 0, -c5)
M.Put(0+1-NU, 1, c5)
M.Put(2+0-NU, 0, c5)
M.Put(1+1-NU, 1, -c5)
M.Put(1+2-NU, 2, -c4)
M.Put(1+3-NU, 3, -c3)
case 1:
M.Put(0+4-NU, 4, c3)
M.Put(2+3-NU, 3, c3)
M.Put(1+4-NU, 4, -c3)
case 2:
M.Put(1+5-NU, 5, -c2)
M.Put(1+6-NU, 6, -c1)
M.Put(0+7-NU, 7, c1)
M.Put(2+6-NU, 6, c1)
M.Put(1+7-NU, 7, -c1)
}
// flags
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
ndim := len(ya)
numjac := false
// structure to hold numerical results
res := ode.Results{Method: method}
// ODE solver
var osol ode.Solver
osol.Pll = true
// solve problem
if numjac {
osol.Init(method, ndim, fcn, nil, &M, ode.SimpleOutput, silent)
} else {
osol.Init(method, ndim, fcn, jac, &M, ode.SimpleOutput, silent)
}
osol.IniH = 1.0e-6 // initial step size
// set tolerances
atol, rtol := 1e-11, 1e-5
osol.SetTol(atol, rtol)
// run
t0 := time.Now()
if fixstp {
osol.Solve(ya, xa, xb, 0.01, fixstp, &res)
} else {
osol.Solve(ya, xa, xb, xb-xa, fixstp, &res)
}
// plot
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
plt.SetForEps(2.0, 400)
args := "'b-', marker='.', lw=1, clip_on=0"
ode.Plot("/tmp/gosl/ode", "hwamplifier_mpi.eps", &res, nil, xa, xb, "", args, func() {
_, T, err := io.ReadTable("data/radau5_hwamplifier.dat")
if err != nil {
chk.Panic("%v", err)
}
for j := 0; j < ndim; j++ {
plt.Subplot(ndim+1, 1, j+1)
plt.Plot(T["x"], T[io.Sf("y%d", j)], "'k+',label='reference',ms=10")
}
})
}
}
// 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
}
// 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
}
// 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
}
// 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
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
myrank := mpi.Rank()
if myrank == 0 {
chk.PrintTitle("Test MUMPS Sol 01b")
}
var t la.Triplet
b := []float64{8.0, 45.0, -3.0, 3.0, 19.0}
switch mpi.Size() {
case 1:
t.Init(5, 5, 13)
t.Put(0, 0, 1.0)
t.Put(0, 0, 1.0)
t.Put(1, 0, 3.0)
t.Put(0, 1, 3.0)
t.Put(2, 1, -1.0)
t.Put(4, 1, 4.0)
t.Put(1, 2, 4.0)
t.Put(2, 2, -3.0)
t.Put(3, 2, 1.0)
t.Put(4, 2, 2.0)
t.Put(2, 3, 2.0)
t.Put(1, 4, 6.0)
t.Put(4, 4, 1.0)
case 2:
la.VecFill(b, 0)
if myrank == 0 {
t.Init(5, 5, 8)
t.Put(0, 0, 1.0)
t.Put(0, 0, 1.0)
t.Put(1, 0, 3.0)
t.Put(0, 1, 3.0)
t.Put(2, 1, -1.0)
t.Put(4, 1, 1.0)
t.Put(4, 1, 1.5)
t.Put(4, 1, 1.5)
b[0] = 8.0
b[1] = 40.0
b[2] = 1.5
} else {
t.Init(5, 5, 8)
t.Put(1, 2, 4.0)
t.Put(2, 2, -3.0)
t.Put(3, 2, 1.0)
t.Put(4, 2, 2.0)
t.Put(2, 3, 2.0)
t.Put(1, 4, 6.0)
t.Put(4, 4, 0.5)
t.Put(4, 4, 0.5)
b[1] = 5.0
b[2] = -4.5
b[3] = 3.0
b[4] = 19.0
}
default:
chk.Panic("this test needs 1 or 2 procs")
}
x_correct := []float64{1, 2, 3, 4, 5}
sum_b_to_root := true
la.RunMumpsTestR(&t, 1e-14, b, x_correct, sum_b_to_root)
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
myrank := mpi.Rank()
if myrank == 0 {
chk.PrintTitle("Test MUMPS Sol 01a")
}
var t la.Triplet
switch mpi.Size() {
case 1:
t.Init(5, 5, 13)
t.Put(0, 0, 1.0)
t.Put(0, 0, 1.0)
t.Put(1, 0, 3.0)
t.Put(0, 1, 3.0)
t.Put(2, 1, -1.0)
t.Put(4, 1, 4.0)
t.Put(1, 2, 4.0)
t.Put(2, 2, -3.0)
t.Put(3, 2, 1.0)
t.Put(4, 2, 2.0)
t.Put(2, 3, 2.0)
t.Put(1, 4, 6.0)
t.Put(4, 4, 1.0)
case 2:
if myrank == 0 {
t.Init(5, 5, 6)
t.Put(0, 0, 1.0)
t.Put(0, 0, 1.0)
t.Put(1, 0, 3.0)
t.Put(0, 1, 3.0)
t.Put(2, 1, -1.0)
t.Put(4, 1, 4.0)
} else {
t.Init(5, 5, 7)
t.Put(1, 2, 4.0)
t.Put(2, 2, -3.0)
t.Put(3, 2, 1.0)
t.Put(4, 2, 2.0)
t.Put(2, 3, 2.0)
t.Put(1, 4, 6.0)
t.Put(4, 4, 1.0)
}
default:
chk.Panic("this test needs 1 or 2 procs")
}
b := []float64{8.0, 45.0, -3.0, 3.0, 19.0}
x_correct := []float64{1, 2, 3, 4, 5}
sum_b_to_root := false
la.RunMumpsTestR(&t, 1e-14, b, x_correct, sum_b_to_root)
}
// 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
}