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Optimal Control of Ripple Formation in Abrasive Water-Jet Cutting - - PowerPoint PPT Presentation

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Abrasive waterjet cutting Instantaneous control Optimal Control of Ripple Formation in Abrasive Water-Jet Cutting Helmut Maurer, Karsten Theien maurer@math.uni-muenster.de ktheissen@dspace.de Wilhelms-Universit at, M unster, Germany

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SLIDE 1

Abrasive waterjet cutting Instantaneous control

Optimal Control of Ripple Formation in Abrasive Water-Jet Cutting

Helmut Maurer, Karsten Theißen maurer@math.uni-muenster.de ktheissen@dspace.de

Wilhelms-Universit¨ at, M¨ unster, Germany

CEA - EDF -INRIA School, May 29 - June 1, 2007

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 1 of 24

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Abrasive waterjet cutting Instantaneous control

Outline

1 Abrasive waterjet cutting

Physical model Mathematical description Numerical results

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 2 of 24

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SLIDE 3

Abrasive waterjet cutting Instantaneous control

Outline

1 Abrasive waterjet cutting

Physical model Mathematical description Numerical results

2 Instantaneous control

Controlling evolution equations into stationary solutions Instantaneous control of an abrasive waterjet cutter

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 2 of 24

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SLIDE 4

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Outline

1 Abrasive waterjet cutting

Physical model Mathematical description Numerical results

2 Instantaneous control

Controlling evolution equations into stationary solutions Instantaneous control of an abrasive waterjet cutter

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 3 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Abrasive waterjet cutter

c van Berkel Technische Berdrijven CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 4 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Abrasive waterjet cutter head

c

  • G. Radons - TU Chemnitz

1 water nozzle 2 abrasive head 3 waterjet 4 abrasive 5 mixing chamber 6 focussing tube 7 abrasive waterjet

c

  • G. Radons - TU Chemnitz

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 5 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Ripple formation

c Clemson University - Geological Sciences c

  • R. Friedrich - Universit¨

at M¨ unster

c JIT Waterjet CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Ripple formation

c Clemson University - Geological Sciences c

  • R. Friedrich - Universit¨

at M¨ unster

c JIT Waterjet CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Ripple formation

c Clemson University - Geological Sciences c

  • R. Friedrich - Universit¨

at M¨ unster

cutting depth y(x, ·) space domain

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

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SLIDE 10

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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SLIDE 11

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

jet profile e.g. Gauß

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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SLIDE 12

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

jet profile e.g. Gauß angle dependence of wear e.g for brittle material: f (y) =

1 1+y2

x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

jet profile e.g. Gauß angle dependence of wear e.g for brittle material: f (y) =

1 1+y2

x

curvature dependence α < 0 ” negative diffusion”

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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SLIDE 14

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

jet profile e.g. Gauß angle dependence of wear e.g for brittle material: f (y) =

1 1+y2

x

curvature dependence α < 0 ” negative diffusion” higher order term β < 0 from Taylor expansion

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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SLIDE 15

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Generalized Kuramoto-Sivashinsky Equation (GKSE)

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

yt = V (x) ·

  • f (y)+ α△y + β△2y
  • − uyx

jet profile e.g. Gauß angle dependence of wear e.g for brittle material: f (y) =

1 1+y2

x

curvature dependence α < 0 ” negative diffusion” higher order term β < 0 from Taylor expansion convective term workpiece material fed into jet with velocity u ≤ 0

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) 1d-domain Ω := (15, 28) endtime T > 0 space-time-cylinder Q := Ω × (0,T) y : Q→ R

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 8 of 24

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SLIDE 17

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) 1d-domain Ω := (15, 28) endtime T > 0 space-time-cylinder Q := Ω × (0,T) y : Q→ R

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 8 of 24

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SLIDE 18

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) 1d-domain Ω := (15, 28) endtime T > 0 space-time-cylinder Q := Ω × (0,T) y : Q→ R

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 8 of 24

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SLIDE 19

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) 1d-domain Ω := (15, 28) endtime T > 0 space-time-cylinder Q := Ω × (0,T) y : Q→ R

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 8 of 24

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SLIDE 20

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) 1d-domain Ω := (15, 28) endtime T > 0 space-time-cylinder Q := Ω × (0,T) y : Q→ R

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 8 of 24

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SLIDE 21

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) jet profile V (x) := e−(x−x0)2 point of impact x0 := 24 angle dependence of wear f (y) :=

1 1+y2

x

α = −1, β = −5.066

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 9 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) jet profile V (x) := e−(x−x0)2 point of impact x0 := 24 angle dependence of wear f (y) :=

1 1+y2

x

α = −1, β = −5.066

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 9 of 24

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SLIDE 23

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) jet profile V (x) := e−(x−x0)2 point of impact x0 := 24 angle dependence of wear f (y) :=

1 1+y2

x

α = −1, β = −5.066

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 9 of 24

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SLIDE 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Mathematical model of abrasive waterjet cutting

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − uyx

in Q initial and boundary conditions y(·, 0) = 0 in Ω y(28, ·) = 0 in (0,T) jet profile V (x) := e−(x−x0)2 point of impact x0 := 24 angle dependence of wear f (y) :=

1 1+y2

x

α = −1, β = −5.066

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 9 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Numerical simulation with u ≡ −0.101 (topview)

cutting depth y(x, ·) space domain x ∈ Ω

u ≡ −0.101 u ≡ −0.065 control pause CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 10 of 24

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SLIDE 26

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Numerical simulation with u ≡ −0.065 (topview)

cutting depth y(x, ·) space domain x ∈ Ω

u ≡ −0.101 u ≡ −0.065 control pause CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 11 of 24

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Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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SLIDE 28

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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SLIDE 29

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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SLIDE 30

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

Problems

1 We do not know if such a stationary solution exists. 2 The GKSE is a nonlinear evolution equation of fourth order. 3 The time intervall (0,T) can be very long.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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SLIDE 31

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

Problems

1 We do not know if such a stationary solution exists. 2 The GKSE is a nonlinear evolution equation of fourth order. 3 The time intervall (0,T) can be very long.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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SLIDE 32

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results

Can we do better with optimal control methods?

Objectives Find a time-dependend feed velocity u(t)...

1 ... which leads to a stationary cutting front with given depth... 2 ... as fast as possible.

Problems

1 We do not know if such a stationary solution exists. 2 The GKSE is a nonlinear evolution equation of fourth order. 3 The time intervall (0,T) can be very long.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 12 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Outline

1 Abrasive waterjet cutting

Physical model Mathematical description Numerical results

2 Instantaneous control

Controlling evolution equations into stationary solutions Instantaneous control of an abrasive waterjet cutter

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 13 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

General optimal control problem

minimize the L2−distance to a given stationary solution z(x)

  • Q

(y(x, t) − z(x))2dx dt dynamic is given by an evolution equation with distributed control yt = g

  • y, Dζ1y, . . . , Dζiy, . . . , Dζly, u
  • in Q

initial and boundary conditions y(·, 0) = y0 in Ω ∂ny + b(y) = 0 in ∂Ω × (0,T) control constraintsy umin ≤ u ≤ umax in Q

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 14 of 24

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SLIDE 35

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

General optimal control problem

minimize the L2−distance to a given stationary solution z(x)

  • Q

(y(x, t) − z(x))2dx dt dynamic is given by an evolution equation with distributed control yt = g

  • y, Dζ1y, . . . , Dζiy, . . . , Dζly, u
  • in Q

initial and boundary conditions y(·, 0) = y0 in Ω ∂ny + b(y) = 0 in ∂Ω × (0,T) control constraintsy umin ≤ u ≤ umax in Q

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 14 of 24

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SLIDE 36

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

General optimal control problem

minimize the L2−distance to a given stationary solution z(x)

  • Q

(y(x, t) − z(x))2dx dt dynamic is given by an evolution equation with distributed control yt = g

  • y, Dζ1y, . . . , Dζiy, . . . , Dζly, u
  • in Q

initial and boundary conditions y(·, 0) = y0 in Ω ∂ny + b(y) = 0 in ∂Ω × (0,T) control constraintsy umin ≤ u ≤ umax in Q

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 14 of 24

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SLIDE 37

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

General optimal control problem

minimize yg(L2)

  • Q

(y(x, t) − z(x))2dx dt subject to Dyqg yt = g

  • y, Dζ1y, . . . , Dζiy, . . . , Dζly, u
  • in Q

subject to IBCy y(·, 0) = y0 in Ω ∂ny + b(y) = 0 in ∂Ω × (0,T) subject to C umin ≤ u ≤ umax in Q

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 14 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Local optimization idea

(outer) time-discretisation 0 = t0 < t1 < . . . < tM−1 < tM = T

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 15 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Local optimization idea

(outer) time-discretisation 0 = t0 < t1 < . . . < tM−1 < tM = T small time intervalls Rm :=]tm, tm+1[, m = 0, . . . M − 1

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 15 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Local optimization idea

equidistant (outer) time-discretisation 0 = t0 < t1 < . . . < tM−1 < tM = T τ := T

M ,

tm := mτ, m = 0, . . . M − 1 small time intervalls Rm :=]tm, tm+1[, m = 0, . . . M − 1

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 15 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Local optimization idea

equidistant (outer) time-discretisation 0 = t0 < t1 < . . . < tM−1 < tM = T τ := T

M ,

tm := mτ, m = 0, . . . M − 1 small time intervalls Rm :=]tm, tm+1[, m = 0, . . . M − 1 Qm := Ω × Rm Σm := ∂Ω × Rm

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 15 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Definition of subproblems (SP)m

minimize

  • Qm

(y(x, t) − z(x))2dx dt subject to yt = g

  • y, Dζ1y, . . . , Dζiy, . . . , Dζly, um
  • in Qm

subject to y(·, mτ) = y∗

m−1(·, mτ)

in Ω ∂ny + b(y) = 0 in Σm subject to umin ≤ um ≤ umax in Qm

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 16 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Optimal control problem for the waterjet cutting model

dynamic yt = V (x) ·

  • f (y) + α△y + β△2y
  • − umyx

in Qm initial and boundary conditions y(·, τm) = y∗

m−1(·, τm)

in Ω y(28, ·) = 0 in Rm control constraints y umin ≤ um ≤ umax in Rm

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 17 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Optimal control problem for the waterjet cutting model

minimize

  • ˜

Q

(y(x, t) − cy)2dx dt subject to yt = V (x) ·

  • f (y) + α△y + β△2y
  • − umyx

in Qm subject to y(·, τm) = y∗

m−1(·, τm)

in Ω y(28, ·) = 0 in Rm subject to umin ≤ um ≤ umax in Rm

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 17 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

What cost functional should be used?

cutting depth y(x, ·) space domain x ∈ Ω

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 18 of 24

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SLIDE 46

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

What cost functional should be used?

cutting depth y(x, ·) space domain x ∈ Ω

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 18 of 24

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SLIDE 47

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Computed instantaneous control

feed velocity u(t) time t

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 19 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Computed instantaneous control (t ∈ [0, 300])

feed velocity u(t) time t

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 19 of 24

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SLIDE 49

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Simulation with computed control (topview)

cutting depth y(x, ·) space domain x ∈ Ω

u ≡ −0.101 u ≡ −0.065 control pause CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 20 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Different cutting depths...

cutting depth y(x, ·) space domain x ∈ Ω

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 21 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

...and their corresponding instantaneous controls

feed velocity u(t) time t

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 22 of 24

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SLIDE 52

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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SLIDE 53

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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SLIDE 54

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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SLIDE 55

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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SLIDE 56

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

  • utlook

This approach... ... may work also for other nonlinear evolution equations (e.g. thin-film processes). ... can easily be implemented for systems which can be simulated by a finite difference scheme.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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SLIDE 57

Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Conclusions and outlook

conclusions For this GKSE it is possible with instantaneous control methods... ... to find unknown stationary solutions of the system. ... to reach a stationary solution faster. ... to reach deeper cutting fronts.

  • utlook

This approach... ... may work also for other nonlinear evolution equations (e.g. thin-film processes). ... can easily be implemented for systems which can be simulated by a finite difference scheme.

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 23 of 24

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Abrasive waterjet cutting Instantaneous control Controlling evolution equations Waterjet cutter

Thank you for your attention!

some references:

  • R. Friedrich, G. Radons, T. Ditzinger, A. Henning

Ripple Formation through an Interface Instability from Moving Growth and Erosion Sources, Physical Review Letters, Vol. 85 No. 23, (2000)

  • G. Radons, R. Neugebauer

Nonlinear Dynamics of Production Systems, Wiley-VCH, ISBN 3527404309, (2004) special thanks to:

  • A. W¨

achter, C. Laird IPOPT (Interior Point OPTimizer)

CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 24 of 24