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

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

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

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

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

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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Abrasive waterjet cutter head 1 water nozzle 2 abrasive head 3 waterjet 4 abrasive 5 mixing chamber 6 focussing tube 7 abrasive waterjet � G. Radons - TU Chemnitz c � G. Radons - TU Chemnitz c CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 5 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Ripple formation � R. Friedrich - Universit¨ c at M¨ unster c � Clemson University - Geological Sciences c � JIT Waterjet CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Ripple formation � R. Friedrich - Universit¨ c at M¨ unster c � Clemson University - Geological Sciences c � JIT Waterjet CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Ripple formation � R. Friedrich - Universit¨ c at M¨ unster cutting depth y ( x , · ) c � Clemson University - Geological Sciences space domain CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 6 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � f ( y )+ α △ y + β △ 2 y y t = V ( x ) · − uy x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � � � f ( y )+ α △ y + β △ 2 y f ( y )+ α △ y + β △ 2 y y t = V ( x ) · y t = V ( x ) · − uy x − uy x jet profile e.g. Gauß CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � � � f ( y )+ α △ y + β △ 2 y f ( y )+ α △ y + β △ 2 y y t = V ( x ) · y t = V ( x ) · − uy x − uy x jet profile e.g. Gauß angle dependence of wear e.g for brittle material: 1 f ( y ) = 1+ y 2 x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � � � f ( y )+ α △ y + β △ 2 y f ( y )+ α △ y + β △ 2 y y t = V ( x ) · y t = V ( x ) · − uy x − uy x jet profile e.g. Gauß curvature dependence α < 0 ” negative diffusion” angle dependence of wear e.g for brittle material: 1 f ( y ) = 1+ y 2 x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � � � f ( y )+ α △ y + β △ 2 y f ( y )+ α △ y + β △ 2 y y t = V ( x ) · y t = V ( x ) · − uy x − uy x jet profile e.g. Gauß curvature dependence α < 0 ” negative diffusion” angle dependence of wear e.g for brittle material: higher order term 1 f ( y ) = 1+ y 2 β < 0 from Taylor expansion x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Generalized Kuramoto-Sivashinsky Equation (GKSE) � � � � f ( y )+ α △ y + β △ 2 y f ( y )+ α △ y + β △ 2 y y t = V ( x ) · y t = V ( x ) · − uy x − uy x convective term workpiece material fed into jet with velocity u ≤ 0 jet profile e.g. Gauß curvature dependence α < 0 ” negative diffusion” angle dependence of wear e.g for brittle material: higher order term 1 f ( y ) = 1+ y 2 β < 0 from Taylor expansion x CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 7 of 24

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions 1d-domain Ω := (15 , 28) y ( · , 0) = 0 in Ω endtime y (28 , · ) = 0 in (0 , T ) 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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions 1d-domain Ω := (15 , 28) y ( · , 0) = 0 in Ω endtime y (28 , · ) = 0 in (0 , T ) 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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions 1d-domain Ω := (15 , 28) y ( · , 0) = 0 in Ω endtime y (28 , · ) = 0 in (0 , T ) 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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions 1d-domain Ω := (15 , 28) y ( · , 0) = 0 in Ω endtime y (28 , · ) = 0 in (0 , T ) 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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions 1d-domain Ω := (15 , 28) y ( · , 0) = 0 in Ω endtime y (28 , · ) = 0 in (0 , T ) 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

Abrasive waterjet cutting Instantaneous control Physical model Mathematical description Numerical results Mathematical model of abrasive waterjet cutting dynamic f ( y ) + α △ y + β △ 2 y � � y t = V ( x ) · − uy x in Q initial and boundary conditions jet profile V ( x ) := e − ( x − x 0 ) 2 y ( · , 0) = 0 in Ω point of impact y (28 , · ) = 0 in (0 , T ) x 0 := 24 angle dependence of wear 1 f ( y ) := 1+ y 2 x α = − 1 , β = − 5 . 066 CEA-EDF-INRIA Maurer, Theißen Optimal Control Methods in Waterjet Cutting 9 of 24