Some Aspects in the Numerics of Nonlinear Acoustics: Time - PowerPoint PPT Presentation

Some Aspects in the Numerics of Nonlinear Acoustics: Time Integration and Open Domain Problems Barbara Kaltenbacher Alpen-Adria-Universit¨ at Klagenfurt RICAM Special Semester on Computational Methods in Science and Engineering Workshop 1: Analysis and Numerics of Acoustic and Electromagnetic Problems

Some Aspects in the Numerics of Nonlinear Acoustics: Time Integration and Open Domain Problems Barbara Kaltenbacher Alpen-Adria-Universit¨ at Klagenfurt RICAM Special Semester on Computational Methods in Science and Engineering Workshop 1: Analysis and Numerics of Acoustic and Electromagnetic Problems joint work with: Rainer Brunnhuber, AAU, Vanja Nikoli´ c, AAU, Christian Clason, U Duisburg-Essen, Manfred Kaltenbacher, TU Vienna, Irena Lasiecka, U Memphis, Richard Marchand, Slippery Rock U, Gunther Peichl, U Graz, Maria K. Pospieszalska, La Jolla Institute, Petronela Radu, U Nebraska at Lincoln, Igor Shevchenko, UCL, Mechthild Thalhammer, U Innsbruck Mathematics of Nonlinear Acoustics

Nonlinear Acoustic Wave Propagation 1

Nonlinear Acoustic Wave Propagation 1

Applications of High Intensity Focused Ultrasound HIFU thermotherapy lithotripsy welding cleaning 2

Outline models time integration nonreflecting boundary conditions 3

modeling 4

Physical Principles main physical quantities: acoustic particle velocity � v ; acoustic pressure p ; mass density ̺ ; decomposition into mean and fluctuating part: v = � v 0 + � v ∼ = � p = p 0 + p ∼ , ̺ = ̺ 0 + ̺ ∼ � v ∼ , 5

Physical Principles main physical quantities: acoustic particle velocity � v ; acoustic pressure p ; mass density ̺ ; decomposition into mean and fluctuating part: v = � v 0 + � v ∼ = � p = p 0 + p ∼ , ̺ = ̺ 0 + ̺ ∼ � v ∼ , governing equations: Navier Stokes equation (under the assumption ∇ × � v = 0) � � � 4 µ V � � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ v 3 equation of continuity ̺ t + ∇ · ( ̺� v ) = 0 state equation � 1 � ̺ ∼ = 1 1 B κ − 1 2 Ap ∼ 2 − c 2 p ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p 5

Derivation of Wave Equation main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , 6

Derivation of Wave Equation main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , governing equations: � � � 4 µ V � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ � v 3 ̺ t + ∇ · ( ̺� v ) = 0 � 1 � 1 − 1 ̺ ∼ = p ∼ B κ 2 Ap 2 c 2 − ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p 6

Derivation of Wave Equation c 2 and ̺ 0 are known parameters main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , governing equations: nht . . . nonlinear and higher order terms � � � 4 µ V � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ � v 3 ̺ t + ∇ · ( ̺� v ) = 0 � 1 � 1 − 1 ̺ ∼ = p ∼ B κ 2 Ap 2 c 2 − ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p 6

Derivation of Wave Equation c 2 and ̺ 0 are known parameters main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , governing equations: nht . . . nonlinear and higher order terms � � � 4 µ V � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ � v 3 v t + ∇ p ∼ = nht ̺ 0 � ̺ t + ∇ · ( ̺� v ) = 0 � 1 � 1 − 1 ̺ ∼ = p ∼ B κ 2 Ap 2 c 2 − ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p 6

Derivation of Wave Equation c 2 and ̺ 0 are known parameters main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , governing equations: nht . . . nonlinear and higher order terms � � � 4 µ V � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ � v 3 v t + ∇ p ∼ = nht ̺ 0 � ̺ t + ∇ · ( ̺� v ) = 0 ̺ ∼ t + ̺ 0 ∇ · � v = nht � 1 � 1 − 1 ̺ ∼ = p ∼ B κ 2 Ap 2 c 2 − ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p 6

Derivation of Wave Equation c 2 and ̺ 0 are known parameters main physical quantities: � v = � p = p 0 + p ∼ , ∇ p 0 = 0 , ̺ = ̺ 0 + ̺ ∼ , ̺ 0 t = 0 v ∼ , governing equations: nht . . . nonlinear and higher order terms � � � 4 µ V � v t + ∇ ( � v · � v ) + ∇ p = + ζ V ∆ � ̺ � v 3 v t + ∇ p ∼ = nht ̺ 0 � ̺ t + ∇ · ( ̺� v ) = 0 ̺ ∼ t + ̺ 0 ∇ · � v = nht � 1 � 1 − 1 ̺ ∼ = p ∼ B κ 2 Ap 2 c 2 − ∼ − p ∼ t ̺ 0 c 4 ̺ 0 c 4 c V c p ̺ ∼ = 1 c 2 p ∼ + nht 6

Derivation of Wave Equation v t + ∇ p ∼ = nht ̺ 0 � ̺ ∼ t + ̺ 0 ∇ · � v = nht ̺ ∼ = 1 c 2 p ∼ + nht 7

Derivation of Wave Equation v t + ∇ p ∼ = nht ̺ 0 � ̺ ∼ t + ̺ 0 ∇ · � v = nht ̺ ∼ = 1 c 2 p ∼ + nht insert line 3 into line 2 to eliminate ̺ ∼ . . . 7

Derivation of Wave Equation ̺ 0 � v t + ∇ p ∼ = nht 1 c 2 p ∼ t + ̺ 0 ∇ · � v = nht 8

Derivation of Wave Equation ̺ 0 � v t + ∇ p ∼ = nht 1 c 2 p ∼ t + ̺ 0 ∇ · � v = nht (This is an evolution with a nice skew-symmetric structure, since ∇· = −∇ ∗ 0 !) 8

Derivation of Wave Equation v t + ∇ p ∼ = nht ̺ 0 � 1 c 2 p ∼ t + ̺ 0 ∇ · � v = nht 9

Derivation of Wave Equation − ∇ · v t + ∇ p ∼ = nht ̺ 0 � ∂ 1 c 2 p ∼ t + ̺ 0 ∇ · � v = nht ∂ t 9

Derivation of Wave Equation − ∇ · v t + ∇ p ∼ = nht ̺ 0 � ∂ 1 c 2 p ∼ t + ̺ 0 ∇ · � v = nht ∂ t ————————— 1 c 2 p ∼ tt − ∆ p ∼ = nht 9

Classical Models of Nonlinear Acoustics I Kuznetsov’s equation [Lesser & Seebass 1968, Kuznetsov 1971] � � B p ∼ tt − c 2 ∆ p ∼ − b ∆ p ∼ t = − 2 A ̺ 0 c 2 p 2 v | 2 ∼ + ̺ 0 | � tt where ̺ 0 � v t = −∇ p for the particle velocity � v and the pressure p , i.e., � 2 A c 2 ( ψ t ) 2 + |∇ ψ | 2 � B ψ tt − c 2 ∆ ψ − b ∆ ψ t = − t since ∇ × � v = 0 hence � v = −∇ ψ for a velocity potential ψ Westervelt equation [Westervelt 1963] � � 1 1 + B p ∼ tt − c 2 ∆ p ∼ − b ∆ p ∼ t = − p 2 ∼ tt ̺ 0 c 2 2 A v | 2 ≈ 1 c 2 ( p ∼ t ) 2 via ̺ 0 | � 10

Classical Models of Nonlinear Acoustics II Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation [Zabolotskaya & Khokhlov 1969] 2 cp ∼ xt − c 2 ∆ yz p ∼ − b β a ̺ 0 c 2 p 2 c 2 p ∼ ttt = ∼ tt x . . . direction of sound propagation Burgers’ equation [Burgers 1974] p ∼ t − b β a 2 c 2 p ∼ ττ = ̺ 0 c 3 p ∼ p ∼ τ τ = t − x c . . . retarded time 11

Advanced Models of Nonlinear Acoustics (Examples) Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979] � � � � ν ∆ ψ tt + (1+ B / (2 A )) ν ν B ∆ 2 ψ t − c 2 ∆ ψ t + c 2 ν Pr∆ 2 ψ ψ ttt − b + b − Pr Pr 2 A Pr 2 Ac 2 ( ψ 2 t ) + |∇ ψ | 2 � � B = − tt 12

Advanced Models of Nonlinear Acoustics (Examples) Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979] � � � � ν ∆ ψ tt + (1+ B / (2 A )) ν ν B ∆ 2 ψ t − c 2 ∆ ψ t + c 2 ν Pr∆ 2 ψ ψ ttt − b + b − Pr Pr 2 A Pr 2 Ac 2 ( ψ 2 t ) + |∇ ψ | 2 � � B = − � B � tt � � t ) + |∇ ψ | 2 ψ tt − c 2 ∆ ψ − b ∆ ψ t 2 Ac 2 ( ψ 2 ( ∂ t − a ∆) − r ∆ ψ t = − tt ν a = Pr. . . thermal conductivity 12

Advanced Models of Nonlinear Acoustics (Examples) Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979] � � � � ν ∆ ψ tt + (1+ B / (2 A )) ν ν B ∆ 2 ψ t − c 2 ∆ ψ t + c 2 ν Pr∆ 2 ψ ψ ttt − b + b − Pr Pr 2 A Pr 2 Ac 2 ( ψ 2 t ) + |∇ ψ | 2 � � B = − � B � tt � � t ) + |∇ ψ | 2 ψ tt − c 2 ∆ ψ − b ∆ ψ t 2 Ac 2 ( ψ 2 ( ∂ t − a ∆) − r ∆ ψ t = − tt ν a = Pr. . . thermal conductivity Jordan-Moore-Gibson-Thompson equation [Jordan 2009, 2014], [Christov 2009], [Straughan 2010] � B � 2 Ac 2 ( ψ t ) 2 + |∇ ψ | 2 τψ ttt + ψ tt − c 2 ∆ ψ − b ∆ ψ t = − t τ . . . relaxation time 12

Advanced Models of Nonlinear Acoustics (Examples) Blackstock-Crighton equation [Brunnhuber & Jordan 2016], [Blackstock 1963], [Crighton 1979] � � � � ν ∆ ψ tt + (1+ B / (2 A )) ν ν B ∆ 2 ψ t − c 2 ∆ ψ t + c 2 ν Pr∆ 2 ψ ψ ttt − b + b − Pr Pr 2 A Pr 2 Ac 2 ( ψ 2 t ) + |∇ ψ | 2 � � B = − � B � tt � � t ) + |∇ ψ | 2 ψ tt − c 2 ∆ ψ − b ∆ ψ t 2 Ac 2 ( ψ 2 ( ∂ t − a ∆) − r ∆ ψ t = − tt ν a = Pr. . . thermal conductivity Jordan-Moore-Gibson-Thompson equation [Jordan 2009, 2014], [Christov 2009], [Straughan 2010] � B � 2 Ac 2 ( ψ t ) 2 + |∇ ψ | 2 τψ ttt + ψ tt − c 2 ∆ ψ − b ∆ ψ t = − t τ . . . relaxation time cf. Kuznetsov’s equation: � B � ψ tt − c 2 ∆ ψ − b ∆ ψ t = − 2 Ac 2 ( ψ 2 t ) + |∇ ψ | 2 t 12

Some Asymptotics Blackstock-Crighton equation: � � � 2 ) + |∇ ψ a | 2 � t + ac 2 ∆ 2 ψ a = − tt − c 2 ∆ ψ a − ( a + b )∆ ψ a t + ad ∆ 2 ψ a ψ a 2 Ac 2 ( ψ a B t t tt Kuznetsov’s equation: � t ) + |∇ ψ | 2 � ψ tt − c 2 ∆ ψ − b ∆ ψ t = − 2 Ac 2 ( ψ 2 B t 13