Application of generalized Convolution Quadrature in Acoustics and - PowerPoint PPT Presentation

Graz University of Technology Institute of Applied Mechanics Application of generalized Convolution Quadrature in Acoustics and Thermoelasticity Martin Schanz joint work with Relindis Rott and Stefan Sauter Space-Time Methods for PDEs Special Semester on Computational Methods in Science and Engineering RICAM, Linz, Austria, November 10, 2016 > www.mech.tugraz.at

Content Generalized convolution quadrature method (gCQM) 1 Quadrature formula Algorithm 2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples 3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

Content Generalized convolution quadrature method (gCQM) 1 Quadrature formula Algorithm 2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples 3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

Content Generalized convolution quadrature method (gCQM) 1 Quadrature formula Algorithm 2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples 3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example Martin Schanz gCQM: Acoustics and Thermoelasticity 2 / 39

Content Generalized convolution quadrature method (gCQM) 1 Quadrature formula Algorithm 2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples 3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example Martin Schanz gCQM: Acoustics and Thermoelasticity 3 / 39

Convolution integral Convolution integral with the Laplace transformed function ˆ f ( s ) t � ˆ � � y ( t ) = ( f ∗ g )( t ) = f ( ∂ t ) g ( t ) = f ( t − τ ) g ( τ ) d τ 0 t 1 � � ˆ e s ( t − τ ) g ( τ ) d τ = f ( s ) d s 2 π i C 0 � �� � x ( t , s ) Integral is equivalent to solution of ODE ∂ ∂ t x ( t , s ) = sx ( t , s )+ g ( t ) x ( t = 0 , s ) = 0 with Implicit Euler for ODE , [ 0 , T ] = [ 0 , t 1 , t 2 ,..., t N ] , variable time steps ∆ t i , i = 1 , 2 ,..., N n n x n ( s ) = x n − 1 ( s ) ∆ t n 1 ∑ ∏ 1 − ∆ t n s + 1 − ∆ t n s g n = ∆ t j g j 1 − ∆ t k s j = 1 k = j Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39

Convolution integral Convolution integral with the Laplace transformed function ˆ f ( s ) t � ˆ � � y ( t ) = ( f ∗ g )( t ) = f ( ∂ t ) g ( t ) = f ( t − τ ) g ( τ ) d τ 0 t 1 � � ˆ e s ( t − τ ) g ( τ ) d τ = f ( s ) d s 2 π i C 0 � �� � x ( t , s ) Integral is equivalent to solution of ODE ∂ ∂ t x ( t , s ) = sx ( t , s )+ g ( t ) x ( t = 0 , s ) = 0 with Implicit Euler for ODE , [ 0 , T ] = [ 0 , t 1 , t 2 ,..., t N ] , variable time steps ∆ t i , i = 1 , 2 ,..., N n n x n ( s ) = x n − 1 ( s ) ∆ t n 1 ∑ ∏ 1 − ∆ t n s + 1 − ∆ t n s g n = ∆ t j g j 1 − ∆ t k s j = 1 k = j Martin Schanz gCQM: Acoustics and Thermoelasticity 4 / 39

Time stepping formula Solution at the discrete time t n y ( t n ) = 1 � ˆ f ( s ) x n ( s ) d s 2 π i C ˆ f ( s )∆ t n f ( s ) x n − 1 ( s ) = 1 1 − ∆ t n s g n d s + 1 � � ˆ 1 − ∆ t n s d s 2 π i 2 π i C C � � f ( s ) x n − 1 ( s ) 1 g n + 1 � =ˆ ˆ 1 − ∆ t n s d s . f ∆ t n 2 π i C Recursion formula for the implicit Euler n n 1 1 � ˆ ∑ ∏ y ( t n ) = f ( s ) ∆ t j g j 1 − ∆ t k s d s 2 π i j = 1 k = j C � � n − 1 n 1 1 1 � = ˆ ˆ ∑ ∏ g n + ∆ t j g j f ( s ) f 1 − ∆ t k s d s ∆ t n 2 π i k = j j = 1 C Complex integral is solved with a quadrature formula Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Time stepping formula Solution at the discrete time t n y ( t n ) = 1 � ˆ f ( s ) x n ( s ) d s 2 π i C ˆ f ( s )∆ t n f ( s ) x n − 1 ( s ) = 1 1 − ∆ t n s g n d s + 1 � � ˆ 1 − ∆ t n s d s 2 π i 2 π i C C � � f ( s ) x n − 1 ( s ) 1 g n + 1 � =ˆ ˆ 1 − ∆ t n s d s . f ∆ t n 2 π i C Recursion formula for the implicit Euler n n 1 1 � ˆ ∑ ∏ y ( t n ) = f ( s ) ∆ t j g j 1 − ∆ t k s d s 2 π i j = 1 k = j C � � n − 1 n 1 1 1 � = ˆ ˆ ∑ ∏ g n + ∆ t j g j f ( s ) f 1 − ∆ t k s d s ∆ t n 2 π i k = j j = 1 C Complex integral is solved with a quadrature formula Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Time stepping formula Solution at the discrete time t n y ( t n ) = 1 � ˆ f ( s ) x n ( s ) d s 2 π i C ˆ f ( s )∆ t n f ( s ) x n − 1 ( s ) = 1 1 − ∆ t n s g n d s + 1 � � ˆ 1 − ∆ t n s d s 2 π i 2 π i C C � � f ( s ) x n − 1 ( s ) 1 g n + 1 � =ˆ ˆ 1 − ∆ t n s d s . f ∆ t n 2 π i C Recursion formula for the implicit Euler n n 1 1 � ˆ ∑ ∏ y ( t n ) = f ( s ) ∆ t j g j 1 − ∆ t k s d s 2 π i j = 1 k = j C � � n − 1 n 1 1 1 � = ˆ ˆ ∑ ∏ g n + ∆ t j g j f ( s ) f 1 − ∆ t k s d s ∆ t n 2 π i k = j j = 1 C Complex integral is solved with a quadrature formula Martin Schanz gCQM: Acoustics and Thermoelasticity 5 / 39

Algorithm First Euler step � � 1 y ( t 1 ) = ˆ f g 1 ∆ t 1 with implicit assumption of zero initial condition For all steps n = 2 ,..., N the algorithm has two steps Update the solution vector x n − 1 at all integration points s ℓ with an implicit Euler step 1 x n − 2 ( s ℓ ) ∆ t n − 1 x n − 1 ( s ℓ ) = + g n − 1 1 − ∆ t n − 1 s ℓ 1 − ∆ t n − 1 s ℓ for ℓ = 1 ,..., N Q with the number of integration points N Q . Compute the solution of the integral at the actual time step t n 2 � � N Q ˆ 1 f ( s ℓ ) y ( t n ) = ˆ ∑ f g n + ω ℓ x n − 1 ( s ℓ ) ∆ t n 1 − ∆ t n s ℓ ℓ = 1 Essential parameter: N Q = N log ( N ) , integration is dependent on q = ∆ t max ∆ t min Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

Algorithm First Euler step � � 1 y ( t 1 ) = ˆ f g 1 ∆ t 1 with implicit assumption of zero initial condition For all steps n = 2 ,..., N the algorithm has two steps Update the solution vector x n − 1 at all integration points s ℓ with an implicit Euler step 1 x n − 2 ( s ℓ ) ∆ t n − 1 x n − 1 ( s ℓ ) = + g n − 1 1 − ∆ t n − 1 s ℓ 1 − ∆ t n − 1 s ℓ for ℓ = 1 ,..., N Q with the number of integration points N Q . Compute the solution of the integral at the actual time step t n 2 � � N Q ˆ 1 f ( s ℓ ) y ( t n ) = ˆ ∑ f g n + ω ℓ x n − 1 ( s ℓ ) ∆ t n 1 − ∆ t n s ℓ ℓ = 1 Essential parameter: N Q = N log ( N ) , integration is dependent on q = ∆ t max ∆ t min Martin Schanz gCQM: Acoustics and Thermoelasticity 6 / 39

Numerical integration Integration weights and points � k 2 � 4 K γ ′ ( σ ℓ ) s ℓ = γ ( σ ℓ ) ω ℓ = 2 π i � n � α T , α = 1 . 5 for N = 25 , T = 5 , t n = N Martin Schanz gCQM: Acoustics and Thermoelasticity 7 / 39

Content Generalized convolution quadrature method (gCQM) 1 Quadrature formula Algorithm 2 Acoustics: Absorbing boundary conditions Boundary element formulation Analytical solution Numerical examples 3 Thermoelasticity: Uncoupled formulation Boundary element formulation Numerical example Martin Schanz gCQM: Acoustics and Thermoelasticity 8 / 39

Absorbing boundary conditions Materials with absorbing surfaces Mechanical modell: Coupling of porous material layer at the boundary Simpler mechanical model: Impedance boundary condition p Z = v · n specific impedance � Z ( x ) = α ( x ) = cos θ 1 − 1 − α S ( x ) � ρ c 1 + 1 − α S ( x ) with density ρ , wave velocity c , and absorption coefficient α S = f ( ω ) Martin Schanz gCQM: Acoustics and Thermoelasticity 9 / 39

Absorbing boundary conditions Materials with absorbing surfaces Mechanical modell: Coupling of porous material layer at the boundary Simpler mechanical model: Impedance boundary condition p Z = v · n specific impedance � Z ( x ) = α ( x ) = cos θ 1 − 1 − α S ( x ) � ρ c 1 + 1 − α S ( x ) with density ρ , wave velocity c , and absorption coefficient α S = f ( ω ) Martin Schanz gCQM: Acoustics and Thermoelasticity 9 / 39

Problem setting Bounded Lipschitz domain Ω − ⊂ R 3 with boundary Γ := ∂ Ω Ω + := R 3 \ Ω − is its unbounded complement. Linear acoustics for the pressure p in Ω σ × R > 0 , ∂ tt p − c 2 ∆ p = 0 in Ω σ , p ( x , 0 ) = ∂ t p ( x , 0 ) = 0 1 ( p ) − σα γ σ c γ σ 0 ( ∂ t p )= f ( x , t ) on Γ × R > 0 with σ ∈ { + , −} , wave velocity c , and α absorption coefficient Martin Schanz gCQM: Acoustics and Thermoelasticity 10 / 39