Are Absorbing Boundary Conditions History Example and Perfectly - PowerPoint PPT Presentation

ABS and PML Martin J. Gander Domain Truncation Are Absorbing Boundary Conditions History Example and Perfectly Matched Layers TBS and ABS Construction Really so Different ? Application: ABCs PML Historical Stretched-Coordinate PML Martin J. Gander Pole Condition Meaning martin.gander@unige.ch Approximations Mathematical Equivalences University of Geneva With PML With ABC Numerical RICAM, November 2011 Experiments Optimization of s 0 Conclusions Joint work with Achim Sch¨ adle

ABS and PML History of Absorbing Boundary Conditions Martin J. Gander Engquist and Majda (1977): Wave Propagation: “In Domain practical calculations, it is often essential to introduce artificial Truncation History boundaries to limit the area of computation. Example Unfortunately [transparent boundary conditions] necessarily have TBS and ABS Construction to be non-local in both space and time and thus are not useful for Application: ABCs PML practical calculations” Historical Stretched-Coordinate PML Bayliss Turkel (1980): Wave-Like Equations: “In the Pole Condition Meaning numerical computation of hyperbolic equations it is not practical Approximations to use infinite domains. Instead, one truncates the domain with an Mathematical Equivalences artificial boundary“ With PML With ABC Numerical Experiments Halpern (Wave Propagation 1982, Diffusion 1987): Optimization of s 0 “. . . one often introduces artificial boundaries with boundary Conclusions conditions chosen so that the problem one gets is well-posed and the solution is ’as close as possible’ to that of the original problem”

ABS and PML Truncation of an Infinite Computational Domain Martin J. Gander Airbus A340 in approach over a city Domain Truncation History Example TBS and ABS Construction Application: ABCs PML Historical Stretched-Coordinate PML Pole Condition Meaning Approximations Mathematical Equivalences With PML With ABC Numerical Experiments Optimization of s 0 Conclusions Truncation of the unbounded domain using ◮ A transparent (or exact) boundary condition (TBC) or an absorbing (or inexact) boundary condition (ABC) ◮ A perfectly matched layer (PML)

ABS and PML Interest for Domain Decomposition Methods Martin J. Gander Computation performed with an optimized Schwarz method Domain on 16 subdomains Truncation History Example TBS and ABS Construction Application: ABCs PML Historical Stretched-Coordinate PML Pole Condition Meaning Approximations Mathematical Equivalences With PML With ABC Numerical Experiments Optimization of s 0 Conclusions An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation (G, Halpern and Magoules, 2006)

ABS and PML An Advice from the Past Martin J. Gander ´ Emile Picard (1893): Sur l’application des m´ ethodes Domain Truncation d’approximations successives ` a l’´ etude de certaines ´ equations History Example diff´ erentielles ordinaires TBS and ABS Construction Les m´ ethodes d’approximation Application: ABCs PML dont nous faisons usage sont Historical Stretched-Coordinate th´ eoriquement susceptibles de PML s’appliquer ` a toute ´ equation, Pole Condition Meaning mais elles ne deviennent vrai- Approximations ment int´ eressantes pour l’´ etude Mathematical Equivalences des propri´ et´ es des fonctions With PML With ABC d´ efinies par les ´ equations Numerical Experiments diff´ erentielles que si l’on ne Optimization of s 0 Conclusions reste pas dans les g´ en´ eralit´ es et si l’on envisage certaines classes d’´ equations.

ABS and PML Construction of Transparent Boundary Conditions Martin J. Gander We consider the model problem Domain ( η − ∆) u = f in Ω = R × (0 , π ) Truncation History u ( x , 0) = 0 Example u ( x , π ) = 0 TBS and ABS Construction Application: ABCs with f compactly supported in Ω int = (0 , 1) × (0 , π ), and u PML bounded at infinity. Historical Stretched-Coordinate y PML Pole Condition π Meaning Approximations Mathematical support Equivalences Γ 0 Γ 1 With PML of f With ABC Numerical Experiments x Optimization of s 0 0 1 Conclusions In order to solve this problem on a computer, the computational domain needs to be truncated in x , and an artificial boundary condition needs to be imposed at x = 0 and x = 1.

ABS and PML Construction of Transparent Boundary Conditions − ∪ Ω int ∪ Ω + Martin J. Gander Based on the decomposition of Ω = Ω y Domain Truncation History π Example TBS and ABS Construction Application: ABCs Ω − Ω + Γ 0 Ω int Γ 1 PML Historical Stretched-Coordinate x PML 0 1 Pole Condition Meaning and the equivalent coupled problems Approximations Mathematical ( η − ∆) v − in Ω − = 0 Equivalences With PML v − = v on Γ 0 With ABC Numerical ∂ n v − ∂ n v = on Γ 0 Experiments Optimization of s 0 ( η − ∆) v = f in Ω int Conclusions ∂ n v + ∂ n v = on Γ 1 v + = v on Γ 1 ( η − ∆) v + in Ω + = 0 with homogeneous conditions at y = 0 and y = π .

ABS and PML Solution of Exterior Problems Martin J. Gander The exterior problems are independent of the data f and can be readily solved using Fourier series in y : for example on Domain Truncation Ω + we have History ( η + k 2 − ∂ xx )ˆ v + = 0 Example TBS and ABS Construction an ordinary differential equation, whose solution is Application: ABCs √ η + k 2 x + B ( k ) e − √ PML η + k 2 x v + ( x , k ) = A ( k ) e ˆ Historical Stretched-Coordinate PML Since v + needs to stay bounded for x → + ∞ , we have Pole Condition A ( k ) = 0 and using the Dirichlet data v + = v at x = 1 we Meaning Approximations obtain Mathematical v (1 , k ) e − √ Equivalences η + k 2 ( x − 1) v + ( x , k ) = ˆ ˆ With PML With ABC Numerical This implies Experiments Optimization of s 0 � v + ( x , k ) | x =1 = − η + k 2 ˆ Conclusions ∂ x ˆ v ( x , k ) | x =1 and similarly on the left boundary Γ 0 , we obtain � v − ( x , k ) | x =0 = − η + k 2 ˆ − ∂ x ˆ v ( x , k ) | x =0

ABS and PML Closing the Interior Problem Martin J. Gander Using these solutions for the interior problem, we get Domain Truncation History ( η − ∆) v = f in Ω int Example F − 1 ( − � η + k 2 ˆ ∂ n v = v ) on Γ 0 TBS and ABS Construction � F − 1 ( − η + k 2 ˆ ∂ n v = v ) on Γ 1 Application: ABCs PML Historical where the inverse tranform in general is a convolution: Stretched-Coordinate PML � ∞ Pole Condition F − 1 ( − � η + k 2 ˆ Meaning v ) = f ( y − ξ ) v ( · , ξ ) d ξ Approximations −∞ Mathematical Equivalences with With PML With ABC � η + k 2 . F ( f ( y )) = − Numerical Experiments Optimization of s 0 By construction, v coincides with u in Ω int , and the Conclusions conditions obtained are called transparent boundary conditions (TBCs). They require a non-local convolution boundary condition.

ABS and PML Using Transparent Boundary Conditions Martin J. Gander Because of their non-local nature, one often approximates Domain TBCs and obtains Absorbing Boundary Conditions (ABCs): Truncation History ◮ by polynomial or rational approximations of the symbol Example TBS and ABS (Enquist, Majda, Halpern, Bruneau-Di Menza, Nataf, Construction Japhet, Szeftel, Shibata . . . ) Application: ABCs PML ◮ by approximation of the convolution kernel (Hairer, Historical Stretched-Coordinate Lubich, Schlichte, Greengard, Strain, Sch¨ adle . . . ) PML Pole Condition ◮ using quadrature rules (Mayfield, Baskakov and Meaning Approximations Popov,. . . ) Mathematical Remark: To couple an incoming field u in from Ω + : Equivalences With PML With ABC Numerical ∂ x ( v − u in ) = F − 1 ( − � η + k 2 (ˆ v − ˆ u in )) on Γ 1 Experiments Optimization of s 0 Conclusions which by linearity is equivalent to ∂ x v −F − 1 ( − � v ) = ∂ x u in −F − 1 ( − � η + k 2 ˆ η + k 2 ˆ u in ) on Γ 1

ABS and PML Perfectly Matched Layers Martin J. Gander J. Berenger (1994): A perfectly matched layer for the absorption of electromagnetic waves Domain Truncation History Split-Field PML: Original idea, split the electromagnetic Example fields into two unphysical fields in the PML region TBS and ABS Construction Application: ABCs y PML Historical π Stretched-Coordinate PML Pole Condition Meaning support Approximations Γ 0 Γ 1 of f Mathematical Equivalences With PML With ABC x Numerical 0 1 Experiments Optimization of s 0 Conclusions S.D. Gedney (1996): An anisotropic perfectly matched layer absorbing media for the truncation of FDTD latices Uniaxial PML (UPML): The PML is described as an artificial anisotropic absorbing material