Solving Time-Harmonic Scattering Problems by the Ultra Weak Variational Formulation

Transcription

1 Introduction Solving Time-Harmonic Scattering Problems by the Ultra Weak Variational Formulation Plane waves as basis functions Peter Monk 1 Tomi Huttunen 2 1 Department of Mathematical Sciences University of Delaware 2 Department of Applied Physics University of Kuopio, Finland Research supported in part by a grant from AFOSR

2 Outline Introduction 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

3 Outline Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

4 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Computational Acoustics Examples [Huttunen] 20 θ = x (mm) y (mm)

5 Acoustic Scattering Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Given the shape and acoustic properties of an object, predict how it interacts with acoustic waves at a single frequency.

6 Acoustic Scattering Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Given the shape and acoustic properties of an object, predict how it interacts with acoustic waves at a single frequency. Incident

7 Acoustic Scattering Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Given the shape and acoustic properties of an object, predict how it interacts with acoustic waves at a single frequency. Incident Scattered

8 Acoustic Scattering Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Given the shape and acoustic properties of an object, predict how it interacts with acoustic waves at a single frequency. Incident Scattered Total

9 Outline Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

10 Helmholtz Equation Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Given a bounded doman Ω the pressure field is P(x, t) = p(x) exp(iωt) where p satisfies ρ 1 p + κ 2 ρ 1 p = 0 in Ω where ρ is the density and the wave number (complex!) is given by κ = ω/c + iα where c is the speed of sound and α is the absorption coefficient. Boundary condition ( ) 1 p ρ n iσp = Q ( ) 1 p ρ n + iσp + g on the boundary Ω where g is data, Q 1 and σ R

11 Introduction A Model Scattering Problem Acoustic Problems The Helmholtz Equation Decisions, decisions... Let Ω R 3 (or R 2 ) with disjoint boundaries Γ and Σ. Approximate u which satisfies u + κ 2 u = 0 in Ω u ν u = g on Γ (Q = 1) iku = 0 on Σ (Q = 0) ABC Γ Scatterer Ω Σ where g describes the incoming plane wave. The region Ω is meshed with tetrahedra and the UWVF applied there. ABC = Absorbing Boundary Condition

12 Outline Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

13 Possible methods Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Integral equations. Handle unbounded media, complex shapes. There are fast solvers but they are difficult to program and complex for penetrable media, coatings, narrow objects... Finite elements. Higher order needed to handle dispersion and becomes expensive at short wavelength. Geometry and complex materials handled. Difficult to solve the linear system and handle unbounded domains.

14 Possible methods Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Integral equations. Handle unbounded media, complex shapes. There are fast solvers but they are difficult to program and complex for penetrable media, coatings, narrow objects... Finite elements. Higher order needed to handle dispersion and becomes expensive at short wavelength. Geometry and complex materials handled. Difficult to solve the linear system and handle unbounded domains.

15 The big decision Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Motivated by medical ultrasound applications (complex structure, short wavelength) we decided on the following: A volume based method (finite element grid) Special shape functions ( basis functions ) that are solutions of the Helmholtz equation on each element.

16 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Methods using special basis functions Partition of unity finite element method = PUFEM (Babuška and Melenk 1997, Keller and Giladi 2001, Huttenen, Gamallo and Astley 2005, Kim et al 2005) Least squares method (Trefftz, Monk and Wang 1999, Desmet 2002) Discontinuous enrichment method (Farhat et al. 2001, 2003, 2005) Plane wave augmented basis in integral equations ( Darrigrand 2001, Perrey-Debain et al. 2002, Chandler-Wilde and Langdon 2004/5,...) Ultra weak variational formulation (Després 1994, Cessenat and Després 1998)

17 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Methods using special basis functions Partition of unity finite element method = PUFEM (Babuška and Melenk 1997, Keller and Giladi 2001, Huttenen, Gamallo and Astley 2005, Kim et al 2005) Least squares method (Trefftz, Monk and Wang 1999, Desmet 2002) Discontinuous enrichment method (Farhat et al. 2001, 2003, 2005) Plane wave augmented basis in integral equations ( Darrigrand 2001, Perrey-Debain et al. 2002, Chandler-Wilde and Langdon 2004/5,...) Ultra weak variational formulation (Després 1994, Cessenat and Després 1998)

18 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Methods using special basis functions Partition of unity finite element method = PUFEM (Babuška and Melenk 1997, Keller and Giladi 2001, Huttenen, Gamallo and Astley 2005, Kim et al 2005) Least squares method (Trefftz, Monk and Wang 1999, Desmet 2002) Discontinuous enrichment method (Farhat et al. 2001, 2003, 2005) Plane wave augmented basis in integral equations ( Darrigrand 2001, Perrey-Debain et al. 2002, Chandler-Wilde and Langdon 2004/5,...) Ultra weak variational formulation (Després 1994, Cessenat and Després 1998)

19 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Methods using special basis functions Partition of unity finite element method = PUFEM (Babuška and Melenk 1997, Keller and Giladi 2001, Huttenen, Gamallo and Astley 2005, Kim et al 2005) Least squares method (Trefftz, Monk and Wang 1999, Desmet 2002) Discontinuous enrichment method (Farhat et al. 2001, 2003, 2005) Plane wave augmented basis in integral equations ( Darrigrand 2001, Perrey-Debain et al. 2002, Chandler-Wilde and Langdon 2004/5,...) Ultra weak variational formulation (Després 1994, Cessenat and Després 1998)

20 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... Methods using special basis functions Partition of unity finite element method = PUFEM (Babuška and Melenk 1997, Keller and Giladi 2001, Huttenen, Gamallo and Astley 2005, Kim et al 2005) Least squares method (Trefftz, Monk and Wang 1999, Desmet 2002) Discontinuous enrichment method (Farhat et al. 2001, 2003, 2005) Plane wave augmented basis in integral equations ( Darrigrand 2001, Perrey-Debain et al. 2002, Chandler-Wilde and Langdon 2004/5,...) Ultra weak variational formulation (Després 1994, Cessenat and Després 1998)

21 Outline Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

22 The Mesh Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Approximate the domain Ω by a tetrahedral finite element mesh consisting of N h tetrahedra Ω k, k = 1,, N h of maximum diameter h.

23 The Mesh Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Approximate the domain Ω by a tetrahedral finite element mesh consisting of N h tetrahedra Ω k, k = 1,, N h of maximum diameter h.

24 The Mesh Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Approximate the domain Ω by a tetrahedral finite element mesh consisting of N h tetrahedra Ω k, k = 1,, N h of maximum diameter h.

25 The Mesh Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Approximate the domain Ω by a tetrahedral finite element mesh consisting of N h tetrahedra Ω k, k = 1,, N h of maximum diameter h. Σ j,k Ω j n j Ω k n k Σ k,j

26 The Mesh Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Approximate the domain Ω by a tetrahedral finite element mesh consisting of N h tetrahedra Ω k, k = 1,, N h of maximum diameter h. Σ j,k Ω j n j Ω k n k Σ k,j Major restriction: ρ and κ must be piecewise constant and constant on each element.

27 Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Required continuity between elements Let p k = p Ωk and p j = p Ωj then since p is a solution of the Helmholtz equation p k = p j and 1 ρ k p k n k = 1 ρ j p j n j on Σ j,k In the UWVF this is achieved by demanding that Robin (one way wave equation) data agree on the interfaces, so on Σ j,k 1 p k + iσp k = 1 p j + iσp j ρ k n k ρ j n j 1 p k iσp k = 1 p j iσp j ρ k n k ρ j n j where σ > 0 is a parameter (function) on Σ j,k (e.g. σ = R(κ)).

28 Outline Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

29 Introduction Variational equations The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF [Cessent and Després] Let ξ k satisfy the adjoint equation ρ 1 ξ k + κ 2 ρ 1 ξ k = 0 in Ω k

30 Introduction Variational equations The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF [Cessent and Després] Let ξ k satisfy the adjoint equation then for σ > 0 (e.g. σ = R(κ)) ρ 1 ξ k + κ 2 ρ 1 ξ k = 0 in Ω k Z Ω k 1 σ! 1 p + iσp ρ n k! 1 ξ k + iσξ k ds = ρ n k Z! 1 1 p iσp Ω k σ ρ n k Z 1 p 2i ξ k 1 Ω k ρ n k ρ! 1 ξ k iσξ k ds ρ n k! ξ k p ds n k

31 Introduction Variational equations The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF [Cessent and Després] Let ξ k satisfy the adjoint equation by Green s Theorem ρ 1 ξ k + κ 2 ρ 1 ξ k = 0 in Ω k Z Ω k 1 σ! 1 p + iσp ρ n k! 1 ξ k + iσξ k ds = ρ n k Z!! 1 1 p 1 ξ k iσp iσξ k Ω k σ ρ n k ρ n k Z 2i 1 Ω k ρ p ξ k 1! ρ ξ k p dv ds

32 Introduction Variational equations The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF [Cessent and Després] Let ξ k satisfy the adjoint equation ρ 1 ξ k + κ 2 ρ 1 ξ k = 0 in Ω k by the Helmholtz and adjoint Helmholtz equations ( ) ( ) 1 1 p 1 ξ k + iσp + iσξ k ds Ω k σ ρ n k ρ n k ( ) ( ) 1 1 p 1 ξ k = iσp iσξ k ds Ω k σ ρ n k ρ n k

33 Introduction Variational Problem Continued The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF = j 1 Ω k σ ( 1 ρ k p k Σ k,j 1 σ + iσp k n k ( 1 p j iσp j ρ j n j ) ( 1 ρ k ξ k ) + iσξ k n k ) ( ξk iσξ k n k ds ) ds Let X k = and let ( 1 ρ k p k n k + iσp k) F k (Y k ) = Ω k and Y k = ( 1 ρ k ξ k n k iσξ k) ( 1 ρ k ξ k n k + iσξ k) Ω k Ω k

34 Introduction Variational Problem Continued The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF Let X k = ( 1 ρ k p k n k + iσp k) Ω k and Y k = ( 1 ρ k ξ k n k + iσξ k) Ω k and let F k (Y k ) = ( 1 ρ k ξ k n k iσξ k) then, for a tetrahedron surrounded by four other tetrahedra 1 Ω k σ X ky k ds = 1 j Σ k,j σ X jf k (Y k ) ds boundary faces are handled using the boundary condition. Ω k

35 Outline Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

36 Introduction The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF The Discrete UWVF For each element Ω k we choose p k directions d j on the unit sphere [Sloan et al.] and define the solution on that element to be a sum of traces of plane waves p k Xk h = j=1 x k j ( 1 ρ k exp(iκd j x) n k + iσ exp(iκd j x)) The test function is, for 1 r p k, Y h k = ( 1 ρ k exp(iκd r x) n k + iσ exp(iκd r x)) In this case F k (Yk h ) is easy to compute: F k (Y h k ) = ( 1 ρ k exp(iκd r x) n k iσ exp(iκd r x)) Ω k Ω k Ω k

37 Introduction Properties of the acoustic UWVF The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF [Huttunen, Monk] The UWVF is a special implementation of the upwind Discontinuous Galerkin method using plane wave basis functions. [Cessenat/Després, 2D] Assume I(κ) = 0. If Q < 1, M = 2µ + 1 X X M L 2 (Γ) Chµ 1/2 u C µ+1 (Ω) The discrete problem has the form (B C)x = b where B is Hermitian positive definite and the eigenvalues of B 1 C lie in the closure of the unit disk excluding 1

38 Outline Introduction 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

39 UWVF results in 2D Introduction Domain Ω is annulus 0.4 r Remesh at each κ to keep λ/h 8 (FEM) relative error relative error wave number κ wave number κ Remesh at each κ to keep 2pλ/h 4.5 relative error relative error (UWVF) wave number κ Dirichlet wave number κ Neumann

40 Conditioning Introduction Basic UWVF uses p directions/element. This can cause bad conditioning for B (e.g. on small elements, if κ changes,...) We use different p k for element Ω k. One possibility: chose p k so that the condition number of the submatrix corresponding to K X Y ds is a desired maximum value. Uniform Basis, Cond( D k ) Non uniform Basis, p k Point Source Ω 2 Ω 1 Γ Domain Cond. No. Non uniform p k Uniform p k

41 Outline Introduction 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

42 Introduction Efficiency of the ABC Logarithm of the modulus of the far field pattern for scattering of a plane wave by a sphere κ = 4, a = 1, wavelength λ 1.6 for different ABC boundary diameter 12 RCS Sphere Classic ; b=2, g=1.75, r=1.5, c= RCS Sphere R=(1;1.25) ; green:ie+fmm0 ; red:ie+fmm Normalized Echo area 4 2 Normalized Echo area Theta ABC at r = 1.25(c), 1.5(r), and r = 1.75(g), 2(b) Note: these are electromagnetic results Theta Exact

43 The PML Introduction The Sommerfeld absorbing boundary condition is not efficient. We want to use the Perfectly Matched Layer (PML) of Bérenger. ABC on outer boundary Modified PML equations Unmodified equations Scatterer G The PML layer absorbs incident waves exponentially rapidly. The only reflection is from the outer boundary (for the continuous problem). PML absorbing layer

44 Introduction UWVF with PML in 3D Let us use the complex stretching of spatial variables x x + i x σ 0 ( x x 0 ) n dx, x x 0, = κ x 0 and define x x = d x. x, x < x 0 By using similar expression for y and z, and requiring p satisfy the Helmholtz equation in primed variables, we obtain a modified Helmholtz equation: ( ) 1 ρ A p + κ2 η 2 ( dy d z p = 0 where A = diag ρ d x, d xd z, d ) xd y. d y d z

45 PML continued Introduction For the PML elements, the boundary function χ k and plane wave basis function are ( ( 1 χ k = n k (A k ) iσ ) ) p k and ϕ k,l = e iκ k d k,l r, ρ k where r = (x, y, z ). Surprisingly, n = 0 works quite well with the UWVF.

46 Introduction Improvement due to the PML 1 Normed amplitude UWVF, σ 0 = 0 UWVF, σ 0 = 500 Rayleigh integral y (mm)

47 Introduction Another Test Example (point source)

48 Introduction Point Source Results Exact σ 0 = 0 σ 0 = 100 z (mm) y (mm) z (mm) y (mm) z (mm) y (mm) Exact σ 0 = 0 σ 0 = 100 σ 0 = 500 σ 0 = z (mm) y (mm) z (mm) y (mm) σ 0 = 500 σ 0 =

49 Outline Introduction 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

50 Introduction The UWVF has been parallelized using domain decomposition (METIS) and MPI. The basic tasks are assembly and the iterative solver (BiCGStab, easily parallelized). Coupling is via faces. Left: METIS decomposition of a mesh around a sphere (!) into 8 parts. Current problem: how to predict the number of directions per element to guarantee good conditioning and accuracy.

51 Introduction Choice of p k [Caryol and Collino] How can we choose p k to ensure accuracy? Idea: Good approximation of a general plane wave is necessary for convergence. In 2D, using p k = 2µ + 1 directions, if h is radius of the element ( ) ( ) 1 µ + 3 kh µ+1 E 1 + (µ + 1)! µ In 2D, to obtain an interpolant with pointwise error ɛ µ κh where W (x) exp(w (x)) = x. ( ( )) 3 1 2/3 2 W 3πɛ 2 (κh) 1/3

52 x Introduction Sphere with radius a = 1. UWVF p z Left: The mesh. Right: The UWVF approximation for a plane wave at κa 63.

53 Introduction Error as a function of the wave number Relative error ( % ) κ a The results are computed in the same mesh using the condition number limit Max(Cond(D k ))< 1e6. Note, the total error includes errors due to UWVF approximation, PML and triangulated surface of the sphere.

54 Introduction Scalability and load distribution CPU Time (s) Normalized storage Number of processors Processor number Left: CPU time as a function of number of processors. Right: Storage on different processors when 12 processors are used.

55 Introduction Iterative solution of the linear system The UWVF linear system can be solved by simple iterative scheme. We use BiCGStab Bi CGStab Relative residual Number of DoF: 3,474, 770 Number of CPUs: 24 (2.8GHz P4) Available Memory: 48GB Switch: 1000BaseT Iteration Solution time is 451s using 25.3 GB memory (109 iterations).

56 Outline Introduction 1 Introduction Acoustic Problems The Helmholtz Equation Decisions, decisions... 2 The Mesh and Continuity Variational Formulation (UWVF) The discrete UWVF 3

57 Introduction Comparison to FEMLAB [Huttunen] FEMLAB P 2 FEM with low order ABC.

58 Introduction Comparison continued FEMLAB (two meshes): f (khz) h (mm) Elem. CPU (s) Error (%) Mem (GB) UWVF (one mesh, variable # directions): f (khz) h (mm) Elem. CPU (s) Error (%) Mem (GB)

59 Introduction FEMLAB implementation The acoustic UWVF code will appear as part of the acoustics module in FEMLAB. The Maxwell and fluid-structure UWVF (in that order!) will also be added later. Please see

60 Submarine: meshes Introduction FEM: UWVF:

61 Introduction Submarine: UWVF calculation in FEMLAB

62 Introduction Submarine: UWVF calculation

63 Introduction Extensions/current work The UWVF can be extended to certain symmetric hyperbolic systems and in particular to Maxwell s equations PML Coupled FMM and UWVF [with Eric Darrigrand, Rennes] Linear elasticity Coupled fluid-solid problem (2D only so far) Comparison with PUFEM [Huttunen, Gamallo and Astley]

64 Summary Introduction The FEM is the best developed volume method for practical computations. High order works best for wave problems with smooth solutions. The UWVF offers an alternative to PUFEM for plane wave bases. We find it competetive to FEM. The UWVF performs well provided the number of directions is chosen carefully and the scatterer is smooth (questions remain about performance near singularities).