INFINITE ELEMENT METHODS FOR HELMHOLTZ EQUATION PROBLEMS ON UNBOUNDED DOMAINS

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1 INFINITE ELEMENT METHODS FOR HELMHOLTZ EQUATION PROBLEMS ON UNBOUNDED DOMAINS Michael Newman Department of Aerospace Engineering Texas A&M University 34 TAMU 744D H.R. Bright Building College Station, TX Supervisor: Dr. Andrzej J. Safjan June 20, 2003 Support of this work by NSF grant # and by ONR grant #N is greatfully acknowledged. Typeset by AMS-TEX

2 Motivating Problem: Scattering of Sonar Waves from a Submarine Hull. TRUNCATION BOUNDARY ASPECT RATIO = 7: Fig. 3: Surrounding a Submarine with an Infinite Element Prolate Spheroid. 2

3 Objectives. To obtain highly-accurate computational solutions to Helmholtz equation problems posed on unbounded domains using infinite elements. 2. To formulate infinite elements which permit a large number of degrees of freedom to be utilized in the radial (infinite) direction. 3. To formulate infinite elements which have numerically well-conditioned stiffness matrices. 4. To formulate infinite elements with h-p-r adaptivity in the radial direction. 5. To formulate infinite elements with the ability to surround boundaries of various aspect ratios. TRUNCATION BOUNDARY 2b 2a Fig. 2: Elliptical Boundary with Aspect Ratio=b/a. 3

4 Three-Dimensional Helmholtz Equation We are interested in solving the following three-dimensional Helmholtz problem posed on the unbounded domain Ω with boundary Ω: u k 2 u =0, u n = g, lim r r Ω x Ω ) ( u r + iku =0 where r is the radial distance from the origin. TRUNCATION BOUNDARY Ω Ω x r ELASTIC SOLID j 0 i n ACOUSTIC FLUID Fig. : Unbounded Domain and Boundary. 4

5 3-D Spherical Multipole Expansion Recall that in spherical coordinates x = r sin θ(cos φi + sin φj)+r cos θk where r 0, 0 θ π, 0 φ<2π, the outward traveling wave solutions of the 3-D Helmholtz equation admit the following eigenfunction expansion u( r, θ, φ )= n n=0 m=0 c nm h (2) n (kr)pn m (cos θ)e imφ where h (2) n (x) are the spherical Hankel functions of the second kind and Pn m (x) are the associated Legendre functions of the first kind. The spherical Hankel functions of the second kind are given by h (2) n (x) = i ( ) x p n e ix, for n =0,, 2,... x where p n (x) are polynomials which satisfy the recurrence relation p n+ (x) =(2n +)xp n (x) p n (x), for n =, 2,... and p 0 (x) =, p (x) =x + i. 5

6 Thus follows the classical multipole expansion theroem by Atkinson and Wilcox [Atkinson, 23], [Wilcox, 24,25]. Theorem. Let u(x) be a radiation function for the region exterior to a sphere x x 0 = c, and let ( r, θ, φ ) be the spherical coordinates for x relative to an origin at x 0. Then u( r, θ, φ )=e ikr n= G n ( θ, φ ) r n where the series converges for r>cand converges absolutely and uniformly in r, θ and φ in any region r c + ɛ>c. The series may be differentiated term by term with respect to r, θ and φ any number of times and the resulting series all converge absolutely and uniformly. 3-D Spheroidal Multipole Expansions [Holford, ] In [] Holford extended the multipole expansion theorem u( ρ, ˆθ, ˆφ )=e ikρ n= G n ( ˆθ, ˆφ ) ρ n of Atkinson and Wilcox to Prolate Spheroidal coordinates x = ρ 2 f 2 sin ˆθ(cos ˆφi + sin ˆφj)+ρcos ˆθk and Oblate Spheroidal coordinates x = ρ sin ˆθ(cos ˆφi + sin ˆφj)+ ρ 2 f 2 cos ˆθk where f 0, ρ f, 0 ˆθ π, 0 ˆφ <2π. 6

7 Weak Formulation of the 3-D Helmholtz Problem Weighted and Conjugated Weak Formulation. where a: U V C π 2π a( u, v )= 0 0 ρ ρ 2 Find u U such that a( u, v )=l(v), v V is the sesquilinear form and l : V C l(v) = ( u v k 2 uv 2 ) u gρρ ρ ρ v J dρd ˆφdˆθ Ω ρ 2 gvds is the antilinear functional. Here U = V = H w(ω) where H w(ω) is the Sobolev space endowed with the norm u H w (Ω) = π 0 2π 0 ρ ρ 2 ( u 2 + u 2 ) J dρd ˆφdˆθ. On the boundary Ω weconsider the function space L 2 ( Ω) endowed with the norm u L 2 ( Ω) = Ω u 2 ds. 7

8 For Prolate Spheroidal coordinates J = g ρρ = (ρ 2 f 2 ρ 2 f ρ 2 f 2 ( + cos 2ˆθ), ) ( + cos 2ˆθ) sin ˆθ >0. Infinite Element Approximation. where Find u h U h such that a( u h,v h )=l(v h ), v h V h U h U, V h V dim U h <, dim V h < 8

9 3-D Multipole Based Infinite Elements [Burnett, 7 0] In [7 0] Burnett developed 3-D infinite elements based on the Prolate and Oblate Spheroidal multipole expansions. The ϕ j ( ρ, ξ, η ), infinite element basis functions are the tensor product of the multipole functions ψ n (ρ) = e ikρ ρ n, for n =, 2,..., ρ ( ρ, ) in the radial direction with χ i ( ξ,η ), Lagrange polynomial shape functions in the angular directions where the transformation from local coordinates to global angular coordinates is given by ˆθ( ξ,η )= i ˆθ i χ i ( ξ,η ), ˆφ( ξ,η )= The corresponding approximate solution is i ˆφ i χ i ( ξ,η ). u h ( ρ, ξ, η )= j u j ϕ j ( ρ, ξ, η ). 9

10 Model Problem. Using the 3-D Global Multipole infinite element we compute the solution to the 3-D Helmholtz equation whose analytic solution is u( r, θ, φ )=h (2) 0 (kr)p 0(cos θ). The domain is taken to be the region outside the sphere Ω={ ( r, θ, φ ) r> }, Ω={ ( r, θ, φ ) r =}. This model problem is solved for k =. INFINITE ELEMENT MESH Fig. 4: Spherical Multipole Infinite Element Mesh. 0

11 CONDITION NUMBER.E+2.E+9.E+7 MULTIPOLE INFINITE ELEMENT.E+5.E+3.E+.E+9.E+7 NE= Number of Infinite Elements in mesh=00 NORMALIZED ERROR e Ω = u u h H w (Ω) u H w (Ω).E+5.E-.E-2.E-3 MULTIPOLE INFINITE ELEMENT NUMBER OF RADIAL DEGREES OF FREEDOM.E-4.E-5 Fig. 5: ILL-Conditioning of the Spherical Multipole Infinite Element.

12 Remedies for the Numerical Ill-Conditioning. A preconditioning method based on a Gram-Schmidt-like process applied to the global multipole functions. 2. Replace the global multipole functions with piecewise multipole functions which have compact support. Preconditioning Method Given the multipole functions ψ n (ρ) = e ikρ ρ n, for n =, 2,..., ρ ( ρ, ) and the sesquilinear form a: V V C associated with the weak formulation of the Helmholtz problem, consider the Gram matrix Γ C n n defined by a( ψ,ψ ) a( ψ,ψ 2 ) a( ψ,ψ n ) a( ψ 2,ψ ) a( ψ 2,ψ 2 ) a( ψ 2,ψ n ) Γ = a( ψ n,ψ ) a( ψ n,ψ 2 ) a( ψ n,ψ n ) Let ψ C n be the vector ψ(ρ) = 2 ψ (ρ) ψ 2 (ρ). ψ n (ρ)

13 and consider a linear transformation of ψ induced by an n n invertible matrix X ˆψ(ρ) =Xψ(ρ). The Gram matrix Γ is transformed into the Gram matrix ˆΓ = XΓX H. Consider now the following decomposition of Γ Γ = LDU = LŨ where D C n n is a diagonal matrix, L C n n is a lower triangular matrix with unit diagonal elements, and U C n n is an upper triangular matrix with unit diagonal elements. Moreover, L = LD 2, Ũ = D 2 U, and D 2 is the positive square root of D computed using the following formula for the positive square root of a complex number: { 2 a + ib = 2 ( a2 + b 2 + a + i a2 + b 2 a), b ( a2 + b 2 + a i. a2 + b 2 a), b < 0 The Gram-Schmidt transformation consists of taking the transformation matrix X to be the lower triangular matrix X = L = D 2 L. Consequently, the transformed Gram matrix ˆΓ = XΓX H H = Ũ L is upper triangular with diagonal elements having unit complex modulus. In other words, the Gram-Schmidt transformation yields transformed fields ˆψ which satisfy the following conditions { a( ˆψ i, ˆψ 0 i<j j ) = i = j. 3

14 The preconditioned infinite element basis functions ˆϕ j ( ρ, ξ, η ) are the tensor product of the Gram-Schmidt transformed multipole functions ˆψ n (ρ) inthe radial direction with the Lagrange polynomial shape functions χ i ( ξ,η )inthe angular directions. The approximate solution is u h ( ρ, ξ, η )= j û j ˆϕ j ( ρ, ξ, η ). 4

15 infinity Ω Ω j 0 i e α3 e α2 e Ω e α Fig. 6: Infinite Element Mesh with Typical Element e. 5

16 CONDITION NUMBER NE=70.E+7 NE=60 NE=50 NE=40.E+6 NE=30 NE=20 NE=0 NORMALIZED ERROR.E+5 e Ω = u u h u.e NE= Number of Infinite Elements in mesh NUMBER OF MULTIPOLE TERMS NE=0.E-5 NE=20 NE=30 NE=40 NE=50 NE=60 NE=70 Fig. 7: Assembled Stiffness Matrix Condition Numbers and Convergence Curves of the Preconditioned Multipole Infinite Element. 6

17 3-D Piecewise Multipole Infinite Element In formulating the 3-D Piecewise Multipole infinite element we replace the global multipole functions ψ n (ρ) = e ikρ ρ n, for n =, 2,..., ρ ( ρ, ) with Lagrange interpolation functions σ p (ρ) inthe variable ρ. Again, the infinite element basis functions ϕ j ( ρ, ξ, η ) are the tensor product of the piecewise multipole functions σ p (ρ) in the radial direction with Lagrange polynomial shape functions χ i ( ξ,η )inthe angular directions. -D Meshing of the Unbounded Radial Domain. We consider a -D meshing of the unbounded radial domain ( ρ, )into element subdomains. Let n e =number of elements in the mesh n =number of nodes per element n n = total number of nodes in the mesh Each element for e =,...,n e isbounded and has n nodes while the element e = n e is unbounded and has n nodes. The total number of radial nodes for this mesh is n n =(n )n e. 7

18 ρ ρ ρ 2 ρ n infinity e e Τ ζ - Fig. 8: -D Meshing of the Unbounded Radial Domain. 8

19 Radial Shape Functions For e =,...,n e the radial shape functions are n ( ) σ p (ρ) = q= q p n ( ρ ρ q )e ikρ, for p =, 2,...,n q= q p ρ p ρ q where ρ,ρ 2,...,ρ n are the nodal coordinates. To perform numerical integrations we introduce the transformation e T :(, ) ( ρ,ρ n ) e T (ζ) = 2 ( ρ n ρ ) ζ + 2 ( ) ρ n + ρ with Jacobian e J(ζ) = ( 2 ( 2 ( ) ρ n ρ ρ n ρ ) ζ + 2 ( )) 2. ρ n + ρ 9

20 For e = n e we take the limit as ρ n. Therefore the shape functions become σ p (ρ) = ρ n q= q p n ( ( ρ ρ q ) ) e ikρ, for p =, 2,...,n. ρ p q= q p ρ p ρ q The transformation e T :(, ) ( ρ,ρ n )becomes and the Jacobian becomes e T (ζ) = 2ρ ζ e J(ζ) = 2ρ ( ζ) 2. 20

21 Distribution of the Radial Nodes in 3-D The distribution of the radial nodes for the 3-D Piecewise Multipole infinite element is based on the transformation T :(, 0) ( ρ, ) T (z) = ρ z.. Uniform Distribution of the Radial Nodes. A uniform partition of the interval (, 0)into n n subintervals yields the radial nodal coordinates ρ q = T ( + q ), for q =, 2,...,n n. n n 2. Non-Uniform Distribution of the Radial Nodes. Using a non-uniform arithmetic partition of the interval (, 0) into n n subintervals yields the radial nodal coordinates ρ q = T ( + ) q( q), for q =, 2,...,n n. n n (n n +) 2

22 TYPICAL INFINITE ELEMENT Fig. 29: Elliptical Mesh of Piecewise Multipole Infinite Elements. 22

23 Model Problem 3. 3-D Numerical Examples Using the Multipole and Piecewise Multipole infinite elements we compute the solution to the 3-D Helmholtz equation whose analytic solution is u( ρ, ˆθ, ˆφ )=h (2) (kr( ρ, ˆθ, ˆφ ))P (cos θ( ρ, ˆθ, ˆφ )) where the transformation from Prolate Spheroidal to Spherical coordinates is given by r = ρ 2 f 2 2 ( cos 2ˆθ), cos θ = ρ cos ˆθ. ρ 2 f 2 2 ( cos 2ˆθ) The domain is taken to be the region outside the spheroid defined by Ω={ ( ρ, ˆθ, ˆφ ) ρ> }, Ω={ ( ρ, ˆθ, ˆφ ) ρ =}. This model problem is solved for k =and aspect ratio of 7:. 23

24 ASPECT RATIO = 7: INFINITE ELEMENT MESH Fig. 2: Prolate Piecewise Multipole Infinite Element Mesh. 24

25 NORMALIZED BOUNDARY ERROR e Ω = u u h L2 ( Ω) u L2 ( Ω).E- GLOBAL MULTIPOLE INFINITE ELEMENT NUMBER OF RADIAL DEGREES OF FREEDOM.E-2 P=.E-3 P=2.E-4.E-5 P= Degree of Radial Shape Functions. NE= Number of Infinite Elements in mesh=00 P=3.E-6 ASPECT RATIO = 7: P=4 P=5 Fig. 5: Comparison of the Boundary Convergence Curves of the Global Multipole and Piecewise Multipole Infinite Elements. Weighted and Conjugated Formulation. Aspect Ratio=7:. Non-uniform Radial Mesh. 25

26 CONDITION NUMBER.E+2.E+9.E+7 GLOBAL MULTIPOLE INFINITE ELEMENT.E+5.E+3.E+ P= Degree of Radial Shape Functions. NE= Number of Infinite Elements in mesh=00 ASPECT RATIO = 7:.E+9.E+7 P= P=2 P=3 P=4 P=5 NORMALIZED ERROR e Ω = u u h H w (Ω) u H w (Ω).E+5.E- GLOBAL MULTIPOLE INFINITE ELEMENT NUMBER OF RADIAL DEGREES OF FREEDOM P=.E-2 P=2.E-3 P=3.E-4 P=4 P=5 Fig. 6: Comparison of the Assembled Stiffness Matrix Condition Numbers and Domain Convergence Curves of the Global Multipole and Piecewise Multipole Infinite Elements. Weighted and Conjugated Formulation. Aspect Ratio=7:. Non-uniform Radial Mesh. 26

27 Unweighted and Unconjugated Weak Formulation. An Unweighted and Unconjugated weak formulation of the 3-D Helmholtz problem is a( u, v )=l(v), v V, u U where a: U V C π 2π ρ ( a( u, v )= lim u v k 2 uv ) J dρd ˆφdˆθ ρ 0 0 ρ lim ρ (ρ2 f 2 ) π 2π is the bilinear form and l : V C l(v) = gv ds 0 Ω 0 u v sin ˆθdˆφdˆθ ρ is the linear functional. Here U = V = H w(ω) where H w(ω) is the Sobolev space endowed with the norm u H w (Ω) = π 0 2π 0 ρ ρ 2 ( u 2 + u 2 ) J dρd ˆφdˆθ. On the boundary Ω weconsider the function space L 2 ( Ω) endowed with the norm u L 2 ( Ω) = Ω u 2 ds. 27

28 NORMALIZED BOUNDARY ERROR e Ω = u u h L2 ( Ω) u L2 ( Ω).E-.E-2 GLOBAL MULTIPOLE INFINITE ELEMENT P=2.E-3 P=3.E-4 P=4.E-5 P= Degree of Radial Shape Functions. P=5.E-6.E-7 NE= Number of Infinite Elements in mesh=00 ASPECT RATIO = 7:.E NUMBER OF RADIAL DEGREES OF FREEDOM Fig. 23: Comparison of the Boundary Convergence Curves of the Global Multipole and Piecewise Multipole Infinite Elements. Unweighted and Unconjugated Formulation. Aspect Ratio=7:. Non-uniform Radial Mesh. 28

29 NORMALIZED ERROR e Ω = u u h H w (Ω) u H w (Ω) GLOBAL MULTIPOLE INFINITE ELEMENT.E+4.E+3.E+2 P= Degree of Radial Shape Functions. NE= Number of Infinite Elements in mesh=00 ASPECT RATIO = 7:.E+.E-.E P=4 P=3 P=2 P=5 NUMBER OF RADIAL DEGREES OF FREEDOM.E-3.E-4.E-5 Fig. 24: Comparison of the Domain Convergence Curves of the Global Multipole and Piecewise Multipole Infinite Elements. Unweighted and Unconjugated Formulation. Aspect Ratio=7:. Non-uniform Radial Mesh. 29

30 CONDITION NUMBER.E+2.E+9 GLOBAL MULTIPOLE INFINITE ELEMENT P= Degree of Radial Shape Functions..E+7 NE= Number of Infinite Elements in mesh=00.e+5 ASPECT RATIO = 7:.E+3 P=4.E+.E+9 P=3 P=2 P=5.E+7.E NUMBER OF RADIAL DEGREES OF FREEDOM Fig. 25: Comparison of the Assembled Stiffness Matrix Condition Numbers of the Global Multipole and Piecewise Multipole Infinite Elements. Unweighted and Unconjugated Formulation. Aspect Ratio=7:. Non-uniform Radial Mesh. 30

31 Effect of Frequency Variations Using the 2-D Piecewise Multipole infinite element we compute the solution to model problem 4 for k =,k =0and k = 00. CONDITION NUMBER.E+5 K= P= Degree of Infinite Element Shape Functions=2.E+4 NE= Number of Infinite Elements in mesh=0 K=0.E+3 K=00 NORMALIZED ERROR e Ω = u u h H w (Ω) u H w (Ω).E NUMBER OF RADIAL DEGREES OF FREEDOM K=.E-5 K=0 K=00 Fig. 32: Assembled Stiffness Matrix Condition Numbers and Convergence Curves for Various Wave Numbers. 3

32 Conclusions. A new piecewise multipole infinite element formulation has been developed. By replacing the global multipole functions with piecewise multipole functions the numerical conditioning of the 2-D and 3-D infinite element global stiffness matrices is greatly improved, enabling an effective use of more than 00 degrees of freedom in the radial direction. The 2-D and 3-D piecewise multipole infinite elements performed well in solving the model problems considered here. 2. The piecewise multipole formulation seems to be insensitive to the value of ka for the same mesh. For ka =, ka =0and ka = 00 the condition number and relative error essentially stay the same. 3. The piecewise multipole formulation gives convergence on the domain Ω and on the boundary Ω when implemented with the UWUC weak formulation of the 3-D Helmholtz equation problem. This contrasts with the global multipole formulation which diverges on Ω when implemented with the UWUC weak formulation. 32

33 Publications Complete details of this research can be found in: () A. Safjan and M. Newman, The Ill-Conditioning of Infinite Element Stiffness Matrices, Computers and Mathematics with Applications, Volume 4, Numbers 0-, May-June 200, pp (2) A. Safjan and M. Newman, On Two-Dimensional Infinite Elements Utilizing Basis Functions with Compact Support, Computers Methods in Applied Mechanics and Engineering, Volume 90, Number 48, September 200, pp (3) A. Safjan and M. Newman, On Three-Dimensional Infinite Elements Utilizing Basis Functions with Compact Support, Computers and Mathematics with Applications, Volume 43, Numbers 8-9, April-May 2002, pp Acknowledgements The support of the National Science Foundation (NSF) grant # and the Office of Naval Research (ONR) grant #N is greatly appreciated. 33