Isogeometric analysis is the concept of using the same functions that describe a geometry in computer-aided design to approximate unknown elds in numerical simulations. This has become a topic of considerable interest to the boundary integral methods community. This paper introduces an eXtended Isogeometric Boundary Element Method (XIBEM), in which isogeometric functions approximating wave potential are enriched using the partition-of-unity method. In this new method, the isogeometric basis is formed from a space of non-uniform rational B-spline (NURBS) functions multiplied by families of plane waves. Using numerical examples, it is shown that this reduces the total number of equations that need to be solved for a given frequency and geometry of problem; this improves the accuracy of and extends the supported frequency range of the boundary element method to include short wave diraction problems

Coates, G.

Peake, M.J.

Trevelyan, J.

English

Durham Research Online

Proceedings of the 9th UK Conference on Boundary Integral Methods, University of Aberdeen,UK, 8-9th July 2013EXTENDED ISOGEOMETRIC BOUNDARY ELEMENT METHOD (XIBEM)FOR ACOUSTIC WAVE SCATTERING PROBLEMSM.J. PEAKE, J. TREVELYAN and G. COATESDurham University, School of Engineering and Computing Sciences, Durham, DH1 3LE, UKe-mail: m.j.peake@durham.ac.ukAbstract. Isogeometric analysis is the concept of using the same functions that describe ageometry in computer-aided design to approximate unknown fields in numerical simulations.This has become a topic of considerable interest to the boundary integral methods community.This paper introduces an eXtended Isogeometric Boundary Element Method (XIBEM), in whichisogeometric functions approximating wave potential are enriched using the partition-of-unitymethod. In this new method, the isogeometric basis is formed from a space of non-uniformrational B-spline (NURBS) functions multiplied by families of plane waves. Using numericalexamples, it is shown that this reduces the total number of equations that need to be solvedfor a given frequency and geometry of problem; this improves the accuracy of and extends thesupported frequency range of the boundary element method to include short wave diffractionproblems.1. INTRODUCTIONMeshing is an important aspect of numerical analysis and procedures that reduce the time re-quired, or improve mesh quality, are of interest to both the academic and industrial communities.One such procedure is isogeometric analysis (IGA), introduced by Hughes et al. [1]. This isthe concept of using the basis functions used to describe a geometry in computer-aided design(CAD) to construct exact geometries for numerical analysis.Early IGA work concentrated on using the finite element method (FEM) for analysis; thispaper focuses on the use of the boundary element method (BEM). The BEM only requiresthe boundaries of scattering surfaces to be meshed; non-uniform rational B-splines (NURBS),functions commonly used in CAD software, also only describe the boundaries of geometries.This makes IGA and BEM a natural combination. Simpson et al. [3] applied this concept toelastostatics, coining the pairing IGABEM; Politis et al. [4] applied it to potential flow problems;Takahashi and Matsumoto [5] combined this with the fast multipole method; Scott et al. [6]presented some three-dimensional problems using T-splines, an alternative to NURBS.Like the FEM, conventional BEM schemes require a mesh to be refined as the wavenumber,k, of a problem increases. For a fixed computational resource, this puts a practical limit on thewavelengths that can be considered for a specified geometry. A number of approaches have beendeveloped to increase this limit [7–10]; in this paper, the partition of unity method (PUM) [11] isused. The PUM, in which the character of the wave propagation is included in the approximatingfunction, was first used for wave scattering with the BEM by de la Bourdonnaye [12] under thename ‘microlocal discretization’. Perrey-Debain et al. [13] showed that the number of degreesof freedom for a given problem could be dramatically reduced using this method.This paper develops an isogeometric, collocation BEM employing PUM for acoustic wavescattering problems; this is named the eXtended Isogeometric BEM (XIBEM).2. FORMULATION OF XIBEM FOR THE HELMHOLTZ EQUATIONLet Ω ⊂ R2 be an unbounded domain containing a smooth scatterer of boundary Γ := ∂Ω.Assuming e−iωt time dependence, the wave equation can be reduced to the well-known Helmholtzequation:∆φ(q) + k2φ(q) = 0, φ ∈ C,q ∈ Ω, (1)where ∆(·) is the Laplacian operator, φ(q) is the unknown wave potential at q, and k is wavenum-ber – related to the wavelength, λ, by k = 2π/λ. The scatterer is impinged by an incident, planewave,φI(q) = AI exp(ikdI · q),∣∣dI∣∣ = 1, (2)where AI is the incident wave amplitude and dI is the direction of propagation.The conventional boundary integral equation (BIE) for the Helmholtz equation [14] is12φ(p) =∫Γ[∂φ(q)∂nG(p,q)− φ(q)∂G(p,q)∂n]dΓ(q) + φI(p), p,q ∈ Γ, (3)where p is an evaluation point and n is the outward-pointing, unit normal at integration pointq. Further, G(p,q) is the fundamental solution (Green’s function), representing the field expe-rienced at q due to a unit source radiating at p (or vice versa).For compact presentation, only one boundary condition is considered: the case of a perfectlyreflecting (“sound-hard”) cylinder, expressed as ∂φ(q)/∂n = 0. A solution to (1) is sought,subject to this boundary condition. The BIE (3) may be then be reformulated as12φ(p) +∫Γ∂G(p,q)∂nφ(q) dΓ(q) = φI(p). (4)2.1 NURBSNURBS are used to discretise (4). An important property of B-splines and NURBS is the knotvector, Ξ; this is a sequence of nondecreasing, real numbers (knots). This paper assume theknot vector has the formΞ = {0, . . . , 0︸ ︷︷ ︸p+1, ξp+1, . . . , ξs−p+1, 1, . . . , 1︸ ︷︷ ︸p+1},where p is the degree of the curve, there are s+ 1 knots, and ξj ≤ ξj+1 for j = 0, . . . , s− 1.The jth B-spline basis function of pth degree, Nj,p, is defined for p = 0 asNj,0(ξ) ={1 if ξj ≤ ξ ≤ ξj+10 otherwise.(5)For p = 1, 2, 3, . . ., it isNj,p(ξ) =ξ − ξjξj+p − ξjNj,p−1(ξ) +ξj+p+1 − ξξj+p+1 − ξj+1Nj+1,p−1(ξ). (6)The number of basis functions, J + 1, is related to the degree of the basis and length ofthe knot vector through J = s− p− 1. For NURBS, each basis function is given an individualweighting, wj ; the jth NURBS basis function of pth degree, Rj,p, is defined asRj,p(ξ) =Nj,p(ξ)wjJ∑i=0Ni,p(ξ)wi. (7)A NURBS curve is defined using NURBS functions and a set of control points Pj :C(ξ) =J∑j=0Rj,p(ξ)Pj . (8)This relationship can provide an analytical geometry given byΓ = {C(ξ) : ξ ∈ [0, 1)} , (9)where C : R→ R2. The mapping between q ∈ Γ and ξ is unique, hence it is assumed that anyfunction f(q) is equivalent to f(ξ).2.2 XIBEMFor IGABEM, NURBS are used to represent the boundary, Γ, but also the variation of velocitypotential along Γ:φ(ξ) =J∑j=0Rj,p(ξ)φj , (10)where φj is the potential associated with each NURBS basis function. XIBEM introduces alinear expansion of plane waves on each NURBS function; (10) is rewritten,φ(ξ) =J∑j=0Rj,pM∑m=0Ajm exp (ikdjm · q) , |djm| = 1, (11)where there are M + 1 plane waves related to each NURBS function with prescribed directionof propagation, djm ∈ R2, and unknown amplitudes, Ajm ∈ C.Substituting (11) into (4) gives12φ(p) +J∑j=0M∑m=0∫ 10∂G(p,q)∂nRj(ξ) exp(ik djm · q)|Jξ|dξ Ajm = φI(p) (12)where |Jξ| is the Jacobian of the mapping in (9). φ(p) is also expanded as in (11).To find the potential on Γ, (12) is collocated at a series of collocation points, p0,p1, . . . ,pZ−1,uniformly-spaced on ξ ∈ [0, 1); Z = (J + 1)(M + 1) is the total number of unknown amplitudes,Ajm, that are sought. This yields a square system of linear equations,[(1/2)P + H]{x} = {b}, (13)where the (usually sparse) square matrix P results from interpolations of the plane waves, φ(p);the right-hand side vector {b} contains the incident wave potentials at the collocation points;and the unknown vector {x} contains the amplitudes, Ajm. The square matrix H is fullypopulated with integrals,hzjm =∫ 10∂G(pz,q)∂nRj(ξ) exp(ik djm · q)|Jξ| dξ. (14)3. NUMERICAL RESULTSFor the numerical examples here, the sound-hard boundary condition is applied; there are nosingular integrals in this case so no regularisation scheme is required. The integrals of (14)are evaluated using Gauss-Legendre quadrature, subdividing the boundary into cells of approx-imately λ/4 in length. To overcome the well-known nonuniqueness problem associated withthe BEM, the authors use the CHIEF method [15]; the system is solved using singular valuedecomposition (SVD). Errors, E , are calculated in a relative L2 norm sense:E =∥∥Φ− Φexact∥∥L2(Γ)‖Φexact‖L2(Γ), (15)where Φ is a vector of potentials along the boundary of the scatterer calculated from the numer-ical simulation; and Φexact is a vector of potentials calculated using either an analytical solutionor an appropriate converged solution.3.1 Unit cylinderConsider a cylinder of radius a = 1, centred at (0, 0), being impinged by a unit-amplitude,incident plane wave propagating in the direction dI = (1, 0). There is an analytical solution [16].The quality of solution of conventional, polynomial BEM and IGABEM simulations is com-pared. Figure 1 shows the errors of solutions by these two methods over a range of ka. Thevariable τ is defined as the number of degrees of freedom per wavelength of the problem. τ ≈ 10for all simulations; as ka increases, the mesh is refined by increasing the number of elements or,for IGABEM, inserting knots. IGABEM clearly provides a greater accuracy of approximation;this is caused by the integration points being mapped to the analytical surface of the cylinderby the NURBS and the approximation of potential being interpolated by the same functions.100 101 102ka10-410-310-210-1Error, EUnit cylinder; τ≈10; dI =(1,0)Piecewise, polynomial BEMIGABEM (NURBS)Figure 1: Comparison of errors of conventional, piecewise BEM and IGABEM.Figure 2 compares the quality of solutions of IGABEM and XIBEM; errors of a piecewisequadratic, partition-of-unity BEM (PU-BEM) [13] are also included. For XIBEM and PU-BEMsimulations, τ ≈ 3 . The IGABEM simulations achieve an accuracy of ∼ 1% while the XIBEMand PU-BEM simulations, with system matrices nine times smaller than the IGABEM, acheivesignificantly greater accuracy.The accuracy of XIBEM and PU-BEM are broadly similar. However, for the PU-BEM sim-ulations, geometry points are not located just by the shape functions; instead, collocation andintegration points are carefully ‘snapped’ to the analytical surface. Without this, the accuracy ofthese simulations is dramatically reduced. A significant benefit of XIBEM is that this mappingis inherent within the NURBS representation.102 103ka10-610-510-410-3Error, EUnit cylinder; dI =(1,0)IGABEM (τ≈10; NURBS)XIBEM (τ≈3; NURBS)PU-BEM (τ≈3; quadratic elements)Figure 2: Comparison of errors of IGABEM, XIBEM and PU-BEM.3.2 Multiple scatterersA second geometry considers multiple scatterers designed to create internal reflections. Figure3 displays the geometry used, consisting of two capsules and a cylinder, and illustrates theabsolute value of the total wave potential; the angle of incidence, θI, is 3π/4 radians. Thereference solution used to calculate errors, E , is a converged solution obtained using the methodof fundamental solutions (MFS) [17].Figure 4 displays the errors of conventional BEM, IGABEM and XIBEM simulations; thevalue of τ for each simulation type is noted in the legend. XIBEM approximations are obtainedusing three times fewers degrees of freedom than used by the other simulations. The IGABEMapproximations are clearly more accurate than those of the conventional BEM; futhermore, theXIBEM approximations have smaller errors than both.Figure 3: Plot of ‖φ‖ illustrating internal reflections of multiple scatterers; ka = 25, θI = 3π/410030 40 50 60 70 80 90 200ka10-610-510-410-310-2Error, EMultiple scatterer; θI =3π/4Conventional BEM, τ=10IGABEM, τ=10XIBEM, τ=3Figure 4: Comparison of errors of conventional, piecewise BEM, IGABEM and XIBEM.4. CONCLUSIONSIsogeometric analysis has significant potential for use with boundary elements methods. Thispaper has demonstrated this for both conventional and enriched methods. XIBEM providesa clear and significant improvement over IGABEM simulations. These improvements are thereduced system size and increased accuracy; this also extends the bandwidth of frequencies overwhich the method can be considered feasible.REFERENCES1. T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs (2005) Isogeometric analysis: CAD, finiteelements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech.Engrg., 194(3941), 4135-4195.2. Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R Hughes, S. Lipton, M.A. Scottand T.W. Sederberg (2010) Isogeometric analysis using T-splines, Comput. Methods Appl.Mech. Engrg., 199(58), 229-263.3. R.N. Simpson, S.P.A. Bordas, J. Trevelyan and T. Rabczuk (2012) A two-dimensionalisogeometric boundary element method for elastostatic analysis, Comput. Methods Appl.Mech. Engrg., 209-212, 87-100.4. C. Politis, A.I. Ginnis, P.D. Kaklis, K. Belibassakis and C. Feurer (2009) An isogeometricBEM for exterior potential-flow problems in the plane, SIAM/ACM Joint Conference onGeometric and Physical Modeling, San Francisco, USA.5. T. Takahashi and T. Matsumoto (2012) An application of fast multipole method to iso-geometric boundary element method for Laplace equation in two dimensions, Eng. Anal.Boundary Elem., 36(12), 1766-1775.6. M.A. Scott, R.N. Simpson, J.A. Evans, S. Lipton, S.P.A. Bordas, T.J.R. Hughes and T.W.Sederberg (2013) Isogeometric boundary element analysis using unstructured T-splines,Comput. Methods Appl. Mech. Engrg., 254, 197-221.7. T. Abboud, J.C. Nédélec and B. Zhou (1995) Integral equation method for high frequencies,C. R. Acad. Sci., Paris, Sér. I, 318(2), 165-170.8. O.P. Bruno, C.A. Geuzaine, J.A. Monro and F. Reitich (2004) Prescribed error toleranceswithin fixed computational times for scattering problems of arbitrarily high frequency: theconvex case, Phil. Trans. R. Soc. A, 362(1816), 629-645.9. V. Domı́nguez, I.G. Graham and V.P. Smyshlyaev (2007) A hybrid numerical-asymptoticboundary integral method for high-frequency acoustic scattering, Numerische Mathematik,106(3), 471-510.10. S. Langdon and S.N. Chandler-Wilde (2006) A wavenumber independent boundary elementmethod for an acoustic scattering problem, SIAM J. Num. Anal., 43(6), 2450-2477.11. J.M. Melenk and I. Babuška (1996) The partition of unity finite element method: Basictheory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314.12. A. de la Bourdonnaye (1994) A microlocal discretization method and its utilization for ascattering problem, C. R. Acad. Sci., Paris, Sér. I, 318, 385-388.13. E. Perrey-Debain, J. Trevelyan and P. Bettess (2003) Plane wave interpolation in directcollocation boundary element method for radiation and wave scattering: numerical aspectsand applications, J. Sound Vibrat., 261(5), 839-858.14. L.C. Wrobel (2002) The Boundary Element Method, Vol. I: Applications in Thermo-fluidsand Acoustics, John Wiley & Sons, Ltd., Chichester, UK.15. H.A. Schenck (1968) Improved integral formulation for acoustic raidation problems, JASA,44(1), 41-58.16. D.S. Jones (1986) Acoustic and Electromagnetic Waves, Clarendon Press, Oxford, UK.17. P.S. Kondapalli, D.J. Shippy and G. Fairweather (1992) Analysis of acoustic scattering influids and solids by the method of fundamental solutions, JASA, 91, 1844-1854.

Extended isogeometric boundary element method (XIBEM) for acoustic wave scattering problems

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Extended isogeometric boundary element method (XIBEM) for acoustic wave scattering problems

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