Iemma U

Marchese V

IEMMA, Umberto

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Meshless Numerical Solution of Boundary IntegralEquations based on Non Uniform RationalBasis-SplinesVincenzo Marchese and Umberto IemmaDepartment of Engineering, Roma Tre University, via Vito Volterra, 62, Roma, Italy(Received 15 August 2013; accepted 11 December 2014The paper deals with the use of Non Uniform Rational Basis-Splines (NURBS) for the global representation ofdomain geometry and unknown functions aimed at the numerical solution of Boundary Integral Equations (BIE).The use of a global NURBS function basis yields a meshless method which does not need the partition of theboundary into elements. The level of the accuracy in the representation of dependent and independent variablescan be changed in each simulation, according to the problem requirements, thanks to the recursive definition ofNURBS. The solving system of equations is assembled by means of the collocation of the integral equation ontothe Greville abscissae in the NURBS parametric space. The unknowns are the locations of the control points inthe vector space the unknown function belongs to. Preliminary numerical results have been obtained in potentialaerodynamics and acoustic scattering. The numerical solution reveals a remarkable level of accuracy in all the testcases analyzed with a convergence rate always higher than the order of the NURBS adopted.1. INTRODUCTIONIn many fields of application, the Boundary Integral Equa-tions (BIE) approach is a well established technique to addressthe solution of Boundary Value Problems (BVP). Using theBIE, it is possible to represent the unknown function at anylocation in the domain as a function of its Cauchy data set.This approach is more recently considered as standard in in-compressible and compressible potential aerodynamics, struc-tural elasticity, heat conduction, electromagnetism, acoustics,and aeroacoustics. The numerical solution typically relies onthe Boundary Element Method (BEM), in all its variants anddeclinations. In classic BEM the boundary of the domain ispartitioned into finite elements, where the dependent and in-dependent variables are approximated using suitable local ba-sis functions. The greatest advantage of the numerical meth-ods based on BIE resides in the reduction of the computa-tional burden required for the numerical solution, due to the re-duced dimensionality of the problem. Although this approachis usually convenient with respect to the so-called field meth-ods, such as Finite Volumes Method (FV), Finite DifferenceMethod (FDM), or Finite Element Method (FEM), in somespecific application the number of boundary elements requiredto capture a specific feature of the phenomenon may becomeextremely high. This is the case, for example, for acousticpropagation and scattering problems, where the wave lengthof the perturbation at high frequencies can be orders of mag-nitude smaller than the characteristic length of the domain ofinterest, thus requiring a huge amount of boundary panels tocorrectly reproduce the scattering and interference effects. Oneof the possible approaches to mitigate this difficulty is improv-ing the accuracy of the local representation of the variablesusing higher-order functions. This allows for the reduction ofthe number of elements needed to achieve the desired level ofaccuracy. In this respect, the literature available is very exten-sive, and an in-depth review is beyond the scope of the paper.Among others, it is worth mentioning the use of third orderpolynomials based on Overhauser1, 2 or Hermite3, 4 elements,recently coupled with Coons patches.5 The typical limitationof all the approaches based on the local representation of thevariables is the restriction of the resulting numerical formula-tion to a single order of accuracy, fixed by the order of the poly-nomial shape functions used. The possibility to overcome thislimitation has already been investigated within the context ofthe finite element method6 using an approach based on general-ized Hermite polynomials. However, a similar attempt for thesolution of BIE is still missing. This goal has been the drivingmotivation in the development of the method presented here.Indeed, using a global representation of the variables basedon NURBS, the order of the basis functions can be improvedwhen needed by the specific application in discussion. This isa consequence of the iterative definition of the NURBS, whichmakes possible the increase of the NURBS degree simply bychanging an input parameter. The use of the NURBS for therepresentation of curves and surfaces is a common techniquein the CAD community for the modeling of complex geome-tries with strict requirements of smoothness and continuity be-tween patches.7, 8 Their use in the numerical solution of BIEis not new, although it is relatively recent. NURBS have beenused to develop boundary element solutions of integral equa-tions in elastostatics,9 in radiation and diffraction problems,10and in potential aerodynamics.11 As already mentioned, thepeculiarity of the present method is in the use of the NURBSfor the development of a global isogeometric approach aimedat the meshless numerical solution of the BVP. The controlpoints used for the representation of the dependent variablesare obtained through the h-refinement of the optimal NURBSInternational Journal of Acoustics and Vibration, Vol. 20, No. 3, 2015 (pp. 143152) 143

Meshless Numerical Solution of Boundary Integral Equations based on Non Uniform Rational Basis-Splines

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