Hybrid boundary element formulation in acoustics L. Gaul, M. Wagner Institute A of Mechanics, University of Stuttgart, Germany. Abstract A symmetric hybrid boundary element method in the frequency domain is introduced for the computation of acoustic radiation and scattering in closed and infinite domains. The Hybrid Stress Boundary Element Method (HSBEM) in a frequency domain formulation is based on the dynamical Hellinger-Reissner potential and leads to a Hermitian, frequency-dependent stiffness equation. As compared to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix are communicated. The field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. This approach results in a linear system of equations with a symmetric stiffness and mass matrix. 1 Introduction The rationale for the search for alternative boundary element methods in dynamics is guided by the demand for simple symmetric formulations, the simplification of field point evaluations and a consistent structure of the spatially discretized equations of motion in boundary element and finite element formulations to ease the treatment of fluid-structure interaction problems For these reasons the authors present a new approach for calculating boundary as well as domain data for dynamic field problems in frequency and time domain.
In contrast to domain discretizations, the suggested procedure is a boundary discretization approach which takes advantage of the reduction of the problem dimension by one. It is not derived from a boundary integral equation (BIE) such as the conventional direct boundary element method (CREM) but based on multifield variational principles which are the underlying principles for hybrid finite elements together with the Hellinger- Reissner and the Hu-Washizu principles. The choice of weighted fundamental solutions as domain approximations selected independently from polynomial approximations of boundary variables, allows us to map domain integrals to boundary integrals. The authors generalized known static formulations to dynamic hybrid boundary element formulations by the use of Hamilton's principle in frequency and time domain for field problems of linear acoustics and elastodynamics [4, 5, 8, 91. Symmetric system matrices are obtained by construction whereas the collocation or Galerkin approach to CBEM lead to nonsymmetric matrices. Double integration of the shape functions in the symmetric Bubnov- Galerkin formulation is avoided and replaced by single integration and weighted superposition of the domain approximation. As the bybrid boundary element method (HBEM) in frequency domain is based on the field approximation with weighted dynamic fundamental solutions, the Sommerfeld radiation condition for the exterior domain is fulfilled. This is why radiating irregular shaped surfaces can be discretized by fluid boundary elements. 2 Hybrid boundary element method Other than BEM based on a weighted residual statement, the hybrid boundary element method is derived from variational principles. In this section the HSBEM [l] is applied to acoustical problems in the frequency domain. Considering a compressible, inviscid fluid with linear changes of variables such as velocity and density, wave propagation in the frequency domain in a fluid domain fl with boundary I' can be described by the Helmholtz equation (qi + /G2p = 0 (1) with the velocity potential 4 as field variable, which is defined by +,i = vi where vi is the particle velocity. Domain sources are neglected in the presented formulation. The wave number is defined as K = wlc where c is the speed of sound. Along the boundary, the outward flux 11, is descri'bed by the normal derivative of the velocity potential 2 = =,ini 11, in which ni is the outward normal vector. Mixed boundary conditions are prescribed on the boundaries I'+ and I'+, respectively, as r$(x) = $(t) on I'+ and $(X) = 4(t) on I'+. (2) The HSBEM is derived from the Hellinger-Reissner potential, which is a complementary energy functional. Thy the derived field variab1.e is
considered as an independent field. In elastodynamics this is the stress tensor in the field and in acoustics it is the gradient of the velocity potential in the domain and the flux 1C, on the boundary. This gradient generates a potential field $v, which is an exclusive function of the gradient and is a-priori independent of the real potential field 4. 2.1 Variational Principle In the following, only steady state time-harmonic motion is considered, thus the field variables can be separated into space and time harmonic functions, where only the real part of the complex representation yields the physical result. Complex variables are denoted by 0. The conjugate complex variable is denoted by ()*. The Hellinger-Reissner potential for acoustics in frequency domain reads with the four independent fields, the potential gradient 4,;, the two independent potential fields $ and $V, and a Lagrangian multiplier introducing the static part of the Helmholtz equation. Additionally, essential boundary conditions have to be considered Note that the potential field 4 is assumed only on the boundary, whereas $,i exists on the boundary and in the domain. Applying the fundamental lemma to the vanishing first variation of the potential gives yields an interpretation of the Lagrangian multiplier as Moreover, the governing field equation Eq. (1) arises. Additionally, natural boundary conditions, as well as compatibility conditions of the two potential fields arise. Considering that the variations on the adjoint boundaries vanish and introducing the interpretation of the Lagrangian multiplier in Eq. (6)) yields
as a starting point for the derivation of the HSBEM. If the Dirichlet boundary condition is enforced as subsidiary condition, Eq. (7) completely ~describes the acoustical field problem. 2.2 Approximation Functions The numerical solution requires approximation functions for the field variables. The approximation of the boundary variable $ is carried out with a shape function vector N and a nodal vector 4 The approximations of the potential $V and gradient $,i in the domain R, as well as the flux 4 on the boundary, are given as a superposition of n fundamental solutions @(T(~), W), obtained as the solution of where W) is the Dirac-Distribution evaluated at the loadpoint $k) that is collocated with the boundary node and T(~) = lx - <ck)l is the Euclidean distance to the loadpoint ~(~1. The fundamental solutions are weighted by unknown parameters yi, i = l..n. Hence, the approximation of the potential and flux fields are obtained as By modifying the domain R such that small spheres with radii E, centered at the load points 6 where the fundamental solutions are singular, are subtracted, the domain R' with boundary I" is introduced. The properties of the Dirac distribution acting at the points now outside of the considered domain let the domain integral in Eq. (7) vanish in the limit R' -+ 0, if the domain approximations in Eq. (10) are introduced 2.3 Matrix Formulation Inserting the approximations given byeqs. (8) and (10) yields a discretized formulation of the vanishing first variation
where OH denotes the complex conjugate transpose. In the absence of domain sources, only boundary integrals remain. The vector f of equivalent nodal fluxes contains the transformations of the given Neumann data to the nodes of the discretization. The matrix H is known as the double-layer potential and will be interpreted as a kinematic transformation matrix in section 3.5.2. The frequency-dependent matrix can be expanded in a static part H. and a frequency-dependent part H, For the following, the feature of a frequency-dependent fundamental solution, that it tends to the behavior of the adjoint static fundamental solution in the limit, when the angular frequency or the distance between field and load point vanish is recalled [6]. Due to this the main diagonal of the static part H. contains a strong singularity. The Hermitian matrix F contains the product of the weakly and strongly singular fundamental solutions at the load points i and j. This matrix can also be expanded in the same way A main diagonal entry Fo,, of the static part is made up of a hypersingular integral over the boundary extension FE, similar to the integral-free term in the conventional BEM, and the remaining boundary P. The second integral can be evalutated as finite-part integral, whereas the entries over FE cannot be evaluated directly, since the transformation to spherical coordinates does not regularize a hypersingularity. The computation of these terms will be discussed in detail in section 2.5. Again applying the fundamental lemma yields a matrix equation for the unknowns 7 and 4. Solving for c$ one obtains ~ c $ = f, with the Hermitian stiffness matrix 2.4 Computation of Field Point Data By solving the boundary-value problem in Eq. (14), the primary unknows of the field problem, the potential $ and the generalized flux 7, become known. The potential in the domain can be determined by Eq. (10) only up to a constant. Thus, a complete description of the potential field can be written as @V =(@i+@cci)ri+rq5c, in R. (17)
where r is a scaling factor of the additional constant potential field @. This field is arbitrary and can be defined in any appropriate way. Eq. (17) can be looked at as establishing two orthogonal subspaces of the space of potential~ W". The subspace W' contains the potentials that purely produce equilibrated fluxes in the domain, since only these are physically meaningful. In this sense, potentials belonging to W' are called admissible herein. The subspace W contains the constant potentials, that yield no flux. The subspaces are complementary, hence W" = W' + W. It cannot be assured beforehand that the generalized fluxes 7 do not contain a certain amount of unequilibrated fluxes. Thus, a vector ci is introduced that maps the constant potential functions @ to the generalized fluxes, compensating for contributions of constant potential. Thus, the first term of Eq. (17) describes only potentials generating equilibrated fluxes and the second term describes constant potentials. As these vectors are in complementary subspaces, the following orthogonality condition for arbitrary yi can be established Eq. (18) yields a way to determine the unknown vector ci. For more detail on the computation of this quantity refer to [2]. One requirement on the field description in Eq. (17) is that it must coincide with the nodal values 4 for compatibility. In a discretized form an orthonormal basis for the N X 1 vector subspace of constant nodal potentials shall be denoted by W. For the nodal values, Eq. (17) reads where 9 is a symmetric N X N matrix of fundamental solutions aij = @(Idi) - [(j)l,w). Generally, the computation of field point data involves no additional integration, hence field point data can be gained with much less effort than in the direct boundary element method [l]. This feature can be combined with the conventional boundary element method to obtain an effective way to compute field point data, if symmetry of the system matrices is not required [7]. 2.5 Orthogonality Properties of the HBEM For the computation of the matrix F the hypersingular main diagonal of the static part F. must be evaluated, where the load points of the Dirichlet and Neumann fundamental solutions coincide. For this reason orthogonality properties of this matrix [l] are used to obtain the main diagonal entries without any further integrations. 2.5.1 Mechanical Coordinate Systems A physical interpretation starts from the virtual work expression for the
static matrix F. lim [614*)~dr = 67H~07 = 67 H-v 4. --to 7 is interpreted as generalized flux and ;pv as generalized potential. This expression yields ;pv = F07, (=I which defines F. as a flexibility matrix [l]. For the further development, two sets of coordinates have to be considered. The external coordinate system consists of the potential field 4 on the boundary, approximated by Eq. (8). The potentials cause a nodal flux f. The internal coordinate system uses the generalized flux 7, introduced in the approximation of the domain, Eq. (10). The adjoint potential field ;pv in the internal system is defined via Eq. (21). Note that this internal system does not physically exist but is an energetically equivalent formulation arising from the Hellinger-Reissner potential. 2.5.2 Consideration of Interior Domains Since constant potentials generate zero flux, it follows for the static stiffness equation that Kow = 0. The matrix KO is singular and W is an orthonormal basis of the nullspace of KO. Considering the virtual work principle, it can be deduced that constant potentials do not produce virtual work, thus the fundamental orthogonality condition holds. In order to solve Eq. (14), relations between the two coordinate systems are necessary. It follows from the first line of Eq. (14), also considering Eq. (21), that -v 4 =Ho~. (23) Thus, the matrix H. transforms potentials from the external into the internal system. Analogously, the second line of Eq. (14) states a transformation relation between the internal and external fluxes. Using this and Eq. (22) shows that the basis of constant potentials also forms the nullspace of the matrix H. How = 0, (24) meaning that constant potentials are not transformed into the internal system, substantiating the fact that the internal system incorporates only equilibrated states. There also exists the left nullspace v of Ho, vtho = 0. This implies that unequilibrated fluxes in the internal system are not transformed into the external system. The conclusion is that if the internal flux belongs to the left nullspace of Ho, 7 E V, it transforms the external flux into the null element of the range of Ho, f E 0. In this case it follows
from Eq. (24) that the external potential 4 belongs to the nullspace of No, 4 E W and therefore the internal potential field transforms into the null element, 4v E 0. Finally, it can be deduced from Eq. (21) that the left and right nullspace of F. are identical to the left nullspace of H. This equation is uniquely solvable in a potential problem for the unkcswn main diagonal entries of Fo. 2.5.3 Orthogonality Properties of i@o In order to apply Eq. (19), the matrix + must be computed. The offdiagonal terms are analytic, but the main diagonal cannot be evaluated due to the singular behaviour of the fundamental solution. As before this singularity is exclusively contained in the static part +o. To compute the main diagonal an orthogonality condition for this matrix is exploited. By construction, the first term in Eq. (19) is set up in such a way that the generalized flux 7 yields only equilibrated potential fields, orthogonal to constant potentials. Then, it can be deduced that unequilibrated generalized fluxes must be orthogonal to the matrix (i@o + wct). Since these are in the subspace V, it follows that as an orthogonality condition. This equation can be used to uniquely compute the yet undetermined main diagonal entries of the static part of the fundamental solution matrix @o. 2.5.4 Consideration of Exterior Domains Finally, taking into account Eq. (26), the second case of an unbounded exterior domain fl can be considered. Only recently orthogonality properties for this case have been published [2, 31. Differently from the interior problem, unequilibrated fluxes yield a completely determined solution for the field problem, due to Sommerfelds radiation condition. Since constant potentials are a valid solution, the field in the domain can be computed by evaluating $V = @,Ti,X E fl contrary to Eq. (17), where an overbar denotes exterior variables. This must also hold for the nodal description, thus 4 = i@ 9. Note that i@ is the same matrix as in the interior case, so that Eq. (26) can be used to compute the singular main-diagonal elements. To obtain and also the boundary solution for the potentials, Eq. (14) for the exterior domain must be solved. The exterior matrices H and P are related to the interior matrices via H+H=I and P+F=9. (27) The first term is a well-known property of the double-layer potential in the direct BEM and the second term can be found by integrating the involved
Boundaq Element Techtlology XIV 12 1 matrices. Still, the main diagonal of the static part is undetermined. For this, the orthogonality properties in Eqs. (25) and (26) are exploited. By adding wct and multiplying v to the second term it follows that as a means to determine the hypersingular main diagonal of PO. Note that neither B. nor F. are singular in the exterior case, as it must be expected, since unbalanced fluxes are a valid solution of the field problem. 3 Numerical Example As a numerical example the 2D model of a car compartment is shown here. The boundary is discretized with 105 linear elements and 1063 field-points are used. The domain is excited by a velocity boundary condition as depicted in Fig. l. This models the vibrations of the engine coupled to the passenger compartment. The remaining boundary carries a homogeneous Neumann condition. Fig. 1 shows the solution of the program hybem de- (4 h$em (b) SYSNOISE Figure 1: Distribution of pressure amplitude in car compartment. Excitation frequency is f = 200Hz. veloped by the authors in comparison with the commercial acoustics code SYSNOISE. The distribution of the pressure amplitude in the car compartment at an excitation frequency of f = 200Hz is displayed. The color scales on both pictures are the same. It can be seen that the results agree very well. Note, that the solution close to the boundary of SYSNOISE are worse. This is due to the fact that only every second evaluated field node on the boundary coincides with a boundary node. Obviously the evaluation close to the singular points with the Somigliana identity as done in SYSNOISE generates more error than the evaluation with the HBEM. 4 Conclusion In the present paper a method for the calculation of interior and exterior acoustical problems with the HSBEM has been presented. The HSBEM
is developed from the dynamical Hellinger-Reissner potential using fundamental solutions for the approximation of the domain variable, giving rise to matrix F with a hyper-singular main diagonal. The main diagonal cannot be evaluated mathematically, thus a criterion for the evaluation of the main diagonal for an exterior domain has been applied, starting from physical interpretations. The presented example of wave propagation shows that the proposed method yields accurate results. Acknowledgements Support by the DFG of project B5 in SFB 404 "multi-field problems in continuum mechanics" at Stuttgart University is gratefully acknowledged. References [l] N. A. Dumont. The hybrid boundary element method. Applied Mechanics Review, 42(11):54-63, 1989. [2] N. A. Dumont and R. A. P. Chaves. The simplified hybrid boundary element method. In CILAMCE XX, Sgo Paulo, Brasilien, 1999. [3] N. A. Dumont, L. Gaul, M. Noronha, and M. Wagner. Hybrid boundary element method in acoustics. In P. Gon~alves et al., editors, 6th PACAM, pages 405-408, Rio de Janeiro, Brasilien, 1999. [4] L. Gaul and M. Wagner. A new hybrid symmetric boundary element method in elastodynamics. In C. Constanda et. al., editors, Integral Methods in Science and Engineering, Pitman Research Notes, pages 3-11, Harlow, UK, 1997. [5] L. Gaul and M. Wagner. Beam response derived from 3d hybrid boundary integral method in elastodynamics. Mechanical Systems and Signal Processing, 11(2):257-268, 1997. [6] L. Gaul and M. Wagner. Zur Dynamik der Fluid-Struktur-Interaktion. Zeitschrift f. angew. Mathematik und Mechanik, 79(S2):289-290, 1999. (71 L. Gaul, M. Wagner, and W. Wenzel. Efficient field point evaluation by combined direct and hybrid boundary element methods. Engineering Analysis with Boundary Element Methods, 21:215-222, 1998. [8] L. Gaul, M. Wagner, and W. Wenzel. Hybrid boundary element methods in frequency and time domain. In 0, von Estorff, editor, Boundary Elements in Acoustics, Southampton, 2000. WIT Press. [g] M. Wagner. Die hybride Randelementmethode in der Akustik und zur Struktur-Fluid-Interaktion. Bericht aus dem Institut A fiir Mechanik 412000, Universitat Stuttgart, 2000.