A Frequency-optimized Discontinuous Formulation for Wave Propagation Problems

Transcription

1 th Flid Dynamics Conference and Ehibit 8 Jne - Jly, Chicago, Illinois AIAA -55 A Freqency-optimized Discontinos Formlation for Wave Propagation Problems Yi Li and Z. J. Wang Department of Aerospace Engineering and CFD Center, Iowa State University, Ames, IA 5 Recently, a new nifying discontinos formlation was developed by Hynh [7], and etended to simple and hybrid meshes by Wang & Gao [] for hyperbolic conservation laws. As with almost all discontinos methods sch as the discontinos Galerkin, spectral volme and spectral difference methods, the new formlation named CPR (Correction Procedre via Reconstrction) employs a piece-wise discontinos polynomial space. In this paper, a hybrid discontinos space inclding polynomial and Forier bases is employed in the CPR formlation to compte broad-band waves. The dispersion and dissipation properties of this method are investigated throgh a Forier analysis. The analysis is also sed to gide the selection of free-parameters to optimize the resoltion of broadband waves and minimize both dissipation and dispersion errors. The reslts are verified with nmerical soltions of the simple scalar advection eqation and the two-dimensional acostic wave eqations. Keywords: CPR (Correction Procedre via Reconstrction), A Hybrid Discontinos Space, Wave Propagation Analysis I. Introdction igh-order methods are highly desired for wave propagation problems inclding aero-acostic and Helectromagnetic waves. As pointed by Tam [], aero-acostic problems are intrinsically nsteady, and the dominant freqency is sally high. They differ from general comptational flid dynamics problems. Many powerfl nmerical algorithms have been developed to solve comptational aeroacostics (CAA) problems, e.g. the dispersion-relation-preserving (DRP) finite difference schemes [], discontinos Galerkin method [6], mltidomain spectral methods [], spectral volme and difference methods [, 9, ], compact schemes [, ] and WENO schemes[, 5, ]. High-order methods capable of handling nstrctred grids are obviosly mch more fleible in dealing with comple geometries. In this paper, we focs on a recent discontinos nstrctred grid formlation called CPR (Correction Procedre via Reconstrction) [7, ], which nifies the DG and SV/SD methods. In most discontinos methods, local spaces based on polynomials are sed. Non-polynomial bases have been stdied in [], and demonstrated for problems with special soltions. In the present stdy, we employ hybrid bases inclding both polynomials and Forier bases to resolve broad-band wave propagation problems. We borrow the ideas from the DRP and pwind DRP schemes in determining the parameters in the Forier bases to maimize the resolvable wave nmber given a certain error threshold. The basic idea of DRP scheme is to optimize the scheme coefficients for the high resoltion of short waves with respect to the comptational grid instead of the trncation error in [,, 5, 6]. A sitable grid density can be determined by the grid reqirement (points-per-wavelength, PPW) following the same procedre as presented in [8, ] withot bondary condition consideration. The dispersive and dissipative errors were investigated to analyze the wave propagation performance and sch a method was sed for finite difference and finite volme methods [9,,, 5, 6]. Van den Abeele [8] performed sch an analysis for the D spectral volme method and H [6] applied it for the discontinos Galerkin method. The CPR method based on hybrid bases is analyzed sing a similar method. This paper is organized as follows. For the sake of completeness, first the CPR method is reviewed in section. One-dimensional wave propagation analysis is given in section. The hybrid bases are optimized by minimizing dispersion and dissipation integrated errors given certain wave nmbers and at the same time satisfying a given resoltion error threshold. Wave propagation properties are investigated by sing Forier analysis for the optimized Gradate Stdent, Department of Aerospace Engineering, yili@iastate.ed, AIAA Stdent Member Professor of Aerospace Engineering, 7 Howe Hall, zjw@iastate.ed, Associate Fellow of AIAA Copyright by Yi Li and Zhijian Wang. Pblished by the, Inc., with permission.

2 hybrid bases. Then the mesh resoltion analysis is given to verify the optimization procedre. In section, nmerical reslts are presented to demonstration the performance of the nmerical method with hybrid bases. Conclding remarks are given in section 5. II. Review of the CPR Formlation The CPR formlation was developed in [7] and []. The basic idea is reviewed here. The conservation eqation is given as + = (.) where is the fl vector. Mltiplying (.) with an arbitrary weighting fnction W and integrating over an element V i, we obtain + = + = (.) Let be an approimate soltion to Q at element i. Becase the approimate soltion is discontinos across the element interface, the face fl term is replaced with a common Riemann fl,, = (.) where is the soltion otside the crrent element V i. Eqation (.) then becomes Applying integration by parts to the last term, we obtain +,, = (.) + +,, = (.5) The term =,, is the normal fl difference and can be viewed as a penalty term. A correction field is introdced sing the following lifting operator, which is defined as We obtain =. (.6) + + = (.7) Under some mild conditions otlined in [], (.7) can be redced to a differential form. Let the degrees of freedom (DOFs) be the soltions at soltion points (SPs) and the eqation (.7) is the tre at the SPs, i.e.,, +, +, = (.8) Eqation (.8) does not involve any eplicit srface or volme integrals, and the lifting operator depends on the choice of weighting fnction W. The performance of this formlation depends on how efficiently the correction field can be compted. III. One-Dimensional Wave Propagation Analysis A. Introdction to wave Propagation Analysis The dispersive and dissipative properties of the spatial discretization of the CPR method will be analyzed. The approach is following the methods by H [6] and Van den Abeele [8]. In this section, D CPR will be analyzed. The D scalar advection eqation with periodic bondary conditions and a harmonic wave as the initial soltion is given as + = (.), = (.)

3 \hwhere is the positive wave speed. The dispersion of the scheme, = represents a sinsoidal wave train with a wave nmber and a freqency. The eact dispersion relation for (.) is =. To (.), a (p+) degree of freedom (DOF) method will be applied on a niform mesh of size. On a local coordinate, for each element i, the approimation =, can be written as a fnction of. On the bondaries between two elements, a Riemann fl is sed, = + (.) In (.), = corresponds to a central fl and = corresponds to the pwind fl. Upwind fl is sed here. Eqation (.) then becomes, +, +, =, =,, + (.) The matri elements are given by the following epressions: =, =,, + (.5) = +, =,, + (.6) is the inversion matri of M, and = (, =,, + ). The non-dimensional wave nmber and freqency are defined as = and Ω =, and the eact dispersion relation is given as Ω =. Sbstitting the epression of a harmonic wave, = into (.), the nmerical dispersion relation determined for pwind fl is given as Ω + + = (.7) The qantity Ω is called the Forier footprint R and R = R + R, and the imaginary part R is a measre of dispersive properties of the scheme, whereas the real part R represents the diffsive behavior which shold be non-positive to keep the scheme stable. For classic finite difference methods with one DOF, the wave range is < <, while with p+ DOF per cell, the wave range is + < < +. B. Hybrid Bases and Optimization of Free-Parameters The piecewise polynomial bases are generally sed as local spaces in most discontinos methods. Nonpolynomial approimation bases sch as eponential fnctions and trigonometric fnctions for the discontinos Galerkin (DG) method were developed in [] to obtain better approimation for specific types of PDEs and initial and bondary conditions. The pwind CPR method based on hybrid bases is stdied, with objective of resolving broad-band wave propagation problems. Hybrid bases are given as T span,,,,, sinα, cosα, sinα, cosα, (.8) where α, α, α are free-parameters. Acostic problems are governed by the same eqations as those in aerodynamics sch as the Eler and Navier- Stokes eqations. However minimm nmerical dispersions and dissipations are reqired to get an accrate amplitde and phase for nmerical calclation of wave propagation []. The optimized schemes sch as central DRP [] and pwind DRP [5, 6] schemes are to assre the transform of the scheme be a good approimation of that the partial derivative over a certain range of wave nmber. The approimation of the first-order spatial derivative / on niform grids for a finite difference method is given by a + j (.9) The basic idea of DRP is that the coefficients are determined by reqiring the Forier transform of the finite difference scheme on the right of (.9) to be a close approimation of the partial derivative on the left. Free-parameters in the hybrid bases for the CPR method are optimized by mimicking the similar idea of DRP [, 5] to maimize the resolvable wave nmber given a certain error threshold. The following two conditions are applied. The optimization process has to allow the normalized vale of Ω N K N and Ω N to be as close to zero as possible for certain integration wave nmbers.

4 E = Ω N K N dk + λ Ω N dk (.) The weight λ is set as. to balance the L norm of the trncated errors of dispersion and dissipation. And e is a predetermined optimized range of wave nmbers. In order to qantify the resoltion of the scheme, set the dispersion and dissipation errors to less than.5%, i.e.[6] Ω K <.5 Ω <.5 (.) Different free-parameters α are fond to minimize the integration error E throgh nmerical searches for a certain wave nmber integration range e with =,, sinα, cosα in Table. And the maimm resolvable non-dimensional wave nmber K are determined sing (.) for each α with respect to a certain integration wave nmber in Table. K increases and then decreases with respect to α which is related to a certain integration range. Table Optimization free-parameter of Forier bases for =,,, Integration range (e) α E - Dispersion E-diffsion E π (.)..77e-9.78e-8 7.6e-9 5*π/ (.9).7.65e-7.9e-6.678e-7 *π/ (.7)..87e-6.8e e-6 7*π/ (5.5). 6.5e e-.65e- *π (6.8).7 6.5e-.e-.85e- Table Maimm resolvable wave nmber for =,,, Integration range α K π (.)..96 5*π/ (.9).7.6 *π/ (.7)..86 7*π/ (5.5)..6 *π(6.8).7.66 α =. is referred as the optimized free-parameter, which minimizes the integration error E over a relatively large wave nmber integration range.7 with the dispersion and dissipation errors to less than.5% in eqation (.). The same procedre is applied for the higher DOF scheme. α =. is the optimized free-parameter with the integration wave nmber range 8.6 for T =,,,, sinα, cosα, and α =.5 and α =. are the optimized free-parameters with the integration wave nmber 9. for T =,, sinα, cosα, sinα, cosα. In Fig, the pwind CPR scheme with optimized base T =,, sin., cos. is compared with polynomial space, Tam & Webb s central DRP and Zhang & Chen s pwind DRP. This optimization scheme has less dispersion errors than the polynomial space and the Tam & Webb s central DRP scheme, bt a little larger dispersion errors than Zhang & Chen s pwind DRP scheme. It is able to resolve the waves with non-dimensional wave nmbers as high as abot. which is very close to the seven stencil scheme, althogh it is a for stencil scheme.

5 (Im(Forier Footprint)- Wave Nmber)/N Dispersion Error Tam & Webb Central DRP Zhang & Chen Upwind DRP,,,,,sin(.*),cos(.*) Wave Nmber/N Fig.. Comparison of dispersion errors Ω K/N and dissipation errors Ω / Re(Forier Footprint)/N Dissipation Error Tam & Webb Central DRP Zhang & Chen Upwind DRP,,,,,sin(.*),cos(.*) Wave Nmber/N C. Mesh Resoltion Analysis In this section, the mesh resoltion analysis is applied to verify the optimization procedre by following the ideas in [8, ]. The nmber of grid points per wavelength (PPW) is presented, with the objective of accrately simlating wave propagation over large distances. The mesh resoltion analysis is based on the principle root, = /, where m is the crrent time step and m+ is the net time step. is the non-dimensional wave nmber, C is the corant nmber and PPW is the points per wave length and PPW = = π/(k ). The local amplitde and phase errors are, respectively Error = σ (.) Error = (.) where = tan (σ /σ ), and σ, σ are the real and imaginary part of σ, respectively. Criterion for comparing schemes is based on the global amplitde and phase errors which are Error = σ / < % (.) Error = n + π<% (.5) where n is the nmber of the wavelength travelled. A very small corant nmber is sed jst for spatial consideration. This is a reasonable measre for selecting a grid density, and reveals the implications of optimization. Grid Points per Wavelength 6,,sin,cos,,sin(*),cos(*) -,,sin(*),cos(*),,sin(*),cos(*) Nmber of Wavelength Travel Grid Points per Wavelength 5 5 Nmber of Wavelength Travel Fig.. Grid resoltion reqirements based on globe amplitde and phase errors In Fig, point per wavelength (PPW) reqirement are presented for pwind CPR with respect to,, sinα, cosα. α = is sperior p to a distance of travel abot 5 wavelengths based on % phase error ,,sin(),cos(),,sin(*),cos(*),,sin(*),cos(*),,sin(*),cos(*)

6 criterion. Sch behavior is typical of optimized schemes. And sally aggressive optimization leads to ecellent performance for small distances of travel bt poor performance for longer distances. This property agrees with the previos analysis that the optimized free-parameter α =., which is close to, shows good dispersive properties with the relatively large wave nmber given a certain resoltion error criterion. The PPW analysis matches the optimization analysis perfectly, and both methods can be sed to verify each other. Becase the schemes are optimized for a given range of wave nmber, they reqired fewer PPW if the propagation distance is relative short. As the nmber of wavelength traveled increase, the advantage of the optimized schemes diminished as the reqired PPW increase. However if fewer PPW is reqired, the se of optimized scheme not only gives most accrate reslts bt also reslts in significant saving of CPU time. IV. Nmerical Tests A. Soltion for D Wave Eqation With Sine Wave as the Initial Condition This problem is sed to verify the performance of the wave propagation characteristics of CPR in terms of space,,, and D convective wave eqation is considered. On the niform mesh with the initial condition is given as follows + = ( = ) (.), = (.) De to Forier spaces, the eact dispersion relation Ω = is eactly satisfied at a certain. For eample if the base T =,, sin, cos is applied, the physical vale Ω = K = k ξ = =. An approimation apace,,, is designed to eactly simlate the eqation. First set cell size and calclate the initial condition s non-dimensional wave nmber K = k = π (k = π). Second choose a space,, sinα, cosα and here K = α ξ ( ξ = ). Finally set K = k α = π /.,,sin(),cos(),,sin(pi*/),cos(pi*/),,sin(*),cos(*),,sin(*),cos(*),,sin(pi*),cos(pi*),,sin(*),cos(*) Fig. (a) α = π ξ = (a), =, t =., T = 6 (b) α = π ξ = π, =, t =., T = 6 Fig shows that the designed spaces eactly nmerically simlate the wave eqations (.) with the initial condition (.). (a) The space,, sinπ /, cos π / is eactly simlating the wave eqation when = (b) The space,, sinπ, cos π is eactly simlating the wave eqation when =. B. A Benchmark Problem CAA Workshop () The governing eqation is the scalar wave eqation with nit wave speed as defined as eqation (.), with the following initial condition, = +, =.7.6 (.) Two different freqencies β =.7 and β =.6 are considered, and set = for an eqivalent DOF. At this grid resoltion, the high freqency wave embedded in the initial condition only has abot.7 and.9 points-per-wave (PPW). It is therefore a challenge for any nmerical scheme to adeqately resolve the high freqency wave (b)

7 ,,,,,sin(.55*),cos(.55*),,sin(*),cos(*) Fig. Nmerical soltion of D wave eqation with the initial condition (5.) (T = 5s, =, th order DOF pwind scheme) - - For β =.7, the approimation space,, sinα, cosα is designed with =.55 and = to eactly simlation the initial wave condition cosβ. Then this approimation space is applied for the wave eqation with the initial condition of eqation (.) in order to get a better simlation. The time integration was carried ot sing a for-stage forth-order Rnge-Ktta scheme and a constant time step of.5s was sed for all cases. In Fig, =.55 shows mch less dissipative behavior among the presented bases.,,,,, ,,,,, 5,,,,sin(.*),cos(.*) Fig 5. Nmerical soltion of D wave eqation with the initial condition (5.) (T = 5, = 5, 6th order DOF pwind) and first row =.7 and second row =.6 In Fig 5, the soltions of CPR pwind scheme with the optimized approimation space,,,, sin., cos. and,, sin.5, cos.5, sin., cos. are compared with the 6 th order polynomial approimation space for =.7.6, and both schemes show better simlations than the polynomial approimation space. And the more Forier bases terms there are, the more accrate the reslts are. C. Two Dimensional Acostic Wave This test case was compted for a D acostic plse. The governing eqations for this test case are the D nonlinear Eler eqations,,,,sin(.*),cos(.*) ,,sin(.*),cos(.*),sin(.5*),cos(.5*) ,,sin(.*),cos(.*),sin(.5*),cos(.5*) where Q, E and F are vectors given by + + = (.) 7

8 =, = +, = + (.5) + + with the mass density, and the velocity components and the pressre. The total energy E is defined by the following eqation = + where is set to. which is the ratio of specific heat to air. The comptations are carried ot on three different strctre grids (55), () and () on a sqare domain [,][,]. The initial soltion is an acostic plse with a Gassian profile and is set as same as one by Kris [5] [6] = +... = + = = And the ambient pressre, mass density and the half-width of the Gassian profile are given as follows (.6) (.7) =, =, =.5 (.8) Y.8.6. Y.8.6. P X X Fig 6. Strctred qadrilateral grids (left) and pressre contors (right) based on tensor prodct basis with approimation spaces,,, The eact soltion of the LEEs for the acostic pressre field is given as,. = =. (.9) with = and J is the zero-th order Bessel fnction of the first kind which is sed as a reference soltion referring to [5] and [6]. Strctred qadrilateral grids are given on left of Fig 6 and pressre contors are given on the right of Fig 6 which is based on tensor prodct basis with approimation spaces,,,. Then the nmerical schemes with th DOF optimized hybrid bases are tested. All nmerical tests are carried ot with =., =., and Gass-Lobatto points are sed as distribtion points for each element for CPR schemes. In Fig 7, errors of the optimized bases,, sin, are smallest among all of approimate bases shown for the grids. This property agrees with the previos analysis that the optimized hybrid bases shows better dispersion and dissipation properties when non-dimensional wave nmbers of the schemes are given in a certain range. In Fig 8 the optimized base still performs better than the corresponding polynomial base for the grids. It is epected that the polynomial bases will perform best when the grids are fine enogh. 8

9 .. E a c t,,s in ( ),c o s ( ),,s in ( * ),c o s ( * ),,s in ( * ),c o s ( * ).. E a c t,, ^, ^,,s in ( * ),c o s ( * ).. P P Fig 7. Pressre distribtion at y =.5 (Tensor prodct bases with th order DOF) on grids... E a c t,,s in ( ),c o s ( ),,s in ( * ),c o s ( * ),,s in ( * ),c o s ( * ). E a c t,, ^, ^,,s in ( * ),c o s ( * ) P P Fig 8. Pressre distribtion at y =.5 (Tensor prodct bases with th order DOF) on grids V. Conclsion The CPR (Correction Procedre via Reconstrction) formlation with a hybrid discontinos space inclding polynomial and Forier bases is stdied. The free-parameters in the hybrid base are optimized based on minimizing both dispersion and dissipation errors and satisfying a certain resoltion of dispersion and dissipation errors. The hybrid bases with optimized free-parameters show good wave propagation properties. A comparison was made with the dispersion and dissipation properties of the central DRP and pwind DRP schemes in D. The for-point stencil optimized hybrid bases is able to resolve waves with non-dimensional wave nmbers as high as the seven-point stencil central and pwind schemes. And the more Forier bases components are sed, the less dispersion and dissipation errors the schemes show. Then the mesh resoltion analysis is given to verify the optimized hybrid bases. The accracy of the hybrid bases depends on the non-dimensional wave nmber. De to the Forier bases inclded in the hybrid bases, the schemes can eactly simlate the wave eqation at some specific non-dimensional wave nmber. This property is verified with the case with a sine wave as the initial condition. The method has been tested for Problem in Category (CP) on benchmark problem in the Forth Comptational Aeroacostics (CAA) Workshop. It is shown that the scheme with optimized Forier bases can resolve waves more accrately than the polynomial bases at.7 PPW. This is followed by the acostic wave problem in D. It is verified again that the advantages of the optimized hybrid bases depend on the non-dimensional wave nmbers. Acknowledgments This stdy has been spported by the Department of Aerospace Engineering in Iowa State University and partially spported by AFOSR. 9

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