Interpolation Techniques for Overset Grids

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Journal of the Arkansas Academy of Science Volume 57 Article 22 2003 Interpolation Techniques for Overset Grids Paul S. Sherman Arkansas State University Nathan B.

1 Journal of the Arkansas Academy of Science Volume 57 Article Interpolation Techniques for Overset Grids Paul S. Sherman Arkansas State University Nathan B. Edgar Arkansas State University Follow this and additional works at: Part of the Other Computer Engineering Commons, and the Theory and Algorithms Commons Recommended Citation Sherman, Paul S. and Edgar, Nathan B. (2003) "Interpolation Techniques for Overset Grids," Journal of the Arkansas Academy of Science: Vol. 57, Article 22. Available at: This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This Article is brought to you for free and open access by It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of For more information, please contact

2 156 Interpolation Techniques for Overset Grids Paul S. Sherman* and Nathan B. Edgar Arkansas State University College of Engineering P.O. Box 1740 State University, AR72467 *Corresponding Author Abstract The use of finite difference schemes in computational aeroacoustics requires the use of structured grids in computational space. Complex geometries in the physical space can be modeled using multiple overlapping grids that are transformed into computational space. In this work, finite difference schemes are used that necessitate the addition of psuedo- or ghost-points in the overlap region of the grids for closure of the difference stencil. The functional values at these ghost points must be approximated from the values at the original grid points. This paper investigates interpolation techniques for these overset grids. Ann th order interpolation scheme using Lagrange polynomials is applied to the one dimensional (ID) wavepropagation problem to test the effects of increasing the interpolation order. This is done for both equal and unequal sized overset grids. Preliminary results from two dimensional (2D) grids willbe presented. Introduction In computational aeroacoustics, finite difference schemes are more commonly used because they are more computationally efficient than the finite element schemes. Finite difference schemes require the use of structured grids incomputational space. An unstructured grid in the physical space can be transformed (Tannehill et al., 1997) into a structured grid in the computational space. However, complex geometries in the physical space may require multiple overlapping grids. In this paper, we discuss the construction of overset grids, problems associated with their OVERLAP GRID POINTS o GHOST-POINTS construction, ID and 2D results which show some of the problems arising from the application of the grids, and some solutions implemented to address these problems. Fig. 1. Example of the overlap regions for IDand 2D grids. For this work, we chose to model the IDand 2D wave equations using a single pair of overlapping structured grids. In general, overlapping grids can have mismatched grid spacing (Petersson, 1999) in each direction and may not share a common point. An example of the overlap regions for ID and 2D grids is provided in Fig. 1. The layout of multiple grids in the physical space willbe dictated by the geometry to be modeled. Overlapping regions will exist wherever multiple grids share a common boundary. Two problems/difficulties must be dealt with in the overlap region. These are closure at the overlap region and the interpolation of overlapping grid points. Additional pseudo- or ghost-points are necessary in the overlap region of the grids in order to provide for closure of the difference stencil. Only the function values are needed at these points for the finite difference schemes to compute the derivatives within the grids. For the schemes used in this work, three ghost-points are necessary for closure. The function values for the ghost-points are determined from the function values of the grid points in the other grid. The values for the ghost-points are determined by an interpolation scheme. Although there are a variety of interpolation schemes available, we chose to use a Lagrange polynomial interpolation technique. Benefits of this technique are ease of programming, computational efficiency, and the fact that the interpolation order can be easily altered and compared. Methods There are three classes of numerical methods used in Published by Arkansas Academy of Science,

157 Interpolation Techniques for Overset Grids this work. The first deals with the approximation of a derivative. The second deals with interpolating a functionis value with n known values.

3 157 Interpolation Techniques for Overset Grids this work. The first deals with the approximation of a derivative. The second deals with interpolating a functionis value with n known values. The third is a digital filter to remove high frequency waves. Finite Difference Schemes.- A numerical solution to a first order partial differential equation requires the approximation of the first derivative based on the function values. A compact finite difference scheme (Lele, 1992) for a uniformly spaced grid is used which can be written in the form where / is the value of the function at x. /'is the derivative of/ with respect to jcat xt Axspacing between x, and xnl a,p,a,b, and care constant coefficients to be determined. Formal truncation errors up to 10th order can be obtained by an appropriate selection of the coefficients a, b, c, a, and p. The general case having non-periodic boundary conditions requires two additional relations appropriate for the near boundary nodes. These can be obtained by setting a and/or J3 equal to zero and by selecting appropriate values for the coefficients a, b, and c. The full set of equations will need the function values at three ghost-points beyond the boundary of the grid. A 10th order penta-diagonal scheme is used for the interior grid points with the followingcoefficients: , a =, fl =, a =, b =, c =. 2 H One of the near boundary relations is an 8th order tridiagonal scheme with the following coefficients: -' 3 «n 25 j. a =, a = 0, a =, b =, c 1= A dispersion-relation-preserving (DRP) scheme (Tam and Webb, 1993) with a 7-point stencil was chosen for the remaining near boundary relation. It has the following coefficients: - a = p=o, = , -= , - = (4) Interpolation.~The function values at the ghost-points can be approximated using Lagrange polynomials (Hoffman, 1992) allowing the function value to be computed as a linear combination of neighboring grid points. The weighting factors are a function only of the positions of the grid points and not the function values at the grid points. Thus, the weighting factors only need to be computed once instead of with each time step. The expression for computing the function value at the ghost-point located at x () is /(* )=2>,(O/(O (5) n(v,,-o where iv(.v u )= (6) n(v,-v,) Filtering.-Numerical techniques use approximations of derivatives to solve partial differential equations. The (!) (21 (3) truncation error of the techniques leads to the introduction of spurious high frequency signals or noise into the results. These spurious signals travel at high wave velocity in the negative direction and can create efalse pulsesi when they interact with grid boundaries or other spurious signals. Lele (1992) proposed using an implicit scheme to filter the computed signal. This can be represented in the following form: &,-,+<&,+f,+0f M +$ = M of, + j(l, +/-, )+f +/-:)+j(fm+/ (^+2 H ) (7, where /is the filtered value /is the unfiltered value. The coefficients are chosen based on the formal accuracy and the desired degree of control on the cutoff frequency. Two schemes, B and C, (Lele, 1992, Fig. 20) were utilized in this work. These methods have a 4 th order formal truncation error and use two parameters to control the cutoff frequency of the filter. Schemes Band C have the followingcoefficients for the interior grid points: a = , /} = , a = = , c = , rf = , a = , )3 = , a = , 6= , = , d = Explicit equations (4 th order formal accuracy and exact filteringof CO = n) for the near boundary nodes are given by the following: lo + iu 4/a-6/,+4/ 4-/ J ) lo >«-7'«+ A ( * +6*-'V.+/.) /3= /3+-^(-/,+4/2 +4/ 4 -/s ). Results and Discussion IDResults.~For all of the IDcases, an equal number of points are chosen on each side of the ghost-point using grid points in both of the overlapped grids as shown in Fig. 2. Also, the initial signal was a Gaussian pulse defined as W =1.0exp[-(ln2)flz! TJ (8) (9) (ii) The first set of tests was performed with the following conditions: a linear interpolation, or 2 terms for the Lagrange polynomial interpolation; Ax for each grid is 1.0; and periodic boundary conditions (provides a wrap-around effect). The amount of overlap varied from 0 to 1 with 0.5 being the 'worst case' when considering the interpolation error. Figure 3 shows the initial pulse and the pulse after propagating for selected amounts of time for each of the overlap values. At overlap values of0 and 1, the propagated pulse shows no discernable error after moving through the overlap region. The worst error occurs with an overlap value of 0.5 where it can be seen that the signal peak has a significant reduction, and there is a presence of high

158 Paul S. Sherman and Nathan B. Edgar 0 0 0 GRID1 0 0 0 GHOST-POINT WHOSE VALUE -^ IS TO BE DETERMINED Fig. 2. Example of IDinterpolation with overlap.

4 158 Paul S. Sherman and Nathan B. Edgar GRID GHOST-POINT WHOSE VALUE -^ IS TO BE DETERMINED Fig. 2. Example of IDinterpolation with overlap. GRID 2 GRID POINTS o GHOST-POINTS Interpolation Techniques Overset Grids Paul S. Sherman* and Nathan B. Edgar Arkansas State University College of Engineering P.O. Box 1740 State University, AR72467 *Corresponding Author Abstract for The use of finite difference schemes in computational aeroacoustics requires the use of structured grids in computational space. Complex geometries in the physical space can be modeled using multiple overlapping grids that are transformed into computational space. In this work, finite difference schemes are used that necessitate the addition ofpsuedo- or ghost-points in the overlap region of the grids for closure of the difference stencil. The functional values at these ghost points must be approximated from the values at the original grid points. This paper investigates interpolation techniques for these overset grids. An n* order interpolation scheme using Lagrange polynomials is applied to the one dimensional (ID) wave propagation problem to test the effects of increasing the interpolation order. This is done for both equal and unequal sized overset grids. Preliminary results from two dimensional (2D) grids willbe presented. Introduction In computational aeroacoustics, finite difference schemes are more commonly used because they are more computationally efficient than the finite element schemes. Finite difference schemes require the use of structured grids in computational space. An unstructured grid in the physical space can be transformed (Tannehill et al., 1997) into a structured grid in the computational space. However, complex geometries in the physical space may require multiple overlapping grids. In this paper, we discuss the construction of overset grids, problems associated with their construction, ID and 2D results which show some of the problems arising from the application of the grids, and some solutions implemented to address these problems. For this work, we chose to model the IDand 2D wave equations using a single pair of overlapping structured grids. In general, overlapping grids can have mismatched grid spacing (Petersson, 1999) in each direction and may not share a common point. An example of the overlap regions for ID and 2D grids is provided in Fig. 1. The layout of multiple grids in the physical space willbe dictated by the geometry to be modeled. Overlapping regions will exist wherever multiple grids share a common boundary. Two problems/difficulties must be dealt with in the overlap region. These are closure at the overlap region and the interpolation of overlapping grid points. Additional pseudo- or ghost-points are necessary in the overlap region of the grids in order to provide for closure of the difference stencil. Only the function values are needed at these points for the finite difference schemes to compute the derivatives within the grids. For the schemes used inthis work, three ghost-points are necessary for closure. The function values for the ghost-points are determined from the function values of the grid points in the other grid. The values for the ghost-points are determined by an interpolation scheme. Although there are a variety of Published by Arkansas Academy of Science,

5 159 Interpolation Techniques for Overset Grids Fig. 3. Comparison of wave propagation results for the IDcase having equal grid spacing and varying amounts of overlap 159

6 160 Paul S. Sherman and Nathan B. Edgar region or the upper boundary and that the ripple triggers a ifalsei pulse at the left boundary. Once again this illustrates the need for a filtering method to remove the unwanted signals. Filtering Tests. -Testing for ID cases of both filtering schemes was achieved by starting with a signal having a high frequency term superimposed on a low frequency term. The first test uses a high frequency with a wave number of 71 (25 cycles/50 grid points). Figure 12 shows that both filtering schemes completely remove the high frequency signal when at constant amplitude. When the amplitude is varied (not shown), additional lower frequency terms are present and are not completely filtered out. The second test uses a high frequency term with a wave number of 2.64 (21 cycles/50 grid points). Figure 13 shows that Scheme B removes more of the unwanted signal, but that near the boundaries, the filtering effect is reduced. The third test uses a high frequency term with a wave number of 2.14 (17 cycles/50 grid points). Figure 14 shows that Scheme C has the same trends as Scheme B but with a lesser filtering effect. The Gaussian pulse used in wave propagation tests, both ID and 2D, contains a continuous spectrum of frequencies. Thus, it represents a "worst case" signal for numerical wave propagation techniques. Both filtering schemes are applied to a Gaussian pulse to determine the errors induced in the signal by the filtering. Figure 15 shows the signal before filteringand the induced errors produced by each filtering scheme. Scheme C, which has a higher cutoff frequency, produces significantly less error, and the dominant error is at the near boundary nodes. spectral-like resolution. J. Comp. Physics 103: Petersson, N. A An algorithm for assembling overlapping grid systems. SIAMJ. Sci. Comput. 20-6: Tarn, C. K.W., andj. C. Webb Dispersion-Relation- Preserving finite difference schemes for computational acoustics. J. Comp. Physics 107: Tannehill, J. C, D. A. Anderson, and R. H. Pletcher Computational fluid mechanics and heat transfer. Taylor and Francis, Philadelphia. 792 pp. Conclusions The results have shown that Lagrange polynomials, of sufficiently high order, can be effectively used to approximate the function values of the ghost-points in the overlap region of ID and 2D finite difference cases. Four and six terms provide barely noticeable errors in the ID cases, and the filtering schemes presented can be used to remove the high frequency ripple that is present. More work stillneeds to be done in applying the filtering to the wave propagation model, especially in the areas near the boundaries and overlap region. The filtering produces conflicting effects. Itreduces the high frequency ripple that is induced by the approximations, and it non-uniformly attenuates a desired signal having high frequency components. Literature Cited Hoffman, J. D Numerical methods for engineers and scientists. McGraw Hill,New York, 825 pp. Lele, S. K Compact finite difference schemes with Published by Arkansas Academy of Science,

$Interpolation Techniques for Overset Grids A EQUAL 7\ /A f\ l\ f\ f\ f~\ f\ q 75.$

7 Interpolation Techniques for Overset Grids A EQUAL 7\ /A f\ l\ f\ f\ f~\ f\ q 75. GRID SIZES / \ / \ / \ / \ / \ / \ / \ / \ _ 05 OVERLAP / \ / \ / \ / \ / \ / \ / \ / \ -j " 5 /VVVYYYY\ 2 TERMS / /\ /\ /\ /\ /\ / \ /\ \ x EQUAL /A /~\ /A /A /A /A /A /A n 7 r - GRID SIZES / \ / \ / \ / \ / \ / \ / \ / \ OVERLAP / \ / \ / \ / \ / \ / \ / \ / \ : 3 TERMS / /\/\/\/\/\/\y\ V U A EQUAL f\ I\ f\ T\ f\ l\ P\ f\ GRID SIZES /\/\/\/\/\/\/\/\ 05 OVERLAP / \ / \ / \ / \ / \ / \ / \ / \ "0.5 /YYVYYYY\ 4 terms / /\/\/\/\/\/\y\ \ o.25 - /AAAAAAAx EQUAL / \ /A / \ / \ /~\ / \ /A / \ GRID SIZES /\/\/\/\/\/\/\/\ OVERLAP / \ / \ / \ / \ / \ / \ / \ / \ : 6 TERMS / / \ / \ / \ / \ y \ / \ y \ V A EQUAL 7\ /A /\ /A /A f\ /A, /A n 7c: GRID SIZES / \ / \ / \ / \ / \ / \ / \ / \ _j OVERLAP / \ / \ / \ / \ / \ / \ / \ / \ /AAAAAAAV 8 TERMS / /\/\/\/\y\/\/\ \ 0 ~i i A^. i^ i, ISO 70^ 80 Fig. 4. Comparison of wave propagation results for the IDcase having equal grid spacing and varying interpolation order

8 162 - Paul S. Sherman and Nathan B. Edgar Fig. 5. Comparison of "long distance" wave propagation results for the ID case having equal grid spacing and varying interpolation order. Published by Arkansas Academy of Science,

9 163 Interpolation Techniques for Overset Grids Fig. 6 Comparison of wave propagation results for the IDcase having unequal grid spacing and varying amounts of overlap 163

10 164 Paul S. Sherman and Nathan B. Edgar Fig. 7. Comparison of wave propagation results for the IDcase having unequal grid spacing and varying interpolation order. Published by Arkansas Academy of Science,

Interpolation Techniques for Overset Grids 0.005 i_ UJ -0.005 0 10 20 30 40 50 60 70 80 0 UJ -0.001 40 80 0.001 0.0005 o UJ -0.0005 40 80 0.001 r EQUAL GRID SIZES 0.0005 - Ul -0.0005 - OVERLAP 0.

11 Interpolation Techniques for Overset Grids i_ UJ UJ o UJ r EQUAL GRID SIZES Ul OVERLAP TERMS Fig. 8. Comparison of the typical errors in a propagated wave for the ID case having equal grid spacing and varying interpolation order

12 166 I Paul S. Sherman and Nathan B. Edgar Fig. 9. Comparison of the typical errors in a propagated wave for the ID case having unequal grid spacing and varying interpolation order. Published by Arkansas Academy of Science,

$167 Interpolation Techniques for Overset Grids r-perform INTERPOLATION ALONG THIS LINE LAST \«" "!$

13 167 Interpolation Techniques for Overset Grids r-perform INTERPOLATION ALONG THIS LINE LAST \«" "! ooo GRID 2 \ ooo««««««* \ _ _ ooo ic~j H pp] «[o] W 1 g ooo GRID1 ooo «I I o I I o I I o I PERFORM INTERPOLATIONS ^^ ALONG THESE LINES FIRST I/ o GHOST-POINTS GRID POINTS RESULTS OF FIRST INTERPOLATIONS ol RESULTS OF LAST INTERPOLATION Fig. 10. Example of 2D interpolation 80 Fig. 11. Contour map of a propagated Gaussian wave after itencounters an overlap region at y = 40 and the upper boundary

14 168 Paul S. Sherman and Nathan B. Edgar ORIGINAL SIGNAL Fig. 12. Comparison of filtering schemes superimposed on a low frequency term. with a high frequency term, having a wave number of K (25 cycles/50 grid points), Published by Arkansas Academy of Science,

15 Interpolation Techniques for Overset Grids ORIGINALSIGNAL 169

16 170 Paul S. Sherman and Nathan B. Edgar ORIGINALSIGNAL Fig. 14. Comparison of filtering schemes superimposed on a low frequency term. with a high frequency term, having a wave number of 2.14 (17 cycles/50 grid points), Published by Arkansas Academy of Science,

I Interpolation Techniques for Overset Grids 1 D 0.8 0.6-0.4-0.

17 I Interpolation Techniques for Overset Grids 1 D E-05 OH O OH OH LLJ 2E05 F FILTERED WITH SCHEME C I 1E " 5 I UJ -1E UO Fig. 15. Comparison of the error induced by filtering Schemes B and C after being applied to a Gaussian pulse

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