COMBINATION OF INTERSECTION- AND SWEPT-BASED METHODS FOR SINGLE-MATERIAL REMAP

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1 Combination of intersetion- and swept-based methods for single-material remap 11th World Congress on Computational Mehanis WCCM XI) 5th European Conferene on Computational Mehanis ECCM V) 6th European Conferene on Computational Fluid Dynamis ECFD VI) E. Oñate, J. Oliver and A. Huerta Eds) COMBINATION OF INTERSECTION- AND SWEPT-BASED METHODS FOR SINGLE-MATERIAL REMAP MATĚJ KLÍMA, MILAN KUCHAŘÍK AND MIKHAIL SHASHKOV Faulty of Nulear Sienes and Physial Engineering FNSPE) Czeh Tehnial University in Prague Břehová 7, Praha 1, , Czeh Republi XCP-4 Group, MS-F644 Los Alamos National Laboratory Los Alamos, NM 87545, USA Key words: Arbitrary Lagrangian-Eulerian Methods, Remap, Flux-based Remap Abstrat. In this paper, we apply the ideas of the hybrid multi-material remapping algorithm in the ase of single-material disrete funtions. We analyze both intersetionand swept-based remapping approahes and show that the swept-based approah an potentially produe higher numerial error and violation of solution symmetry. To fix this, we test two swithes piking one or the other method, depending on the funtion derivatives. We demonstrate properties of both original methods and their ombination on a seleted numerial example. 1 INTRODUCTION In the numerial simulations of fluid flows, the hoie of the omputational mesh is ruial. Traditionally, there have been two viewpoints utilizing the Lagrangian or the Eulerian framework, eah with its own advantages and disadvantages. In a pioneering paper [1], Hirt et al. developed the formalism for a mesh whose motion an be determined as an independent degree of freedom, and showed that this general framework an be used to ombine the best properties of the Lagrangian and Eulerian methods. This lass of methods has been termed Arbitrary Lagrangian-Eulerian or ALE. Many authors have desribed the ALE strategies to optimize their auray, robustness, or omputational effiieny. It isusual toseparatethealesheme into threedistint stages: 1)aLagrangianstage, in whih the fluid quantities and the omputational mesh are advaned to the next time level; 2) a rezoning stage, in whih the geometrial quality of the omputational mesh 5977

2 is improved; and 3) a remapping stage, in whih all fluid quantities are onservatively transferred from the Lagrangian to the rezoned mesh. In this paper, we fous on the last part of the ALE sheme the remapping stage in 2D Cartesian geometry. For realisti simulations, multi-material ALE is typially used. In this ase, more than one material is allowed in eah ell of the omputational mesh. Multi-material remapping typially involves intersetions of the pure material polygons with the neighboring ells to onstrut the mass fluxes. In the series of papers [2, 3, 4], the idea of hybrid remapping is presented. The expensive intersetions are used only in the viinity of material interfaes, while a heaper swept-based method is used in pure-material regions. In this paper, we analyze both intersetion-based and swept-based remapping methods and show that the swept-based approah an bring additional error to the remapped values in ertain situations. It is therefore possible to apply the same methodology in the single-material ase in order to redue the numerial error and improve the symmetry of the remapped quantity field. We suggest two swithes whih alter the methods in a unified flux-based pseudo-hybrid framework and demonstrate their performane on a seleted numerial example. The rest of the paper is organized as follows. In Setion 2, the standard intersetionbased and swept-based remapping methods are reviewed and the idea of pseudo-hybrid remapping is introdued. In Setion 3, the error analysis of both approahes is performed in a onsiderably simplified ase and the partiular terms responsible for additional numerial error of the swept-based approah are identified. In Setion 4, we suggest two swithes based on the first and seond derivatives, hoosing the appropriate methods in different parts of the domain. Both methods are ompared with their pseudo-hybrid extension on a seleted numerial example in Setion 5. Finally, the whole paper is onluded in Setion 6. 2 REMAPPING METHODS Remapping represents a onservative transfer of all fluid quantities from the original Lagrangian) omputational mesh to the new rezoned) one. In this paper, we only fous on the remap of a single ell-entered quantity density ρ. For more details on the remap of the full set of fluid quantities in the same framework, see [5]. If both original and new meshes have the same topology, suh as shown in Figure 1 a), the remapping of mass in ell an be written in a flux form m = m + C) F, m, 1) where represents the new ell, represents a ell from the neighborhood C) of ell, and F m, represents the mass flux from to. The omputation of suh fluxes depends on the partiular remapping method. Finally, the remapped density is obtained just by diving by the new ell volume, ρ = m /V. 5978

3 F m, F m, a) b) ) Figure 1: a) Original red) and rezoned blak) omputational mesh. b) Intersetion-based fluxes between and its neighbors shown in different olors. ) Swept-based fluxes between and its edgeneighbors shown in different olors. 2.1 Intersetion-based remap The most straightforward approah for quantity remapping is the intersetion-based method, also known as overlays. For details, see for example [6, 5]. This approah an be used in the flux form 1), where the mass fluxes are omputed as F, m = ρ x,y)dxdy ρ x,y)dxdy. 2) For an example, see Figure 1 b). Here, the first term represents the part of the flux whih needs to be added to the ell as the partiular piee of the domain was added to during the rezoning stage. Similarly, flux orresponding to the removed piees of is represented by the seond term here. The density reonstrution ρ x,y) in ell is performed by a standard pieewise-linear reonstrution proess suh as desribed in [7]. This approah is known to be onservative, onsistent, linearity-preserving, seond order aurate, and in the ase of limited reonstrution also loal-bound-preserving. On the other hand, the intersetions need to be onstruted, whih inreases the omputational ost of this method signifiantly. 2.2 Swept-based remap In the swept-based approah, the fluxes are approximated using swept regions instead of onstrution of the intersetions [6, 8]. Swept region is defined by the motion of a partiular ell edge during the rezoning stage, so only fluxes belonging to the edgeneighbors exist. For explanation, see Figure 1 ). The flux form 1) an then be redued to m = m + Fe m, 3) e E) 5979

4 where the flux through edge e is onstruted as Fe m = ρ x,y)dxdy. 4) Ω e Here, Ω e represents the swept region orresponding to edge e and the reonstrution is taken either from ell or from its neighbor over the edge e, depending on the sign of the swept region area, { if VΩe < 0 = if V Ωe 0. 5) This approah is signifiantly more effiient as no intersetions are required. It is onservative, onsistent, linearity-preserving, and seond order aurate. On the other hand, the loal-bound-preservation is not guaranteed if no additional mehanism is applied [8]. 2.3 Pseudo-hybrid remap In a series of papers [2, 3, 4], we have developed the framework of hybrid remapping for the multi-material quantity remapping. The intersetion-based remap an be generalized for multiple materials in a straightforward way, while this generalization is not lear for the swept-based approah. The main idea of the hybrid remapping tehnique is following: perform the expensive intersetions only in the viinity of material interfaes, while heaper swept-based remap is done in the rest of the domain. In this paper, we inorporate the same idea for the single-material ase. However, the main motivation is slightly different here. In the next Setion, we analyze the errors of the intersetion- and swept-remapped density values for a simplified ell motion during the rezoning stage. We demonstrate that the error of the swept-based remap ontains additional terms in ertain ases whih an result in a higher numerial error and espeially symmetry violation of the remapped values. Therefore, eah of these methods an be more suitable in different parts of the omputational domain. Here, we adopt the simplest twostep approah of hybrid remap [3] and ombine both remapping approahes by a different riteria than the presene of the material interfae funtion derivatives, deviation of the funtion reonstrution in the neighborhood, pathologial motion of the ell in the rezoning stage, et. The two-step hybrid remapping method does the omplete Eulerian part of the ALE algorithmrezoning+remapping) twie. First, all mesh nodes are marked as pureall their adjaent ells are pure and ontain the same material) or mixed. In the first step, only pure nodes are rezoned. This annot affet any mixed ells so the swept-based remapping method an be used as only pure material fluxes are omputed. In the seond step, the remaining mixed nodes are rezoned and the intersetion-based remapping method is used, whih treats the multi-material fluxes properly. Let us note, that this ombination keeps onsisteny, linearity-preservation, and the seond order of auray of both remapping methods. Typially, it improves the omputational ost of the remapper. On the other 5980

5 hand, the overhead of the method involves marking of the nodes and remapping by both methods in the buffer region of ells ontaining both pure and mixed nodes, whih an derease the effiieny of the method in the ase of more omplex material interfaes. In the pseudo-hybrid method proposed in this paper, we suggest similar ombination of the intersetion-based and swept-based remapping methods. However, the swith marking the nodes annot be based on material interfaes, as we fous on single-material quantity here. Two possible swithes are suggested in Setion 4. 3 ERROR ANALYSIS OF REMAPPING METHODS In papers ontaining the omparison of the intersetion-based and swept-based remapping methods suh as [6, 8, 9], the numerial errors of both approahes are omparable and the swept-based remap an be even slightly better in some ases. In [6], a basi error-analysis of both approahes is performed. The von Neumann analysis for a partiular flux is performed in [10], explaining the slightly-lower numerial error of the sweptbased method in the ase of the traverse flows. Here, we perform similar analysis for a full quadrilateral omputational ell by remapping a higher-order Taylor series using its pieewise-linear reonstrution. Let us assume that we have a polynomial funtion fx,y) = C 0 +C x x+c y y C x 2x2 +C xy xy C y 2y C x 3x C x 2 yx 2 y C xy 2xy C y 3 y3, 6) representing the third-order Taylor series entered in the origin, where the C oeffiients represent the funtion values/derivatives. Let us onstrut a 5 5 equidistant uniform mesh overing the entire 5 2 l x, 5 2 l x 5 2 l y, 5 2 l y domain, where l x and l y stand for the ell size in eah diretion. Let us perform the reonstrution in eah ell by the pieewiselinear funtion in a least-squares manner, as desribed in [5]. Let us move the nodes of the entral ell by same vetor [dx,dy], as shown in Figure 2. Now, we perform remapping of the funtion to the new mesh using both intersetion-based and swept-based approahes. We have implemented the pieewise-linear funtion reonstrution and both remapping approahes in the omputer algebra system Maple and are able to perform the full remap with general parameters [l x,l y ] and [dx,dy]. Next to it, we have onfirmed the following results in a standard numerial remapping ode. The 5 5 mesh is large enough to avoid any boundary effets to appear in the entral ell. The error in ell for the intersetion-based remap an be omputed as the differene between the remapped value and the analyti funtion evaluated in the entroid of the new ell, E int = ρ int fx,y ) = C x 3 lx 4 dx2 l2 x 12 dx 1 6 dx3 ) +C y 3 ly 4 dy2 l2 y 12 dy 1 ) 6 dy3. 7) 5981

6 ~ dx,dy ly lx Figure 2: Original equidistant blak) and new red) mesh. All verties of the entral ell are shifted by [dx,dy]. As we an see, the error ontains only the third-order derivatives, so the intersetionbased remap will be exat for a paraboli funtion in the speial ase of ell translation. Moreover, no terms ontaining mixed derivatives are present in the error. Same approah an be applied for the swept-based error onstrution, E swept = ρ swept fx,y ) = E int lx +C x 2 y 4 dxdy 1 ) 2 dx2 dy +C xy 2 ly 4 dxdy 1 ) 2 dxdy2. 8) As we an see, the error ontains exatly the same terms as in the ase of the intersetionbased error and additional terms ontaining the mixed derivatives. Unfortunately, the magnitude of these error formulas depends not only on the polynomials in dx and dy with the partiular ell parameters l x, l y ), but also on the C oeffiients, whih an be both positive and negative, and depend on the partiular funtion fx,y). On the other hand, the swept-based error formula ontains the same ommon terms as the intersetions-based error formula and the extra terms whih an inrease its magnitude if the C oeffiients have the right sign worst ase senario). In other words, for C x 3 + C y 3 C x 2 y + C xy 2, both methods have omparable numerial errors. If their signsareopposite, C x 3 +C y 3 C x 2 y +C xy 2), theswept-based methodalwaysprodues higher numerial error. This onlusion is supported by the plot of the error terms shown in Figure 3 for a speial ase of l x = l y = 1. It is simple to investigate that for our given mesh) the absolute value of the extra term if always slightly larger than the absolute value of the ommon term. Let usnote, that infigure3, thevalueofdy = 0.5worst ase) is assumed. For smaller values of dy, the magnitude of the extra term will derease as it is linear in dy. On the other hand, for dy small, term opposite from 8) with dy and dx swithed) will beome dominant and the extra term will beome signifiant again. The onlusion therefore is that in the ase of a omputational ell translation, the terms orresponding 5982

7 ommon term extra term 0.01 error terms !"!!$!"!# dx Figure 3: Plot of the ommon error term dx2 4 dx 12 dx3 6 appearing in both intersetions-based and swept-based error formulas blue line) and the additional error term dxdy 4 dx2 dy 2 for a hosen value of dy = 0.5 representing the maximum motion) appearing only in the swept-based error formula red line) as funtions of dx. to the mixed derivatives of the remapped funtion an bring signifiant ontribution to the total error of the swept-based remapping approah, whih do not show up for the intersetion-based approah. Let us remind, that this simple analysis only holds for a very speial ase of ell translation. For a general ase of arbitrary ell deformation, we are not able to analyze the situation properly as the error formulas are too ompliated. However, we are able to analyze few more simplified ases, suh as ell ompression/expansion, single node motion, or hourglass-type edge deformation, for whih similar onlusions an be done. 4 SWITCHES OF REMAPPING METHODS In this paper, we use two different swithes for ombining the intersetion-based and swept-based remapping methods in different parts of the omputational domain in the two-step hybrid remapping manner [3]. Let us note that these swithes are onstruted for eduational purposes only, in reality, one an onstrut a swith adapted for the partiular needs. We did some preliminary tests with the swithes deteting standard deviation of the density error in the surrounding ells, hourglass detetor, or detetor of mixed derivative magnitude, for preliminary results see [11]. The first swith is inspired by the Harten-Zwas sheme [12] and utilizes the first derivative of the density funtion. The motivation is lear: it will trigger the intersetion-based remap in regions with high gradients, deteting the funtion disontinuities where the swept-based remap is expeted to produe maximal error. The swith is omputed as s = 1 ρ max ρ x ) 2 + ) 2 ρ, 9) y whereρ max standsforthemaximal densityintheentiredomainandthedensityderivatives in ell are omputed by a standard pieewise-linear reonstrution tehnique. The swith 5983

8 is ompared with a threshold, typial values are between 0.1 and 10. The seond swith is based on the magnitude of the seond derivative in order to detet areas of high urvature of the disrete funtion. It does not false trigger in regions of onstant gradients. The funtion urvature in ell is desribed by its Hessian matrix, H = 2 ρ ) x ) 2 2 ρ y x ) 2 ρ x y 2 ρ y ) 2, 10) where the seond derivatives are omputed numerially from the first derivatives, using the same standard least-squares-based reonstrution tehnique. A loal extremum is present if the matrix is positive or negative definite. To hek this, we onstruted a swith based on the determinant of this matrix, s = 1 ρ max ) 2 ρ x 2 ) 2 ρ y 2 2 ρ x y ) 2. 11) The higher this value is, the higher urvature is present. If a partiular threshold value is reahed, the intersetion-based method is performed. 5 NUMERICAL EXAMPLE In this Setion, we demonstrate the performane of the remapping methods on a partiular example taken from [3]. In this test, the initial ell masses are set aording to a disontinuous radially-symmetri density funtion { x 1/2) 1+e y 1/2) 2 if x 1/2) fx,y) = 2 +y 1/2) 2 1/4 12) x 1/2) 1+e 6 2 +y 1/2) 2 1/4 otherwise in the entire 0,1 2 domain overed by an equidistant orthogonal omputational mesh ontaining 25 2, 50 2, 100 2, 200 2, or ells. To enhane the effets of eah remapping method, we perform the yli remapping test using the tensor-produt mesh motion [6], x n i = x 0 i 1 dn )+ xi) 0 3 d n, yi n = yi 0 1 dn )+ ) yi 0 2 d n, 13) d n = sin2πtn ), t n = n/n max, 2 14) where x n i,yn i ) denotes the nodal position of node i in the atual remapping step n, n max stands for the total number of steps, and x 0 i,yi) 0 denotes the nodal position in the initial mesh. The number of remapping steps inreases with the mesh size, n 25 max = 50, n 50 max = 100, n100 max = 200, n200 max = 400, n400 max = 800. This mesh motion ontains a lot of ell translations analyzed in Setion 3) ombined with ell ompressions and expansions, 5984

9 having similar effets. The same series of remapping steps is performed by all available remapping methods intersetion-based, swept-based, and the pseudo-hybrid remapping with the swithes based on the first and the seond derivatives from Setion 4. The threshold values used in the simulations are S 1stder = 8 and S 2ndder = 37. These values were hosen heuristially to minimize the number of ells treated by the intersetion-based approah while keeping good symmetry properties of the solution. In general, this test represents remapping of a symmetri funtion over a series of non-symmetrially moving meshes. Suh situation an appear in real hydrodynami simulations if the mesh rezoner does not take the symmetri nature of the problem into aount. Table 1: L 1 numerial error of eah remapping method intersetion-based, swept-based, and pseudohybrid with swithes based on the first/seond derivatives) for different mesh resolutions. Mesh size Intersetions Swept PH 1st deriv. PH 2nd deriv In Table 1, we an see the total numerial error of eah simulation. As the initial and final meshes oinide, the error an be omputed as L 1 = m m 0, 15) m0 where m stands for the final mass in ell after the series of remaps and m 0 is the initial mass. As we an see, both intersetion-based and swept-based methods exhibit omparable errors, in general, the swept-based approah is performing slightly better. This orresponds to the observations from [6, 8, 9, 10]. The density profiles are shown in Figure 4. We an see the symmetry violation of the swept-based approah, whih does not show up in the intersetion-based approah. This violation an be seen also for the pseudo-hybrid remaps, but it is not so signifiant. To onfirm this observation, we alulate the relative standard deviation of the L 1 errors to express its variability over the four quadrants of the domain, σ Ll = q=1 Lq 1 L 1) 2 L 1, 16) where L q 1 represents the total mass error for a partiular quadrant q and L 1 stands for the total mass error over the entire domain. The results for all remapping methods are shown 5985

10 Initial profile Profile in n = n max /4 Intersetion Swept Pseudo-hybrid, 1st deriv. swith Pseudo-hybrid, 2nd deriv. swith Figure 4: Initial, intermediate maximal mesh deformation), and final density profiles for all remapping methods intersetion-based, swept-based, and pseudo-hybrid with swithes based on the first/seond derivatives) of double-exponential funtion on 25 2 mesh. in Table 2. As we an see, the error deviation is signifiantly lower for the intersetionbased method, so this remapping approah preserves symmetri funtion better than the swept based approah. This symmetry violation is aused by the terms ontaining mixed derivatives in equation 8). We an also see that the pseudo-hybrid methods show omparable error deviations with the intersetion-based approah for higher mesh resolutions). 5986

11 Table 2: Error deviation σ L1 in %) of eah remapping method intersetion-based, swept-based, and pseudo-hybrid with swithes based on the first/seond derivatives) for different mesh resolutions. Mesh size Intersetions Swept PH 1st deriv. PH 2nd deriv Let us also omment on the omputational effiieny of the remapping methods. This analysis is strongly implementation-dependent and an differ signifiantly in different odes. In general, the intersetion-based approah is the slowest one as it requires onstrution of the expensive ell intersetions, while the swept-based approah is fastest. The pseudo-hybrid approahes ontain ertain overhead, as desribed in Setion 2.3. In our ode, the intersetion based approah is about 4.2-times slower than the swept-based approah for the finest mesh. The pseudo-hybrid methods were about 1.8-times slower, whih is signifiantly better than the intersetion-based approah while keeping its symmetry properties. 6 CONCLUSIONS In this paper, we analyzed the standard intersetion-based and swept-based remapping tehniques and identified the mixed-derivative terms responsible for the numerial error and espeially symmetry violation in the ase of the swept-based method. We followed the idea of hybrid remapping and suggested a ombination of both approahes, whih dereases these errors and improves symmetry of the remapped funtion. Performane of all methods was demonstrated on a stati-remapping numerial test. 7 ACKNOWLEDGMENTS This work was performed under the auspies of the National Nulear Seurity Administration of the US Department of Energy at Los Alamos National Laboratory under Contrat No. DE-AC52-06NA25396 and supported by the DOE Advaned Simulation and Computing ASC) program. The authors aknowledge the partial support of the DOE Offie of Siene ASCR Program. This work was partially supported by the Czeh Tehnial University grant SGS13/220/OHK4/3T/14, the Czeh Siene Foundation projet S and by the Czeh Ministry of Eduation projet RVO

12 REFERENCES [1] Hirt, C.W., Amsden, A.A., and Cook, J.L. An arbitrary Lagrangian-Eulerian omputing method for all flow speeds. J. Comput. Phys. 1974) 14: [2] Kuharik, M., Breil, J., Galera, S., Maire, P.-H., Berndt, M., and Shashkov, M. Hybrid remap for multi-material ALE. Comput. Fluids 2011) 46: [3] Berndt, M., Breil, J., Galera, S., Kuharik, M., Maire, P.-H., and Shashkov, M. Twostep hybrid onservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods. J. Comput. Phys. 2011) 230: [4] Kuharik, M. and Shashkov, M. One-step hybrid remapping algorithm for multimaterial arbitrary Lagrangian-Eulerian methods. J. Comput. Phys.2012) 231: [5] Kuharik, M. and Shashkov, M. Conservative multi-material remap for staggered multi-material arbitrary Lagrangian-Eulerian methods. J. Comput. Phys. 2014) 258: [6] Margolin, L.G. and Shashkov, M. Seond-order sign-preserving onservative interpolation remapping) on general grids. J. Comput. Phys. 2003) 184: [7] Barth, T.J. Numerial methods for gasdynami systems on unstrutured meshes. An introdution to Reent Developments in Theory and Numeris for Conservation Laws, Proeedings of the International Shool on Theory and Numeris for Conservation Laws. Springer, [8] Kuharik, M., Shashkov, M., and Wendroff, B. An effiient linearity-and-boundpreserving remapping method. J. Comput. Phys. 2003) 188: [9] Loubere, R. Private ommuniation, 2013). [10] Lauritzen, P.H., Erath, Ch., and Mittal, R. On simplifying inremental remap -based transport shemes. J. Comput. Phys. 2011) 230: [11] Klima, M. Pseudo-hybrid remapping. Tehnial report, Czeh Tehnial University in Prague, 2013). [12] Laney, C.B. Computational Gasdynamis. Cambridge University Press, 1998). 5988