Application of Quaternions for Mesh Deformation

Transcription

1 NASA/TM Application of Quaternions for Mesh Deformation JamshiA.Samareh Langley Research Center, Hampton, Virginia April 2002

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3 NASA/TM Application of Quaternions for Mesh Deformation Jamshi A. Samareh Langley Research Center, Hampton, Virginia National Aeronautics an Space Aministration Langley Research Center Hampton, Virginia April 2002

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5 Abstract Anewthree-imensionalmesheformationalgorithm,baseonquaternionalgebra,is introuce.abriefoverviewofquaternionalgebraisprovie,alongwithsomepreliminary resultsfortwo-imensionalstructureanunstructureviscousmesheformation. Introuction Mesh eformation is an important element in the analysis of moving boies an shape optimization. The lack of robust an efficient mesh eformation tools is still a major barrier to routine applications of high-fielity tools such as computational flui ynamics (CFD) an computational structural mechanics (CSM) for multiisciplinary analysis an optimization. For example, CFD application for shape optimization requires a robust, automatic, an efficient tool to propagate the bounary eformation into the fiel mesh. For the graient-base optimization, the efficiency is particularly crucial where in aition to the bounary eformation the sensitivity of the bounary coorinates must be propagate into the fiel mesh. Figure 1 shows an example of a bounary perturbation, where the bounary has been eforme, rotate, an translate. Y Uneforme Y Deforme X X Fig. 1 Uneforme an eforme Meshes The bounary eformation is efine as δ = r r (1) u i i i u where ri are the uneforme bounary coorinates, an ri are the eforme bounary coorinates. There are two basic techniques to propagate the bounary perturbations into the fiel mesh: 1) mesh regeneration, an 2) mesh eformation. The next two subsections provie an overview of these techniques for structure an unstructure meshes. StructureMesh Most structure gri regeneration an eformation techniques are base on transfinite interpolation (TFI). Gaitone an Fies have provie a mesh regenerating technique base on TFI with exponential blening functions [1]. The choice of blening functions has a consierable influence on the quality an robustness of the fiel mesh. Soni has propose a set of blening functions base on arclength [2]; such a set is extremely effective an robust for mesh regeneration an eformation. Jones an Samareh have presente an algorithm for general multiblock mesh regeneration an eformation base on Soni's blening functions [3]. 1

6 Hartwich an Agrawal have use a variation of the TFI metho [4]. They have introuce two new techniques: the use of the ì slave-masterî concept to semiautomate the process, an the use of a Gaussian istribution function to preserve the integrity of meshes in the presence of multiple boy surfaces. Wong et al. have use Algebraic an Iterative Mesh 3D (AIM3D), which is base on a combination of algebraic an iterative methos [5]. Leatham an Chappell have use a Laplacian technique more commonly use for unstructure mesh eformation [6]. UnstructureMesh For unstructure meshes with large geometry changes, a new mesh may nee to be regenerate at the beginning of each optimization cycle. Botkin has introuce a local remeshing proceure that operates only on the specific eges an faces associate with the esign variable changes [7]. Similarly, Koiyalam, Kumar, an Finnigan have use a mesh regeneration technique base on the assumption that the soli moel topology stays fixe for small perturbations [8]. Soli moel topology comprises the number of mesh-points, eges, an faces. Any change in the topology will cause the moel regeneration to fail. To avoi such a failure, a set of constraints among esign variables must be satisfie, in aition to constraints on their bouns. During shape optimization, the bounary mesh may unergo many small eformations; it woul be too costly to regenerate the mesh in response to these eformations. In aition, the new, regenerate mesh may not have the same number of mesh points an/or the same connectivity. Either of these situations will result in iscontinuous sensitivity erivatives. Batina has presente a mesh eformation algorithm that i alleviate the nee for mesh regeneration. Batinaís approach moels mesh eges with springs [9]. The spring stiffness k jk for a given ege jk is taken to be inversely proportional to the element ege length. Then, the fiel mesh movement is compute through the static equilibrium equations: δ k δ n jk n m =, where k jk = k jk rj rk m The summation is over all the eges of the elements. The coefficient k jk is relatively large for small cells. Therefore these small cells, which are usually near the bounary of the boy, ten to unergo rigi boy movement. This rigi boy movement avois rapi variations in eformation, thus eliminating the possibility of small cells having very large changes in volume. These large changes coul lea to negative cell volumes. Blom [10] has provie a etaile analysis for the spring metho an raws an analogy between the spring metho an an elliptic ifferential equation approach for structure mesh generation. Zhang an Belegunu have propose an algorithm similar to the spring analogy that can hanle large mesh eformation [11]. They have use the ratio of the cell Jacobian to the cell volume for the spring stiffness. Crumpton an Giles have foun the spring analogy inaequate an ineffective for large mesh eformations [12] an propose a formulation base on the heat conuction equation with the coefficient of thermal conuctivity inversely proportional to cell volume. They attribute their success to the choice of cell volume use in the criteria for a vali (2) 2

7 mesh. In contrast, the spring analogy uses only eges, which are not irectly linke to the mesh valiity. Farhat et al. [13] have propose a moification to the spring analogy algorithm to inclue aitional torsional spring to control mesh skewness an foling. For two-imensional applications, they emonstrate that the moifie algorithm has avantages in terms of robustness, quality, an performance. Tezuyar an Behr [14] have propose an algorithm base on linear elasticity, which inclues full stress tensor. Cavallo et al. [15] have applie this metho to mesh eformation for aero/propulsive flowfiel calculations. They note that the metho preserves the mesh quality, an it prouces a better mesh than the spring analogy metho. The linear elasticity approach requires solving the complete stress tensor. In contract, the spring analogy represents only the iagonal elements of the stress tensor. Cavallo et al. have conclue that the elasticity approach is consierably more expensive. Roleofbounaryorientationinmesheformation The traitional eformation algorithms, such as interpolation an spring analogy, use bounary translation to eform the fiel mesh. However, the bounary eformation alters the bounary position as well as the bounary orientation (i.e., rotation angle) as shown in Fig. (2). The traitional mesh eformation algorithms o not use this aitional information on the changes in the bounary orientation. Morton, Melville, an Visbal [16] have propose a TFI algorithm to interpolate the bounary eformation as well as the changes in the orientation through Euler angle. They conclue the inclusion of Euler angle preserves the mesh orthogonality for significant eformations. They successfully applie the algorithm to a two-imensional structure CFD mesh. n u n θ u Y Y X X Uneforme Deforme Fig. 2 Orientation changes cause by eformation Extension of the approach of Morton, Melville, an Visbal [16] to three-imensional applications requires the irect interpolation of the changes in the orientation (three Euler angles). Historically, Euler angle representation is the most popular interpolation technique for orientation [17]. Euler angles ignore the interaction of multiple rotations about the separate axes which coul lea to ì gimbal lockî as escribe by Watt an Watt [17]. 3

8 The three Euler rotations can be accomplishe by a single rotation about a vector. This single rotation simplifies the interpolation process, but it has the inherent problem of non-smooth interpolation an the so-calle ì gimbal-lock.î To avoi these problems, quaternions are use to represent the changes in the bounary orientation. A brief introuction of quaternion algebra is presente in the next section. Then, a general threeimensional mesh eformation algorithm base on quaternion algebra is presente. Whatarequaternions? Only brief review of quaternion algebra is provie here; reaers are referre to the work of Altmann [18], Shoemake [19], an Philips, Hailey, an Gerbert [20] for more etails. There is some controversy on who invente quaternion algebra. The articles by Altman [18] an Philips et al. [20] provie a very interesting history of quaternion algebra. A quaternion is a generalize complex number (hypercomplex number) that is compose of one real an three imaginary numbers ( Q= q0 + qi 1 + q2j+ q3k), where ii = jj = kk = 1, ij = ji = k, jk = kj = i, ki = ik = j. The following is a set quaternion properties that will be use later: * Conjugate of a quaternion, Q = q qi q j q k Magnitue of a quaternion, Q = QQ = q + q + q + q * Unit quaternion, Q = 1 Associative, ( QQ 1 2) Q3 = Q1( QQ 2 3) Not commutative, QQ 1 2 QQ * * Inverse of a quaternion, Q = Q /( QQ ) 1 * For unit quaternions, Q = Q A quaternion can be interprete as a scalar together with a vector (irection), Q= [, s v], s = q0, v = ( q1, q2, q3) In this notation, quaternion multiplication has the particularly simple form QQ 1 2 = [ s1, v1][ s2, v2] = [ ss 1 2 v1 v2, s1v2 + s2v1+ v1 v 2] where enotes the vector ot prouct, an enotes the vector cross prouct. Quaternions are ieal for moeling rotations. The last three components of a quaternion represent the axis aroun which the rotation occurs, an the first component represents the magnitue of the rotation. There are three steps involve in rotating a point, p, about a unit vector, u, by an angle, θ. First, a quaternion is constructe for the point as P = [0, p ]. Secon, a quaternion is constructe for the rotation as θ θ Q= [, s v], s = cos, v = u sin (3) Thir, the point is rotate as Protate = QPQ. If Q is a unit quaternion, then we can use the * conjugate of the quaternion to perform the rotation, = QPQ. Multiple rotations can be Protate 4

9 simplifie by using a single quaternion. For example, if Q1 an Q 2 are unit quaternions representing two rotations, the two rotations can be combine as Q2( QPQ 1 1 ) Q2 = ( Q2Q1) P( Q1 Q2 ) = ( Q2Q1) P( Q! 2Q1) "#$ Quaternion coorinates represent rotation as Cartesian coorinates represent translation as a single vector. This characteristic has been fully exploite in representing attitue of aircraft kinematics [20]. Quaternion coorinates are best for interpolation of orientation as use in computer animation. Shoemake has presente a robust an efficient application of quaternions for BÈzier interpolation of orientation use in computer animation [19]. QuaternionsanMeshDeformation This section presents a technique to moel the bounary eformation by quaternion algebra. When these bounary quaternions are applie to the uneforme bounary mesh, they prouce the eforme bounary mesh an orientation. The eformation vectors, δ, represent the bounary translation, which is efine in the Eucliian space. In traitional mesh eformation algorithms, these vectors are use to propagate the eformation into the fiel mesh. In a similar manner, we will use the bounary quaternions to propagate the eformation into the fiel mesh. The process of etermining bounary quaternions is ivie into three steps, as shown in Fig. (3). n u n 0 0 Uneforme Deforme nu n n n Step 1 Step 2 Step 3 Fig. 3 Process of bounary quaternion construction 5

10 In step 1, the mesh points for the uneforme bounary ( r ), the eforme bounary ( r ), an the neighboring points ( r u an u i r ) are translate to the origin. u1 u u 1 r = r ri, r = r r i (4) u1 In the secon step, r is rotate so that the uneforme bounary normal vector aligns with the eforme bounary normal vector. This rotation is moele with a quaternion. First, the normal vector of a plane share by both eforme an uneforme normal vectors share (efine as u= nu n ) an the angle α between two normal vectors is etermine. Then, a quaternion is u efine for the rotation as Q1 [cos α, sin α u1 = n n ]. Points r are rotate by quaternion to 2 2 u2 form r, such that r = Q[0, r ] Q (5) u2 u u2 In the thir step, points r are rotate about the eforme bounary normal vector to minimize the angle between corresponing neighboring points. The optimum rotation angle, θ, is efine u2 as the average angle between corresponing eges of r an the eges of eforme bounary. Another quaternion can then be efine for this rotation, Q2 = [cos θ, n sin θ ]. 2 2 These two quaternions are combine to form a single quaternion as Qi = QQ 1 2. The total translation vector for the bounary can now be efine as i = ri i, where 1 [,0] Q[0, r ] Q. i = i i i Quaternions an total translation vectors for all bounary mesh points have been compute. The translation vectors account for the translation, an quaternions account for the changes in the bounary orientation. The translation vectors an quaternions are propagate into the fiel mesh by one of the traitional eformation algorithms such as TFI or the spring analogy. Then, the fiel mesh is upate base on the fiel values for the translation vectors an quaternions as Results R = +Q [0, R ] Q (6) u 1 fiel fiel fiel fiel fiel The results are presente for structure an unstructure viscous mesh eformations. Figure (4) shows a viscous structure mesh with 257x65 mesh points. The uneforme mesh lines are orthogonal to the bounaries. The bounary mesh is eforme, rotate, an translate to simulate aeroelastic eformation. Figure (4) shows comparisons of TFI (left sie of figure) an quaternion approach (right sie of figure). Unlike the traitional TFI, the quaternion approach can clearly preserve the bounary orthogonality. Because the bounary quaternions are base on the changes in the bounary mesh point positions as well as the orientations, the algorithm can guarantee that the mesh near the bounary has the same characteristics as the uneforme mesh. Figure (4) clearly emonstrates this important property. i 6

11 Next, the quaternion approach is applie to an unstructure viscous mesh, where the flap has been rotate. The spring analogy was use to propagate the bounary quaternions to the fiel mesh. The results are shown in Fig. (5). Again, the use of quaternion has preserve the mesh characteristics. Conclusions A new three-imensional mesh eformation algorithm base on quaternion algebra has been presente. These preliminary two-imensional results inicate the traitional algorithms such as TFI an spring analogy can be easily augmente with the quaternions to preserve mesh quality near the viscous bounary. We plan to apply this metho for three-imensional structure an unstructure viscous meshes. We also plan to evaluate the quality of meshes eforme by quaternion approach by means of CFD applications. 7

12 (a) Deforme Uneforme Interpolation Quaternion (b) Interpolation Quaternion Fig. 4 Deformation comparison for structure mesh Uneforme Deforme Fig. 5 Deformation comparison for unstructure mesh 8

13 [1] Gaitone, A. L., an Fies, S. P., ì Three-Dimensional Moving Mesh Metho for the Calculation of Unsteay Transonic Flows,î Aeronautical Journal, Vol. 99, No. 984, 1995, pp. 150ñ160. [2] Soni, B. K., ì Two- an Three-Dimensional Gri Generation Internal Flow Applications,î AIAA Paper , January [3] Jones, W. T., an Samareh, J. A., ì A Gri Generation System for Multiisciplinary Design Optimization,î AIAA Paper , June (Available online at [4] Hartwich, P. M., an Agrawal, S., ì Metho for Perturbing Multiblock Patche Gris in Aeroelastic an Design Optimization Applications,î AIAA Paper , [5] Wong, A. S. F., Tsai, H. M., Cai, J., Zhu, Y., Liu, F., ì Unsteay Flow Calculations with a Multi-Block Moving Mesh Algorithm,î AIAA , January [6] Leatham, M., an Chappell, J. A., ì On the Rapi Regeneration of Hybri Gris Due to Design Driven Geometry Perturbation,î Sixth International Conference on Numerical Gri Generation in Computational Fiel Simulation, Mississippi State University, 1998, pp. 533ñ542. [7] Botkin, M. E., ì Three-Dimensional Shape Optimization Using Fully Automatic Mesh Generation,î AIAA Journal, Vol. 30, No. 5, 1992, pp. 1932ñ1934. [8] Koiyalam, S., Kumar, V., an Finnigan, P., ì Constructive Soli Geometry Approach to Three- Dimensional Structural Shape Optimization,î AIAA Journal, Vol. 30, No. 5, 1992, pp. 1408ñ1415. [9] Batina, J. T., ì Unsteay Euler Airfoil Solutions Using Unstructure Dynamic Meshes,î AIAA Journal Vol. 28, No. 8, 1990, pp [10] Blom, F. J., ì Consierations on the Spring Analogy,î International Journal for Numerical Methos in Fluis, Vol. 32, 2000, pp [11] Zhang, S., an Belegunu, A. D., ì A Systematic Approach for Generating Velocity Fiels in Shape Optimization,î Structural Optimization, Vol. 5, No. 1ñ2, 1993, pp. 84ñ94. [12] Crumpton, P. I., an Giles, M. B., ì Implicit Time-Accurate Solutions on Unstructure Dynamic Gris,î International Journal for Numerical Methos in Fluis, Vol. 25, No. 11, 1997, pp. 1285ñ1300. [13] Farhat, C., Degan, C., Koobus, B., Lesoinne, M., ì Torsional Springs for Two-Dimensional Dynamics Unstructure Flui Meshes,î Computer Methos in Applie Mechanics an Engineering, Vol. 163, 1998, pp [14] Tezuyar, T. E., Behr, M., ì A New Strategy for Finite-Element Computations Involving Moving Bounaries an Interfacesó The Deforming-Spatial-Domain/Space-Time Proceure: I. The Concept an the Preliminary Numerical Tests,î Computer Methos in Applie Mechanics an Engineering, Vol. 94, 1992, pp [15] Cavallo, P. A., Hosangai, A., Lee, T. A., Dash, S. M., ì Dynamics Unstructure Gri Methoology with Application to Aero/Propulsive Flowfiels,î AIAA Paper , [16] Morton, S. A., Melville, R. B., Visbal, M. R., ì Accuracy an Coupling Issues of Aeroelastic Navier-Stokes Solutions on Deforming Meshes,î Journal of Aircraft, Vol. 35, No. 5, September-November 1998, pp [17] Watt, A., Watt, M., ì Avance Animation an Renering Techniques: Theory an Practice,î Aison- Wesley Publishing Company, New York, 1992, Chap. 17. [18] Altmann, S. L., ì Rotations, Quaternions, an Double Groups,î Clarenon Press, Oxfor, [19] Shoemake, K., ì Animating Rotation with Quaternion Curves,î SIGGRAPH 1985 Conference Proceeings, July 1985, pp [20] Philips, W. F., Hailey, C. E., Gerbert, G. A., ì Review of Attitue Representation Use for Air Kinematics,î Journal Aircraft, Vol. 38, No. 4, July-August 2001, pp

14 REPORT DOCUMENTATION PAGE Form Approve OMB No Public reporting buren for this collection of information is estimate to average 1 hour per response, incluing the time for reviewing instructions, searching existing ata sources, gathering an maintaining the ata neee, an completing an reviewing the collection of information. Sen comments regaring this buren estimate or any other aspect of this collection of information, incluing suggestions for reucing this buren, to Washington Heaquarters Services, Directorate for Information Operations an Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA , an to the Office of Management an Buget, Paperwork Reuction Project ( ), Washington, DC AGENCY USE ONLY (Leave blank) 2. REPORT DATE April TITLE AND SUBTITLE Application of Quaternions for Mesh Deformation 3. REPORT TYPE AND DATES COVERED Technical Memoranum 5. FUNDING NUMBERS AUTHOR(S) Jamshi A. Samareh 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, VA PERFORMING ORGANIZATION REPORT NUMBER L SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics an Space Aministration Washington, DC SPONSORING/MONITORING AGENCY REPORT NUMBER NASA/TM SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassifie-Unlimite Subject Category 64 Distribution: Nonstanar Availability: NASA CASI (301) b. DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 wors) A new three-imensional mesh eformation algorithm, base on quaternion algebra, is introuce. A brief overview of quaternion algebra is provie, along with some preliminary results for two-imensional structure an unstructure viscous mesh eformation. 14. SUBJECT TERMS quaternion, shape optimization, mesh, gri, eformation 15. NUMBER OF PAGES PRICE CODE 17. SECURITY CLASSIFICATION OF REPORT Unclassifie 18. SECURITY CLASSIFICATION OF THIS PAGE Unclassifie 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassifie 20. LIMITATION OF ABSTRACT UL NSN Stanar Form 298 (Rev. 2-89) Prescribe by ANSI St. Z