COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2001 00:1{7 [Verion: 2000/03/22 v1.0] A tabilied peudo-hell approach for urface parametriation in CFD deign problem O. Soto,R.Lohner and C. Yang CSI/Laboratory for Computational Fluid Dynamic - George Maon Univerity, MS 4C7 4400 Univerity Drive - Fairfax, VA 22030-4444, USA SUMMARY A urface repreentation for Computational Fluid Dynamic (CFD) hape deign problem i preented. The urface repreentation i baed on the olution of a implied peudo-hell problem on the urface to be optimied. A tabilied nite element formulation i ued to perform thi tep. The methodology ha the advantage of being completely independent of the CAD repreentation. Moreover, the uer doe not have to predene any et of hape function to parameterie the urface. The cheme ue a reaonable dicretiation of the urface to automatically build the hape deformation mode, byuingthe peudo-hell approach and the deign parameter poition. Almot every point of the urface grid can be choen a deign parameter, which lead to a very rich deign pace. Mot of the deign variable are choen in an automatic way, which make the cheme eay to ue. Furthermore, the urface grid i not ditorted through the deign cycle which avoid remehing procedure. An example i preented to demontrate the propoed methodology. Copyright c 2001 John Wiley & Son, Ltd. key word: Surface parametriation deign hape optimiation CFD 1. INTRODUCTION Mot of the urface repreentation algorithm commonly ued for hape deign problem are baed on CAD entitie uch a NURBS, Coon patche, Beier and B-Spline curve and urface. Thee are contructed before, or after, a urface grid i obtained to dene the hape deformation mode [7, 4]. In general, uch hape function are dened by patche, and the deformation of the urface i controlled by ome control point or knot. Another common approach i to ue global hape function dened over the complete urface, uch a Hick- Henne function, tandard NACA function or mapping [6, 3]. The deformation mode are then controlled by parameter aociated with thee function. Correpondence to: Orlando Soto - CSI/Laboratory for Computational Fluid Dynamic - George Maon Univerity, MS 4C7-4400 Univerity Drive - Fairfax, VA 22030-4444, USA - e-mail: oto@c.gmu.edu - web: http://www.c.gmu.edu/~oto Copyright c 2001 John Wiley & Son, Ltd.
2 O. SOTO, R. L OHNER AND C. YANG In thee approache, the deign pace i retricted to the control or knot point that dene the deformation mode (CAD approach), or to the function parameter dening the global hape function. The key idea of the preent work i a cheme in which almot every point onthe urface can be choen a deign parameter. In thi way, the deign pace will be only contrained by the urface grid denity. In addition, the cheme generate urface free of ingularitie which can degrade or inhibit the convergence of the elected optimiation methodology. 2. PSEUDO-SHELL SCHEME The main goal of any urface parametriation cheme for hape deign problem i to generate uciently mooth geometrie during the whole optimiation proce. In general, a CAD or global hape function repreentation aure that not only the normal movement of the point on the urface are continuou, but alo their rt (rotation) and econd (curvature) derivative with repect to the urface tangential coordinate (C 2 continuity). Let be the urface to parameterie and B(x) :=ft 1 t 2 ng an orthonormal bai dened on each point x 2, with t 1 and t 2 two unitvector tangent to and n the normal one. If ome normal deection are impoed to a et of point x 2,aC 2 continuou diplacement eld may be computed by olving the following PDE problem (the uual ummation convention i implied in the whole paper): l ; @u n @t l =0 for l =1 2 n =0 @ @u n @t l l ; =0 for l =1 2: (1) @t l In (1), i the rotation eld on, l = t l it l tangential component, n = n it normal one, u the diplacement eld on and u n = u n it normal component. The boundary condition of (1) are given by the impoed normal deection and by geometrical retriction. Problem (1) aure the required continuity ofthe normal urface deection. However, it can not upport tangential diplacement eld, which are required in mot deign cae. To takeinto accountuch on-plane mode, the problem (1) i enriched with the following elaticity problem on : @ul @uk @ @t k h @t k + @u k @t l i + @ @t l h 1 ; 2 @t k i =0 for l k =1 2 (2) where u l = u t l i the l tangential component of the diplacement eld, and and are the material hear modulu and Poion ratio repectively. In thi work the value of thee propertie were xed to = 1 and = 0:499, to obtain a quai-incompreible on-plane deformation. Thi reduce the grid ditortion acro the deign cycle and, therefore, the need for remehing. Problem (1) and (2), together with compatible boundary condition given by the impoed diplacement and by the geometrical contraint, have to be olved in a coupled manner to obtain mooth urface movement. 3. STABILIED FINITE ELEMENT SOLUTION Equation (2) can be approximated uing a tandard nite element approximation without major obtacle. However, the tandard Galerkin aproximation of problem (1) may yield
A STABILIED PSEUDO-SHELL APPROACH FOR SURFACE PARAMETRIATION 3 undeirable inetabilitie due to the divergence contraint (third equation in (1)). Thi pathology i very well known by author working in the nite element olution of Reiner- Mindlin bending plate and incompreible ow problem [2]. From a mathematical point of view, the nite element interpolation ued to approximate the rotation () and the deection (u n ) in (1) mut atify a compatibility condition (or Babuka-Brei [1] condition), which tranlate in the ue of mixed interpolation. Such hortcoming can be circumvented by uing a tabilied formulation that allow the ame interpolation for the two eld, and the ue of low-order linear element, two highly deirable numerical feature from a computational point of view. The tability of the problem (1) may be fullled by employing an algebraic ubgrid cale (ASGS) type formulation [2] a follow: Find ( l u n ) 2 V l W n uch that, (v @w n l ; )( @u n @t l l ; )d+ (v @t @w n l l ; )( @u n @t l l ; )d= w @t ; l n ( @u n l ; @t l )tn l d; (3) 8(v l w n ) 2 V l W n and for l =1 2. Here V l and W n are the tandard nite element pace where the l rotation component and the normal deection are interpolated, repectively, the tabiliing parameter that depend on the element ie h, andt n l the l component of the unit vector tangent to and normal to it boundary ;. The rt left hand ide and the right hand ide term are the tandard Galerkin contribution obtained after integrating by part the Laplacian term, and the lat left hand ide term i the ASGS tabiliing contribution. Note that the formulation i conitent in the ene that the olution of (1) i olution of (3). The right hand ide term i equal to ero, becaue at ; or u n i precribed, which implie w n = 0, or the rotation i equal to the deection gradient l = @u n =@t l (natural boundary condition). At thipoint, the only value to dene in (3) i. It i eay to ee that (1) i totally analog to the problem of a thin plate in which the hear term ha been neglected [2]. The ASGS tability analyi dictate that for an innitely thin plate turn out to be: = k h 2 ; 1 with k = c 1 12(1 ; ) and c 1 a poitive contant with an optimal value of c 1 = 4. The value ued for the material propertie and are the ame ued for the on-plane deformation equation (2). In ummary, the nal dicrete form of the peudo-hell problem may be written a: Find ( u) 2 V W uch that, + v n n d + @wl @uk @t l 1 ; 2 k h 2 (v l ; @w n @t l )( l ; @u n @t k @t l )d+ @w l @t k @ul @t k + @u k @t l d d = 0 8(v w) 2 V W and for l =1 2: (5) Again, v =(v 1 v 2 v n )and w =(w 1 w 2 w n )arethevectorial tet function for the rotation and diplacement eld, repectively. The lat two term of (5) are the tandard Galerkin dicrete form aociated to problem (2), after integrating by part and auming only Dirichlet and natural boundary condition. A a nal remark, it can be oberved that the econd term of (5) ha the form of a penalty contribution. The value of k=h 2 i everal order of magnitude larger than the other formulation contant for uual meh ie. Such term enforce in a weak form a mooth variation of the rotation eld, and in thi ene, the required C 2 continuity for the normal diplacement. (4)
4 O. SOTO, R. L OHNER AND C. YANG 3.1. Boundary Condition and Deign Parameter Selection The boundary condition for (5) in a hape optimiation problem are dictated by the deign parameter diplacement and the geometrical retriction. Even though for mot cae uch a et of contraint i enough to olve the peudo-hell problem (5), numerical experience ha hown that a better control over the whole urface movement i obtained if the boundary of, i.e. ;,imoved following a et of cubic B-pline curve. Thi movement i totally compatible with the allowable deformation given by the peudo-hell approach. Given a nite element partition of, the general procedure to olve (5) i: Algorithm 1 (i) Select a et of deign parameter. (ii) Compute the boundary ;. (iii) Adjut ; to a et of cubic B-pline curve. (iv) Compute the diplacement of the deign parameter with the optimiation algorithm. (v) Move the deign parameter on ;, and all ; following the B-pline curve. (vi) Solve (5) uing the diplacement of ;, the diplacement of the interior deign parameter (not in ; ) and the geometric retriction, a boundary condition. All the tep of Algorithm 1 may be done in a quai-automatic way. Thi i better explained in the example decribed below. D c x y Deign variable poition A B Figure 1. Surface meh of the Wigley hull Figure 1 how the urface meh of a Wigley hull. Due to ymmetry only half of it i analyed. The unique deign parameter that have been choen by the uer, and therefore their deformation direction, are located at the four corner (A-D). The boundary ; can be computed automatically from the dicretiation of the hull urface (meh edge that belong to jut one element). After obtaining ;, the B-pline curve were dened by joining the egment of ; between corner. One deign parameter on the middle of each B-pline curve wa added, and the deformation mode were automatically dened in uch a way that they fulll following geometric retriction: Point on the plane =0(hull' top) can be moved only on the plane, and point on the plane y = 0 can alo have only on-plane diplacement. Finally, interior deign point were automatically elected allowing minimum three edge of eparation among them, and among them and any B-pline point. The deformation direction of uch interior deign parameter i normal to. Thee point have been highlighted in Figure 1.
A STABILIED PSEUDO-SHELL APPROACH FOR SURFACE PARAMETRIATION 5 4. DESIGN ALGORITHM Thi ection decribe how the propoed peudo-hell urface repreentation may be ued in an optimiation environment. The deign algorithm utilied for thi purpoe i hown below. Algorithm 2 Evaluate the objective function I in the original geometry. FOR k =1 n d * Perturb the coordinate of the k deign variable in it deformation direction by a mall. The ret of deign parameter are not moved. * Move ; uing the B-pline curve if it i neceary. * Solve the peudo-hell problem (5) uing the boundary condition decribed in tep (vi) of Algorithm 1. Then, a perturbed geometry 0 i obtained. * Evaluate the objective function I 0 in the perturbed geometry 0. * Obtain the gradient of the objective function repect to the k deign variable by nite dierence a (I ; I 0 )=. END FOR Make a line earch in the negative gradient direction to nd a minimum. Perform a new deign cycle beginning with the nal geometry of the line earch procedure. In Algorithm 2, the peudo-hell problem ha to be olved n d time, where n d i the number of deign variable, to compute the gradient, and another everal time during the line earch procedure. Thi may performed in a very fat manner if the linear ytem of equation aociated to (5) i olved by uing a direct LU decompoition, and the LU matrice are computed and aved once at the beginning. The olution of the peudo-hell problem i then performed by very fat backward-forward ubtitution. However, the right hand ide of the ytem ha to be rebuilt each time due to the change in the boundary condition value. Again, thi can be performed in a very fat manner if the original matrix and right hand ide, without impoed boundary condition, are tored a well. The amount of memory needed to ave the LU matrice and the original matrix and RHS i very mall (the problem i practically 2D) compared with the capacity of modern computer. 5. NUMERICAL EXAMPLE In thi ection the wave drag minimiation of the Wigley hull preented in Figure 1 i performed. The problem retriction were: the volume encloed by the hull and the plane y =0mut remain contant, and the initial waterplane ( = 0) moment of inertia hould be conerved too. The problem may be written a the minimiation of the following objective function: C D jv ; V j jj ; Jj I = w 1 C + w 2 + w D V 3 (6) J where C D and C D are the wave drag and it initial value, V and V the encloed volume and it initial value, J and J the waterplane moment of inertia and it repective initial value, and w 1, w 2 and w 3 are relative weight. The eroth-order lender-hip approximation [7] wa
6 O. SOTO, R. L OHNER AND C. YANG Figure 2. Example of deformation mode: Top corner, bottom corner and interior deign variable ued to evaluate the wave drag. The exceptional implicity of thi calculation method render it ideally uited for routine hull form deign and optimiation. The procedure decribed in the ection 3.1 generated 52 deign variable (ee Figure 1). In Figure 2 the deformation mode of ome of them are hown. After eight deign cycle, a 90% drag reduction wa obtained while the volume and moment of inertia contraint were fullled. The nal hull i hown in Figure 3. The CPU time for the entire deign proce wa of 385 econd uing a ingle R1000 Silicon Graphic proceor. The initial LU decompoition for the peudo-hell approach pent 1.13 econd of CPU, while each ubequent backward-forward ubtitution took only 0.05 econd.
A STABILIED PSEUDO-SHELL APPROACH FOR SURFACE PARAMETRIATION 7 a) x b) c) y x y x Figure 3. a) Initial hull (dahed line) and nal hull (continuou line) b) Initial hull c) Final hull 6. CONCLUSIONS A very fat peudo-hell approach that produce mooth ingularity-free hape wa preented to parameterie urface in CFD optimiation problem. The uer ha to generate only the original urface meh and a few deign variable. The ret of the deign parameter and their repective deformation mode can be generated automatically by the method. Thi make the cheme eay to ue. Given that almot every point on the urface i choen a deign variable, the deign pace i very rich, leading to fat convergence to a good optimal olution. Due to the way the cheme i contructed, meh ditortion i minimied, avoiding remehing procedure. The propoed peudo-hell repreentation i currently being ued in more elaborated optimiation algorithm baed on an adjoint approach to compute the gradient and the Navier-Stoke equation for the ow olution [5]. REFERENCES 1. F. Brei and M. Fortin. Mixed and hybrid nite element method. Springer-Verlag, 1991. 2. R. Codina. On tabilied nite element method for linear ytem of convection-diuion-reaction equation. Computer Method in Applied Mechanic and Engineering, 188:61{82, 2000. 3. A. Jameon. Optimum aerodynamic deign uing CFD and control theory. AIAA-95-1729-CP, 1995. 4. J. Samareh. Geometry modeling and grid generation for deign and optimiation. ICASE/LaRC/NSF/ARO WORKSHOP ON COMPUTATIONAL AEROSCIENCES IN THE 21t CENTURY, Hampton VA, USA., 1998. 5. O. Soto and R. Lohner. General methodologie for incompreible ow deign problem. AIAA-2001-1061, 2001. 6. C. Sung and J. Kwon. Aerodynamic deign optimiation uing the navier-toke and adjoint equation. Paper AIAA 2001-0266, 2001. 7. C. Yang, F. Noblee, and R. Lohner. Practical hydrodynamic optimiation of a wave cancellation multihull hip. 2001 Annual and International Maritime Expoition, Orlando FL, USA, 2001.