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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13, July 2002 Computational Design Optimization Using RANS J.C. Newmar~ ID, R. Pankajakshar~, D.~. Whitfield, and L.K. Taylor (Computational Simulation arid Design Center, Mississippi State University, USA) ABSTRACT Sensitivity analysis and computational design capabilities have been developed and incorporated into a Reynolds Averaged Navier-Stokes code referred to as UNCLE. Sensitivity analysis techniques are reviewed and a novel method for computing derivatives of real functions using complex variable numerical differentiation is presented. Three exploratory hydrodynamic optimization examples are performed to access the computational design methodology. These optimizations include the redesign of a modern marine propeller for improved cavitation performance, redesign of a bow bulb on a surface combatant to reduce downstream vortices, and the redesign of a rudder in the wake of a propeller to reduce outboard surface cavitation. INTRODUCTION Through relentless experimental and computational field simulation (CFS) studies, resourceful design engineers have produced near optimal (marine, aeronautical, automotive, etc.) configurations. To further improve these designs, where the margin for improvement is small, designers will require additional information such as sensitivity derivatives. Sensitivity derivatives, which provide a measure of how the system output will respond to a change in system input, are an invaluable commodity to the designer and may be used to make informed decisions about possible directions for improving existing designs. This information may additionally be used to expedite the design of new engineering systems for which there is no vast experimental or computational database, thus, reducing design cycle time and cost. Sensitivity derivatives may also be used in other fields such as parameter estimation and-uncertainty analysis, as well as provide additional understanding of the physical process at hand. As an example, the additional understanding of the physical phenomena may lead to new or improved scaling laws and, thus, address issues between model and full scale experiments. These needs are the impetus for the development of efficient and accurate sensitivity analysis procedures based on well-validated CFS software. To maximize the benefits of these procedures, they must have the capability of resolving both the physics and the geometric complexities of practical configurations (i.e., high-fidelity CFS). High-fidelity CFS may be used to evaluate the performance of hydrodynamic components such as propellers, as well as resolve interaction effects between components (e.g., propeller and the ship hull). Using the propeller and ship hull as an example, it is expected that the efficiency and cavitation characteristics of a propeller designed in open water will differ from a propeller operating in the wake of a ship hull, it may then be concluded that the interaction must be considered. Numerous propeller and ship hull geometry, at various operational conditions (e.g., advance ratio, Reynolds number, etc.), can be quickly investigated with CFS (as compared to the time required to fabricate and experimentally test the same number of variations). From these candidate designs, only the most promising will then require experimental verification. Coupling high-fidelity CFS codes with numerical optimization software can be used to produce these promising designs, and may provide solutions to current design deficiencies or shortcomings. Computational design has become an active area of research in the hydrodynamic community. To this end, a brief review of the literature shall be included. Although simulation has been used in the design of hydrodynamic configurations in the past, only those that utilize sensitivity derivatives and gradient-based optimization methods will be considered in this review. A detailed and concise overview of sensitivity analysis methods and aerodynamic design optimization research may be found in Newman et al. (19991. Hino (1999) used the Navier-Stokes equations coupled with a sequential quadratic programming (SQP) method to minimize the wave and viscous drag on ship hull forms. In that work, Hino adopted a discrete-adjoint variable approach to compute sensitivity derivatives and redesigned a hypothetical tanker hull and a Series 60 hull for minimum resistance. Subsequently, based on Jameson's (1988) work, Cowles and Martinelli (2000) described a control- theory (continuous) approach to sensitivity analysis for incompressible, turbulent viscous flows and applied this approach to match target pressure distributions (inverse design) on finite span wings and sails. Tahara et al.
(2000) used a Navier-Stokes code, with finite- difference gradients, for CFD-based design of the bow bulb on a surface combatant. In Tahara no free surface effects were considered, and the objective was to minimize the downstream vorticity in the vicinity of the bulbous bow. Ragab (2001 ) describes the use of a continuous-adjoint approach in free-surface potential flow and applies the approach to redesign a base-line ship for minimum wave resistance, and to match target pressure distributions. Soto and Lohner (2001) proposed an incomplete-gradient adjoins formulation based on the continuous approach to sensitivity analysis where only the adjoins on the boundary of the domain is computed. In their approach the derivatives of the cost or objective function were computed using finite- difference. Soto and Lohner, using the incompressible Euler equations, redesigned 3D hydrofoils to maximize the minimum pressure on pressure and suction surfaces, and to redesign the bow bulb on a surface ship for minimum wave drag. Dreyer and Martinelli (2001) utilize a continuous-adjoint approach for target pressure matching of propulsor configurations using the psuedo- compressible Euler equations in a rotating frame. In this work, the current computational design methodology is demonstrated on three, nontrivial hydrodynamic components. Presented herein is a discussion of the flow analysis code, the methods for computing sensitivity derivatives, the parameterization of the design surfaces, and the exploratory optimizations performed. MOW ANALYSIS The flow analysis code used to perform the simulations, and into which sensitivity analysis and design capabilities have been incorporated, is referred to as UNCLE - UNsteady Computation of fieLD Equations (Whitfield et al. 1994, Pankajakshan et al. 2002~. UNCLE is a parallel, multiblock, multigrid structured grid code which solves the time dependent Reynolds Averaged Navier-Stokes (RANS) equations in either a rotating or an absolute reference frame. Additionally this software has linear and nonlinear free- surface capabilities, and incompressible and arbitrary Mach number (Taylor et al., 2001) versions. UNCLE has been extensively validated and is currently used by the Navy and industry for hydrodynamic simulations. More information on UNCLE may be found in the cited literature. SENSITIVITY DERIVATIVES Background For sensitivity analysis and design optimization based on CFS the field state equation to be solved is usually a system of partial differential equations (PDE). In the current hydrodynamic design methodology this system is represented by the RANS equations. Differentiation of this system of PDE (i.e., sensitivity analysis) can be performed at one of two levels. In the first method, termed the continuous or variational approach, the PDE are differentiated prior to discretization, either directly or by introducing Lagrange multipliers that are defined as a set of continuous linear equations adjoins to the governing PDE. Subsequently, these directly differentiated or adjoins equations are discretized and solved. In the second method, termed the discrete approach, the PDE are differentiated after discretization. The discrete approach may also be cast in either the direct or an adjoins formulation. A more detailed discussion on the discrete and the continuous formulation, as applied to aerodynamic shape optimization, may be found in Hou et al. (1994) and Jameson (1988), respectively. With gradient-based optimization methods, the search direction is determined using the first derivatives of the objective and constraint functions with respect to the vector of independent or design variables (i.e., sensitivity derivatives). This is not to say that the search direction is solely based on first-derivative information; it is possible to estimate higher-order derivatives using the computed first derivatives. In general, the objective and constraint functions may be expressed as Fi(Q,X ,pk ~ . Here, Q represents the disciplinary state vector which is produced from the CFS, X is the computational mesh over which the PDE are discretized, and ink the vector of design variables. The sensitivity derivatives of these functions may be obtained by direct differentiation with respect to implicit and explicit dependencies as aF (aF )T aQ (aF )r aX `1' To compute the sensitivity derivatives in Eq. 1 the sensitivity of the state vector, ac/~3k' and the grid sensitivity terms, ax/ask ~ are needed. This approach to sensitivity analysis is referred to as the direct differentiation method. Since these derivatives require the sensitivity of the state vector, the number of linear systems needing to be solved will be equal to the number of design variables. On the other hand, if in the design problem under consideration the sum of the objective and constraint functions is less than the number of design variables, a more efficient alternative approach may be formulated. This method is referred to as the adjoins variable approach, and may be written as T VFi = at +(3X ) ask Fi [ 3 (2)
where OF are adjoins vectors defined in such a way as to eliminate the dependence of the objective and constraint functions on the sensitivity of the state vector, and R represents the disciplinary state equation. The number of linear systems needing to be solved is equal to the number of functions required by the problem formulation. As previously discussed, two methods exist to perform the sensitivity analysis - continuous and discrete approaches. Either may be cast into a direct (Eq.1) or an adjoins (Eq.2) formulation. From the fact that the derivatives produced by a continuous approach are not consistently discrete with the CFS solver, optimization routines using these derivatives may fail to converge or even diverge. Furthermore, derivation of numerical boundary conditions for the continuous adjoins have been found to be difficult and time consuming. Thus, in the current work, the discrete approach to sensitivity analysis is used. Sensitivity Analysis For the discrete-direct approach (Eq.1) the sensitivity of the field variables ac/a'~k are required, and for the discrete-adjoint approach (Eq.2) the adjoins vectors OF are needed. To obtain these, the discrete residual vector from the CFS for a steady-state solution may be written as R(Q~k ),X ark cook )= 0 ~ ~ where the explicit and implicit dependencies of the residual on the state vector Q. the computational mesh X, and the design variables ink are asserted. In the discrete-direct approach, Eq.3 is directly differentiated with respect to the design variables to produce the following linear equation dR = aR an + aR ax + aR =0 <4y Ask I Opk OX S0k 3pk or, rearranging aR aQ ~aR ax aR~ _ = _- + use aQ apk ax ark al3k where aR/aQ and aR/ax are the Jacobian matrices evaluated with-a converged (steady-state) solution, and ax/apk are the grid sensitivity terms. The solution of Eq.S poses the difficulty of solving a large linear system of equations for each design variable. Furthermore, because this equation is a linear system, the linearizations of the residual wth respect to the state vector and the grid (i.e., the Jacobian matrices) must be exact. The detriments of using inexact linearizations in Eq.S have been explored by Newman et al. (1995~. Solving these systems, however, is made more tractable when the above equations are recast into what has been termed the incremental iterative form (Korivi, 1994) as follows A^n~&Q)=_~OR(3Q ~f+&R ax + aR ~ (bar ink ) LaQ~pk ) ax ark ink ~ (afk ) (ark ) +/\n\) (6b) where A may be any convenient approximation to the higher-order Jacobian that converges the linear system. An approximation is possible because the equations are now cast in delta form, with the physics contained in the right-hand-side vector. Is has been found that the first-order Jacobian works well for use in the coefficient matrix of Eq.6a; most CFS codes also use the first-order Jacobian for this purpose. Two particularly attractive features of the incremental iterative strategy are that (i) a more diagonally dominant matrix may be used to drive the solution of the linear system (as opposed to the sometimes ill-conditioned higher-order Jacobian), and (ii) the higher-order Jacobian now resides on the right- hand-side of the equations and may be dealt with in an explicit manner. When in this form, only the k-vectors resulting from the matrix-vector product of (aR/~Q)(~Q/apk~ are of concern. Hence, CPU time and memory efficient methods for constructing the exact matrix-vector product can be utilized. To this end, higher-order spatially accurate discrete-direct sensitivity analysis procedures for aerodynamic shape optimizations have been developed (Newman et al., 1997~. The discrete-adjoint variable formulation begins by combining Eq.4 from the direct differentiation method with the sensitivity derivatives in Eq.1. From this adjoins vectors may be conveniently defined such that the sensitivity of the field variables are no longer needed. Nevertheless, the end result requires the solution of the following linear systems for the adjoins vectors T(a3R ) Gil = aaFi (7) The adjoins variable approach may also be recast in incremental iterative form, however, for brevity these details shall be omitted. It should be noted that all the same linearizations required by the direct approach are required by the adjoins variable method; they are simply transposed and used at different stages in the computation of the sensitivity derivatives. Hence, a sensitivity analysis code which was developed for
either of the discrete approaches may be modified to produce the other. The task of constructing exactly or analytically all of the required linearizations and derivatives by hand for the discrete approaches, and then building the software for evaluating these terms can be extremely tedious. This problem is compounded by the inclusion of even the most elementary turbulence model (for viscous flow) or the use of sophisticated grid generation packages for adapting (or regenerating) the computational mesh to the latest design. One solution to this problem has been found in the use of a technique known as automatic differentiation. Application of this technique to an existing source code, that evaluates output functions, automatically generates another source code that evaluates both output functions and derivatives of those functions with respect to specified code input or internal parameters. A pre-compiler software tool, called ADIFOR (Automatic Differentiation of FORtran, Bischof et al., 1992), has been developed and utilized with much success to obtain complicated derivatives from advanced CFS and grid generation codes for use within aerodynamic design optimization procedures (Green et al., 1993, 1996 and Taylor et al. 1997~. The use of ADIFOR produces code that, when executed, evaluates these derivatives via a discrete-direct approach, referred to as forward-mode automatic differentiation. More recently, automatic differentiation software has emerged that enables the derivatives to be evaluated with a discrete- adjoint approach (Mohammadi, 1997 and Carte et al., 1998~. This type of automatic differentiation is known as reverse-mode. A detailed and concise review on the use of sensitivity analysis in aerodynamic shape optimization has been reported by Newman et al. (1999a); the reader is directed to this source for discussion on the methods presented thus far. Complex Taylor Series Expansion (CTSE) Method The methods previously discussed will require differentiation of the CFS software, either by hand or with pre-compiler software. Other methods to obtain sensitivity derivatives are based on numerical techniques. The simplest numerical technique is the finite-difference approximation. Another is a relatively new numerical technique, developed by Newman et al. (1998) for -performing disciplinary and multi- disciplinary sensitivity analysis, Rich uses complex variables to approximate derivatives of real functions. This method is based on ideas that where explored over three decades ago by Lyness and Moler (1967) and Lyness (1967), and recently revisited by Squire and Trapp (19981. Both numerical approaches will be discussed below. For a central finite-difference approximation to the derivative, one may expand the function in a Taylor series about a given point using a forward and a backward step, and then subtracting to yield df tf (x+h)- f (x-h~ _ h d f _ h d f _... `8 dx 2h 3! dx3 5 ! dx5 This expression for the derivative has a truncation error of O(h2~. The advantage of the finite-difference approximation to obtain sensitivity derivatives is that any existing code may be used without modification. The disadvantages of this method are the computational time required and the possible inaccuracy of the derivatives. The former is due to the fact that for each design variable, two well-converged CFS solutions are required to evaluate the central finite-difference. In the case of nonlinear fluid flow, for example, these solutions may become prohibitively expensive for a large number of design variables. The latter is attributed to the sensitivity of the derivatives to the choice of step size. To minimize the truncation error one selects a smaller step size, however, an exceedingly small step size may produce significant subtractive cancellation errors. The optimal choice for the step size is not known a priori, and may vary from one function to another, and from one design variable to the next. Instead of the finite-difference approximation, consider expanding the function in a Taylor series using a complex step as f(X +hi)= f~x)+h' df _ h d f h3' d3f h4 d4f _ + + 3! dX3 4! dx4 (9) where ~ = `/~. Solving this expression for the a imaginary part of the function yields df hn[f (X + h ~ ~ + h d f _ h d f + . .. (10) Tic h 3 ! dX3 5 ! d~c5 This expression for the derivative also has a truncation error of O(h2~. By evaluating the function with a complex argument, both the function and its derivative are obtained, without subtractive terms, and thus cancellation errors are avoided. The real part is the function value to second order. The advantages of the complex variable approximation, referred to as the Complex Taylor Series Expansion (CTSE) method, are numerous. First, like the f~nite-difference approximation to the derivatives, very little modification to the software is required. All the original features and capabilities of the software are retained. Thus, user experience is not lost and ongoing advancements and enhancements can be readily introduced into subsequent versions without extensive modifications or re-differentiation. This is in
direct contrast to hand or automatically differentiated sensitivity analysis codes where any modification to the original software will require re-differentiation. This advantage is extremely useful in the problem formulation stages of the design process when new objective and constraint functions are being explored. Second, this method is equivalent to a discrete-direct approach, either from automatic differentiation or hand differentiated codes solved in incremental iterative form, in the way that the state vector and its derivatives are being solved for simultaneously. When solving the state equation, the state vector resides in the real part and the derivatives in the imaginary part. Unlike the finite-difference approximation, fully converged flow solutions are not required to obtain derivatives of sufficient accuracy for design. Finally, the CTSE method is not sensitive to step size selection and only requires step sizes that avoid excessive truncation error; thus, it has been shown that this method demonstrates true second-order accuracy (Newman et al., 1998 and Anderson et al., 2000~. Additionally, the CTSE technique can be used to compute second derivative information using available data, but these computations are subject to cancellation errors. The only disadvantage of the CTSE method is the increased runtime required by the evaluating routines when run with complex arguments. RESULTS Three examples analysis exploratory hydrodynamic optimization were performed to access the sensitivity and computational design capabilities developed in UNCLE. Each is representative of typical components that have been recently redesigned for the DDG51 fleet. These examples include redesign of a marine propeller, redesign of the bow bulb, and redesign of the rudder. Figure 1: P5 158 Geometry and tip vortices. P5168 Propeller For this study a propeller of typical, modern design was selected, and is designated as Propeller 5168 (P5168~. P5168 is a five bladed controllable pitch propeller, and was tested in the DTMB, CDNSWC 36" water tunnel (Jessup, 1996~. This propeller was used by ONR for validation purposes of CFS codes in predicting tip vortex flows about Navy surface ship propellers. The ultimate goal of the ONR was to demonstrate the usefulness of CFS codes in ranking various candidate blade tip geometries to suppress tip vortex cavitation. The geometry and tip vortices, as predicted via UNCLE, are shown in Fig. 1. To reduce tip cavitation, the objective of the redesign was to maximize the weighted sum of average and minimum pressure on the suction and pressure surfaces. The design surface was selected as the outer most 10% of the blade span as shown in Fig. 2. The ores sure and suction design surfaces where parameterized with a Bezier surface that controlled the normal thickness variation as N M In = ~ ~ Pij i=0 j=0 N! ui (1-U)N-' i! (N-i)! | M! vj (1 - v)M -i 1 (11) where Pij are the Bezier control points that were used as the design variables, and u, v represent the parametric variables in the chordwise and spanwise directions, respectively. A two-dimensional example of this parameterization is shown in Fig. 3 for an initially symmetric hydrofoil. Geometric constraints were enforced such that slope and curvature would remain unchanged between the pressure and suction surfaces. With these constraints, the total number of design variables reduced to 8. Transfinite interpolation (TFI) of surface deformations into the volume mesh was used to modify the computational mesh to the latest design. Figure 2: P5 168 design surface.
\ h4~n comb - LIn. ~~0itI" sud=~ / Twins ~ ~ 44 case 5~a''~ i: VAst] ~: I' OF use - Dig 4s HE .~ ~ . iiO.0 1 ND it: - t:~ ~ ~~ W.~¢'4~S~ f ~ Hi t 1. 07 Figure 3: Sample 2D normal thickness variation. The analysis used the incompressible version of UNCLE and the viscous simulation ms performed in parallel with 10 blocks. The flow Reynolds number was 4.26M with an advance ratio of 1.1. Sensitivity derivatives where obtained via the direct approach with the CTSE method used to construct all linearizations. Since this was an exploratory study, only design improvement over the original P5168 propeller was sought. To this end, one design cycle was performed (i.e., only one search direction was computed and traversed) with 23 line-search steps taken. The results of this design cycle are shown in Fig. 4. As seen, the minimum and average pressure on the suction design surface has been increased by 5.44% and 0.40%, respectively, and the minimum and average pressure on the pressure design surface has been increased by 5.20% and 22.28%, respectively. The blade shape for this design cycle became more bulbous on the pressure side while introducing a cusp near the trailing edge on the suction side. The normal thickness variation for the pressure and suction surfaces are shown in Fig. Sa and b, respectively. The pressure on the suction surface and the change in pressure between optimized and original are depicted in Fig. 6. As seen, the lowest surface pressure occurs at the trailing edge tip and, thus, this is the region of greatest increase. It should be noted that as additional design cycles are performed, the shape of the propeller will continue to evolve as the optimum is approached. The propeller shape after this first design cycle may or may not be indicative of the final design. During each design cycle further improvements will be observed. However, if the shape is progressing towards a design that exhibits undesirable features, or manufacturability difficulties, the optimization problem can be reformulated with additional constraints and/or objective functions. .£ ~ ~, . ,i s .. . s ~ ~ ~~ ~ < Ajar ~~~ t~ ~~ l,'~:~s: >~i¢~~s.4 ~ ,A< ~-i: ~~.:.:f' I. 1; ~ ~0 US ~ 1~-( .' ~ Pours S~rF~ ~ it > j' ~ Sew S~ M~u~ t! _, s=~u sU§ I._ ~C== ' ' ' ' ~ ' ' ' ' 1'O' ' ' ' 1t ' ' ' :'0 ' ' Let ~ ea~ Sew Figure 4: Optimization results for the P5 168 redesign. 4 U OFF p~ ~~ ~ NonnalTnic~n.Ss Vad~on MOD 0~17 6~ 6~ 4~6 g~3 (a) Pressure surface. Palm t - _ . Normal Thickness V - anion ~ 4013 NEWS O~ O Suction surface. Figure 5: Results of normal thickness variation (u=0 is leading edge, v=1 is tip).
- Pi~ureon Such SuH~ ~20 0,St41 S - opt Path:: ~.02g7: 0~. Of Figure 6: Suction surface pressure and pressure change. Bow Bulb Redesign The second exploratory study consisted of the redesign of the bow bulb (or sonar dome) on a modified 5415 ship hull. Typical sonar arrays that reside in the bulbous bow are illustrated in Fig. 7. The design surface, which is shown in Fig. 8, was parameterized with the normal thickness variation given in Eq. 11. Volume constraints where enforced such that the largest sonar array that could reside in the original geometry would not be penetrated. The objective was to minimize the swirl of the flow in the region around and downstream of the bulbous bow. To this end, a weighted sum of the average and maximum swirl parameter (Remotigue, 1999) was minimized. _ Figure 7: Typical bow bulb and sonar arrays. Figure 8: Design surface for the bow bulb redesign. The analysis used the arbitrary Mach number version of UNCLE and the viscous simulation used the ~-e turbulence model. The flow Reynolds number was 14M with a Froude number of zero (i.e., no free surface effects were considered). Sensitivity derivatives where obtained via the direct approach with the CTSE method used to construct all linearizations. Due to the fact that the ~rameterization of the sonar dome resulted in 24 shape design variables, and that the analysis and sensitivity analysis were performed sequentially on a single block, only two design cycles were performed. As seen via the width contours for the original and modified bow shown in Fig. 9, the resulting design after two cycles became more bulbous in the rear portion of the sonar dome. The resulting effect on the swirl parameter on a downstream cutting plane is illustrated in Fig. 10. It can be observed that the swirl has been reduced, and thus design improvement has been achieved.
(a) Oricinal width contours I. - ~ ~ ~ =l ¢~-1 ~,`4 '.~' {~ ~ ~ (b) (change (optimized-original) width contours. Figure 9: Width contours for the original and change in width contours for bow bulb redesign. Twisted Rudder Redesign The final optimization example was the redesign of the twist distribution of a rudder placed in the wake of a propeller. This design was originally performed and patented by Shen (1997) and is considered a tremendous success for cavitation reduction (Krueger, 2001~. The experimental apparatus used for testing this design by Shen is shown in Fig. 11; additional details on the propeller placement and the rudder geometry may be found in the cited literature. The original design by Shen was inspired by the fact that during full scale trials to access the hydroacoustic and hydrodynamic performance of redesigned propellers, severe surface cavitation was observed on the outboard surface of the rudders. Subsequently, drydock inspections confirmed cavitation erosion on the outboard rudder surface, while none was present on the inboard surface (Jiang 19951. Shen then conducted a computational and experimental design of the rudder twist distribution and found that by aligning the local rudder twist to the incoming propeller induced flow angles, cavitation on the outboard surface could be significantly reduced. Since the rudder profile is symmetric, the objective used in the current work was to design the rudder for zero side force at zero rudder deflection. The analysis and design was conducted using the arbitrary Mach number version of UNCLE. The propeller was modeled as a body force propulsor, and the fluid simulated as inviscid flow. The twist distribution was parameterized with a Bezier curve. Constraints were enforced to allow no twist at the root or tip, and zero twist derivative at the root. With these constraints, the parameterization resulted in 4 design variables. Once again, the CTSE technique was used to construct all linearizations required by the direct method. (a) Downstream cutting plane. ;~ ~) Swirl parameter for original geometry. (c) Swirl parameter for redesigned geometry. Figure 10: Results of the bow bulb swirl minimization on a downstream cutting plane. Swirl parameter contours are shown with the same scale for the original and redesigned geometry.
(a) Side view. (hi Front view. Figure 11: Experimental apparatus for twisted rudder of Shen (1997~. _ c ot$;hen (~) ~ fit, : h ~ . S. . : . e me, ~ i/ Norma sun Figure 12: Results of the twisted rudder redesign. (a) Front view of initial and final twisted rudders. - ,,,,,, (b) Pressure for initial and final twisted rudders. (c) Pressure for initial and final twisted rudders. Figure 13: Pressure distributions on untwisted and twisted rudders. For this design, the analyses required in the design cycles and the computation of the sensitivity derivatives where performed in parallel over 16 blocks and 4 design variables. Hence the optimization was performed using a total of 64 processors. After 3 design cycles the side force was reduced by three orders of magnitude. Results of this optimization are shown in Fig. 12 and compared with the design produced by Shen (19971. As seen, the developed computational design procedure produced nearly the same twist distribution. However, the current computational design code took less than one day to produce this distribution. The front view of the original and redesigned rudders, shaded by pressure, illustrating the twist along the span of the rudder, are shown in Fig. 1 3a. The pressure change on the pressure (inboard) and suction (outboard) surfaces are shown in Figs. 13b and c, respectively. The minimum pressure on the twisted rudder rose 15.44% from the original.
CONCLUSIONS The primary objectives of this paper were to detail the development of sensitivity analysis and computational design capabilities incorporated into UNCLE, and to introduce a novel technique for computing derivatives of Hal functions using complex variables to the hydrodynamic community. Sensitivity analysis provides an additional level of information about the flow physics and how such phenomena will change with variations in geometric and non-geometric parameters. In the current work, this sensitivity analysis has been used to demonstrate design enhancement for three physically and geometrically different hydrodynamic configurations. Complete design optimization was not performed, the objective was to demonstrate that design improvements over the baseline geometry could be achieved using numerical optimization. For these design results to be meaningful, experienced design engineers would need to formulate the optimization problem. To this end, the redesign of the twisted rudder represents the most realistic design example. This optimization was formulated based on the design conducted by Shen (1997) and, thus, very close agreement was observed. Additionally, this demonstrates that computational design may be a viable tool to aid in the design process and ultimately reduce design cycle time and costs. ACKNOWLEDGE This work was sponsored by Dr. L. Patrick Purtell of the Office of Naval Research. This support is gratefully acknowledged. REFERENT Anderson, W.K., Newman III, J.C., Whitfield, D.L., and Nielsen, E.J., "Sensitivity Analysis for the Navier- Stokes Equations on Unstructured Meshes Using Complex Variables," AIAA J., Vol. 39, No. 1, Jan. 2000, pp. 56-63. Bischof, C., Carle, A., Corliss, G., Grienwank, A., and Hovland, P., "ADIFOR: Generating Derivative Codes from Fortran Programs," Scientific Programming, Vol. 1, No. 1, pp. 11-29, 1992. Carte, A., Pagan, M., and Green, L.L., "Preliminary Results From the Application of Automated Adjoint Code Generation to CFL3D," AIAA Paper 984807, Sept. 1998. Cowles, G., and Martinelli, L., "A Control-Theory Based Method for Shape Design in Incompressible Viscous Flow using RANS," AIAA Paper 00-2544, June 2000. Dreyer, J.J., and Martinelli, L., "Hydrodynamic Shape Optimization of Propulsor Configurations Using a Continuous Adjoint Approach," AIAA Paper 01-2580, June 2001. Green, L.L., Bischof, C., Griewank, A., Haigler, K., and Newman, P.A., "Automatic Differentiation of Advanced CFD Codes With Respect to Wing Geometry Parameters for MDO," Proceedings of the 26 U.S. National Congress on Computational Mechanics Washington, D.C., Aug. 1993. Green, L.L., Newman, P.A., and Haigler, K.J., "Sensitivity Derivatives for Advanced CFD Algorithms and Viscous Modeling Parameters Via Automatic Differentiation," J. Comp. Physics, Vol. 125, 1996, pp. 313-324. Hino, T., "Shape Optimization of Practical Ship Hull Forms Using Navier-Stokes Analysis," Proceedings of the 7th International Conference on Numerical Ship Hvdrodvnamics, Nates, France, July 19-22, 1999. Hou, G.J.-W., Taylor III, A.C., and Korivi, V.M., "Discrete Shape Sensitivity Equations for Aerodynamic Problems," Int. J. Num. Meth. Engr., Vol. 37, 1994, pp. 2251-2266. Jameson, A., "Aerodynamic Design via Control Theory " J. Sci. Comp. Vol. 3 1988 pp. 233-260. 9 ~ ~ ~ Jessup, S., Private Communications, July 1996. Jiang, I.C., Remmers, K.D., and Shen, Y.T., "Rudder Cavitation Studies at DTMB Large Cavitation Channel," International Symposium on Cavitation, Deauville, France, 1995. Korivi, V.M., Taylor III, A.C., Newman, P.A., Hou, GJ.-W., and Jones, H.E., "An Approximate Factored Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives," J. Comp. Physics, Vol. 113, 1994, pp. 336-346. Krueger, K., "Twisted Rudder: A Navy Success Story," Wavelengths, Carderock Division, Naval Surface Warfare Center, September 2001. Lyness, J.N., "Numerical Algorithms Based on the Theory of Complex Variables," Proceedings of the ACM 22n~ National Conf., Thomas Book Co., Washington, DC, 1967, pp. 12~134. Lyness, J.N., and Moler, C.B., "Numerical Differentiation of Analytic Functions," SIAM J. Numer. Anal., Vol. 4, 1967, pp. 202-210. Mohammadi, B., "Optimal Shape Design, Reverse Mode of Automatic Differentiation and Turbulence," AIAA Paper 97-0099, Jan. 1997. Newman III, J.C., Taylor III, A.C., and Burgreen, G.W., "An Unstructured Grid Approach to Sensitivity Analysis and Shape Optimization Using the Euler
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