ERCODO2004_238. ERCOFTAC Design Optimization: Methods & Applications. Sequential Progressive Optimization Using Evolutionary and Gradient Algorithms

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1 ERCOFTAC Design Optimization: Methods & Applications International Conference & Advanced Course Athens, Greece, March 31- April 2, 2004 Conference Proceedings Editors: K.C. Giannakoglou (NTUA), W. Haase (EADS-M) ERCODO2004_238 Sequential Progressive Optimization Using Evolutionary and Gradient Algorithms L.A. Catalano A. Dadone V. Daloiso G. Anaclerio Politecnico di Bari, ITALY

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3 Design Optimization International Conference, March 31- April , Athens, Greece, SEQUENTIAL PROGRESSIVE OPTIMIZATION USING EVOLUTIONARY AND GRADIENT ALGORITHMS Luciano A. Catalano, Andrea Dadone, Vito S. E. Daloiso, Giancarlo Anaclerio Dipartimento di Ingegneria Meccanica e Gestionale, & Centro di Eccellenza in Meccanica Computazionale, Politecnico di Bari, via Re David 200, I-70125, Bari, ITALY, catalano@poliba.it Key words: Fluid Dynamics, Progressive Optimization, Efficient Finite-Difference Sensitivities, Evolutionary Algorithms, Sequential Optimization. Abstract. An efficient progressive optimization strategy, based on the simultaneous convergence of the design process, of the flow analysis and of the mesh fineness, is firstly combined with a state-of-the-art genetic algorithm to improve its efficiency and to demonstrate that evolutionary methods are able to locate the near-optimum regions even on coarse grids and with partially converged flow solutions. Then, on the basis of this result, the paper proposes a sequential, genetic/gradient-based, progressive optimization strategy, which exploits the capability of the genetic algorithm to identify one or more near-optimum regions, while retaining the high efficiency of the progressive gradient-based technique. All proposed optimization strategies have been tested versus the inverse design of an airfoil in inviscid and laminar flow conditions: both the gradient-based and the sequential progressive optimization strategies exhibit robustness and comparable high efficiency, the converged design optimizations being obtained in computational times equal to those required by 10 and multigrid flow analyses on the finest grid, respectively. 1 INTRODUCTION Among many available optimization techniques, two main methodologies are considered suitable for the application to fluid-dynamic problems, by both scientific and industrial researchers: evolutionary algorithms and gradient-based techniques. As widely recognized, these two competitive methodologies are characterized by complementary advantages and disadvantages: the genetic algorithms are very efficient in case of multidisciplinary or constrained problems, where more than one optimal values are usually found and an effective global search is required; however, the cost of a genetic procedure is usually very high. On the contrary, a gradient-based technique is very efficient in local search, but can fail in finding more than one optimal values. Joining the research activity on the gradient-based techniques, the first two authors have contributed to the development of an adjoint progressive optimization strategy for the fluiddynamic design of transonic airfoils and cascades [1], [2]. The procedure is based on the idea of the simultaneous convergence of the design process, of the adjoint equation solution, and of the flow analysis; the last one includes also the global refinement from a coarse to a sufficiently fine 1

4 mesh. The computational time for the evaluation of the sensitivity derivatives is strongly reduced by defining the adjoint problem on the coarsest mesh employed. In Refs. [1], [2], it has been shown that the method allows to perform the inverse design of a transonic aerodynamic profile in a computational time equal to that required by 2 5 single-grid flow analyses. More ingredients have been recently added: a new method has been proposed in Ref. [3] for the definition of a set of orthogonal shape functions, which can provide a good representation of complex and innovative aerodynamic shapes. Moreover, since a very cumbersome analytical or symbolic differentiation is needed in case of more complex flow models and the definition of the adjoint system is even impossible when using black-box commercial codes, the authors have recently developed a method [3] for computing the sensitivity derivatives by means of finite differences of the flow solution as efficiently as for the adjoint method. The new method exploits some important multigrid concepts: the same multigrid subroutines are used to compute and store (an approximate value of) the relative local truncation error between the finest- and the coarsestgrid solutions for the current configuration; the sensitivity derivatives are then evaluated using finite differences of the objective function (eventually smoothed in case of shocks), obtained by perturbing the geometry and computing the solution correction on the coarsest level employed, using the relative local truncation error, previously stored. It is noteworthy that this new approach i) is very simple to program if a multigrid technique is already implemented, ii) is very efficient, since only coarse-grid solutions are involved and an easy parallelization is possible, iii) avoids very cumbersome differentiations of the discretized flow equations, and iv) can be applied also to black-box commercial codes, which cannot exploit the high efficiency of the adjoint approach. Trying to abandon the competitivity between the genetic and the gradient-based approaches, this paper aims at combining them so as to develop a robust and efficient optimization procedure which exploits the advantages of both techniques. The first contribution consists in the combination of the aforementioned progressive strategy with a state-of-the-art genetic algorithm, to demonstrate that evolutionary methods are able to locate the near-optimum regions even on coarse grids and with partially converged flow solutions, and thus can exploit the efficiency enhancement provided by progressivity, without losing in robustness. The second contribution is an effective sequential combination of the genetic and gradient-based optimization techniques, which aims at retaining the high efficiency of the progressive gradient-based strategy and at exploiting the global-search capability of the genetic algorithm. The proposed progressive optimization starts with the application of the genetic algorithm on coarse grids only, so as to identify one or more regions where one or more optimal values could be located, in an acceptable CPU time; then, one or more gradient-based optimizations with increasing mesh fineness and convergence of the flow solution are performed to refine both the optimal design parameters and the accuracy of the objective function. All employed methodologies are described in detail and finally tested versus the inverse design of an airfoil in inviscid and laminar flow conditions. 2 FLOW SOLVER An unstructured cell-vertex triangular grid is used to discretize the 2D Euler and Navier- Stokes equations governing the flows considered in this paper. A left state and a right state 2

5 are linearly reconstructed on the two sides of each interface (ij), obtained by connecting either the barycenters or the circumcenters of two neighbouring triangles. Similarly to the 1-D case, a unique left-neighbouring cell is used to define the flow gradient employed in the reconstruction [2], [4], [5]: it is defined as the cell C ji which contains the prolongation of the side (ji), plotted as a dot-dashed line in Figure 1. Standard one-dimensional limiters are also applied straightforwardly. The flux-difference-splitting of Roe [6] is then used to solve the Riemann problem defined at each interface. A standard finite-element Galerkin discretization is used for the viscous terms interface (ij) C ji j lji l ij i b y x Figure 1: Higher-order reconstruction. Figure 2: Orthogonal base functions. The discretized governing equations are solved by means of a four-stage Runge-Kutta scheme, coupled with an Implicit Residual Smoothing procedure: Reference [7] fully describes the technique here employed to define the smoothing lines on cell-vertex unstructured grids. A standard V-cycle Full MultiGrid (FMG) [8] has been also implemented both to accelerate convergence to steady state and to compute multigrid-aided finite-difference sensitivities. Finer grids are created during the nested iteration by means of a global uniform refinement, improved by a grid-point adjustment. 3 GRADIENT-BASED OPTIMIZATION STRATEGY Many currently used gradient-based techniques employ an adjoint system to compute the sensitivity derivatives efficiently. However, a cumbersome analytical, symbolic or numerical differentiation of the flow equations is needed when using complex models and/or discretizations. For such a reason, the authors have recently developed an alternative technique, named Multigrid-Aided Finite-Difference (MAFD) procedure, which allows to compute the sensitivity derivatives as efficiently as with the adjoint method, but independently of the flow model and discretization [3], [9]. The simplest way to compute the gradient of the objective function I consists of perturbing each parameter defining the geometry, ξ j, computing the corresponding flow solution, and finally 3

6 evaluating I/ ξ j by means of finite differences. This approach is strongly time-consuming, in particular when a high number of design variables is involved and a scalar processor is used. Its main advantages are the simplicity, the straightforward parallelization, the invariance with respect to the flow modelization and/or discretization, and the possible combination with blackbox commercial codes. The Multigrid-Aided Finite-Difference (MAFD) procedure, developed in [3], [9] and employed in this paper, aims at reducing the computational required by the computation of the perturbed geometries, while maintaining the advantages cited above. According to the so-called dual viewpoint of the multigrid (MG) technique, a constant term is computed at each MG cycle and added to the right-hand side on the coarser grid level: as known, this term represents an approximate value of the relative local truncation error (RLTE) between the finer grid and the coarser one [8]. Accordingly, the MAFD method is based on the following important considerations: i) the MG strategy solves the flow equations on coarser grid levels with the same accuracy of the finer level, thanks to the addition of the RLTE term. Moreover, ii) a correct choice of the design parameters should give a smooth perturbation of the blade profile, that can be seen effectively on a coarser level. Finally, iii) the approximate RLTE, which mainly represents the difference of accuracy between two nested grid levels, is not affected by a small, smooth perturbation of one design parameter. On the basis of these three considerations, the proposed method allows to compute the difference between the flow solutions of two perturbed geometries using a coarser grid level and a value of the RLTE computed only once, using the unperturbed geometry. Centered finite differences have been preferred with respect to one-side differences for robustness, rather than for accuracy. However, an exhaustive comparison between the two possible approaches has not been performed yet. It is noteworthy that the coarse-grid evaluation of the perturbed flow fields is very efficient, while preserving the fine-grid accuracy: the coarser grid levels have a much lower number of cells and allow the use of a higher time step (the time step is at least doubled at each coarsening). Clearly, the required computational still depends on the number of design parameters, N ξ, when using a scalar processor; however, the entire procedure can be easily parallelized: in such a case, the required computational would become negligible and independent of N ξ. To date, the MAFD procedure has been tested versus 2- and 3-D inviscid and 2-D laminar flows, using either structured or unstructured flow solvers [3], [9]. 4 THE GENETIC ALGORITHM The genetic code employed in this paper is the one provided by D.L. Carroll [10]: the encoded individual is a binary string which links the N ξ design parameters, each of them being represented by 10 bits; in all proposed applications, the population contains N p = 20 individuals. The socalled tournament selection has been chosen as reproduction operator, and is supported by a shuffling technique which allows to choose random pairs for mating. Other additional operators employed are jump and creep mutation, uniform cross-over and elitism. Some minor modifications have been applied to improve the efficiency of the genetic algorithm in the proposed applications: the variation range of the design parameters has been fixed so as to limit the chance of generating non-physical profiles, which are eventually discarded by a suitable check; moreover, the objective function is computed and stored only when a new individual is generated, so as to save CPU time when the same individual appears more than once. 4

7 5 PROGRESSIVE OPTIMIZATION STRATEGY A very effective method to reduce the CPU time needed by the optimization run is the socalled progressive optimization strategy [1], which consists in the simultaneous convergence of the design process and of the flow analysis, also including the global refinement from a coarse to a sufficiently fine mesh. This approach has been widely employed by the authors in the frame of a gradient-based optimization technique, with converged design optimizations produced in CPU times comparable to that required by the analysis of a single geometry, using either the adjoint or the MAFD procedures. Using the norm of the gradient as parameter for detecting the convergence level of the optimization procedure, less accurate sensitivity derivatives (i.e. with partially converged flow solutions computed on coarser levels) are used when the geometry is far from the optimal one; then, the convergence of the flow solution and the number of mesh points are increased accordingly with the reduction of the gradient. Starting the optimization on coarser grids and using partially converged flow solutions drastically reduce the computational cost of the entire optimization procedure, without affecting its robustness and capability of finding the optimum, as demonstrated by the large number of applications proposed so far [2], [3], [9], [11]. Full details of the gradient-based progressive optimization strategy can be found in the cited References. A first contribution of this paper consists in the application of the progressive strategy to the genetic algorithm: the main difficulty of this combination relies in the non-existence of a parameter detecting the convergence level of genetic algorithms, which usually employ the maximum number of generations as stop criterion. Here, an alternative parameter, f n, has been defined to evaluate the convergence level at the generation n, namely the average of the best-fitness variations, weighted with respect to the generation number: nj=2 (j f j ) f n = f nj=2 max j, where f j = f j f j 1 is the variation (between two consecutive generations) of the best fitness, f, and f max is its maximum variation: f max = max{ f 2, f 3,..., f n }. When applying the progressive optimization strategy, the minimum flow residual is decreased according with the value of f n ; moreover, the (two) grid refinements are set at f n < 0.5 and f n < 0.1, respectively. 6 SEQUENTIAL GENETIC/GRADIENT-BASED PROGRESSIVE STRATEGY The second contribution of the present paper is an effective sequential combination of the genetic and gradient-based optimization techniques: the optimization starts with the application of the genetic algorithm on coarse grids only, so as to identify one or more regions where one or more optimal values could be located, in an acceptable CPU time; at this stage, in fact, an approximate but efficient optimization can be considered satisfactory to define one or more starting geometries for the one or more following gradient-based applications, which aim at refining the optimal values, as well as the accuracy of the objective function. During the entire 5

8 sequential optimization, all concepts of progressivity are retained, namely the minimum flow residual follows the convergence level (namely f n or the norm of the gradient) and also the mesh is refined accordingly. The proposed sequential optimization allows to retain the high efficiency of the progressive gradient-based strategy and to exploit the capability of the genetic algorithm to identify one or more regions where optimal values could be located. The switch to the gradient-based method has to be performed when the genetic algorithm has located one or more regions where the population density is over a threshold value. Accordingly, the gradient-based initial profiles are the ones with minimum or low values of dispersion, the dispersion index related to the generic individual i being defined as: where is the square error of individual i and Np j=1 σ i = d2 ij N p d 2, max N ξ d 2 ij = (ξ k,i ξ k,j ) 2, k=1 N ξ d max = (ξ k,max ξ k,min ) 2 k=1 is the maximum width of the search domain. In the proposed applications, the switch to the gradient-based method is performed when the minimum dispersion at the generation n is reduced to half the value assumed at the first generation, namely when: P n P 1 < 0.5, where: P n = min{σ i } n i = 1,..., N p. 7 RESULTS The numerical applications proposed in this section aim at i) demonstrating the robustness of the progressive genetic procedure and the efficiency improvement with respect to its standard version; ii) comparing the performances of the progressive sequential approach proposed in this paper with those resulting from the stand-alone application of the MAFD progressive optimization strategy. The sequential approach should exhibit its best performances in the case of constrained or multidisciplinary problems, where it should be capable of identifying more than one near-optimum regions at a low cost. On the contrary, the sequential approach should suffer from the comparison 6

9 with a gradient-based technique in case of simple unconstrained problems. For such a reason, a simple unconstrained problem, namely the inverse design of an airfoil in both inviscid (transonic) and laminar flow conditions, has been preferred to more complex applications. Moreover, the proposed test-cases have known solutions: a reasonable reference (or target) airfoil has been preliminarly defined by combining the four orthogonal (two symmetric and two antisymmetric) base functions [3] provided in Figure 2, with weights (or design parameters) ξ 1 = 1.2, ξ 2 = 0.5, ξ 3 = 0.1 and ξ 4 = 0.1, see the solid line plotted in Figure 3. The reference airfoil has been then used to compute the prescribed (target) pressure distribution on the airfoil. This makes possible to monitor the convergence of all optimization procedures by means of the magnitude of the objective function, I. target initial optimal Figure 3: Target, initial and optimal profiles for the MAFD progressive optimization. Figure 4: Mach number contours for the inviscid transonic flow past the reference airfoil ( M = 0.05). Figure 5: Mach number contours for the laminar flow past the reference airfoil ( M = 0.05; levels added at wall to show separation). 7

10 Both the MG and the progressive optimization employ three grid levels, and the finest mesh is composed of nodes and triangles (392 nodes on the airfoil). The Mach number contours for the inviscid (M = 0.8, α = 1 ) and the laminar (M = 0.8, α = 10, RE = 500) reference flows are provided in Figure 4 and in Figure 5, respectively. -1 gradient-based sequential genetic -2 TRANSONIC -1-2 gradient based sequential genetic LAMINAR TE -3-3 Log 10 (I) -4-5 Log 10 (I) Figure 6: Convergence histories for the inviscid transonic test-case Figure 7: Convergence histories for the laminar test-case. Both test-cases have been approached by using the three progressive optimization strategies described so far, namely: i) the MAFD technique, ii) the genetic algorithm, iii) the sequential approach. Figures 6 and 7 provide the corresponding convergence histories, namely the magnitude of the objective function plotted versus the computational in a bi-logarithmic diagram, so as to contain all runs ξ 1 ξ 2 ξ 3 ξ ξ 1 ξ 2 ξ 3 ξ Figure 8: Inviscid transonic test-case: convergence histories of the design parameters using the gradient-based algorithm Figure 9: Laminar test-case: convergence histories of the design parameters using the gradient-based algorithm. 8

11 The MAFD progressive optimization strategy is able to drive an arbitrarily-guessed, initial airfoil, to the (known) target one, in a computational time comparable to that required by the analysis of the reference profile. In both test-cases, the initial profile, plotted as a dashed line in fig. 3, is defined by the design parameters ξ 1 = 3.0, ξ 2 = 0.0, ξ 3 = 0.0 and ξ 4 = 0.0. The whole optimization process is stopped when log 10 I 5.0 at the finest level which corresponds to a log 10 (I) 7.0. One unit is defined as the computational time required to run one converged MG analysis on the finest mesh. It is noteworthy that the MG acceleration procedure has not been used in most of the previous applications [1, 2, 11], where the unit was defined with reference to one single-grid analysis, unless the optimization procedure took advantage of the grid refinement. On the contrary, the present MG analysis takes the same advantages of the optimization code. Since a different measure is used, the optimization process will require an increased number of units with respect to all non-mg applicationsfound in the cited references. The optimization procedure is firstly applied on the coarsest level, which cannot take advantage of the MAFD procedure. Taking as example the transonic test-case, the first grid refinement is performed after 0.8 units. At the second grid level, the MAFD technique employs the coarser mesh to compute the sensitivity derivatives: the second refinement is then located at 1.6. Clearly, the spent on the coarser levels becomes more relevant with respect to previous applications, since the advantages of the coarse-grid evaluation of the perturbed flow-fields are reduced or even missed. Figures 6 and 7 indicate that the required to obtain the more than satisfactory convergence level of -5.0, on the finest mesh, is about 10 for the transonic flow and 15 for the laminar flow. This convergence level is even excessive for engineering applications: for the transonic test-case, for example, the optimization has found the optimal design parameters ξ 1 = , ξ 2 = , ξ 3 = and ξ 4 = , but a satisfactory convergence was already obtained after few units, as demonstrated by the convergence histories of the design parameters provided in Figures 8 and 9. The optimal airfoil (symbols) is perfectly superposed to the target configuration (solid line), see Figure ξ 1 ξ 2 ξ 3 ξ ξ 1 ξ 2 ξ 3 ξ Figure 10: Inviscid transonic test-case: convergence histories of the design parameters using the genetic algorithm. Figure 11: Laminar test-case: convergence histories of the design parameters using the genetic algorithm. 9

12 As expected, the performances of the genetic progressive optimization are much worse with respect of the gradient-based technique, see the dot-dashed lines in Figures 6 and 7. However, the use of progressivity has significantly increased the efficiency of the genetic algorithm, thus allowing to complete the run in an acceptable time. On the contrary, a complete run of the standard method could not been achieved due to the unacceptable CPU time required. Moreover, the genetic algorithm has located the near-optimum region even using coarse grids and partially converged flow solutions gradient-based sequential TRANSONIC -1-2 gradient based sequential LAMINAR TE -3-3 Log 10 (I) -4-5 Log 10 (I) Figure 12: Comparison between the MAFD and the sequential progressive optimizations: zoom of the convergence histories for the inviscid transonic test-case Figure 13: Comparison between the MAFD and the sequential progressive optimizations: zoom of the convergence histories for the laminar test-case. The application of the sequential progressive optimization to the inviscid and to the laminar test-cases provides the convergence histories plotted as dashed lines in Figures 6 and 7, respectively. The progressive sequential procedure consists of an initial research of the near-optimum region using the genetic algorithm on the coarsest level (with partially converged flow solutions), and of a further optimization refinement by means of the MAFD progressive technique. Clearly, the convergence histories of the sequential approach and of the previous genetic run coincide on the coarsest level. For clarity, Figures 12 and 13 provide a zoom of the convergence histories of the gradient-based and of the sequential strategies, using a linear scale for the computational. It can be observed that the switch criterion has been satisfied after about 4 and 7.5 units for the inviscid and the laminar test-cases, respectively. Correspondingly, a good initial profile has been assigned to the gradient-based optimization in both cases, see the dashed lines of Figures 14 and 15, which correspond to ξ 1 = , ξ 2 = , ξ 3 = , ξ 4 = , for the inviscid transonic test-case and to ξ 1 = , ξ 2 = , ξ 3 = , ξ 4 = , for the laminar test-case, respectively. The computations on the intermediate and on the fine grid levels are then performed by means of the efficient MAFD progressive procedure, rather than using the expensive genetic algorithm. The computational s required by the MAFD progressive optimization and by the sequential progressive strategy are comparable: the converged design optimizations have been obtained in computational times equal to those required by 10 10

13 Target profile GB initial profile Target profile GB initial profile Figure 14: Inviscid transonic test-case: target and gradient-based initial profiles. Figure 15: Laminar test-case: target and gradient-based initial profiles. and multigrid flow analyses on the finest grid, respectively. For completeness, Figures 16 and 17 provide the convergence histories of the design parameters ξ 1 ξ 2 ξ 3 ξ ξ 1 ξ 2 ξ 3 ξ Figure 16: Inviscid transonic test-case: convergence histories of the design parameters using the sequential procedure Figure 17: Laminar test-case: convergence histories of the design parameters using the sequential procedure. 8 CONCLUSIONS An efficient progressive optimization strategy, based on the simultaneous convergence of the design process, of the flow analysis and of the mesh fineness, has been combined with a genetic algorithm to improve its efficiency and to demonstrate that evolutionary methods are able to locate the near-optimum regions, even on coarse grids and with not fully converged flow solutions. On the basis of this result, the paper has presented a new type of progressive 11

14 optimization strategy, based on the sequential application of a genetic algorithm with partially converged flow solutions on coarse grids, and of a gradient-based technique with increasing mesh fineness and convergence level. The proposed numerical applications demonstrate that the sequential progressive approach is as efficient as the progressive gradient-based strategy; its furhter advantage consists in its capability of preliminarly identifying more than one nearoptimum regions, which is crucial in case of constrained or multidisciplinary design problems. ACKNOWLEDGEMENTS This has been supported by MIUR (Ministero dell Istruzione, dell Università e della Ricerca). REFERENCES [1] Dadone, A., Grossman, B., Progressive Optimization of Inverse Fluid Dynamic Design Problems, Computers and Fluids, 29, pp. 1-32, [2] Catalano, L.A., Dadone, A., Progressive optimization for the efficient design of 3D cascades, AIAA Paper , [3] Catalano, L.A., Dadone, A., Daloiso, V.S.E., Mele, G., Progressive optimization using orthogonal shape functions and efficient finite-difference sensitivities, AIAA Paper , [4] Catalano, L.A., A new reconstruction scheme for the computation of inviscid compressible flows on 3D unstructured grids, Int. J. Numer. Meth. Fluids, 40 (1-2), pp , [5] Catalano, L.A., Daloiso, V.S.E., Accurate computation of 2D turbulent compressible flows on unstructured grids, AIAA Paper , [6] Roe, P.L., Characteristic based schemes for the Euler equations, Annual Review of Fluid Mechanics, 18, pp , [7] Catalano, L.A., Daloiso, V.S.E., Upwinding and implicit residual smoothing on cell-vertex unstructured grids, 7 th ICFD Conference, Oxford, United Kingdom, 2004, also submitted to Int. J. Numer. Meth. Fluids. [8] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31, pp , [9] Catalano, L.A., Dadone, A., Daloiso V.S.E., Progressive optimization on unstructured grids using multigrid-aided finite-difference sensitivities, 7 th ICFD Conference, Oxford, United Kingdom, 2004, also submitted to Int. J. Numer. Meth. Fluids. [10] Carroll, D.L., Fortran Genetic Algorithm Driver, version 1.7a, available on the web site [11] Dadone, A., Grossman, B., Fast convergence of viscous airfoil design problems, AIAA Paper ,