Large-scale CFD Optimization based on the FFD ...

Large-scale CFD Optimization based on the FFD Parametrization using the Multipoint Approximation Method in an HPC Environment Yury M. Korolev1 and Vassili V. Toropov 2 Queen Mary University of London, London, E1 4NS, UK Shahrokh Shahpar3 Rolls-Royce plc, Derby, DE24 8BJ, UK

Abstract. An adaptation of the Multipoint Approximation Method (MAM) to a highperformance computing (HPC) environment is presented and demonstrated by high-fidelity CFDbased design optimization of a highly-loaded transonic rotor, significantly improving its efficiency in a much reduced design time. MAM is incorporated into the Rolls-Royce SOPHY design system, used in this study, that includes the free-form deformation (FFD) parametrization of the blade shape, automatic meshing, CFD analysis, and post-processing.

T

I. Introduction

urbomachinery applications present a number of challenges for optimization. Evaluating model responses, which are provided by state-of-the-art CFD codes, is usually a computationally expensive task. The run time for one response evaluation may be several hours or even days. Moreover, these responses contain numerical noise, and are subject to occasional simulation failures, the latter being caused by random factors as well as non-physical designs examined by the optimizer during the search. A ‘good’ feature of some of turbomachinery applications is that sensitivities of the responses may be available via adjoint CFD. The accuracy of the sensitivities, however, may be lower than that of the responses. All these features need to be taken into account by the optimization method, which becomes even more challenging when the dimensionality of the problem (i.e. the number of design variables) grows in order to address more challenging problems of turbomachinery design. The multipoint approximation method (MAM) has been successfully used to solve optimization problems in turbomachinery1,2,3. The multipoint approximation method, as reported in Refs. 4-6, is an iterative optimization technique based on mid-range approximations built in trust regions. A trust region is a sub domain of the design space in which a set of design points, treated as a small-scale design of experiments (DoE), are evaluated. These and a subset of previously evaluated design points are used to build approximations of the objective and constraint functions that are considered to be valid within a current trust region. The trust region will then translate and change size as the optimization progresses. The trust region strategy has gone through stages of development to account for the presence of numerical noise in the response function values7,8 and occasional simulation failures9. The mid-range approximations used in the trust regions, as originally suggested in Ref. 4 for structural optimization problems, are intrinsically linear functions (i.e. nonlinear functions that can be led to a linear form by a simple transformation) for individual sub-structures, and assembly of them for the whole structure. This was enhanced by the use of gradientassisted approximations,6 use of simplified numerical models that is also termed a multi-fidelity approach,10 and the use of analytical models derived by Genetic Programming11,3. One of the recent developments12 involved the use of approximation assemblies, i.e. a two stage approximation building process that is conceptually similar to the original one used in Ref. 4 but free from the limitation that lower level approximations are linked to individual substructures. The Moving Least Squares Method (MLSM) was proposed in Ref. 13 for smoothing and interpolation of scattered data and later used in the mesh-free form of the FEM14. As described in Ref. 15, it can be used as a technique for surrogate modelling and used in MDO frameworks. The MLSM is a weighted least squares method 1

Postdoctoral Research Fellow, School of Engineering and Materials Science Professor of Aerospace Engineering, School of Engineering and Materials Science, AIAA Associate Fellow, FRAeS 3 RR Engineering Associate Fellow - Aerothermal Design Systems, AIAA Associate Fellow, FRAeS 1 American Institute of Aeronautics and Astronautics 2

where the weights depend on the Euclidian distance from a sample point to where the surrogate model is to be evaluated. The weight value for a certain sample point decays as the distance increases. Describing the weight decay with a Gaussian function tends to be the most useful option even though many others have been evaluated in Ref. 16. As demonstrated in Ref. 17, the cross-validated MLSM can be used both for design variable screening and for surrogate modelling. In order to create an efficient MDO framework for problems with disparate discipline attributes Ref. 18 extended the optimization approach to the use of local DOEs and MLS approximations built in different subspaces of the total design variable space corresponding to the individual disciplines. The subspaces are finally combined into the total design variable space in which the resulting MDO problem is solved. The computational complexity of the response evaluations motivates extensive use of parallel computations in turbomachinery optimization. Despite of significant work on implementation of MAM in a parallel computing environment,1 this method still did not fully exploit the paradigm of parallel computations. At the end of each iteration there was a ‘bottleneck’, when only one processor had to be used at a time to evaluate the responses at the proposed centre of the next trust region. Some recent work on parallelization of the multipoint approximation method was presented in Ref. 19. The improved trust region strategy adapts the search to the available computational resources and enables their uniform use. In this implementation, MAM receives the number of available processors (NAP) as input and adjusts its calls for the external (CFD) software accordingly, i.e. it only submits jobs in bunches of a multiples of NAP.

II. The Multipoint Approximation Method A typical formulation of a constrained optimization problem that MAM works with is as follows: (1)

where is the vector of design parameters, and respectively, is the objective function, and

are the lower and upper bounds for the design variables, are the constraints. The numbers of design variables and

constraints are and , respectively. MAM attempts to solve this problem by using approximations of the objective function and constraints in a series of trust regions. The trust region strategy seeks to ‘zoom in’ on the region where the constraint minimum is achieved. It aims at finding a trust region that is sufficiently small for the approximations to be sufficiently good for the design improvement and that contains the point of the constrained minimum as its interior point. The main loop of MAM is organized as follows. 1. Initialization: choose a starting point

and initial trust region

such that

2. At the k-th iteration the current approximation to the constrained minimum is .

.

, the current trust region is

a.

Design of Experiments (DoE): a set of points is chosen that will be used to build approximations. Responses are evaluated at the DoE points and approximations are built using the obtained values. Currently, the pool of approximation methods available in MAM consists of Metamodel Assembly12 and the moving least-squares metamodels13,14. Other approximation methods could be used as well. Denote the approximate objective function and constraints by and , respectively.

b.

The original optimization problem (1) is replaced by the following problem: (2)

c.

We solve the approximate problem (2) using Sequential Quadratic Programming20,21 (SQP) with various starting points and obtain a number of local minima. These points (and, depending on availability of computational resources, a number of other, randomly generated points) are candidates for the center of the next trust region. Original responses are evaluated at the candidate points. A point with the best combination of objective value and constraint violation is selected. Let us denote this point by . 2 American Institute of Aeronautics and Astronautics

d.

The centre of next trust region is positioned at . The size of the next trust region depends on the quality of approximations at the previous iteration, on the history of the points , and on the size of the current trust region. The quality of approximations is assessed using the values of original responses at all candidate points that have been obtained at the previous step. e. The termination criterion is checked (it is a part of the trust region strategy and depends on the position of the point , the size of the current trust region and the quality of approximations). If the termination criterion is satisfied, the algorithm proceeds to step 3. Otherwise, it returns to step 2. 3. Optimization terminates. The obtained approximation to the solution of the problem (1) is . As we have pointed out in the introduction, parallel computations play an important role in modern turbomachinery optimization. Therefore, it is very important to bring together the framework of MAM and the paradigm of parallel computations. Despite the sequential nature of MAM, it can be done effectively by parallelizing the calls to the external software, since a response evaluation typically takes much longer than any MAM-related computations (Design of Experiments, metamodeling or solving approximate optimization problems). External software is called at two phases of the MAM algorithm: during the Design of Experiments phase and the candidate point assessment phase. Details of our approach to these phases will be explained below. The main idea is to submit jobs to the external software in multiples of the number of available processors (NAP) only. This number is received by MAM as an input parameter and remains unchanged during the whole optimization run. It is worth noting that our approach implicitly assumes that the evaluation times for the responses are similar regardless of the design variables. Whilst it may not always be true, we do not have (or, at least, do not assume to have) any a priori knowledge about the dependence of the response evaluation time on the design variables. In this situation, the assumption we make seems quite natural.

III. Design of Experiments The quality of the Design of Experiments (DoE) is crucial for the performance of any metamodeling-based method. The DoE points should have a reasonably good spread in the trust region in order to ensure good exploration of the latter. The DoE should also be non-collapsible, i.e. its the projection onto any subspace of the design space should itself be a non-degenerate DoE (some motivation for that is given, for example, in Ref. 22). The iterative and trust region-based nature of MAM sets some further requirements for the DoE. It can happen that the current trust region contains points that have been evaluated earlier. Including them into the DoE would increase the efficiency of the algorithm. Therefore, the DoE should allow to distribute new points in the trust region taking into account the existing ones. In order to ensure uniform usage of the available computational resources, the method should be able to produce designs of an arbitrary size to account for any possible number of available processors. Another desirable feature is that the number of DoE points in high dimensions should only linearly depend on n so that the computation time for the while DoE should be affordable for sufficiently large n (say, for ). For this purpose a non-collapsible randomized Design of Experiments has been developed. Suppose that the current trust region is the hypercube (one can always achieve this by a linear transformation). Suppose that points already exist in the trust region. New points will be added one-by-one, therefore, it is sufficient to describe the procedure that is used to add one point to an existing DoE. In order to ensure non-collapsibility of the achieved design, it is necessary to avoid the regions where at least in one dimension,

, and for at least one existing point

,

(

is a

constant). In order to ensure a reasonable spread of the points, it is also necessary to avoid regions where for at least one existing point

,

(here

indicates any norm in

, for instance, the

Euclidean norm, and is a constant). These ‘forbidden’ regions are represented as ‘stripes’ and circles in Fig. 1. A straightforward way to avoid the `forbidden` regions would be to sample points randomly from the uniform distribution on and reject all points that happen to be in the `forbidden` areas. However, this procedure would not be efficient in producing non-collapsible designs, as demonstrated later. Consider the case . First note that if , it can happen that the stripes cover the whole area. Therefore, the largest value of that remains useful is such that

, for example,

. If

and the stripes don't overlap, the probability that a point

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randomly picked from a uniform distribution avoids all the stripes is . This quantity tends to zero as , indicating that that the chance of avoiding all the stripes will become arbitrarily small as the number of variables grows, and the acceptance rate in the reject-accept procedure described above will be prohibitively low. We overcome this problem by using the following sampling strategy. Let us fix a coordinate . The ‘allowed’ area in this coordinate is the interval less all the intervals of length centered at the values of the -th coordinate of all existing DoE points. This set is non-empty if . Denote the total length of all the intervals in the allowed set by . If a random point is sampled from a uniform distribution on , it can be uniquely transformed to a point in the allowable set. This way all the stripes can be avoided so that the ‘acceptance rate’ of this procedure is 100%.

Figure 1: An example of ‘forbidden areas’ for a DoE in 2D Let us describe the mappings between the interval and an allowable set, which consists of a number of intervals in of total length . The allowable set may be coded as a vector that contains the lower bounds of the allowable intervals at positions 1,3,…, and the upper bounds of these intervals at positions 2,4,… The first allowable interval is , the second one is , etc.. Let represent the original coordinate, , and represent its image in the interval . Then the transformation can be described as follows. For a given , define . Then

The inverse transformation

is also easily defined. It can be represented by the following pseudo code:

The above procedure allows to avoid all stripes but not the balls, see Figure 1. To avoid the balls, we can employ a reject-accept procedure. Let us find out which values of the parameter are acceptable. We will focus on the 4 American Institute of Aeronautics and Astronautics

case when distances are measured by means of the supremum norm, i.e.

. It is clear that, to

achieve a better spread, one would prefer higher values of the parameter . However, one needs a guarantee that the balls will not cover the whole volume in the hypercube that is left after `cutting out` the stripes. Simple geometric considerations (see Figure 2) suggest that the additional `forbidden` volume at an existing DoE point is . The total additional forbidden volume is less than or equal to (equality holds if the balls don't overlap).

Figure 2: Additional ‘forbidden’ volumes The total volume left after the stripes are cut out is given by

. Therefore, the volume of the feasible region

is guaranteed to be non-zero if the additional volume given by the balls is less than

The latter condition defines an upper bound on

that guarantees non-emptiness of the feasible region. The value

is recommended. In practice, the process starts with higher values of the parameter relaxes it towards

, i.e.

and

if a point does not become acceptable after a given number of trials (typically, 100). This

improves the uniformity of the spread of the DoE points in the trust region. There is also a guarantee that the minimum distance between any two DoE points will not be smaller than .

IV. Trust Region Strategy Suitable for an HPC Environment The ‘mid-range’ nature of the approximations used in MAM makes it less likely to fall into the nearest local minimum than a purely gradient-based technique. However, it has been attempted to further improve the non-local property of the optimization search with a modified trust region strategy while also attempting to use computational resources more uniformly through all stages of the search. In this approach, several candidates for the centre of the next trust region are proposed at each iteration. The number of candidate points is typically chosen equal to the number of available processors which ensures an optimal use of the computational resources later, when the responses will be evaluated at the candidate points. 5 American Institute of Aeronautics and Astronautics

In order to find the desired number of distinct candidates, an approximated optimization problem (2) is solved using the Sequential Quadratic Programming (SQP) method starting from several different initial points. There are many possibilities to generate the set of starting points (for example, Sobol sequences22,23). In the developed implementation the optimization runs start from different DoE points in the current trust region. It may happen that this procedure will produce a number of distinct candidates which is less than the desired number (that is a multiple of the number of available processors NAP). In fact, it is typical that the number of local minima in the problem (2) is smaller than NAP and therefore a desired number of candidates cannot be found using this procedure only (this will be the case, for example, if all the approximated responses are convex in the current trust region as in such a case there is only one local minimum for the problem (2)). In order not to waste available computational resources, and to allow for a better exploration of the design space, more points are added up to the total of NAP by the same approach that is used to choose the DoE points (see Section III). Once the candidate points have been chosen, the original responses are evaluated at these (multiple of) NAP points. Next, one of them is to become the centre of the next trust region. A point with the best combination of the objective function value and satisfaction of constraints will be chosen. Of course, it is easy to compare two designs if one of them has a lower value of the objective function and does not violate any constraints more than the other one, but often candidates will not be comparable in this manner, and a scalar quality criterion needs to be established. For this purpose, an exterior penalty function approach is used. The candidate solutions are compared by means of a penalized objective function

where

is the value of the original objective function at the point

, the values

,

are the values of the violation of different constraints at the point , the function penalizes the violation of constraints, and is a constant. The function can be chosen in several different ways. It is essential that it is monotone with respect to each constraint function value (i.e. if the value of one of constraint functions gets smaller while the remaining constraint functions are unchanged, the value of the penalty should not increase). Currently, the pool of penalty functions in MAM consists of the following ones:

ψ1 (t) = ( max t i ) β , i=1,...,M

M

ψ 2 (t) = ∑ t i β , i=1

M

€

where is a constant, typically between 1 and 2, and t ∈ R+ is a vector of constraint violations. The local (within the current trust region) quality of approximation of the original responses by metamodels is an important indicator in the trust region strategy. To assess the approximation quality, we need to use points in the current trust region for which values of the original responses are available and which have not been used to build metamodels. The latter requirement can be illustrated by the fact that, for example, in the moving least squares € method, an appropriate choice of the closeness of fit parameter can force the approximations to be exact at all the DoE points. Assessing the quality of approximations at the DoE points would in this case indicate zero discrepancy regardless of the actual quality of approximations. Therefore, the points, that have been used to build metamodels, cannot be used to assess their quality. In the previous version of MAM12, the quality of approximations was assessed using the original and approximate responses at the solution of the approximate optimization problem in the current trust region (the center of the next trust region). Of course, measuring the discrepancy at one point does not really provide comprehensive information about the closeness of the original and approximate responses in the current trust region, but it was all that could be afforded without a significant additional computational effort. Now that multiple candidates for the center of the next trust region are evaluated at each iteration, all of them can be used to assess the quality of approximations. Since these candidates are reasonably spread in the current trust region, an integral assessment of the approximation quality can be established. In particular, the following quantities are calculated

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⎛ P ⎞1/ 2 j j 2 ˜ ⎜ ∑ (Fi (x ) − Fi (x )) ⎟ j =1 ⎟ , δ i = ⎜ P ⎜ ⎟ ∑ (Fi (x j ))2 ⎟ ⎜ ⎝ j =1 ⎠ where

and

,

i = 0,1,..., M,

are the original and approximate responses, respectively, and

are the candidate points used to assess the quality of approximations.

€ V. Case Study: NASA Rotor 37 Introduction. NASA rotor 37 is a rotor blade for an axial core compressor that was developed at the NASA Lewis Research Centre24. It has a relatively high design speed pressure ratio of 2.1 with a tip speed of 454 m⁄s, yielding a peak Mach number of approximately 1.4. It was part of a broad program on axial flow compressors with the objective of attaining high-pressure ratios and efficiencies well within stall margins, with minimal number of stages. NASA rotor 37 has been extensively used for simulation by various researchers, see e.g. Ref. 25, 26. Most CFD codes over-predict the total pressure ratio well beyond the experimental uncertainty, whilst total temperature ratios are higher than those obtained from the experiment. Specifically, towards the hub and the tip regions, the total temperatures are far lower, some researchers attribute this to a hub leakage ahead of the rotor, e.g. see detailed uncertainty analysis conducted in Ref. 26. Unfortunately, further experimental studies could not be carried out on rotor 37 with leakage, as the blade has been broken during a prior experiment. Depending on predictions of the total temperature and the total pressure ratios (PR), adiabatic efficiency η variation between the various simulation codes is significant. However, in general, codes that obtain reasonable predictions of the total pressure and temperature under-predict the efficiency. In this study, simulations are carried out using the Rolls-Royce in-house codes – PADRAM27,28 for mesh and geometry generations and HYDRA29 for 3D-RANS computations. Numerical validations of these codes for NASA Rotor 37 have been previously documented in Ref. 30.

Figure 3: Computational domain Meshing Methodology. The Rolls–Royce in-house mesh generator PADRAM27,28 is a multi-block, algebraic grid generator that uses a C-O-H grid topology for creating both multistage and multi-passage structured grids. The computational domain considered in this study is shown in Figure 3. It comprises stationary upstream and downstream centre bodies and a rotating disk, upon which the rotor is mounted. A hub fillet is incorporated as per the geometry specifications26 using an O mesh that blends with the passage upper and lower H meshes. At the tip, a butterfly H mesh is used with 20 layers between the blade tip and the casing. The tip gap here is 0.356 mm. Detailed mesh sensitivity analysis has indicated that a mesh-independent solution can be obtained with a total mesh size of 4.2 million, see Figure 4 for its details.

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Figure 4: The mesh used in the CFD analysis Design Space. A novel design space definition provided by the PADRAM and used here is based on the FreeForm Deformation31 (FFD). This is a method first introduced within the computer graphics and animation industry to enable flexible deformation of geometric models. A one-to-one correspondence is established between the points within the original and the deformed volumes of space. Objects embedded within the original volume are thus deformed by mapping the point-set representing the object to their corresponding points in the deformed volume. In PADRAM, a FFD surface-orientated algorithm has been applied in the blade-to-blade aerofoil sections by means of a 2D control grid distribution. The design variables are defined in terms of the coordinates of the vertices defining this deformation grid and are allowed to move within 10% of their original locations, see Fig 5.

Figure 5: FFD definition of design variables As this figure indicates, it is possible to move any of the FFD coordinates to the LE of the aerofoil automatically and then rotate along the blade stagger. In the current optimisation, a 3x3 grid is employed that covers the whole of the aerofoil domain, see Figure 6.

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Figure 6: The FFD 3x3 grid covering the aerofoil domain Along the radial span, the FFD grid is applied at three different blade-to-blade sections (0%, 50%, and 100%). This means that for each aerofoil the displacements of 54 control points are computed. An interpolation based on the cubic splines is then used to recreate the whole 3D geometry. Flow Solver. The 3-D RANS equations are solved using the Rolls–Royce in-house general-purpose CFD code HYDRA32. HYDRA is an unstructured solver that uses an edge-based data structure with the flow data stored at the cell vertices. A flux-differencing algorithm based on the monotone upstream cantered schemes for conservation laws (MUSCL)33 is used for the space discretization. For the steady-state solution, the preconditioning of the discrete flow equations is applied using a block Jacobi pre-conditioner and a five-stage Runge–Kutta scheme. An element collapsing multi-grid algorithm is used to accelerate the convergence to the steady state. In the present study, the two-equation k-ω shear stress transport turbulence model is used, assuming fully turbulent boundary layers (i.e., no transition modelling), and with wall functions. Radial distributions of the total temperature and the total pressure are

˙ T0 /P0 prescribed at the rotor inlet with their values obtained from24. At the rotor exit, a fixed exit capacity m boundary condition is enforced. This effectively represents a constant operating line and is a reasonable simulation of how a blade row will actually run in a multistage environment. Periodic boundaries are enforced at the upper and lower walls.   The optimization problem is defined as follows: € minimize

100 - µ (η),

s.t.

PR1 < PRdesign < PR2 m1 < mdesign < m2

where the bounds denoted by the subscripts 1 and 2 represent a 0.95% and 1.05% limit on the pressure ratio and mass-flow rate (i.e. the values at the cruise point). This problem was solved with the Multipoint Approximation Method available in the Rolls-Royce in-house optimization toolkit SOFT (Smart Optimization for Turbomachinery34).

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Failed cases

Red: infeasible

Blue: best result found

Figure 7: Optimisation convergence history Figure 7 shows the history of convergence for the MAM optimisation run. Every red or green points represents a CFD simulation, the blue point is the best design found. It is remarkable that for the large design space considered here, despite the initial CFD failures (e.g. because the FFD-generated geometry was invalid), the optimiser has learnt to drive the design to a feasible region only within a limited number of runs (of the order of 500 runs for 54 design variables was needed) to produce an efficiency improvements of the order of 2.5% (from 85.58% to 88.12%). This is higher of all the other externally reported improvements for this rotor in the open literature. When a particular design considered in the optimization process results in a simulations failure, a message is passed to MAM that treats such a design as a failed one.

Figure 8: Optimisation convergence history excluding the failed states 10 American Institute of Aeronautics and Astronautics

When the failed designs are excluded from the optimisation history, the optimization trajectory in Figure 8 is obtained, indicating how the trust region shrinks as the optimiser homes on a promising part of the design space. Figures 9-11 show the front and back views of the optimized blade (shown in blue) superimposed on the datum geometry (shown in red). The optimized blade looks sensible and does not exhibit extreme lent or twist along the blade.

Figures 9-11: The optimized blade (blue) and datum geometry (red) Figure 12 shows the circumferentially averaged efficiency plots along the span. This figures indicates a good increase in efficiency is obtained on all the section at various heights, however, the most significant increase is at around 70-80% height. Close examination of the blade pressure values on the suction surface indicate that the shock-wave has moved further downstream near the tip, forming an oblique (weaker) lambda type shock wave, reducing the shock-boundary-layer interaction losses.

Figure 12: Circumferentially averaged efficiency plots along the span

VI. Conclusion The Multipoint Approximation Method has been adapted to a high-performance computing environment. The built-in Design of Experiments has been improved to achieve both uniformity and non-collapsibility, and the trust region strategy has been enhanced to uniformly utilize all available computational resources and avoid bottlenecks. 11 American Institute of Aeronautics and Astronautics

Performance of MAM was demonstrated on an optimization problem for a transonic rotor using the FFD parameterization of the shape of a blade.

Acknowledgments This research was supported by the UK Technology Strategy Board and Rolls-Royce project SILOET-II (Strategic Investment in Low-Carbon Engine Technology). The authors would like to thank Rolls–Royce plc for permission to publish this work.

References 1

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13 American Institute of Aeronautics and Astronautics