Struct Multidisc Optim (2012) 46:129–136 DOI 10.1007/s00158-011-0757-1
INDUSTRIAL APPLICATION
Enhanced POD projection basis with application to shape optimization of car engine intake port Manyu Xiao · Piotr Breitkopf · Rajan Filomeno Coelho · Catherine Knopf-Lenoir · Pierre Villon
Received: 26 August 2011 / Revised: 2 December 2011 / Accepted: 26 December 2011 / Published online: 13 January 2012 c Springer-Verlag 2012
Abstract In this paper we present a rigorous method for the construction of enhanced Proper Orthogonal Decomposition (POD) projection bases for the development of efficient Reduced Order Models (ROM). The resulting ROMs are seen to exactly interpolate global quantities by design, such as the objective function(s) and nonlinear constraints involved in the optimization problem, thus narrowing the search space by limiting the number of constraints that need to be explicitly included in the statement of the optimization problem. We decompose the basis into two subsets of orthogonal vectors, one for the representation of constraints and the other one, in a complementary space, for the minimization of the projection errors. An explicit algorithm is presented for the case of linear objective functions. The proposed method is then implemented within a bi-level ROM and is illustrated with an application to the multi-objective shape optimization of a car engine intake port for two competing objectives: CO2 emissions and engine power. We show that optimization using the proposed method produces Pareto dominant and realistic solutions for the flow fields within the combustion chamber, providing further insight into the flow properties. Keywords Proper Orthogonal Decomposition · Reduced order modeling · Intake port · Optimization
M. Xiao · P. Breitkopf (B) · C. Knopf-Lenoir · P. Villon Laboratoire Roberval, Université de Technologie de Compiègne, Compiègne, France e-mail:
[email protected] R. Filomeno Coelho BATir, Université Libre de Bruxelles, Brussels, Belgium
1 Introduction Using high-fidelity simulation models to predict the response of structures for design optimization and uncertainty quantification often leads to an unacceptable computational cost, thus motivating the research of techniques to extract features from complex physical fields using a reduced number of full-order numerical experiments. The Proper Orthogonal Decomposition (Berkooz et al. 1993) is of particular interest in the optimization of engineering problems, where a set of scalar objective/constraint functions that depends on the values of design variables, is evaluated such as a surface/volume integration of a velocity/stress/. . . field obtained from a finite element/volume simulation. Literature reviewed thus far shows a large volume of work recently published on improving the precision of POD-like ROMs and on interpolation between these ROMs. One of the research directions focuses on minimizing the number of full-scale analyses by including information about the gradients. Weickum et al. (2009) enriched a POD for the transient response of linear elastic structures by using the gradients of the POD modes with respect to the design/random parameters for robust shape optimization. In the same spirit (Carlberg and Farhat 2011; Hay et al. 2010) extended the concept of POD snapshots to include derivatives of the state variables with respect to system input parameters. In order to avoid generating additional sampling points, an effort has been made to develop interpolating strategies between the ROMs. Missoum (2008) used Lagrange interpolation to control the relative contributions of individual modes in order to perform a random-field-based probabilistic optimization of a tube impacting a rigid wall. Mathelin and Le Maitre (2010) proposed polynomial transformation
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of the POD projection coefficients over a coarse time-step, for application to the 2D flow past a circular cylinder in asymptotic and transient cases. Amsallem et al. (2009) interpolated the ROM data in a tangent space to the manifold of symmetric positive-definite matrices, and mapped the result back to the manifold for the dynamic characterization of a parameterized structural model so as to evaluate its response to a given input. Degroote et al. (2010) compared spline interpolation of the reduced-order system matrices in the original space as well as in the tangent space to the Riemannian manifold with Kriging interpolation of the predicted outputs for a steady-state thermal design problem and probabilistic analysis via Monte Carlo simulation of an unsteady contaminant transport problem. Our approach in this paper targets the development of multi-objective/multi-disciplinary optimization techniques using high-quality ROMs. In this work we improve the bilevel reduced-order model strategy (Filomeno Coelho et al. 2007, 2008) based on the POD of the initial data set and on kriging/RBF/MLS/. . . approximation of the projection coefficients. In Xiao et al. (2010) we have introduced the constrained POD, which consists basically of the adaptation of POD coefficients in order to interpolate quantities of interest (objective/constraint functions). Here, we focus on further tailoring the POD technique in order to modify the basis vectors rather than the coefficients in order to interpolate global quantities exactly. The benefits are a better physical meaning of the adapted POD modes, a lower number of basis vectors and a deeper insight into the postprocessed optimization results. This approach, focusing on the improvement of the precision of the ROM by an appropriate choice of basis vectors, may be used in local as well as in global versions of the POD. The paper is organized in three parts: in the first section, we revisit the standard POD, reformulated here as a minimization problem. This presentation allows the introduction of additional constraints, aiming to enhance the projection coefficients of a standard POD basis (Xiao et al. 2010). Section 2 is the central part of the paper, where the basis vectors are considered as variables of a constrained minimization problem and an algorithm is presented for the explicit computation of additional modes. In the third section, we present the data analysis and the results obtained for the multi-objective shape optimization results of a car engine intake port. We close with some concluding remarks and prospects for future work.
2 Proper Orthogonal Decomposition T Let v(k) = v1(k) , . . . , vn(k) ; k = 1, . . . , M be a set of snapshots of the velocity field obtained by running a “high
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fidelity” model on a representative sample of M points in the design space, where n is the size of a snapshot and corresponds typically to the number of degrees of freedom associated with a finite element model. The Principal Component Analysis of the data allows us to express the snapshots in terms of an average snapshot v and the set of basis vectors v˜ (k) = v + α (k)
(1)
The basis is usually presented (after mean centering the data) as a set of eigenvectors of the covariance matrix, or singular value decomposition of the data matrix. The coefficients α are calculated by the projection of these snapshots on the basis α (k) = v(k) , , k = 1 . . . M. (2) The reduced order model is obtained by using only a subset ,m of the first m
