Manyu Xiao

Constrained Proper Orthogonal Decomposition based on QR-factorization for aerodynamical shape optimization

While performing numerical simulations for design optimization, one of the major issues is reaching a good compromise between accuracy and computational effort. Simulation methods such as finite elements, finite volumes, etc. provide high fidelity numerical solutions at a cost which may be prohibitive in optimization problems requiring frequent calls to the discretized equations solver. The Proper Orthogonal Decomposition (POD) constitutes an economical and efficient option to decrease the cost of the solution; however, the truncation of the POD basis implies an error in the calculation of the global quantities used as objectives and optimization constraints, which in turn might bias the optimization results. Our idea is thus to improve the snapshot POD by means of a reduced basis constructed to provide an exact interpolation of the quantities of interest obtained by integration of the physical fields. To this end, we reformulate the POD as a minimization problem where the desired properties are expressed as a set of constraints impacting the calculation of both the modes and the coefficients. The main contribution of this paper is to provide a detailed mathematical justification for our constrained variant of the POD, including a graphical interpretation of the proposed approach. The constrained POD method is then applied to the problem of representing the pressure field around a 2D wing, and is compared with the traditional POD.

Proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization

Shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations. Reduced-order models based on proper orthogonal decomposition (POD) constitute an economical alternative. However the truncation of the POD basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results. In our former contribution (Xiao and Breitkopf, 2013), we have introduced a constrained POD projector allowing for exact linear constraint verification for a reduced order model. Nevertheless, this approach was limited a to relatively low numbers constraints. Therefore, in the present paper, we propose an approach for a high number of constraints. The main idea is to extend the snapshot POD by introducing a new constrained projector in order to reduce both the physical field and the constraint space. This allows us to search for the Pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine-tuning of the reduced model for a particular purpose. We illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables.

Extended Co-Kriging interpolation method based on multi-fidelity data

Abstract The common issue of surrogate models is to make good use of sampling data. In theory, the higher the fidelity of sampling data provided, the more accurate the approximation model built. However, in practical engineering problems, high-fidelity data may be less available, and such data may also be computationally expensive. On the contrary, we often obtain low-fidelity data under certain simplifications. Although low-fidelity data is less accurate, such data still contains much information about the real system. So, combining both high and low multi-fidelity data in the construction of a surrogate model may lead to better representation of the physical phenomena. Co-Kriging is a method based on a two-level multi-fidelity data. In this work, a Co-Kriging method which expands the usual two-level to multi-level multi-fidelity is proposed to improve the approximation accuracy. In order to generate the different fidelity data, the POD model reduction is used with varying number of the basis vectors. Three numerical examples are tested to illustrate not only the feasibility and effectiveness of the proposed method but also the better accuracy when compared with Kriging and classical Co-Kriging.

On-line Metamodel-Assisted Optimization with Mixed Variables

The optimization of complex civil engineering structures remains a major scientific challenge, mostly because of the high number of calls to the finite element analysis required by the complete design process. To achieve a significant reduction of this computational effort, a popular approach consists in substituting the high-fidelity simulation by a lower-fidelity regression model, also called a metamodel. However, most metamodels (like kriging, radial basis functions, etc.) focus on continuous variables, thereby neglecting the large amount of problems characterized by discrete, integer, or categorical data. Therefore, in this chapter, a complete metamodel-assisted optimization procedure is proposed to deal with mixed variables. The methodology includes a multi-objective evolutionary algorithm and a multiple kernel regression model, both adapted to mixed data, as well as an efficient on-line enrichment of the metamodel during the optimization. A structural benchmark test case illustrates the proposed approach, followed by a critical discussion about the generalization of the concepts introduced in this chapter for metamodel-assisted optimization.

Sizing Optimization of Lightweight Multilayer Thermal Protection Structures for Hypersonic Aircraft

In the context of hypersonic aircraft design, the major issue of the thermal protection system (TPS) is to obtain the optimal thickness of the insulation layers due to intense aerodynamic heating during re-entering earth’s atmosphere. In this study, an idea combining a transient heat transfer model and an efficient optimization model is introduced for multi-layer insulation of TPS. The TPS geometric dimensions in the thickness direction are particularly considered as the design variables and the objective function is the mass of the thermal protection structure with the limitation of the extreme temperatures of the hypersonic vehicle structure. In order to decrease the computational complexity, the Globally Convergent Method of Moving Asymptotes (GCMMA) method is specially used to search the optimal solution. The results show that the usage of multilayer insulation materials for the TPS can save more than 25% weight compared to a single-layer TPS. The detailed analysis and comparison indicate the advantages of the presented optimization model.Copyright

Investigation of Three Genotypes for Mixed Variable Evolutionary Optimization

While the handling of optimization variables directly expressed by numbers (continuous, discrete, or integer) is abundantly investigated in the literature, the use of nominal variables is generally overlooked, despite its practical interest in plenty of scientific and industrial applications. For example, in civil engineering, the designers of a structure made out of beams might have to select the best cross-section shapes among a list of available geometries (square, circular, rectangular, etc.), which can be modeled by nominal data. Therefore, in the context of single- and multi-objective evolutionary optimization for mixed variables, this study investigates three genetic encodings (binary, real, and real-simplex) for the representation of mixed variables involving both continuous and nominal parameters. The comparison of the genotypes combined with the instances of crossover is performed on six analytical benchmark test functions, as well as on the multi-objective design optimization of a six-storey rigid frame, showing that for mixed variables, real (and to a lesser extent: real-simplex) coding provides the best results, especially when combined with a uniform crossover.

Investigation of three genotypes for mixed variable evolutionary optimization

While the handling of optimization variables directly expressed by numbers (continuous, discrete, or integer) is abundantly investigated in the literature, the use of nominal variables is generally overlooked, despite its practical interest in plenty of scientific and industrial applications. For example, in civil engineering, the designers of a structure made out of beams might have to select the best cross-section shapes among a list of available geometries (square, circular, rectangular, etc.), which can be modeled by nominal data. Therefore, in the context of singleand multi-objective evolutionary optimization for mixed variables, this study investigates three genetic encodings (binary, real, and real-simplex) for the representation of mixed variables involving both continuous and nominal parameters. The comparison of the genotypes combined with the instances of crossover is performed on six analytical benchmark test functions, as well as on the multi-objective design optimization of a six-storey rigid frame, showing that for mixed variables, real (and to a lesser extent: real-simplex) coding provides the best results, especially when combined with a uniform crossover.

High-Performance Computing Strategies for Complex Engineering Optimization Problems

1School of Mechanical Engineering, Northwestern Polytechnical University, Shaanxi, Xi’an, China 2Department of Mathematics, University of Rome “La Sapienza”, Piazzale Aldo Moro 5, 00185 Rome, Italy 3Department of Mechanical Engineering, Technical University of Denmark, Building 403, Room 115, 2800 Kongens Lyngby, Denmark 4Civil Engineering Laboratoire de Genie Civil et Genie Mecanique INSA Rennes, France 5Department of Applied Mathematics, Northwestern Polytechnical University, Shaanxi, Xi’an 710072, China

A parallel algorithm based on Galerkin theory for block-tridiagonal linear systems

The work presented in this paper focuses on parallel iterative method for solving block-tridiagonal linear systems. Being based on Galerkin theory predetermining Wm=Vm, and forming a suitable matrix Vm, a parallel iterative method on distributed-memory multi-computer is established. Then Eq. (3.1)-(3.3) are provided for carrying out computation required by our parallel algorithm. Furthermore, convergence is proved when the coefficient matrix A is a symmetric positive definite matrix, and the sufficient condition is given. In the end, two illustrative examples implemented on HP rx2600 cluster show that our algorithms parallel acceleration rates and efficiency are higher.