POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification

In this paper, a reduced-order model (ROM) based on the proper orthogonal decomposition and the discrete empirical interpolation method is proposed for efficiently simulating time-fractional partial differential equations (TFPDEs). Both linear and nonlinear equations are considered. We demonstrate the effectiveness of the ROM by several numerical examples, in which the ROM achieves the same accuracy of the full-order model (FOM) over a long-term simulation while greatly reducing the computational cost. The proposed ROM is then regarded as a surrogate of FOM and is applied to an inverse problem for identifying the order of the time-fractional derivative of the TFPDE model. Based on the Levenberg–Marquardt regularization iterative method with the Armijo rule, we develop a ROM-based algorithm for solving the inverse problem. For cases in which the observation data is either uncontaminated or contaminated by random noise, the proposed approach is able to achieve accurate parameter estimation efficiently.     

Enhanced POD projection basis with application to shape optimization of car engine intake port

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.     

Sugeno fuzzy PID tuning,by genetic-neutral for AVR in electrical power generation

We report a novel design method for determining the optimal proportional-integral-derivative (PID) controller parameters of an automatic voltage regulator (AVR) system, using a combined genetic algorithm (GA), radial basis function neural network (RBF-NN) and Sugeno fuzzy logic approaches. GA and a RBF-NN with a Sugeno fuzzy logic are proposed to design a PID controller for an AVR system (GNFPID). The problem for obtaining the optimal AVR and PID controller parameters is formulated as an optimization problem and RBF-NN tuned by GA is applied to solve the optimization problem. Whereas, optimal PID gains obtained by the proposed RBF tuning by genetic algorithm for various operating conditions are used to develop the rule base of the Sugeno fuzzy system and design fuzzy PID controller of the AVR system to improve the system's response (∼0.005 s). The proposed approach has superior features, including easy implementation, stable convergence characteristic, good computational efficiency and this algorithm effectively searches for a high-quality solution and improve the transient response of the AVR system (7E−06). Numerical simulation results demonstrate that this is faster and has much less computational cost as compared with the real-code genetic algorithm (RGA) and Sugeno fuzzy logic. The proposed method is indeed more efficient and robust in improving the step response of an AVR system.     

Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load

This paper presents an efficient structural design optimization strategy that combines the reduced basis method (RBM) with the equivalent static load (ESL). In dynamic response optimization using ESL, the computation of a static system optimization is repeatedly executed under multiple static loads. In this process, we propose parametrizing the static system and employing the RBM with global proper orthogonal decomposition (POD). In general, the snapshots for the sampling procedure under multiple loads increase proportionally to the number of loads, which results in an inefficient computational procedure. Thus, we propose taking snapshots with a proper orthogonal mode (POM) of the multiple loads rather than for the multiple loads themselves. The number of snapshots then decreases, and the original system can be efficiently reduced. We directly employ the framework of the proposed RBM with the POM of multiple loads to ESL-based design optimization, and the results indicate that the proposed method is more efficient than conventional ESL-based optimization as well as a full order model. Various numerical examples, including comparisons of relative errors and the dynamic response optimizations, support the strength of the proposed strategy.     

Optimal earthquake-resistant design: A reliability-based,global cost approach

This paper presents a methodology for incorporating into the decision-making process of design the explicit consideration of possible future performance of the designed structure in the presence of uncertainty in both loading and structural parameters assumed in the design process. Design is viewed as an optimization problem, taking into account as a merit function terms related to building cost as well as possible future damage. Randomness of loads, load-effects, structural resistances and structural parameters affecting response are considered. In this framework a method suited both to linear as well as nonlinear structural behavior is presented. Approximate expressions of the probabilities of the above random variables as functions of the central moment of their distributions are introduced. Together with a firstorder expansion of the input-output response function characterizing the structural model, this procedure gives computational feasibility for actual design applications. An example is included to clarify the methodology proposed and to illustrate typical features of problem formulation and solution.     

A wave propagation approach for reduced dynamic modeling of distillation columns: Optimization and control

Reduced models enable real-time optimization of large-scale processes. We propose a reduced model of distillation columns based on multicomponent nonlinear wave propagation (Kienle 2000). We use a nonlinear wave equation in dynamic mass and energy balances. We thus combine the ideas of compartment modeling and wave propagation. In contrast to existing reduced column models based on nonlinear wave propagation, our model deploys a hydraulic correlation. This enables the column holdup to change as load varies. The model parameters can be estimated solely based on steady-state data. The new transient wave propagation model can be used as a controller model for flexible process operation including load changes. To demonstrate this, we implement full-order and reduced dynamic models of an air separation process and multi-component distillation column in Modelica. We use the open-source framework DyOS for the dynamic optimizations and an Extended Kalman Filter for state estimation. We apply the reduced model in-silico in open-loop forward simulations as well as in several open- and closed-loop optimization and control case studies, and analyze the resulting computational speed-up compared to using full-order stage-by-stage column models. The first case study deals with tracking control of a single air separation distillation column, whereas the second one addresses economic model predictive control of an entire air separation process. The reduced model is able to adequately capture the transient column behavior. Compared to the full-order model, the reduced model achieves highly accurate profiles for the manipulated variables, while the optimizations with the reduced model are significantly faster, achieving more than 95% CPU time reduction in the closed-loop simulation and more than 96% in the open-loop optimizations. This enables the real-time capability of the reduced model in process optimization and control.     

Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.     

Robust optimization of 2D airfoils driven by full Navier-Stokes computations

A new approach to the constrained design of aerodynamic shapes is suggested. The approach employs Genetic Algorithms (GAs) as an optimization tool in combination with a Reduced-Order Models (ROM) method based on linked local data bases obtained by full Navier-Stokes computations. The important features of the approach include: (1) a new strategy for efficient handling of non-linear constraints in the framework of GAs (2) scanning of the optimization search space by a combination of full Navier-Stokes computations with the ROM method (3) multilevel parallelization of the whole computational framework.The method was applied to the problem of one-point transonic profile optimization with non-linear constraints. The results demonstrated that the approach combines high accuracy of optimization (based on full Navier-Stokes computations) and efficient handling of various non-linear constraints with high computational efficiency and robustness. A significant computational time-saving (in comparison with optimization tools fully based on Navier-Stokes computations) allowed the method to be used in a demanding engineering environment.     

Likelihood-based representation of epistemic uncertainty and its application in robustness-based design optimization

In this paper, we propose a new likelihood-based methodology to represent epistemic uncertainty described by sparse point and/or interval data for input variables in uncertainty analysis and design optimization problems. A worst-case maximum likelihood-based approach is developed for the representation of epistemic uncertainty, which is able to estimate the distribution parameters of a random variable described by sparse point and/or interval data. This likelihood-based approach is general and is able to estimate the parameters of any known probability distributions. The likelihood-based representation of epistemic uncertainty is then used in the existing framework for robustness-based design optimization to achieve computational efficiency. The proposed uncertainty representation and design optimization methodologies are illustrated with two numerical examples including a mathematical problem and a real engineering problem.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

Bi-level model reduction for coupled problems

In this work a methodology is proposed for the optimization of coupled problems, and applied to a 3D flexible wing. First, a computational fluid dynamics code coupled with a structural model is run to obtain the pressures and displacements for different wing geometries controlled by the design variables. Secondly, the data are reduced by Proper Orthogonal Decomposition (POD), allowing to expand any field as a linear combination of specific modes; finally, a surrogate model based on Moving Least Squares (MLS) is built to express the POD coefficients directly as functions of the design variables. After the validation of this bi-level model reduction strategy, the approximate models are used for the multidisciplinary optimization of the wing. The proposed method leads to a reduction of the weight by 6.6%, and the verification of the solution with the accurate numerical solvers confirms that the solution is feasible.     

Fast simulation‐driven antenna design using response‐feature surrogates

In this article, a computationally efficient procedure for electromagnetic (EM)‐simulation‐driven design of antennas is presented. Our methodology is based on local approximation models of the antenna response, established using a set of suitably selected characteristic features rather than the entire response (such as reflection versus frequency). The approximation model is utilized to verify the level of satisfying/violating given performance requirements, and to guide the optimization process towards a better design. By exploiting the fact that the dependence of the response features on the designable parameters of the antenna of interest is simple (close to linear or quadratic), the feature‐based optimization converges faster than conventional optimization of frequency‐based EM‐simulated responses. In order to further speed up the design, coarse‐discretization simulations are utilized to estimate the feature gradients with respect to adjustable parameters of the problem at hand. The optimization algorithm is embedded in the trust‐region framework for safeguarding convergence. The proposed technique is demonstrated using two antenna examples. In both the cases, the optimum design is obtained at the computational cost corresponding to a few high‐fidelity EM antenna simulations. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:394–402, 2015.     

Coupling the cross-entropy with the line sampling method for risk-based design optimization

An algorithm for risk-based optimization (RO) of engineering systems is proposed, which couples the Cross-entropy (CE) optimization method with the Line Sampling (LS) reliability method. The CE-LS algorithm relies on the CE method to optimize the total cost of a system that is composed of the design and operation cost (e.g., production cost) and the expected failure cost (i.e., failure risk). Guided by the random search of the CE method, the algorithm proceeds iteratively to update a set of random search distributions such that the optimal or near-optimal solution is likely to occur. The LS-based failure probability estimates are required to evaluate the failure risk. Throughout the optimization process, the coupling relies on a local weighted average approximation of the probability of failure to reduce the computational demands associated with RO. As the CE-LS algorithm proceeds to locate a region of design parameters with near-optimal solutions, the local weighted average approximation of the probability of failure is refined. The adaptive refinement procedure is repeatedly applied until convergence criteria with respect to both the optimization and the approximation of the failure probability are satisfied. The performance of the proposed optimization heuristic is examined empirically on several RO problems, including the design of a monopile foundation for offshore wind turbines.     

Innovative seismic design optimization with reliability constraints

Performance-Based Design (PBD) methodologies is the contemporary trend in designing better and more economic earthquake-resistant structures where the main objective is to achieve more predictable and reliable levels of safety and operability against natural hazards. On the other hand, reliability-based optimization (RBO) methods directly account for the variability of the design parameters into the formulation of the optimization problem. The objective of this work is to incorporate PBD methodologies under seismic loading into the framework of RBO in conjunction with innovative tools for treating computational intensive problems of real-world structural systems. Two types of random variables are considered: Those which influence the level of seismic demand and those that affect the structural capacity. Reliability analysis is required for the assessment of the probabilistic constraints within the RBO formulation. The Monte Carlo Simulation (MCS) method is considered as the most reliable method for estimating the probabilities of exceedance or other statistical quantities albeit with excessive, in many cases, computational cost. First or Second Order Reliability Methods (FORM, SORM) constitute alternative approaches which require an explicit limit-state function. This type of limit-state function is not available for complex problems. In this study, in order to find the most efficient methodology for performing reliability analysis in conjunction with performance-based optimum design under seismic loading, a Neural Network approximation of the limit-state function is proposed and is combined with either MCS or with FORM approaches for handling the uncertainties. These two methodologies are applied in RBO problems with sizing and topology design variables resulting in two orders of magnitude reduction of the computational effort.     

An efficient procedure for structural reliability-based robust design optimization

This paper focuses on the development of an optimization tool with the aim to obtain robust and reliable designs in short computational time. The robustness measures considered here are the expected value and standard deviation of the performance function involved in the optimization problem. When using these robustness measures combined, the search of optimal design appears as a robust multiobjective optimization (RMO) problem. Reliable design addresses uncertainties to restrict the structural probability of failure. The mathematical formulation for the reliability based robust design optimization (RBRDO) problem is obtained by adding a reliability based constraint into the RMO problem. As both, statistics calculations and the reliability analysis could be very costly, approximation technique based on reduced-order modeling (ROM) is also incorporated in our procedure. The selected ROM is the proper orthogonal decomposition (POD) method, with the aim to produce fast outputs considering structural non-linear behavior. Moreover, to obtain RBRDO designs with reduced CPU time we propose others developments to be added in the integrated tool. They are: Probabilistic Collocation Method (PCM) to evaluate the statistics of the structural responses and, also, an approximated reliability constraints procedure based on the Performance Measure Approach (PMA) for reliability constraint assessment. Finally, Normal-Boundary Intersection (NBI) or Normalized Normal-Constraint (NNC) multiobjective optimization techniques are employed to obtain fast and even spread Pareto robust designs. To illustrate the application of the proposed tool, optimization studies are conducted for a linear (benchmark) and nonlinear trusses problems. The nonlinear example consider different loads level, exploring the material plasticity. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches.     

Optimization of passive vibration isolators mechanical characteristics

The contribution of optimization has been essential to the more recent developments in design of new mechanical structures and materials. The objective of this work is to apply the models of material and structural optimization to the design of passive vibration isolators. A computational tool to identify the optimal viscoelastic characteristics of a nonlinear one-dimensional isolator was developed. The cost functional involves the minimization of a weighted average of the maximum transient and steady state response amplitudes for a set of predefined dynamic loads. The optimal isolator behaviour is obtained by a simulated annealing method. The solutions obtained are analyzed and discussed concerning their dependence on the applied forces and objective function selection. The results obtained can facilitate the design of elastomeric materials with improved behaviour in terms of dynamic stiffness for passive vibration control.     

Reliability based optimization in aeroelastic stability problems using polynomial chaos based metamodels

In this work, reliability based design optimization (RBDO) of two aeroelastic stability problems is addressed: (i) divergence, which arises in static aeroelasticity, and (ii) flutter, which arises in dynamic aeroelasticity. A set of design variables is considered as random variables, and the mean mass is minimized for a given set of constraints — including the probability of failure by divergence or flutter. The optimization process requires repeated evaluation of reliability, which is a major contributor to the total computational cost. To reduce this cost, a polynomial chaos expansion (PCE)-based metamodel is created over a grid in the parameter space. These precomputed PCEs are then interpolated for reliability calculation at intermediate points in the parameter space, as demanded by the optimization algorithm. Two new modifications are made to this method in this work. First, the Gauss quadrature rule is used — instead of statistical simulation — to estimate the chaos coefficients for higher computational speed. Second, to increase this computational gain further, a non-uniform grid is chosen instead of a uniform one, based on relative importance of the design parameters. This relative importance is found from a global sensitivity analysis. This new modified method is applied on a rectangular unswept cantilever wing model. For both optimization problems, it is observed that the proposed method yields accurate results with a considerable computational cost reduction, when compared to simulation based methods. The effect of grid spacing is also explored to achieve the best computational efficiency.     

Reliability-based design optimization with cooperation between support vector machine and particle swarm optimization

Reliability-based design optimization (RBDO) is concerned with designing an engineering system to minimize a cost function subject to the reliability requirement that failure probability should not exceed a threshold. Conventional RBDO methods are less than satisfactory in dealing with discrete design parameters and complex limit state functions (nonlinear and non-differentiable). Methods that are flexible enough to address the concerns above, however, come at a high computational cost. To enhance computational efficiency without sacrificing model flexibility, we propose a new RBDO framework: PS2, which combines Particle Swarm Optimization (PSO), Support Vector Machine (SVM), and Subset Simulation (SS). SS can efficiently estimate small failure probabilities, based on which SVM is adopted to evaluate the reliability of candidate solutions using binary classification. PSO is employed to solve the discrete optimization problem. Primary emphasis is placed upon the cooperation between SVM and PSO. The cooperation is mutually beneficial since the SVM classifier helps PSO evaluate the feasibility of solutions with high efficiency while the optimal solutions obtained by PSO assist in retraining the SVM classifier to attain better accuracy. The PS2 framework is implemented to find the optimal design of a ten-bar truss, whose component sizes are selected from a commercial standard. The reliability constraints are non-differentiable with two failure modes: yield stress and buckling stress. The interactive process between PSO and SVM contributes greatly to the success of the PS2 framework. It is shown that in various trials the PS2 framework consistently outperforms both the double-loop and single-loop approaches in terms of computational efficiency, solution quality, and model flexibility.