Efficient solution for Galerkin-based polynomial chaos expansion systems

Iterative solvers and preconditioners are widely used for handling the linear system of equations arising from stochastic finite element method (SFEM) formulations, e.g. galerkin-based polynomial chaos (G-P-C) Expansion method. Especially, Preconditioned Conjugate Gradient (PCG) solver and the Incomplete Cholesky (IC) preconditioner are shown to be adequate choices within this context. In this study, approaches for the automated adjustment of the input parameters for these tools are to be introduced. The proposed algorithms aim to enable the use of the PCG solver and IC preconditioner in a black-box fashion. As a result, the requirement of the expertise for using these tools is removed to a certain extend. Furthermore, these algorithms can be used also for the implementation purposes of SFEM’s within general purpose software by increasing the ease of the use of these tools and hence leading to an improved user-comfort.     

Efficient design of experiments for sensitivity analysis based on polynomial chaos expansions

Global sensitivity analysis aims at quantifying respective effects of input random variables (or combinations thereof) onto variance of a physical or mathematical model response. Among the abundant literature on sensitivity measures, Sobol indices have received much attention since they provide accurate information for most of models. We consider a problem of experimental design points selection for Sobol’ indices estimation. Based on the concept of D-optimality, we propose a method for constructing an adaptive design of experiments, effective for calculation of Sobol’ indices based on Polynomial Chaos Expansions. We provide a set of applications that demonstrate the efficiency of the proposed approach.     

Kinetic feedback design for polynomial systems

New computational methods are proposed in this paper to construct polynomial feedback controllers for the stabilization of polynomial systems with linear input structure around a positive equilibrium point. Using the theory of chemical reaction networks (CRNs) and previous results on dynamical equivalence, a complex balanced or weakly reversible zero deficiency closed loop realization is achieved by computing the gain matrix of a polynomial feedback using optimization. It is shown that the feedback resulting in a complex balanced closed loop system having a prescribed equilibrium point can be computed using linear programming (LP). The robust version of the problem, when a convex set of polynomial systems is given over which a stabilizing controller is searched for, is also solvable with an LP solver. The feedback computation for rendering a polynomial system to deficiency zero weakly reversible form can be solved in the mixed integer linear programming (MILP) framework. It is also shown that involving new monomials (complexes) into the feedback does not improve the solvability of the problems. The proposed methods and tools are illustrated on simple examples, including stabilizing an open chemical reaction network.     

A genetic tool for optimal design sequencing in complex engineering systems

Methods in multidisciplinary design optimization rely on computer tools to manage the large amounts of information involved. One such tool is DeMAID (DEsign Manager's Aide for Intelligent Decomposition), which incorporates planning and scheduling functions to analyse the effect of the information coupling between design tasks in complex systems on the efficiency of the design process. Scheduling involves the formation of circuits of interdependent design tasks, and the minimization of feedbacks within these circuits. Recently there has been interest in the incorporation of other considerations in the sequencing of tasks within circuits. This study presents the program Gendes (GENetic DEsign Sequencer), a sequencing tool based on a genetic algorithm. The program currently has the capability to minimize feedbacks as well as crossovers (intersections in the flow of design information which obscure straightforward evaluation), and allows the potential for other considerations in the sequencing function.This paper presents the development of this tool and the methods used. The results of computational studies to determine the most effective settings of the genetic algorithm for the task sequencing problem are presented, including population size, objective function weighting for the tradeoff between feedbacks and crossovers, mutation rate, and choice of selection operator and fitness function form. The incorporation of Gendes into the DeMAID scheduling function is explored, and the method is applied to two test systems to show its feasibility.     

A gradient-based uncertainty optimization framework utilizing dimensional adaptive polynomial chaos expansion

To improve the efficiency of solving uncertainty design optimization problems, a gradient-based optimization framework is herein proposed, which combines the dimension adaptive polynomial chaos expansion (PCE) and sensitivity analysis. The dimensional adaptive PCE is used to quantify the quantities of interest (e.g., reliability, robustness metrics) and the sensitivity. The dimensional adaptive property is inherited from the dimension adaptive sparse grid, which is used to evaluate the PCE coefficients. Robustness metrics, referred to as statistical moments, and their gradients with respect to design variables are easily derived from the PCE, whereas the evaluation of the reliability and its gradient require integrations. To quantify the reliability, the framework uses the Heaviside step function to eliminate the failure domain and calculates the integration by Monte Carlo simulation with the function replaced by PCE. The PCE is further combined with Taylor’s expansion and the finite difference to compute the reliability sensitivity. Since the design vector may affect the sample set determined by dimension adaptive sparse grid, the update of the sample set is controlled by the norm variations of the design vector. The optimization framework is formed by combining reliability, robustness quantification and sensitivity analysis, and the optimization module. The accuracy and efficiency of the reliability quantification, as well as the reliability sensitivity, are verified through a mathematical example, a system of springs, and a cantilever beam. The effectiveness of the framework in solving optimization problems is validated by multiple limit states example, a truss optimization example, an airfoil optimization example, and an ONERA M6 wing optimization problem. The results demonstrate that the framework can obtain accurate solutions at the expense of a manageable computational cost.

     

MEMS stochastic model order reduction method based on polynomial chaos expansion

Modeling and simulation of MEMS devices is a very complex task which involve the electrical, mechanical, fluidic and thermal domains, and there are still some uncertainties need to be accounted because of uncertain material and/or geometric parameters factors. According to these problems, we put forward to stochastic model order reduction method under random input conditions to facilitate fast time and frequency domain analyses, the method firstly process model order reduction by Structure Preserving Reduced-order Interconnect Macro Modeling method, then makes use of polynomial chaos expansions in terms of the random input and output variables for the matrices of a finite element model of the system; at last we give the expected values and standard deviations computing method to MEMS stochastic model. The simulation results verify the method is effective in large scale MEMS design process.     

Efficient uncertainty quantification with the polynomial chaos method for stiff systems

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear system of equations at every time step. Using the Galerkin approach the size of the system state increases from n to S × n, where S is the number of PC basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n³) to O(S³n³). The S³-fold increase can make the computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with no negative impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code.     

Conceptual design of a human error tolerant interface for complex engineering systems

Various surveys and compilations have led to the conclusion that “human error” is a primary cause of most major accidents in aviation, power production and process control. This conclusion has led to a variety of efforts to reduce or possibly eliminate human error. While such efforts to reduce human error are important, they can, if taken to an extreme, be very short-sighted. A strategy that is more likely to be successful is one that tolerates the occurrence of errors, but avoids their consequences. Error tolerance can be achieved in three complementary ways: (1) feedback about current consequences, (2) feedback about future consequences, and (3) intelligent error monitoring. These approaches are complementary and can be viewed as providing multiple levels of support relative to the consequences of human error. This paper elaborates on each of these approaches and then suggests how they might be integrated in terms of a human error-tolerant interface for complex engineering systems. A conceptual design for such an interface is presented. Also, the practical implications and limitations of implementing this design are considered.     

Expert systems for engineering design

Abstract: An expert system embodies a humun expert's domain-specific knowledge and skill, acquired and refined over years of experience. A number of problems in diagnosis and engineering design can be solved by Using current expert-system techniques. This paper enumerates the main components of such problems and the steps that are taken in solving them. A few prototypical articificial intellegence systems embody techniques that can be applied to engineering problems. These are surveyed, and their relevance to components of design problems is discussed. Some expert systems in design domains are summarized, with emphasis on aspects that can illustrate wider applicability of the techniques. A number of avenues of further research are evident, and the area of engineering design offers rich opportunities for advancing the state-of-the-art in expert systems. An annotated selective bibliography is included     

Nonlinear state feedback design for continuous polynomial systems

The objective of this paper is to simplify the complexity of practical implementation of the input-state feedback linearization technique for the control of input-affine systems. A polynomial approach which makes use of the Taylor series expansion and the Kronecker product is developed. Our work aims to address the problem of synthesizing a polynomial control via a nonlinear analytical coordinates transformation. To check the effectiveness of the investigated approach, we consider the control problem of a series DC motor. A comparative study with the input-state feedback linearization control is developed.     

Identification and prediction of stochastic dynamical systems in a polynomial chaos basis

Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust.     

A web services-based multidisciplinary design optimization framework for complex engineering systems with uncertainties

With the increased complexity of complex engineering systems (CES), more and more disciplines, coupled relationships, work processes, design data, design knowledge and uncertainties are involved. Currently, the MDO is facing unprecedented challenges especially in dealing with the CES by different specialists dispersed geographically on heterogeneous platforms with different analysis tools. The product design data integration and data sharing among the participants and the workflow optimization hamper the development and applications of MDO in enterprises seriously. Therefore, a multi-hierarchical integrated product design data model (MiPDM) supporting the MDO in web environment and a web services-based MDO framework considering aleatory and epistemic uncertainties are proposed in this paper. With the enabling technologies including web services, ontology, workflow, agent, XML, and evidence theory, the proposed framework enables the designers geographically dispersed to work collaboratively in the MDO environment. The ontology-based workflow enables the logical reasoning of MDO to be processed dynamically. Finally, a proof-of-concept prototype system is developed based on Java 2 Platform Enterprise Edition (J2EE) and an example of supersonic business jet is demonstrated to verify the web services-based MDO framework.     

Recursive maximum likelihood parameter estimation for state space systems using polynomial chaos theory

This paper combines polynomial chaos theory with maximum likelihood estimation for a novel approach to recursive parameter estimation in state-space systems. A simulation study compares the proposed approach with the extended Kalman filter to estimate the value of an unknown damping coefficient of a nonlinear Van der Pol oscillator. The results of the simulation study suggest that the proposed polynomial chaos estimator gives comparable results to the filtering method but may be less sensitive to user-defined tuning parameters. Because this recursive estimator is applicable to linear and nonlinear dynamic systems, the authors portend that this novel formulation will be useful for a broad range of estimation problems.     

Robust dynamic optimization in heterogeneous multiscale catalytic flow reactors using polynomial chaos expansion

This paper explores the application of optimal design and operational strategies under uncertainty to a transient multiscale catalytic flow reactor system. The catalytic reactor is modeled using a spatially-dependent multiscale model that comprises lattice-based kinetic Monte Carlo (kMC) models coupled with continuum partial differential equations (PDEs) to account for the fine-scale and the macroscale reactor behaviour, respectively. This work compares two uncertainty propagation techniques, power series expansion (PSE) and polynomial chaos expansion (PCE), to assess their performance in multiscale process systems. The analysis reveals that PCE provides accurate results at minimal computational cost for the multiscale catalytic reactor model under the conditions considered in this work. PCE is subsequently used to perform robust dynamic optimization studies on the catalytic reactor system under uncertainty. The first study determines the optimal temperature trajectories that maximize the reactor’s performance under uncertainty. The second study aims to identify the optimal design and operating policies that allow the reactor, under uncertainty in the multiscale model parameters, to meet targeted performance specifications within a level of confidence. Both studies illustrate the benefits of performing dynamic optimization studies to improve performance for multiscale process systems under uncertainty.