Intrusive Polynomial Chaos Expansion Research Articles

Stochastic simulation of magnetically controlled convection in porous media with random porosity via Karhunen-Loève expansion and intrusive polynomial chaos expansion

Understanding and quantifying uncertainty factors are crucial for accurately predicting thermomagnetic convection phenomena. This study presented a mathematical model and algorithm framework for uncertainty analysis of thermomagnetic convection in porous media with random porosity. The proposed method combined Karhunen-Loève expansion and intrusive polynomial chaos expansion to represent the input random parameters and output response, respectively. The Galerkin projection method was employed to decouple the stochastic governing equations of thermomagnetic convection into deterministic governing equations that can be efficiently solved using the finite volume method. By solving the polynomial chaos expansion of the output response and employing the stochastic projection method to solve the associated decoupled governing equations, the temporal evolution and statistical characteristics of the output response were obtained. The study revealed that porosity uncertainty in the porous medium affects the thermomagnetic convection of paramagnetic fluid, exhibiting significant chaos effects. Monte Carlo simulations were performed to validate the accuracy of the proposed approach and demonstrate its computational efficiency compared to traditional Monte Carlo methods. This research provides valuable insights into the uncertainty analysis of thermomagnetic convection and offers a promising methodology for analyzing heat transfer in various porous media applications.

Modeling of spatial permittivity variations using Karhunen–Loève expansion for stochastic electromagnetic problems

Fabrication tolerance often causes spatial variations in material properties in electromagnetic systems. In this paper, these spatial variations are modeled using Karhunen–Loéve expansion, which is truncated for a finite dimensional representation. Uncertainty quantification is performed by incorporating this model into an intrusive polynomial chaos expansion based spectral stochastic finite element method. Using a numerical example, it has been shown that the stochastic response obtained by the proposed approach is accurate and computationally efficient. The impact of correlation length of the covariance function used in the spectral expansion is analyzed.

Uncertain trajectory planning integrating polynomial chaos and convex programming

The present work explores the robust trajectory optimization scheme considering both the initial state disturbance and multiple constraints. An uncertain multi-constraint optimization model has been first established. Due to the existence of the stochastic disturbance, standard numerical trajectory planning algorithms cannot be directly applied to address the considered issue. Hence, based on the intrusive polynomial chaos expansion, we present a deterministic quantification for the stochastic state and constraints, so that the transformed optimization model becomes solvable for standard numerical optimization methods. To obtain the enhanced computational performance, an hp pseudo-spectral sequential convex programming procedure combined with a penalty function and backtracking search is proposed. This is achieved by discretizing and convexifying the nonlinear dynamics/constraints using hp quadrature collocation and successive linearization, respectively, and by adjusting the confidence region manually in the iteration. The simulation of a three-dimensional interception with the specific impact angle is conducted to verify the effectiveness. The simulation results show that the initial solutions are insensitive to the convex optimization, and the control commands generated by the proposed algorithm are effective against the initial state disturbance.

Development and assessment of an intrusive polynomial chaos expansion‐based continuous adjoint method for shape optimization under uncertainties

AbstractThis article contributes to the development of methods for shape optimization under uncertainties, associated with the flow conditions, based on intrusive Polynomial Chaos Expansion (iPCE) and continuous adjoint. The iPCE to the Navier–Stokes equations for laminar flows of incompressible fluids is developed to compute statistical moments of the Quantity of Interest which are, then, compared with those obtained through the Monte Carlo method. The optimization is carried out using a continuous adjoint‐enabled, gradient‐based loop. Two different formulations for the continuous adjoint to the iPCE PDEs are derived, programmed, and verified. Intrusive PCE methods for the computation of the statistical moments require mathematical development, derivation of a new system of governing equations and their numerical solution. The development is presented for a chaos order of two and two uncertain variables and can be used as a guide to those willing to extend this development to a different set of uncertain variables or chaos order. The developed method and software, programmed in OpenFOAM, is applied to two optimization problems pertaining to the flow around isolated airfoils with uncertain farfield conditions.

基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化

An intrusive polynomial chaos expansion method and a finite element program framework were proposed to quantify the uncertainty of flow and heat transfer problems under stochastic boundary conditions. In this method, the Karhunen-Loeve expansion was used to express the stochastic boundary condition, and the polynomial chaos expansion method was used to express the output random field. At the same time, the control equation was transformed into a set of deterministic equations with the spectral decomposition technology, and each polynomial chaos was solved to obtain the statistical characteristics of the numerical solution. Compared with the Monte Carlo method, this method can accurately and efficiently predict the uncertainty characteristics of flow and heat transfer problems under stochastic boundary conditions, and can save a lot of computing resources.

Stochastic design of high altitude propellers

High-Altitude Platform Stations or High-Altitude Pseudo-Satellites (HAPS) use propulsion systems which are commonly based on propellers. In this paper, an algorithm for the design of those propellers considering uncertainties is developed and applied. The algorithm is based on the non intrusive polynomial chaos expansion scheme, which converts the stochastic design problem into an equivalent deterministic one. Two uncertainties are studied and characterized: 1) the stratospheric wind fluctuations using reanalysis datasets and 2) the variability of the aerodynamic coefficients caused by the low Reynolds number. The results of the method are analyzed to tackle how relevant the uncertainties are in the propulsion of the stratospheric platforms. The case of study is an ideal stratospheric airship that operates at a mean wind speed of 9 m/s and requires a thrust of 100 N, both uncertain magnitudes. The propeller is built on NACA4412 airfoils and the cost function to be maximized is the mean net propulsion efficiency. The new method provides a relevant gain in the mean efficiency when compared with the deterministic optimization.

An Intrusive PCE Extension of the Contour Integral Method and its Application in Electrical Engineering

AbstractFor the study of electromagnetic waves in a 2D environment, the Helmholtz equation can be reformulated in boundary integral form by the so‐called Contour Integral Method (CIM). The influence of uncertainty in the geometry is studied via Polynomial Chaos Expansion (PCE). A distinction is made between the so‐called intrusive and non‐intrusive PCE. Both methods are applied to the CIM and compared to each other and to Monte‐Carlo simulation (MC).

A painless intrusive polynomial chaos method with RANS-based applications

This paper presents a method for the quantification of uncertainty propagation using intrusive Polynomial Chaos Expansion (iPCE). Contrary to commonly implemented intrusive methods built on a problem specific-approach that depends on the governing PDEs, the proposed method, although intrusive in nature, can easily be implemented to any system of equations. A proper mathematical framework is developed that performs the derivation and numerical solution of the iPCE equations with little additional effort, avoiding the laborious mathematical and software development commonly associated with intrusive approaches. Computational cost, convergence and stability properties are analyzed in detail and discussed. The proposed uncertainty quantification (UQ) method is applied to the Reynolds-Averaged Navier–Stokes (RANS) equations, coupled with the Spalart–Allmaras turbulence model and results are compared with those of the non-intrusive PCE (niPCE).

Scalable domain decomposition solvers for stochastic PDEs in high performance computing

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. Although these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. Parallel sparse matrix–vector operations are used to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.

Modeling Spatially Varying Uncertainty in Composite Structures Using Lamination Parameters

An approach is presented for modeling spatially varying uncertainty in the ply orientations of composite structures. Lamination parameters are used with the aim of reducing the required number of random variables. Karhunen–Loève expansion is employed to decompose the uncertainty in each ply into a sum of random variables and spatially dependent functions. An intrusive polynomial chaos expansion is proposed to approximate the lamination parameters while preserving the separation of the random and spatial dependency. Closed-form expressions are derived for the expansion coefficients in two case studies; an initial example in which uncertainty is modeled using random variables, and a second random field example. The approach is compared against Monte Carlo simulation results for a variety of layups as well as closed-form expressions for the mean and covariance. By summing the polynomial chaos basis functions through the laminate thickness, the separation of the random and spatial dependency may be preserved at a laminate level and the number of random variables reduced for some minimum number of plies. The number of variables increases nonlinearly with the number of Karhunen–Loève expansion terms, and as such, the approach is only beneficial in low-order expansions using relatively few Karhunen–Loève expansion terms.

Efficient uncertainty quantification of large two-dimensional optical systems with a parallelized stochastic Galerkin method.

It is well-known that geometrical variations due to manufacturing tolerances can degrade the performance of optical devices. In recent literature, polynomial chaos expansion (PCE) methods were proposed to model this statistical behavior. Nonetheless, traditional PCE solvers require a lot of memory and their computational complexity leads to prohibitively long simulation times, making these methods non-tractable for large optical systems. The uncertainty quantification (UQ) of various types of large, two-dimensional lens systems is presented in this paper, leveraging a novel parallelized Multilevel Fast Multipole Method (MLFMM) based Stochastic Galerkin Method (SGM). It is demonstrated that this technique can handle large optical structures in reasonable time, e.g., a stochastic lens system with more than 10 million unknowns was solved in less than an hour by using 3 compute nodes. The SGM, which is an intrusive PCE method, guarantees the accuracy of the method. By conjunction with MLFMM, usage of a preconditioner and by constructing and implementing a parallelized algorithm, a high efficiency is achieved. This is demonstrated with parallel scalability graphs. The novel approach is illustrated for different types of lens system and numerical results are validated against a collocation method, which relies on reusing a traditional deterministic solver. The last example concerns a Cassegrain system with five random variables, for which a speed-up of more than 12× compared to a collocation method is achieved.

A stochastic Galerkin approach to uncertainty quantification in poroelastic media

Recent concerns over the safety of oil and natural gas extraction, fracking, and carbon sequestration have driven the need to develop methods for uncertainty quantification for coupled subsurface flow and deformation processes. Traditional Monte Carlo methods are versatile but exhibit prohibitively slow convergence. In this work, we develop an intrusive polynomial chaos expansion method for Biot’s poroelasticity equations based on the Galerkin projection with uniform and log-normally distributed material parameters. We analyze accuracy and efficiency of our method and compare it to the Monte Carlo method. We verify exponential convergence of the stochastic Galerkin approach.

Stochastic Modeling of Incident Gust Effects on Aerodynamic Lift

The intrusive formulation of polynomial chaos expansion is implemented to determine uncertainty in aerodynamic loads on a rigid airfoil due to imprecise parameters that characterize an incoming gust. The results yield the sensitivity of the lift coefficient to variations in intensities and integral length scales of the gust fluctuations. The results show that lift coefficient fluctuations about the mean of each time step are affected primarily by the intensity ofthe fluctuations of the vertical velocity component, which should be expected. Next in order of importance is the integral length scale of the vertical velocity component. This implementation of the intrusive polynomial chaos expansion provides guidance for future efforts to quantify uncertain gust loads on micro air vehicles with higher fidelity models.

Uncertainty analysis near bifurcation of an aeroelastic system

Variations in structural and aerodynamic nonlinearities on the dynamic behavior of an aeroelastic system are investigated. The aeroelastic system consists of a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We follow two approaches to determine the effects of variations in the linear and nonlinear plunge and pitch stiffness coefficients of this aeroelastic system on its stability near the bifurcation. The first approach is based on implementation of intrusive polynomial chaos expansion (PCE) on the governing equations, yielding a set of nonlinear coupled ordinary differential equations that are numerically solved. The results show that this approach is capable of determining sensitivity of the flutter speed to variations in the linear pitch stiffness coefficient. On the other hand, it fails to predict changes in the type of the instability associated with randomness in the cubic stiffness coefficient. In the second approach, the normal form is used to investigate the flutter (Hopf bifurcation) boundary that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing LCO. The results show that this mathematical approach provides detailed aspects of the effects of the different system nonlinearities on its dynamic behavior. Furthermore, this approach could be effectively used to perform sensitivity analysis of the system's response to variations in its parameters.