Model reduction for supersonic cavity flow using proper orthogonal decomposition(POD)and Galerkin projection

The reduced-order model （ROM） for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition （POD） and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption, valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes （N-S） equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation （DNS） results, the proposed model predicts flow dynamics （e.g., dominant frequency and amplitude） accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.

Radial Basis Function Interpolation and Galerkin Projection for Direct Trajectory Optimization and Costate Estimation

This work presents a novel approach combining radial basis function(RBF)interpolation with Galerkin projection to efficiently solve general optimal control problems.The goal is to develop a highly flexible solution to optimal control problems,especially nonsmooth problems involving discontinuities,while accounting for trajectory accuracy and computational efficiency simultaneously.The proposed solution,called the RBF-Galerkin method,offers a highly flexible framework for direct transcription by using any interpolant functions from the broad class of global RBFs and any arbitrary discretization points that do not necessarily need to be on a mesh of points.The RBF-Galerkin costate mapping theorem is developed that describes an exact equivalency between the Karush-Kuhn-Tucker(KKT)conditions of the nonlinear programming problem resulted from the RBF-Galerkin method and the discretized form of the first-order necessary conditions of the optimal control problem,if a set of discrete conditions holds.The efficacy of the proposed method along with the accuracy of the RBF-Galerkin costate mapping theorem is confirmed against an analytical solution for a bang-bang optimal control problem.In addition,the proposed approach is compared against both local and global polynomial methods for a robot motion planning problem to verify its accuracy and computational efficiency.

Reliability analysis of structure with random parameters based on multivariate power polynomial expansion

A new method for calculating the failure probabilityof structures with random parameters is proposed based onmultivariate power polynomial expansion, in which te uncertain quantities include material properties, structuralgeometric characteristics and static loads. The structuralresponse is first expressed as a multivariable power polynomialexpansion, of which the coefficients ae then determined by utilizing the higher-order perturbation technique and Galerkinprojection scheme. Then, the final performance function ofthe structure is determined. Due to the explicitness of theperformance function, a multifold integral of the structuralfailure probability can be calculated directly by the Monte Carlo simulation, which only requires a smal amount ofcomputation time. Two numerical examples ae presented toillustate te accuracy ad efficiency of te proposed metiod. It is shown that compaed with the widely used first-orderreliability method （ FORM） and second-order reliabilitymethod （ SORM）, te results of the proposed method are closer to that of the direct Monte Carlo metiod,and it requires much less computational time.

A reduced model for particle calcination for use in DEM/CFD simulations

We treat the accurate simulation of the calcination reaction in particles,where the particles are large and,thus,the inner-particle processes must be resolved.Because these processes need to be described with coupled partial differential equations(PDEs)that must be solved numerically,the computation times for a single particle are too high for use in simulations that involve many particles.Simulations of this type arise when the Discrete Element Method(DEM)is combined with Computational Fluid Dynamics(CFD)to investigate industrial systems such as quicklime production in lime shaft kilns.We show that,based on proper orthogonal decomposition and Galerkin projection,reduced models can be derived for single particles that provide the same spatial and temporal resolution as the original PDE models at a considerably reduced computational cost.Replacing the finite volume particle models with the reduced models results in an overall reduction of the reactor simulation time by about 40%for the sample system treated here.

Convergence Rate for Galerkin Approximation of the Stochastic Allen–Cahn Equations on 2D Torus

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen–Cahn equations driven by space-time white noise on T^(2). First we prove that the convergence rate for stochastic 2D heat equation is of order α-δ in Besov space C^(-α) for α∈(0, 1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen–Cahn equations of order α-δ in C^(-α) for α∈(0, 2/9) and δ > 0 arbitrarily small.

A Novel Iterative Penalty Method to Enforce Boundary Conditions in FiniteVolume POD-Galerkin Reduced Order Models for Fluid Dynamics Problems

A Finite-Volume based POD-Galerkin reduced ordermodel is developed for fluid dynamics problems where the(time-dependent)boundary conditions are controlled using two different boundary control strategies:the lifting function method,whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor.The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation.The boundary control methods are compared and tested for two cases:the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel.The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields.Finally,the reduced order models are 270-308 times faster than the full ordermodels for the lid driven cavity test case and 13-24 times for the Y-junction test case.