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Jung J. Choi - One of the best experts on this subject based on the ideXlab platform.
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Hybrid Spectral Difference/Embedded MPWENO Method for Conservation Laws
53rd AIAA Aerospace Sciences Meeting, 2015Co-Authors: Jung J. ChoiAbstract:Recently, interest has been increasing towards applying high-order methods to engineering applications with complex geometries. As a result, a family of discontinuous high-order methods, such as Discontinuous Galerkin (DG), Spectral Volume (SV) and Spectral Difference (SD) methods, are under active development. These methods provide spectral-like results and are highly parallelizable due to local Solution Reconstruction within each cell. But, these methods suffer from Gibbs phenomenon near discontinuities. Artificial viscosity and sub-cell shock capturing method have been developed circumventing this problem. As an attempt towards applying a discontinuous high-order method for large scale engineering applications involving discontinuities in flows with complex geometries, a hybrid SD/embedded FV method is introduced by Choi. In this hybrid approach, structured finite volume cells are embedded in hexahedral elements containing discontinuity and highorder shock capturing scheme is used to overcome Gibbs phenomenon. In smooth flow regions away from discontinuities, the spectral difference method is employed. In this paper, the hybrid SD/embedded FV method is further investigated with a suite of test cases. In addition, the idea of embedding structured FV elements employed in the hybrid SD/embedded FV method is further extended to unstructured hexahedral grid and is introduced as the embedded structured element (ESE) framework for high-order method using unstructured hexahedral grid. The embedded structured element framework is workin-progress, but it shows promising results for applying high-order method for complex geometries. The error analysis and a suite of 1D and 2D test cases are presented further investigating the hybrid SD/embedded FV method using structured grid. One example employing the ESE framework is also included and discussed.
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Hybrid spectral difference/embedded finite volume method for conservation laws
Journal of Computational Physics, 2015Co-Authors: Jung J. ChoiAbstract:Recently, interests have been increasing towards applying the high-order methods to various engineering applications with complex geometries 30]. As a result, a family of discontinuous high-order methods, such as Discontinuous Galerkin (DG), Spectral Volume (SV) and Spectral Difference (SD) methods, is under active development. These methods provide high-order accurate Solutions and are highly parallelizable due to the local Solution Reconstruction within each element. But, these methods suffer from the Gibbs phenomena when discontinuities are present in the flow fields. Various types of limiters 43-45] and artificial viscosity 46,48] have been employed to overcome this problem.A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the reSolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate Solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method 56]. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field.The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of Solution DOF is considered in both SD and FV elements.
Clinton P. T. Groth - One of the best experts on this subject based on the ideXlab platform.
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A second-order maximum-entropy inspired interpolative closure for radiative heat transfer in gray participating media
Journal of Quantitative Spectroscopy and Radiative Transfer, 2020Co-Authors: Joachim A. R. Sarr, Clinton P. T. GrothAbstract:Abstract A new interpolative-based approximation to the second-order maximum-entropy, M2, moment closure for predicting radiative heat transfer in gray participating media is proposed and described. In addition to preserving many of the desirable mathematical properties of the original M2 closure, the proposed interpolative approximation provides significant reductions in computational costs compared to the costs of the original M2 closure by avoiding repeated numerical Solution of the corresponding optimization problem for entropy maximization. Theoretical details of the proposed interpolative-based closure, along with a description of an efficient Godunov-type finite-volume scheme that has been developed for the numerical Solution of the resulting system of hyperbolic moment equations, are presented. The finite-volume method makes use of limited linear Solution Reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton-Krylov-Schwarz (NKS) iterative method for Solution of the resulting non-linear algebraic equations arising from the spatial discretization procedure. The predictive capabilities of the proposed interpolative M2 closure are assessed by considering a number of model problems involving radiative heat transfer within one- and two-dimensional enclosures, the results for which are compared to Solutions of the first-order maximum entropy, M1, moment closure, as well as those of the more commonly adopted spherical harmonic moment closure techniques (first-order P1 and third-order P3) and the popular discrete ordinates method (DOM). The latter is used as a benchmark for comparisons, whenever exact Solutions are not available. The numerical results illustrate the promise of the proposed M2 closure, with the closure outperforming the M1, P1 and P3 closures for virtually all cases considered.
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A High-Order Finite-Volume Scheme for Large-Eddy Simulation of Turbulent Premixed Flames
52nd Aerospace Sciences Meeting, 2014Co-Authors: Luiz Tobaldini Neto, Clinton P. T. GrothAbstract:A novel, parallel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is developed and applied to large-eddy simulation (LES) of turbulent premixed flames. The high-order CENO finite-volume scheme is applied to the Solution of the Favre-filtered Navier-Stokes equations governing turbulent flows of a fully-compressible reactive mixture on three-dimensional, multi-block, body-fitted, computational mesh consisting of hexahedral volume elements. Unlike standard ENO schemes, which require Solution Reconstruction on multiple stencils, the CENO method uses a hybrid Reconstruction approach based on a fixed central stencil, thereby avoiding the complexities of other ENO schemes while providing high-order accuracy at relatively lower computational cost. The CENO discretization of the inviscid fluxes combines an unlimited high-order k-exact leastsquares Reconstruction technique based on the optimal central stencil with a monotonicitypreserving, limited, linear, Reconstruction algorithm. Switching in the hybrid procedure is determined by a smoothness indicator such that the unlimited high-order Reconstruction is retained for smooth Solution content that is fully resolved and reverts to the limited lower-order scheme, enforcing Solution monotonicity, for regions with abrupt variations (i.e., discontinuities and under-resolved regions). The high-order viscous fluxes are computed to the same order of accuracy as the hyperbolic fluxes based on a k-order accurate cell interface gradient derived from the unlimited, cell-centered, Reconstruction. The proposed cell-centered finite-volume scheme is formulated for three-dimensional multi-block mesh consisting of generic hexahedral cells and applied to LES of premixed flames.For the reactive cases flows of interest, a flamelet-based subfilter-scale (SFS) model is used to describe the unresolved influences of interaction between the turbulence and combustion. This SFS combustion model is based on a presumed conditional moment (PCM) approach in conjunction with flame prolongation of intrinsic low-dimensional manifold (FPI) tabulated chemistry. Numerical results are discussed for a freely propagating flame in an isotropic turbulence field and for a laboratory-scale lean premixed methane-air Bunsentype flame. The performance of the proposed high-order scheme for turbulent reactive flows is discussed.
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High-Order CENO Finite-Volume Schemes for Multi-Block Unstructured Mesh
20th AIAA Computational Fluid Dynamics Conference, 2011Co-Authors: Sean Mcdonald, Marc R.j. Charest, Clinton P. T. GrothAbstract:High-order discretization techniques remain an active area of research in Computational Fluid Dynamics (CFD) since they offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lowerorder methods. In spite of the successes to date, efficient, universally-applicable, highorder discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes. A novel, high-order, Central Essentially Non Oscillatory (CENO), cell-centered, finite-volume scheme is examined for the Solution of the conservation equations of inviscid, compressible, gas dynamics on multi-block unstructured meshes. This scheme was implemented for both two- and three-dimensional meshes consisting of triangular and tetrahedral computational cells, respectively. The CENO scheme is based on a hybrid Solution Reconstruction procedure that combines an unlimited high-order k-exact, least-squares Reconstruction technique with a monotonicity preserving limited piecewise linear least-squares Reconstruction algorithm. Fixed central stencils are used for both the unlimited high-order k-exact Reconstruction and the limited piecewise linear Reconstruction. In the proposed hybrid procedure, switching between the two Reconstruction algorithms is determined by a Solution smoothness indicator that indicates whether or not the Solution is resolved on the computational mesh. This hybrid approach avoids the complexities associated with Reconstruction on multiple stencils that other essentially non-oscillatory (ENO) and weighted ENO schemes can encounter. As such, it is well suited for Solution Reconstruction on unstructured mesh. The CENO scheme for unstructured mesh is described and analyzed in terms of accuracy, computational cost, and parallel performance. In particular, the accuracy of reconstructed Solutions for arbitrary functions and idealized flows is investigated as a function of mesh reSolution. The ability of the scheme to accurately represent Solutions with smooth extrema while maintaining robustness in regions of under-resolved and/or nonsmooth Solution content (i.e., Solutions with shocks and discontinuities) is demonstrated for a range of problems.
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Three-Dimensional MHD on Cubed-Sphere Grids: Parallel Solution-Adaptive Simulation Framework
20th AIAA Computational Fluid Dynamics Conference, 2011Co-Authors: Lucian Ivan, Scott Northrup, H. De Sterck, Clinton P. T. GrothAbstract:and scalable cubed-sphere grid framework is described for simulation of magnetohydrodynamic (MHD) space-physics ows in domains between two concentric spheres. The unique feature of the proposed formulation compared to existing cubed-sphere codes lies in the design of a cubed-sphere framework that is based on a genuine and consistent multi-block implementation, leading to ux calculations, adaptivity, implicit solves, and parallelism that are fully transparent to the boundaries between the six grid root blocks that correspond to the six sectors of the cubed-sphere grid. Crucial elements of the proposed approach that facilitate this exible design are: an unstructured connectivity of the six root blocks of the grid, multi-dimensional k-exact Reconstruction that automatically takes into account information from neighbouring cells, and adaptive division of the six root blocks into smaller blocks of varying reSolution that are all treated exactly equally for ghost cell information transfers, ux calculations, adaptivity, implicit solves and parallel distribution. The approach requires signicant initial investment in developing a general and sophisticated adaptive multi-block implementation, with the added complexity of unstructured root-block connectivity, but once this infrastructure is in place, a simulation framework that is uniformly accurate and easily scalable can be developed naturally, since blocks that are adjacent to sector boundaries or sector corners are not treated specially in any way. The general design principles of the adaptive multi-block approach are described and, in particular, how they are used in the implementation of the cubed-sphere framework. The nite-volume discretization, parallelization, and implicit solves are also described. The adaptive mesh renement (AMR) algorithm uses an upwind spatial discretization procedure in conjunction with limited linear Solution Reconstruction and Riemann-solver based ux functions to solve the governing equations on multi-block
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Parallel Adaptive Mesh Refinement Scheme for Three-Dimensional Turbulent Non-Premixed Combustion
46th AIAA Aerospace Sciences Meeting and Exhibit, 2008Co-Authors: X. Gaoand, Clinton P. T. GrothAbstract:A parallel adaptive mesh refinement (AMR) algorithm is described for predicting turbulent non-premixed gaseous combusting flows in three space dimensions. The Favreaveraged Navier-Stokes equations governing a reactive mixture of thermally perfect gases, the two transport equations of the k-! turbulence model, and the time-averaged species transport equations, are all solved using a fully coupled finite-volume formulation on bodyfitted multi-block hexahedral mesh. The numerical algorithm adopts a cell-centred upwind finite-volume discretization procedure and uses limited Solution Reconstruction, approximate Riemann solver based flux functions to determine the inviscid (hyperbolic) flux at cell interfaces. The viscous (elliptic) components of the cell face flux are evaluated by employing a hybrid average gradient-diamond path approach. For the treatment of near-wall turbulence, both low-Reynolds-number and wall-function formulations of the k-! model are used, with a procedure for automatically switching from one to the other, depending on mesh reSolution. A flexible block-based hierarchical octree data structure is used to maintain the connectivity of the Solution blocks in the multi-block mesh and facilitate automatic Solution-directed mesh adaptation according to physics-based refinement criteria. This AMR approach allows for anisotropic mesh refinement and the block-based data structure readily permits efficient and scalable implementations of the algorithm on multi-processor architectures. Numerical results for turbulent non-premixed methane-air diffusion flames are described to demonstrate the validity and potential of the parallel AMR approach for predicting complex combusting flows.
Carl Ollivier-gooch - One of the best experts on this subject based on the ideXlab platform.
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Stability analysis and improvement of the Solution Reconstruction for cell-centered finite volume methods on unstructured meshes
Journal of Computational Physics, 2019Co-Authors: Reza Zangeneh, Carl Ollivier-goochAbstract:Abstract The purpose of this paper is to develop a framework in which one can identify and predict the numerical instability of the steady state Solution due to the Solution Reconstruction for cell-centered finite volume methods on unstructured meshes and to stabilize the problem by optimizing the Reconstruction stencil. In this work, we first develop and extend a mathematical method, introduced by Haider and his colleagues, to measure the stability impact of the Reconstruction phase for both linear and nonlinear problems regardless of the Solution. Second order and third order accurate advection and Burgers problems as well as second order Euler problems are used to present detailed practical results and discussion around the use of the local Reconstruction map for stability analysis. This method shows that for a range of different physical problems, increasing the stencil size will usually lead to more stable problems. Additionally, an empirical study is performed which sheds light on connections between the mesh properties and the stability of the Reconstruction, which in turn helps choose the Reconstruction stencil more wisely. Secondly, we propose a systematic approach to optimize both the shape and the size of the Reconstruction stencil for better numerical stability through eigenvalue analysis. In this approach, one can directly optimize the Solution Reconstruction stencil for every control volume to obtain better numerical stability and convergence properties for steady state problems. A second order accurate Euler problem as well as a third order accurate laminar Navier-Stokes problem are used to showcase the applicability of the algorithm.
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High-order ENO schemes for unstructured meshes based on least-squares Reconstruction
35th Aerospace Sciences Meeting and Exhibit, 1997Co-Authors: Carl Ollivier-goochAbstract:High-order accurate schemes for conservation laws for unstructured meshes are not nearly so well advanced as such schemes for structured meshes. Consequently, little or nothing is known about the possible practical advantages of high-order discretization on unstructured meshes. This article is part of an ongoing effort to develop high-order schemes for unstructured meshes to the point where meaningful information can be obtained about the trade-offs involved in using spatial discretizations of higher than second-order accuracy on unstructured meshes. This article describes a high-order accurate ENO Reconstruction scheme, called DD-L{sub 2}-ENO, for use with vertex-centered upwind flow Solution algorithms on unstructured meshes. The Solution of conservation equations in this context can be broken naturally into three phases: (1) Solution Reconstruction, in which a polynomial approximation of the Solution is obtained in each control volume. (2) Flux integration around each control volume, using an appropriate flux function and a quadrature rule with accuracy commensurate with that of the Reconstruction. (3) Time evolution, which may be implicit, explicit, multigrid, or some hybrid.
Natalia Petrovskaya - One of the best experts on this subject based on the ideXlab platform.
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algorithms of Solution Reconstruction on unstructured grids in computational aerodynamics impact on aircraft design at the boeing company
2016Co-Authors: Natalia PetrovskayaAbstract:We describe work that demonstrated the benefits achieved when the mathematical and computational aspects of a fluid dynamics problem were brought together to work on real-world aerodynamic applications. The research into Solution Reconstruction on adaptive grids was required by The Boeing Company in order to help them to design an efficient and accurate discretization of the governing equations that have to be solved numerically for the generation of aerodynamic data for various flow regimes. While earlier insight into the Solution Reconstruction problem was purely based on empirical intuition, research conducted by the author under a contract with Boeing has resulted in the development of the necessary synthetic judgement in which the importance of accurate Reconstruction on unstructured grids has been fully recognised by the CFD researchers at Boeing and has helped them to make an informed decision on the choice of a discretization method in their CFD code. Efficient use of CFD in the design of new aircraft has allowed The Boeing Company to further strengthen their core operations, improve their execution and competitiveness and leverage their international advantage.
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A quadratic least‐squares Solution Reconstruction in a boundary layer region
International Journal for Numerical Methods in Biomedical Engineering, 2009Co-Authors: Natalia PetrovskayaAbstract:A local weighted least-squares (LS) method is often used to approximate a Solution function in computational aerodynamics problems. In our paper we study LS approximation by a quadratic polynomial on unstructured grids that have the high cell aspect ratio. It will be shown in the paper that an LS method degrades to unacceptable accuracy on stretched meshes and weighting of distant stencil points does not result in a more accurate Reconstruction. A concept of numerically distant points will be employed to explain the reasons behind the method's poor performance and an approach will be discussed that allows one to improve the results of a quadratic LS Reconstruction in a boundary layer region. Copyright © 2009 John Wiley & Sons, Ltd.
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Numerical Methods and Applications - Solution limiters and flux limiters for high order discontinuous Galerkin schemes
Numerical Methods and Applications, 1Co-Authors: Natalia PetrovskayaAbstract:We analyze a general concept of limiters for a high order DG scheme written for a 1-D problem. The limiters, which are local and do not require extended stencils, are incorporated into the Solution Reconstruction in order to meet the requirement of monotonicity and avoid spurious Solution overshoots. A limiter β will be defined based on the Solution jumps at grid interfaces. It will be shown that β should be 0 < β < 1 for a monotone approximate Solution.
Ioannis G Kevrekidis - One of the best experts on this subject based on the ideXlab platform.
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extended dynamic mode decomposition with dictionary learning a data driven adaptive spectral decomposition of the koopman operator
Chaos, 2017Co-Authors: Qianxiao Li, Felix Dietrich, Erik M Bollt, Ioannis G KevrekidisAbstract:Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the Solution Reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorit...
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extended dynamic mode decomposition with dictionary learning a data driven adaptive spectral decomposition of the koopman operator
Chaos, 2017Co-Authors: Felix Dietrich, Erik M Bollt, Ioannis G KevrekidisAbstract:Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the Solution Reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good Reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.
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extended dynamic mode decomposition with dictionary learning a data driven adaptive spectral decomposition of the koopman operator
arXiv: Dynamical Systems, 2017Co-Authors: Felix Dietrich, Erik M Bollt, Ioannis G KevrekidisAbstract:Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the Solution Reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good Reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.