Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

Altmann, Robert; Heiland, Jan

doi:10.1007/11221_2018_2

Cited by 5 publications

(5 citation statements)

References 56 publications

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“…The solution X of the reduced-order Riccati equation (5) or equivalent (14) can be applied to approximate the solution X of the Riccati equation (2). Applying the inverse projection and plugging the eigenvalue decomposition, the desired factored solution Z can be attained to approximate X as X = ZZ T .…”

Section: Estimation Of the Factored Solution Of The Riccati Equationmentioning

confidence: 99%

“…These models can be formed by linearizing incompressible Navier-Stokes equations by the mixed finite element method in terms of space and time variables [1]. In this discretization technique, the variable layout within the system will be kept invariant [2]. The linearized incompressible Navier-Stokes models have a sparse inputoutput structure embedding the block matrix setup as…”

Section: Introductionmentioning

confidence: 99%

“…As a result, the dimensions of the system matrices grow over time, and hence the simulations incorporating them became infeasible due to massive computational costs involving memory requirement and greedy time-span. So, contemporary computational tools and conventional solvers are not compatible with the simulation of the large dimensional Riccati equation (2). Consequently, the efficiency of the Riccati-based boundary feedback stabilization of the system (1) will be afflicted drastically [9][10].…”

Section: Introductionmentioning

confidence: 99%

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Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

Uddin,

Khan

2024

Contemp. Math.

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In this article, we are proposing an updated form of Krylov subspace-based interpolatory projection techniques for the stabilization of incompressible Navier-Stokes flows. In the proposed techniques, we utilize the reduced-order modelling approach implicitly, where reduced-order matrices need not be gathered explicitly. To estimate the aimed optimal feedback matrix, only the factored solution of the desired continuous-time algebraic Riccati equation (CARE) needs to be stored through the classical eigenvalue decomposition. The sparse structure of the target systems will remain invariant within the matrix-vector operations for generating the bases of the projector matrices through Krylov subspace techniques, where a cohesive projection scheme will be incorporated to ensure the potency of the projector matrices. So, the proposed techniques will be feasible for memory allocation and enhance the rapid convergence of the simulation. Analysis of the target systems’ transient characteristics, such as eigenvalues and step-responses, will be used to ascertain the competence and reliability of the proposed techniques. Necessary computation will be done numerically through MATLAB. Stabilization of the transient behaviors and minimization of the simulation time are the prime concerns in this work. Eventually, by comparing with the contemporary techniques the advancement of the proposed techniques will be confirmed.

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Section: Estimation Of the Factored Solution Of The Riccati Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

Uddin,

Khan

2024

Contemp. Math.

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“…Generally, the projection needs not be computed explicitly as it will be implicitly realized during the time integration; [29]. However, if only the velocity is of interest, the model could be trained to best approximate T N(υ)υ which resides on a submanifold of dimension (n v −rank ).…”

Section: Semi-discretization Divergence-free Coordinates and Boundary...mentioning

confidence: 99%

Convolutional Neural Networks for Very Low-Dimensional LPV Approximations of Incompressible Navier-Stokes Equations

Heiland

Benner

Bahmani

2022

Front. Appl. Math. Stat.

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The control of general nonlinear systems is a challenging task in particular for large-scale models as they occur in the semi-discretization of partial differential equations (PDEs) of, say, fluid flow. In order to employ powerful methods from linear numerical algebra and linear control theory, one may embed the nonlinear system in the class of linear parameter varying (LPV) systems. In this work, we show how convolutional neural networks can be used to design LPV approximations of incompressible Navier-Stokes equations. In view of a possibly low-dimensional approximation of the parametrization, we discuss the use of deep neural networks (DNNs) in a semi-discrete PDE context and compare their performance to an approach based on proper orthogonal decomposition (POD). For a streamlined training of DNNs directed to the PDEs in a Finite Element (FEM) framework, we also discuss algorithmical details of implementing the proper norms in general loss functions.

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“…Details of the Navier-Stokes equations with the discretization can be found in references [1,2]. Linearizing the Navier-Stokes equations in space and time variables by a mixed fnite element method without altering the time variable converts them into linear time-invariant systems [3]. Te incompressible Navier-Stokes fow can be written as the following diferential-algebraic equations:…”

Section: Introductionmentioning

confidence: 99%

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

Islam

Uddin

et al. 2022

Mathematical Problems in Engineering

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In this paper, we explore the Riccati-based boundary feedback stabilization of the incompressible Navier–Stokes flow via the Krylov subspace techniques. Since the volume of data derived from the original models is gigantic, the feedback stabilization process through the Riccati equation is always infeasible. We apply a ℋ 2 optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system. An extended form of the Krylov subspace-based two-sided iterative algorithm (TSIA) is implemented, where the computation of an equivalent Sylvester equation is included for minimizing the computation time and enhancing the stability of the reduced-order models with satisfying the Wilson conditions. Inverse projection approaches are applied to get the optimal feedback matrix from the reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing works.

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Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

Cited by 5 publications

References 56 publications

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

Convolutional Neural Networks for Very Low-Dimensional LPV Approximations of Incompressible Navier-Stokes Equations

Sparsity-Preserving Two-Sided Iterative Algorithm for Riccati-Based Boundary Feedback Stabilization of the Incompressible Navier–Stokes Flow

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