Showing papers on "Model order reduction published in 2019"

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

Abstract: In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly nonintrusive since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate and can be used as a robust predictive tool for nonintrusive model order reduction of complex fluid flows.In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered trul...

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

Abstract: This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable...

A deep learning enabler for non-intrusive reduced order modeling of fluid flows

Abstract: In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various deep neural network architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equations, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a non-intrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly non-intrusive, since any prior information about the underlying governing equations is not required for generating the reduced order model. Our \textit{a posteriori} analysis shows that the proposed data-driven approach is remarkably accurate, and can be used as a robust predictive tool for non-intrusive model order reduction of complex fluid flows.

Model Order Reduction for Problems with large Convection Effects

Abstract: The reduced basis method allows to propose accurate approximations for many parameter dependent partial differential equations, almost in real time, at least if the Kolmogorov n-width of the set of all solutions, under variation of the parameters, is small. The idea is that any solutions may be well approximated by the linear combination of some well chosen solutions that are computed offline once and for all (by another, more expensive, discretization) for some well chosen parameter values. In some cases, however, such as problems with large convection effects, the linear representation is not sufficient and, as a consequence, the set of solutions needs to be transformed/twisted so that the combination of the proper twist and the appropriate linear combination recovers an accurate approximation. This paper presents a simple approach towards this direction, preliminary simulations support this approach.

Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks

Abstract: A new model order reduction approach is proposed for parametric steady-state nonlinear fluid flows characterized by shocks and discontinuities whose spatial locations and orientations are strongly parameter dependent. In this method, solutions in the predictive regime are approximated using a linear superposition of parameter dependent basis. The sought after parametric reduced-basis are obtained by transporting the snapshots in a spatially and parametrically dependent transport field. Key to the proposed approach is the observation that the transport fields are typically smooth and continuous, despite the solution themselves not being so. As a result, the transport fields can be accurately expressed using a low-order polynomial expansion. Similar to traditional projection-based model order reduction approaches, the proposed method is formulated mathematically as a residual minimization problem for the generalized coordinates. The proposed approach is also integrated with well-known hyper-reduction strategies to obtain significant computational speed-ups. The method is successfully applied to the reduction of a parametric 1-D flow in a converging-diverging nozzle, a parametric 2-D supersonic flow over a forward facing step and a parametric 2-D jet diffusion flame in a combustor.

Dynamically Closed-Loop Controlled Soft Robotic Arm using a Reduced Order Finite Element Model with State Observer

Abstract: This paper presents a computationally efficient method to model and simulate soft robots. Finite element methods enable us to simulate and control soft robots, but require us to work with a large dimensional system. This limits their use in real-time simulation and makes those methods less suitable for control design tools. Using model order reduction, it is possible to create a reduced order system for building controllers and observers. Model reduction errors are taken into account in the design of the low-order feedback, and it is then applied to the large dimensional, unreduced model. The control architecture is based on a linearized model of the robot and enables the control of the robot around this equilibrium point. To show the performance of this control method, pose-to-pose and trajectory tracking experiments are conducted on a pneumatically actuated soft arm. The soft arm has 12 independent interior cavities that can be pressurized and cause the arm to move in three dimensions. The arm is made of a rubber material and is casted through a lost-wax fabrication technique.

A dynamic closure modeling framework for model order reduction of geophysical flows

Abstract: In this paper, a dynamic closure modeling approach has been derived to stabilize the projection-based reduced order models in the long-term evolution of forced-dissipative dynamical systems. To simplify our derivation without losing generalizability, the proposed reduced order modeling (ROM) framework is first constructed by Galerkin projection of the single-layer quasigeostrophic equation, a standard prototype of large-scale general circulation models, onto a set of dominant proper orthogonal decomposition modes. We then propose an eddy viscosity closure approach to stabilize the resulting surrogate model considering the analogy between large eddy simulation (LES) and truncated modal projection. Our efforts, in particular, include the translation of the dynamic subgrid-scale model into our ROM setting by defining a test truncation similar to the test filtering in LES. The a posteriori analysis shows that our approach is remarkably accurate, allowing us to integrate simulations over long time intervals at a nominally small computational overhead.

Sliding-Window-Based Real-Time Model Order Reduction for Stability Prediction in Smart Grid

Abstract: In this paper, a new real-time model order reduction technique for stability prediction in the smart grid is proposed. The proposed method uses an online proper orthogonal decomposition algorithm. A snapshot matrix on a sliding sampling window is used for extracting the main components of the system states by performing a randomized singular value decomposition. After reducing the order of the system, a local linear model is estimated for this snapshot matrix. Then, the state of the system is predicted in a sliding prediction window. Finally, a suitable stability index is calculated and the stability of the system is forecasted in this prediction window. The proposed method is capable of predicting the transient stability, unstable/critical machines and the stability limit. In addition, it can be used for the first swing and multiswing instability detection. The simulations on three test systems show that the proposed technique can predict system stability with the high precision in real time. The computational burden and the length of prediction horizon is suitable for practical applications and the proposed algorithm has significant advantages in case of large-scale power systems.

Learning slosh dynamics by means of data

Abstract: In this work we study several learning strategies for fluid sloshing problems based on data. In essence, a reduced-order model of the dynamics of the free surface motion of the fluid is developed under rigorous thermodynamics settings. This model is extracted from data by exploring several strategies. First, a linear one, based on the employ of Proper Orthogonal Decomposition techniques is analyzed. Second, a strategy based on the employ of Locally Linear Embedding is studied. Finally, Topological Data Analysis is employed to the same end. All the three distinct possibilities rely on a numerical integration scheme to advance the dynamics in time. This thermodynamically consistent integrator is developed on the basis of the General Equation for Non-Equilibrium Reversible–Irreversible Coupling, GENERIC [M. Grmela and H.C Oettinger (1997). Phys. Rev. E. 56 (6): 6620–6632], framework so as to guarantee the satisfaction of first principles (particularly, the laws of thermodynamics). We show how the resulting method employs a few degrees of freedom, while it allows for a realistic reconstruction of the fluid dynamics of sloshing processes under severe real-time constraints. The proposed method is shown to run faster than real time in a standard laptop.

Inverse Compensator for A Simplified Discrete Preisach Model Using Model-Order Reduction Approach

Abstract: The classical Preisach model, which is built by the superposition of a great number of relay operators, is one of the most popular models to represent the hysteretic behaviors in various applications, such as the smart materials-based actuators. However, the construction of the inverse compensator for the classical Preisach model is very challenging for some reasons, first, the analytical inverse of the classical Preisach model is not available, and, second, due to a huge amount of the relay operators the implementation of the inverse compensator is troublesome and causes heavy computational burden. To overcome these drawbacks, a simplified discrete Preisach model is developed in this paper. The simplified model has an explicit expression with respect to the input of the model, thus it is simple to construct its inverse compensator using the inverse multiplicative structure approach. To reduce the computational effort in implementing the inverse compensator, the model-order reduction method is employed to reduce the complexity of the inverse compensator. Experimental tests are carried out to validate the effectiveness of the proposed approach.

Model Order Reduction by Using the Balanced Truncation and Factor Division Methods

Abstract: The aim of this paper is to construct a new model order reduction method for linear dynamic systems. In this technique, the denominator polynomial of the reduced order model (ROM) is obtain...

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

Abstract: This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of in...

Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems

Abstract: In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.

Symplectic model order reduction with non-orthonormal bases

Abstract: Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.

Order Reduction of Linear Dynamic Systems by Improved Routh Approximation Method

Abstract: In this paper, a new simplified Routh approximation technique is proposed for the model order reduction (MOR) of large scale linear time-invariant systems. In reduced order modeling, the Ro...

Comparing four model-order reduction techniques, applied to lithium-ion battery-cell internal electrochemical transfer functions

Abstract: Physics-based models of lithium-ion battery dynamics are developed from fundamental electrochemical principles and describe cell internal electrochemical variables in addition to terminal voltage. Real-time estimates of the values taken on by internal cell variables provided by such models might be leveraged by future battery-management systems to control fast-charging and routine use of a battery pack to maximize performance but minimize aging. These models are most naturally described as sets of coupled partial-differential equations (PDEs), and so the greatest obstacle to their adoption stems from the computational complexity involved in finding solutions to the model equations. To make a feasible physics-based model for battery management, we must construct reduced-order approximations to these PDE models. In this paper, we present four methods to find high-fidelity discrete-time state-space reduced-order models (ROMs) that approximate infinite-order transcendental transfer functions that model the PDE relationships of all electrochemical variables of interest. These four methods are compared for a single cell based on speed, memory usage, robustness, and accuracy of the predictions of the resulting reduced-order models with respect to precise numerical simulations of the PDEs. We find that all four methods produce ROMs that match the linearized PDEs closely in the frequency domain and that yield time-domain simulations that match those from the nonlinear PDEs as well, but that each xRA method has distinct features so that different applications might prefer one method versus another.

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Abstract: We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A classical reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the manifolds of associated residuals. This approach exploits the fact that the manifolds of parameter-dependent matrices and vectors involved in the full order model are contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems.

Model Order Reduction of Linear Systems via the Cross Gramian and SVD

Abstract: On the basis of the cross Gramian and the singular value decomposition, this brief investigates model order reduction of linear time-invariant systems in the view of the symmetric case and the nonsymmetric case. It guarantees the reduced system for the nonsymmetric system to be data-sparse. The asymptotical stability of the reduced system is discussed and an output error bound is provided. The correlation of the Hankel singular values in the proposed method and the balanced truncation method is presented. Finally, numerical results indicate that the reduced system of the symmetric system is equivalent to that resulting from balanced truncation, while the reduced system of the nonsymmetric system has a good performance.

Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction

Abstract: In this work, we present an efficiently computational approach for designing material micro-structures by means of topology optimization. The central idea relies on using the isogeometric analysis integrated with the parameterized level set function for numerical homogenization, sensitivity calculation and optimization of the effective elastic properties. Design variables, which are level set values associated with control points, are updated from the optimizer and represent the geometry of the unit cell. We further improve the computational efficiency in each iteration by employing reduced order modeling when solving linear systems of the equilibrium equations. We construct a reduced basis by reusing computed solutions from previous optimization steps, and a much smaller linear system of equations is solved on the reduced basis. Two- and three-dimensional numerical results show the effectiveness of the topology optimization algorithm coupled with the reduced basis approach in designing metamaterials.

Analytic prediction of isolated forced response curves from spectral submanifolds

Abstract: We show how spectral submanifold (SSM) theory can be used to provide analytic predictions for the response of periodically forced multi-degree-of-freedom mechanical systems. These predictions include an explicit criterion for the existence of isolated forced responses that will generally be missed by numerical continuation techniques. Our analytic predictions can be refined to arbitrary precision via an algorithm that does not require the numerical solutions of the mechanical system. We illustrate all these results on low- and high-dimensional nonlinear vibration problems. We find that our SSM-based forced response predictions remain accurate in high-dimensional systems, in which numerical continuation of the periodic response is becoming computationally expensive.

Order Reduction in Linear Dynamical Systems by Using Improved Balanced Realization Technique

Abstract: In this article, a new model order reduction scheme is proposed for the simplification of large-scale linear dynamical models. The proposed technique is based on the balanced realization method, in which the steady-state gain problem of the balanced truncation is circumvented. In this method, the denominator coefficients of the reduced system are evaluated by using the balanced realization, and the numerator coefficients are obtained by using a simple mathematical procedure as given in the literature. The proposed technique has been illustrated through some standard large-scale systems. This method gives the least performance error indices compared to some other existing system reduction methods. The time response of the approximated system, evaluated by the proposed method, is also shown which is the excellent matching of the response of the actual model when compared to the responses of other existing techniques.