Showing papers on "Model order reduction published in 2020"

Machine learning for nonintrusive model order reduction of the parametric inviscid transonic flow past an airfoil

Abstract: Fluid flow in the transonic regime finds relevance in aerospace engineering, particularly in the design of commercial air transportation vehicles. Computational fluid dynamics models of transonic flow for aerospace applications are computationally expensive to solve because of the high degrees of freedom as well as the coupled nature of the conservation laws. While these issues pose a bottleneck for the use of such models in aerospace design, computational costs can be significantly minimized by constructing special, structure-preserving surrogate models called reduced-order models. In this work, we propose a machine learning method to construct reduced-order models via deep neural networks and we demonstrate its ability to preserve accuracy with a significantly lower computational cost. In addition, our machine learning methodology is physics-informed and constrained through the utilization of an interpretable encoding by way of proper orthogonal decomposition. Application to the inviscid transonic flow past the RAE2822 airfoil under varying freestream Mach numbers and angles of attack, as well as airfoil shape parameters with a deforming mesh, shows that the proposed approach adapts to high-dimensional parameter variation well. Notably, the proposed framework precludes the knowledge of numerical operators utilized in the data generation phase, thereby demonstrating its potential utility in the fast exploration of design space for diverse engineering applications. Comparison against a projection-based nonintrusive model order reduction method demonstrates that the proposed approach produces comparable accuracy and yet is orders of magnitude computationally cheap to evaluate, despite being agnostic to the physics of the problem.

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

Abstract: We propose a general---i.e., independent of the underlying equation---registration method for parameterized model order reduction. Given the spatial domain $\Omega \subset \mathbb{R}^d$ and the man...

Model order reduction assisted by deep neural networks (ROM-net)

Abstract: In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

VarNet: Variational Neural Networks for the Solution of Partial Differential Equations

Abstract: In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. Given this loss function, we also propose an approach to optimally select the space-time samples, used to train the NN, that is based on the feedback provided from the PDE residual. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction (MOR) of PDEs. We demonstrate the performance of our method through extensive numerical experiments for the advection-diffusion PDE as an important case-study.

POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

Abstract: In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

Model order reduction with neural networks: Application to laminar and turbulent flows

Abstract: We investigate the capability of neural network-based model order reduction, i.e., autoencoder (AE), for fluid flows. As an example model, an AE which comprises of a convolutional neural network and multi-layer perceptrons is considered in this study. The AE model is assessed with four canonical fluid flows, namely: (1) two-dimensional cylinder wake, (2) its transient process, (3) NOAA sea surface temperature, and (4) $y-z$ sectional field of turbulent channel flow, in terms of a number of latent modes, a choice of nonlinear activation functions, and a number of weights contained in the AE model. We find that the AE models are sensitive against the choice of the aforementioned parameters depending on the target flows. Finally, we foresee the extensional applications and perspectives of machine learning based order reduction for numerical and experimental studies in fluid dynamics community.

A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries

Abstract: A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM), recently proposed in Main and Scovazzi, J Comput Phys [17]. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

Projection-based model reduction with dynamically transformed modes

Abstract: We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction.

Abstract: This work deals with optimal control problems as a strategy to drive bifurcating solutions of nonlinear parametrized partial differential equations towards a desired branch. Indeed, for these governing equations, multiple solution configurations can arise from the same parametric instance. We thus aim at describing how optimal control allows to change the solution profile and the stability of state solution branches. First of all, a general framework for nonlinear optimal control problems is presented in order to reconstruct each branch of optimal solutions, discussing in detail the stability properties of the obtained controlled solutions. Then, we apply the proposed framework to several optimal control problems governed by bifurcating Navier-Stokes equations in a sudden-expansion channel, describing the qualitative and quantitative effect of the control over a pitchfork bifurcation and commenting in detail the stability eigenvalue analysis of the controlled state. Finally, we propose reduced order modeling as a tool to efficiently and reliably solve parametric stability analysis of such optimal control systems, which can be unbearable to perform with standard discretization techniques such as the Finite Element Method.

Conservative Model Order Reduction for Fluid Flow

Abstract: In the past decade, model order reduction (MOR) has been successful in reducing the computational complexity of elliptic and parabolic systems of partial differential equations (PDEs). However, MOR of hyperbolic equations remains a challenge. Symmetries and conservation laws, which are a distinctive feature of such systems, are often destroyed by conventional MOR techniques which result in a perturbed, and often unstable reduced system. The importance of conservation of energy is well-known for a correct numerical integration of fluid flow. In this paper, we discuss model reduction, that exploits skew-symmetry of conservative and centered discretization schemes, to recover conservation of energy at the level of the reduced system. Moreover, we argue that the reduced system, constructed with the new method, can be identified by a reduced energy that mimics the energy of the high-fidelity system. Therefore, the loss in energy, associated with the model reduction, remains constant in time. This results in an, overall, correct evolution of the fluid that ensures robustness of the reduced system. We evaluate the performance of the proposed method through numerical simulation of various fluid flows, and through a numerical simulation of a continuous variable resonance combustor model.

A Subspace-Splitting Moment-Matching Model-Order Reduction Technique for Fast Wideband FEM Simulations of Microwave Structures

Abstract: This article describes a novel model-order reduction (MOR) approach for efficient wide frequency band finite-element method (FEM) simulations of microwave components. It relies on the splitting of the system transfer function into two components: a singular one that accounts for the in-band system poles and a regular part that has no in-band poles. In order to perform this splitting during the reduction process, the projection basis is formed of two sets of orthogonal vectors that must be computed in sequence. The first set to be computed consists of the in-band eigenvectors that are associated with the dynamics of the electromagnetic field, while the second set uses the block moments of the original system, which are computed in the orthogonal complement to the subspace spanned by the in-band eigenvectors. The advantage of this method is that it results in a more compact reduced-order model than a method that employs only moment matching for the projection basis computations.

Enhanced model-order reduction approach via online adaptation for parametrized nonlinear structural problems

Abstract: With regard to the parameterized projection-based reduced-order model, it is significant to consider the computational efficiency as well as its capability for the parametric variation. The proposed approach is based on the online adaptive procedure to improve the accuracy and stability of the reduced-order model. Achieving efficient computation in online adaptation, a matrix version of the discrete empirical interpolation method is employed to approximate the nonlinear finite element matrix, independently. The proposed approach is applied to analysis of a structure with geometric and material nonlinearities. As a result, the computational efficiency during the offline/online steps of the proposed approach is significantly improved, compared to other existing approaches. Moreover, within the present numerical examinations, it is found that the proposed approach is capable of accurately addressing broad parametric variations by using only ten percent of the number of data used in the conventional ROM from the preliminary computation.

Reduced-Order State Space Model for Dynamic Phasors in Active Distribution Networks

Abstract: The rapid proliferation of distributed generation (DG) in active distribution networks (ADNs) increases the power system modeling complexity, which requires more efficient numerical methods to analyze dynamic interactions among DGs in ADN. This paper proposes a reduced-order dynamic phasor (DP) modeling method which is applied to large-scale ADNs with DGs for reducing the modeling scale and allowing larger time steps in ADN analyses. First, a unified state-space DP model, referred to as full-order DP model for linear circuits, is developed on the abc -axis to provide a comprehensive technique for interfacing DGs with other electric devices in large ADNs. The model order reduction (MOR) is applied to this full-order model to reduce the computation burden for large-scale ADNs. The paper develops a state-space DP model on positive and negative sequence synchronous rotating reference frames for three-phase DGs, and a stationary frame for single-phase DGs. The accuracy and the efficiency of the proposed models are verified using the modified IEEE 123-node test feeder and IEEE European low voltage test feeder with DGs.

A new model order reduction method for the design of compensator by using moment matching algorithm

Abstract: The aim of this paper is the construction of a new model reduction technique for large scale stable linear dynamic systems. It is principally focused on the dominant modes and time moments retentio...

Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

Abstract: An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed It aims to automatize the POD-Greedy algorithm combined with empirical interpolation At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction The proposed technique is able to derive a suitable match between the reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model possible This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm An efficient output error indicator plays a key role in the adaptive scheme We also propose an improved output error indicator based on previous work Upon convergence of the POD-Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable reduced-order model can be constructed The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering

Cross-Gramian Model Reduction Approach for Tuning Power System Stabilizers in Large Power Networks

Abstract: Poorly damped inter-area modes of oscillations represent a major concern to power system operation since they detain the power transfer capability of transmission networks. This situation becomes more stringent as the tie-lines are heavily stressed and/or large amounts of renewable energy resources are installed. To overcome this issue, a detailed mathematical model is proposed in this paper to reduce the linearized model of a large power system using the cross-Gramian technique. The presented approach divides the system into a study area which contains one generation unit with installed power system stabilizer (PSS) and an external one which comprises the rest of generation units in the system. Model order reduction is only applied to the external area with the objective of maintaining the characteristics of the original model. Meanwhile, the dynamics of the study area are preserved to provide the required damping through the designed PSS. In addition, an online tuning methodology is also presented to provide robust damping performance in response to changes in the system operating conditions. The deployed cross-Gramian model order reduction alleviates the computational burden and time associated with the online PSS tuning when original power system models are used. The effectiveness of the proposed approach is tested using the New-England 39-bus system in addition to another practical system which resembles the Northern Regional Power Grid India test system.