scholarly journals Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Shobhit Jain ◽  
Paolo Tiso
Keyword(s):  
Finite Element ◽  
Model Reduction ◽  
Computational Cost ◽  
Galerkin Projection ◽  
Dimensional Linear Subspace ◽  
Reduction Techniques ◽  
Energy Conserving ◽  
Reduction Methods ◽  
Wing Structure ◽  
Nonlinear Manifolds

Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.

Application of projection-based model reduction to finite-element plate models for two-dimensional traveling waves

2016 ◽  
Vol 28 (14) ◽  
pp. 1886-1904 ◽  
Author(s):  
Vijaya VN Sriram Malladi ◽  
Mohammad I Albakri ◽  
Serkan Gugercin ◽  
Pablo A Tarazaga
Keyword(s):  
Finite Element ◽  
Model Reduction ◽  
Traveling Waves ◽  
Large Scale ◽  
Fe Model ◽  
Plate Model ◽  
Reduction Techniques ◽  
State Frequency ◽  
Scale Down ◽  
The Stability

A finite element (FE) model simulates an unconstrained aluminum thin plate to which four macro-fiber composites are bonded. This plate model is experimentally validated for single and multiple inputs. While a single input excitation results in the frequency response functions and operational deflection shapes, two input excitations under prescribed conditions result in tailored traveling waves. The emphasis of this article is the application of projection-based model reduction techniques to scale-down the large-scale FE plate model. Four model reduction techniques are applied and their performances are studied. This article also discusses the stability issues associated with the rigid-body modes. Furthermore, the reduced-order models are utilized to simulate the steady-state frequency and time response of the plate. The results are in agreement with the experimental and the full-scale FE model results.

scholarly journals Implementing model reduction to the JuliaFEM platform

2018 ◽  
Vol 51 (1) ◽  
pp. 36-54 ◽  
Author(s):  
Marja Liisa Rapo ◽  
Jukka Aho ◽  
Hannu Koivurova ◽  
Tero Frondelius
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Model Reduction ◽  
Open Source ◽  
Degrees Of Freedom ◽  
Mass Matrices ◽  
Dynamic Condensation ◽  
Dynamic Analyses ◽  
Reduction Methods ◽  
Large Model

JuliaFEM is an open source finite element method solver written in the Julia language. This paper presents an implementation of two common model reduction methods: the Guyan reduction and the Craig-Bampton method. The goal was to implement these algorithms to the JuliaFEM platform and demonstrate that the code works correctly. This paper first describes the JuliaFEM concept briefly after which it presents the theory of model reduction, and finally, it demonstrates the implemented functions in an example model. This paper includes instructions for using the implemented algorithms, and reference the code itself in GitHub. The reduced stiness and mass matrices give the same results in both static and dynamic analyses as the original matrices, which proves that the code works correctly. The code runs smoothly on relatively large model of 12.6 million degrees of freedom. In future, damping could be included in the dynamic condensation now that it has been shown to work.

scholarly journals Advances in Hierarchical Model Reduction and combination with other computational reduction methods

2018 ◽  
Author(s):  
Giulia Meglioli ◽  
Matteo Zancanaro ◽  
Francesco Ballarin ◽  
Simona Perotto ◽  
Gianluigi Rozza
Keyword(s):  
Hierarchical Model ◽  
Computational Cost ◽  
Order Reduction ◽  
Reduced Basis ◽  
Element Discretization ◽  
Modal Expansion ◽  
Reduction Techniques ◽  
Computational Performance ◽  
Reduction Methods ◽  
Computational Reduction

In this work we present address the combination of the Hierarchical Model (Hi-Mod) reduction approach with projection-based reduced order methods, exploiting either on Greedy Reduced Basis (RB) or Proper Orthogonal Decomposition (POD), in a parametrized setting. The Hi-Mod approach, introduced in, is suited to reduce problems in pipe-like domains featuring a dominant axial dynamics, such as those arising for instance in haemodynamics. The Hi-Mod approach aims at reducing the computational cost by properly combining a finite element discretization of the dominant dynamics with a modal expansion in the transverse direction. In a parametrized context, the Hi-Mod approach has been employed as the high-fidelity method during the offline stage of model order reduction techniques based on RB or POD. The resulting combined reduction methods, which have been named Hi-RB and Hi-POD, respectively, will be presented with applications in diffusion-advection problems, fluid dynamics and optimal control problems, focusing on the approximation stability of the proposed methods and their computational performance.

Using Model Reduction Methods within Incremental Four-Dimensional Variational Data Assimilation

2008 ◽  
Vol 136 (4) ◽  
pp. 1511-1522 ◽  
Author(s):  
A. S. Lawless ◽  
N. K. Nichols ◽  
C. Boess ◽  
A. Bunse-Gerstner
Keyword(s):  
Data Assimilation ◽  
Model Reduction ◽  
Variational Data Assimilation ◽  
Superior Performance ◽  
Water Model ◽  
Computationally Efficient ◽  
Full System ◽  
Ocean Data Assimilation ◽  
Reduction Techniques ◽  
Reduction Methods

Abstract Incremental four-dimensional variational data assimilation is the method of choice in many operational atmosphere and ocean data assimilation systems. It allows the four-dimensional variational data assimilation (4DVAR) to be implemented in a computationally efficient way by replacing the minimization of the full nonlinear 4DVAR cost function with the minimization of a series of simplified cost functions. In practice, these simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. In this paper, a new method is proposed for deriving the simplified problems in incremental 4DVAR, based on model reduction techniques developed in the field of control theory. It is shown how these techniques can be combined with incremental 4DVAR to give an assimilation method that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.

scholarly journals Fast Multiscale Reservoir Simulations With POD-DEIM Model Reduction

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2141-2154 ◽  
Author(s):  
Yanfang Yang ◽  
Mohammadreza Ghasemi ◽  
Eduardo Gildin ◽  
Yalchin Efendiev ◽  
Victor Calo
Keyword(s):  
Velocity Field ◽  
Model Reduction ◽  
Finite Volume ◽  
Computational Cost ◽  
Solution Space ◽  
Mass Conservation ◽  
Galerkin Projection ◽  
Local Model ◽  
Fine Grid ◽  
Reservoir Simulations

Summary We present a global/local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media. Our approach identifies a low-dimensional structure in the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that achieves a high compression of the model. This compression is achieved because the velocity field is conservative for any low-order reduced model in our framework, whereas a typical global model reduction that is based on proper-orthogonal-decomposition (POD) Galerkin projection cannot guarantee local mass conservation. The lack of mass conservation can be observed in numerical simulations that use finite-volume-based approaches. The discrete empirical interpolation method (DEIM) approximates fine-grid nonlinear functions in Newton iterations. This approach delivers an online computational cost that is independent of the fine-grid dimension. POD snapshots are inexpensively computed with local model-reduction techniques that are based on the generalized multiscale finite-element method (GMsFEM) that provides (1) a hierarchical approximation of the snapshot vectors, (2) adaptive computations with coarse grids, and (3) inexpensive global POD operations in small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology provides an error bound in simulations. Our numerical results, by use of a two-phase immiscible flow, show a substantial speedup, and we compare our results with the standard POD-DEIM in a finite-volume setup.

Model Reduction of Contact Dynamics Simulation Using Modified Lyapunov Balancing Method

2011 ◽  
Author(s):  
Jianxun Liang ◽  
Ou Ma
Keyword(s):  
Finite Element ◽  
Model Reduction ◽  
Contact Stiffness ◽  
Influential Factors ◽  
Contact Dynamics ◽  
Dynamics Simulation ◽  
Integration Time ◽  
Time Step ◽  
Reduction Techniques ◽  
Dynamics Simulations

Finite element models can accurately simulate impact-contact dynamics response of a multibody dynamical system. However, such a simulation remains very inefficient because very small integration time step must be used when solving the involved differential equations. Although many model reduction techniques can be used to improve the efficiency of finite element based simulations, most of these techniques cannot be readily applied to contact dynamics simulations due to the high nonlinearity of the contact dynamics model. This paper presents a model reduction approach for finite-element based multibody contact dynamics simulation, based on a modified Lyapunov balanced truncation method. An example is presented to demonstrate that, by applying the model reduction the simulation process is significantly speeded up and the resulting error is bounded within an acceptable level. The performance of the method with respect to some influential factors such as element size, shape and contact stiffness is also investigated.

A Modified Discrete Empirical Interpolation Method for Reducing Non-Linear Structural Finite Element Models

2013 ◽  
Author(s):  
Paolo Tiso ◽  
Rob Dedden ◽  
Daniel Rixen
Keyword(s):  
Finite Element ◽  
Interpolation Method ◽  
Computational Cost ◽  
Order Reduction ◽  
Galerkin Projection ◽  
Alternative Formulation ◽  
Discrete Empirical Interpolation Method ◽  
Empirical Interpolation Method ◽  
The Cost ◽  
Discrete Empirical Interpolation

Model Order Reduction (MOR) in nonlinear structural analysis problems in usually carried out by a Galerkin projection of the primary variables on a sensibly smaller space. However, the cost of computing the nonlinear terms is still of the order of the full system. The Discrete Empirical Interpolation Method (DEIM) is an effective algorithm to reduce the computational cost of the nonlinear terms of the discretized governing equations. However, its efficiency is diminished when applied to a Finite Element (FE) framework. We present here an alternative formulation of the DEIM that suits FE discretized problems and preserves the efficiency and the accuracy of the original DEIM method.

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