An Extension of the Classical Subset of Nominal Modes Method for the Model Order Reduction of Gyroscopic Systems

2018 ◽  
Vol 141 (5) ◽  
Author(s):  
Christian U. Waldherr ◽  
Damian M. Vogt
Keyword(s):  
Computation Time ◽  
Order Reduction ◽  
The Finite Element Method ◽  
Model Order ◽  
Bladed Disk ◽  
Coriolis Effects ◽  
Wide Range ◽  
Numerical Effort ◽  
Further Development ◽  
Gyroscopic Systems

In the structural dynamics design process of turbomachines, Coriolis effects are usually neglected. This assumption holds true if no pronounced interaction between the shaft and disk occurs or if the radial blade displacements are negligible. For classical rotordynamic investigations or for machines where the disk is comparatively thin or weak, Coriolis effects as well as centrifugal effects like stress stiffening and spin softening have to be taken into account. For the analysis of complex structures, the finite element method is today the most commonly used modeling approach. To handle the numerical effort in such an analysis, the aim of the present work is the further development of an existing reduced order model, which also allows the consideration of Coriolis effects without the loss of accuracy for a wide range of rotational speeds. In addition to the investigation of the tuned design of the bladed disk using cyclic boundary conditions, the described method is also appropriate to investigate mistuning phenomena including Coriolis effects. Due to the fact that the computation time can be reduced by two orders of magnitude, the method also opens up the possibility for performing probabilistic mistuning investigations including Coriolis effects.

An Extension of the Classical Subset of Nominal Modes Method for the Model Order Reduction of Gyroscopic Systems

2018 ◽  
Author(s):  
Christian U. Waldherr ◽  
Damian M. Vogt
Keyword(s):  
Computation Time ◽  
Order Reduction ◽  
The Finite Element Method ◽  
Model Order ◽  
Bladed Disk ◽  
Coriolis Effects ◽  
Wide Range ◽  
Numerical Effort ◽  
Further Development ◽  
Gyroscopic Systems

In the structural dynamics design process of turbomachines, Coriolis effects are usually neglected. This assumption holds true if no pronounced interaction between the shaft and disk occurs or if the radial blade displacements are negligible. For classical rotordynamic investigations or for machines where the disk is comparatively thin or weak, Coriolis effects as well as centrifugal effects like stress stiffening and spin softening have to be taken into account. For the analysis of complex structures the finite element method is today the most commonly used modeling approach. To handle the numerical effort in such an analysis, the aim of the present work is the further development of an existing reduced order model, which also allows the consideration of Coriolis effects without the loss of accuracy for a wide range of rotational speeds. In addition to the investigation of the tuned design of the bladed disk using cyclic boundary conditions, the described method is also appropriate to investigate mistuning phenomena including Coriolis effects. Due to the fact that the computation time can be reduced by two orders of magnitude, the method also opens up the possibility for performing probabilistic mistuning investigations including Coriolis effects.

Model of induction brazing of nonmagnetic metals using model order reduction approach

2018 ◽  
Vol 37 (4) ◽  
pp. 1515-1524 ◽  
Author(s):  
Pavel Karban ◽  
David Pánek ◽  
Ivo Doležel
Keyword(s):  
Heat Treatment ◽  
Model Order Reduction ◽  
Computation Time ◽  
Order Reduction ◽  
Nonlinear Problems ◽  
Model Order ◽  
Content Type ◽  
Weakly Nonlinear ◽  
Induction Brazing ◽  
Multiphysics Problems

Purpose A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing. Design/methodology/approach The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM. Findings The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability. Research limitations/implications The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated. Practical implications The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials. Originality/value Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

scholarly journals Study of methods for dimension reduction of complex dynamic linear systems models

2018 ◽  
Vol 226 ◽  
pp. 04036
Author(s):  
Yuriy M. Manatskov ◽  
Torsten Bertram ◽  
Danil V. Shaykhutdinov ◽  
Nikolay I. Gorbatenko
Keyword(s):  
Linear Systems ◽  
Main Idea ◽  
Computation Time ◽  
Order Reduction ◽  
Complex Dynamic ◽  
Model Order ◽  
Reduced Order ◽  
Advantages And Disadvantages ◽  
Optimization Stage ◽  
Linear Systems Of Equations

Complex dynamic linear systems of equations are solved by numerical iterative methods, which need much computation and are timeconsuming ones, and the optimization stage requires repeated solution of these equation systems that increases the time on development. To shorten the computation time, various methods can be applied, among them preliminary (estimated) calculation or oversimple models calculation, however, while testing and optimizing the full model is used. Reduced order models are very popular in solving this problem. The main idea of a reduced order model is to find a simplified model that may reflect the required properties of the original model as accurately as possible. There are many methods for the model order reduction, which have their advantages and disadvantages. In this article, a method based on Krylov subspaces and SVD methods is considered. A numerical experiments is given.

The Galerkin Lumped Parameter Method for Thermal Problems

2012 ◽  
Author(s):  
Alfredo Bermúdez ◽  
Francisco Pena
Keyword(s):  
Reduction Method ◽  
Order Reduction ◽  
Parameter Method ◽  
The Finite Element Method ◽  
Reduced Basis ◽  
Model Order ◽  
Lumped Parameter ◽  
Lumped Parameter Models ◽  
Order Reduction Method ◽  
Lumped Parameter Method

In this contribution, we present a method called Galerkin lumped parameter (GLP) method, as a generalization of the lumped parameter models used in engineering. This method can also be seen as a model-order reduction method. Similarities and differences are discussed. In the GLP method, introduced in [1], domain is decomposed into several sub-domains and a time-independent adapted reduced basis is calculated solving elliptic problems in each sub-domain. The method seeks a global solution in the space spanned by this basis, by solving an ordinary differential system. This approach is useful for electric motors, since the decomposition into several pieces is natural. Numerical results concerning heat equation are presented. Firstly, the comparison with an analytic solution is shown to check the implementation of the numerical algorithm. Secondly, the thermal behavior of an electric motor is simulated, assuming that the electric losses are known. A comparison with the solution obtained by the finite element method is shown.

scholarly journals Symplectic Model Order Reduction with Non-Orthonormal Bases

2019 ◽  
Vol 24 (2) ◽  
pp. 43
Author(s):  
Patrick Buchfink ◽  
Ashish Bhatt ◽  
Bernard Haasdonk
Keyword(s):  
Minimization Problem ◽  
Model Order Reduction ◽  
Order Reduction ◽  
Global Optimum ◽  
Strong Nonlinearity ◽  
Model Order ◽  
Reduced Order ◽  
Orthonormal Bases ◽  
Wide Range ◽  
Computational Resources

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.

Model Order Reduction of 1D Diffusion Systems Via Residue Grouping

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Kandler A. Smith ◽  
Christopher D. Rahn ◽  
Chao-Yang Wang
Keyword(s):  
Model Order Reduction ◽  
State Space Model ◽  
Lithium Ion ◽  
Computation Time ◽  
Order Reduction ◽  
Optimization Procedure ◽  
Model Order ◽  
Diffusion Systems ◽  
Spatially Distributed ◽  
Order Reduction Method

A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly 1∕20 the computation time of the full order models and require 1000–5000 times less CPU time for numerical identification compared to the optimization procedure.

scholarly journals Direct Verification of Linear Systems with over 10000 Dimensions

10.29007/dwj1 ◽  
2018 ◽  
Author(s):  
Stanley Bak ◽  
Parasara Sridhar Duggirala
Keyword(s):  
Linear Systems ◽  
Computation Time ◽  
Order Reduction ◽  
Direct Analysis ◽  
Principle Of Superposition ◽  
Basis Matrix ◽  
Time Step ◽  
Model Order ◽  
Large Linear Systems ◽  
Core Part

We evaluate a recently-proposed reachability method on a set of high-dimensional lin- ear system benchmarks taken from model order reduction and presented in ARCH 2016. The approach uses a state-set representation called a generalized star set and the principle of superposition of linear systems to achieve scalability. The method was previously shown to have promise in terms of scalability for direct analysis of large linear systems. For each benchmark, we also compare computing the basis matrix, a core part of the reachabil- ity method, using numerical simulations versus a matrix exponential formulation. The approach successfully analyzes systems with hundreds of dimensions in minutes, and can scale to systems that have over 10000 dimensions with a computation time ranging from tens of minutes to tens of hours, depending on the desired time step.

scholarly journals On the Energy Stable Approximation of Hamiltonian and Gradient Systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Herbert Egger ◽  
Oliver Habrich ◽  
Vsevolod Shashkov
Keyword(s):  
Numerical Approximation ◽  
Nonlinear Partial Differential Equations ◽  
Galerkin Approximation ◽  
Order Reduction ◽  
Time Discretization ◽  
Underlying Structure ◽  
Model Order ◽  
Wide Range ◽  
Discretization Schemes ◽  
Theoretical Results

AbstractA general framework for the numerical approximation of evolution problems is presented that allows to preserve an underlying dissipative Hamiltonian or gradient structure exactly. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov–Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results, and connections to other discretization approaches will be highlighted.

Application of TEM to Deformation, Wear, and Fracture of Ceramics

1974 ◽  
Vol 32 ◽  
pp. 472-473
Author(s):  
B. J. Hockey
Keyword(s):  
Wear Resistance ◽  
High Temperature ◽  
Mechanical Behavior ◽  
Brittle Materials ◽  
High Temperature Strength ◽  
Wide Range ◽  
Corrosive Environments ◽  
Further Development

Ceramics, such as Al2O3 and SiC have numerous current and potential uses in applications where high temperature strength, hardness, and wear resistance are required often in corrosive environments. These materials are, however, highly anisotropic and brittle, so that their mechanical behavior is often unpredictable. The further development of these materials will require a better understanding of the basic mechanisms controlling deformation, wear, and fracture.The purpose of this talk is to describe applications of TEM to the study of the deformation, wear, and fracture of Al2O3. Similar studies are currently being conducted on SiC and the techniques involved should be applicable to a wide range of hard, brittle materials.

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