An enhanced AMLS method and its performance

doi:10.1016/j.cma.2015.01.004

Computer Methods in Applied Mechanics and Engineering

Volume 287, 15 April 2015, Pages 90-111

https://doi.org/10.1016/j.cma.2015.01.004 Get rights and content

Abstract

In this paper, we present an effective new component mode synthesis (CMS) method based on the concept of the automated multi-level substructuring (AMLS) method. Herein, the original transformation matrix of the AMLS method is enhanced by considering the residual mode effect, and the resulting unknown eigenvalue in the formulation is approximated by employing the idea of the improved reduced system (IRS) method. Using the newly defined transformation matrix, we develop an enhanced AMLS method by which original finite element (FE) models can be more precisely approximated by reduced models, and their solution accuracy is significantly improved. The formulation details of the enhanced AMLS method is presented, and its accuracy and computational cost is investigated through numerical examples.

Introduction

While computation capability has increased rapidly, the demand for large scale finite element (FE) models has increased even more rapidly. Therefore, it has always been an important issue to reduce computational cost. A variety of model reduction methods have been developed and widely used in many engineering fields [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. The focus in model reduction is on reducing computational cost with the least possible loss in accuracy.

Within the structural dynamics community, component mode synthesis (CMS) is a popular and effective finite element (FE) model reduction method [5], [6], [7], [8], [9], [10], [11], [12]. In CMS methods, an original (global) FE model is partitioned into smaller substructures, substructural eigenvalue problems are solved, and a reduced model constructed by retaining only dominant substructural modes is used for calculations, instead of the much larger original FE model. For this reason, CMS methods can significantly reduce overall computational cost required for many applications (e.g., controller design for multi-body dynamics systems, structural health monitoring, structural design optimization, model identification).

In the 1990s, the automated multi-level substructuring (AMLS) method, a computer-aided CMS method, was proposed in the field of applied mathematics [13], [14], [15], [16]. Due to its computational efficiency, involving recursive partitioning and matrix reordering processes, the AMLS method has become popular for reduced-order modeling. Recently, Bennighof and Lehoucq [17] proposed a well-defined formulation of the AMLS method based on the concept of the Craig–Bampton (CB) method [6], [17]. The AMLS method has been also used as a solver of eigenvalue problems in many commercial FE software.

In the original CB and AMLS methods, a transformation matrix is constructed by retaining only dominant substructural modes. Using the transformation matrix, original FE models can be transformed into reduced models, which approximate the original models. With this procedure, residual substructural modes are simply truncated without further consideration. However, when the residual mode effect is considered, the accuracy of the original transformation matrix can be improved. That is, the original (global) models can be more precisely approximated. This approach has been used for flexibility based CMS methods, in which, unlike for the CB and AMLS methods, substructures are connected with a free interface [7], [10], [11], [12].

In this study, we derive a new transformation matrix for the AMLS method enhanced by considering the residual mode effect. One difficulty is the fact that the enhanced transformation matrix contains an unknown eigenvalue. In order to approximate the unknown eigenvalue, we adopt O’Callahan’s idea, which was originally proposed to develop the improved reduced system (IRS) method by improving Guyan reduction [18]. Finally, the enhanced transformation matrix is defined without the unknown eigenvalue, and by using the newly defined transformation matrix, an enhanced AMLS method is proposed. The reduced FE models obtained from the enhanced AMLS methods have the same size as those obtained from the original AMLS method. However, compared to the original AMLS method, the enhanced AMLS method can provide significantly improved reduced-order models.

In the following sections, we present the general framework of CMS methods in Section 2, and briefly review the original AMLS method in Section 3. In Section 4, the formulation details of the enhanced AMLS method are presented, and its performance and computational cost are tested in Sections 5 Numerical examples, 6 Computational cost, respectively. The conclusions are given in Section 7.

Section snippets

Component mode synthesis

In this section, the general framework of component mode synthesis (CMS) is briefly presented. In structural dynamics, the linear dynamics equations of a global (non-partitioned) FE model can be expressed as $M_{g} {\ddot{u}}_{g} + K_{g} u_{g} = f_{g},$ where $M_{g}$ and $K_{g}$ are the global mass and stiffness matrices, respectively, and $u_{g}$ and $f_{g}$ are the global displacement and force vectors, respectively. Subscript $g$ denotes the global structure.

Considering a free harmonic vibration $(f_{g} = 0)$ , from Eq. (1), the following eigenvalue

Original AMLS method

Since the AMLS method proposed by Bennighof and his coworkers [17], [19], [20] is based on the Craig–Bampton (CB) method [6], substructures are connected at a fixed interface boundary, see Fig. 1(b). However, unlike for the CB method, the interface boundary DOFs are also considered as substructures in the AMLS method. The interior DOFs are considered as the bottom level substructures and the interface boundary DOFs are considered as the higher level substructures or highest level

Enhanced AMLS method

In the original AMLS method, the reduced transformation matrix ${\bar{T}}_{0}$ can be constructed retaining dominant modes only. However, when residual modes are appropriately considered, the reduced transformation matrix ${\bar{T}}_{0}$ can be enhanced.

The transformation matrix $T_{0}$ defined in Eq. (5) can be represented as $T_{0} = \hat{Ψ} Φ with \hat{Ψ} = [\begin{matrix} I \\ ⋱ \\ I & {\hat{Ψ}}_{i, j} \\ 0 & ⋱ \\ I \end{matrix}], Φ = [\begin{matrix} Φ_{1} \\ ⋱ & 0 \\ Φ_{i} \\ 0 & ⋱ \\ Φ_{N_{s}} \end{matrix}], for i = 1, 2, \dots, N_{s} and j = i + 1, i + 2, \dots, N_{s},$ where $\hat{Ψ}$ is the multi-level constraint mode matrix, and $Φ$ is the eigenvector matrix that contains all the

Numerical examples

In this section, we compare the performance of the enhanced AMLS method to the original AMLS method. The original and enhanced AMLS methods were implemented using MATLAB. Four structural problems are considered: rectangular plate, cylindrical solid, bench corner structure and hyperboloid shell problems, in which, for finite element modeling, 4-node MITC shell [23], [24], [25], [26] and 8-node brick finite elements are used.

The frequency cut-off mode selection method is used to select the

Computational cost

In order to investigate the computational cost required for the enhanced AMLS method, computation times are measured, and compared with those of the original AMLS method. A sparse matrix computation with MATLAB is used in a personal computer (Intel core (TM) i7-3770, 3.40 GHz CPU, 16GB RAM). Note that, of course, computation times vary depending on implementation details of the computer codes, as well as on the performance of the computers used. Therefore, the results discussed in this section

Conclusions

In this paper, we presented a new component mode synthesis (CMS) method developed by improving the automated multi-level substructuring (AMLS) method. Unlike for the original AMLS method, the residual mode effect is considered in constructing the transformation matrix. As a result, the original AMLS transformation matrix is enhanced by the residual flexibility, in which the unknown eigenvalue is approximated using O’Callahan’s approach from the improved reduced system (IRS) method.

The enhanced

Acknowledgments

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2014R1A1A1A05007219), and the Human Resources Development (No. 20134030200300) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy.

References (33)

M.I. Friswell et al.
Model reduction using dynamic and iterated IRS techniques
J. Sound Vib.
(1995)
R.H. MacNeal
Hybrid method of component mode synthesis
Comput. Struct.
(1971)
D.J. Rixen
A dual Craig–Bampton method for dynamic substructuring
J. Comput. Appl. Math.
(2004)
F. Bourquin et al.
Intrinsic component mode synthesis and plate vibrations
Comput. Struct.
(1992)
F. Bourquin et al.
Numerical study of an intrinsic component mode synthesis method
Comput. Methods Appl. Mech. Engrg.
(1992)
Y. Lee et al.
Improving the MITC3 shell finite element by using the Hellinger–Reissner principle
Comput. Struct.
(2012)
Y. Lee et al.
The MITC3+ shell finite element and its performance
Comput. Struct.
(2014)
J.G. Kim et al.
Estimating relative eigenvalue errors in the Craig–Bampton method
Comput. Struct.
(2014)
J.G. Kim et al.
An accurate error estimator for Guyan reduction
Comput. Methods Appl. Mech. Engrg.
(2014)
A. Ibrahimbegovic et al.
Nonlinear dynamics of flexible multibody systems
Comput. Struct.
(2003)

R. Guyan

Reduction of stiffness and mass matrices

AIAA J.

(1965)

B.M. Irons

Structural eigenvalue problems: elimination of unwanted variables

AIAA J.

(1965)

Y. Xia et al.

A new iterative order reduction (IOR) method for eigensolutions of large structures

Internat. J. Numer. Methods Engrg.

(2004)

W. Hurty

Dynamic analysis of structural systems using component modes

AIAA J.

(1965)

R.R. Craig et al.

Coupling of substructures for dynamic analysis

AIAA J.

(1965)

W.A. Benfield et al.

Vibration analysis of structures by component mode substitution

AIAA J.

(1971)

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View full text

An enhanced AMLS method and its performance

Abstract

Introduction

Section snippets

Component mode synthesis

Original AMLS method

Enhanced AMLS method

Numerical examples

Computational cost

Conclusions

Acknowledgments

J. Sound Vib.

Comput. Struct.

J. Comput. Appl. Math.

Comput. Struct.

Comput. Methods Appl. Mech. Engrg.

Comput. Struct.

Comput. Struct.

Comput. Struct.

Comput. Methods Appl. Mech. Engrg.

Comput. Struct.

Reduction of stiffness and mass matrices

AIAA J.

Structural eigenvalue problems: elimination of unwanted variables

AIAA J.

A new iterative order reduction (IOR) method for eigensolutions of large structures

Internat. J. Numer. Methods Engrg.

Dynamic analysis of structural systems using component modes

AIAA J.

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AIAA J.

Vibration analysis of structures by component mode substitution

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