Automated Multi-level Substructuring Research Articles

Error estimation for the automated multi‐level substructuring method

SummaryIn this study, we propose an effective method to estimate the reliability of finite element models reduced by the automated multi‐level substructuring (AMLS) method. The proposed error estimation method can accurately predict relative eigenvalue errors in reduced finite element models. A new, enhanced transformation matrix for the AMLS method is derived from the original transformation matrix by properly considering the contribution of residual substructural modes. The enhanced transformation matrix is an important prerequisite to develop the error estimation method. Adopting the basic concept of the error estimation method recently developed for the Craig–Bampton method, an error estimation method is developed for the AMLS method. Through various numerical examples, we demonstrate the accuracy of the proposed error estimation method and explore its computational efficiency. Copyright © 2015 John Wiley & Sons, Ltd.

An enhanced AMLS method and its performance

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.

Improving eigenpairs of automated multilevel substructuring with subspace iterations

This paper improves the eigenpair approximations obtained from the automated multilevel substructuring (AMLS) method by subspace iterations. Two variants of AMLS hybrid Subspace Iteration Method (AMLS-SIMa and AMLS-SIMb) are proposed. AMLS-SIMa is a derivative of the basic subspace iteration by utilizing the AMLS approximations as initial vectors. AMLS-SIMb further takes advantage of the AMLS transformed block diagonal stiffness matrix to avoid factorization of the original stiffness matrix. Numerical experiments show that: (a) the error of AMLS approximate eigenpairs can be significantly reduced with just a few iteration steps; (b) AMLS-SIMb is more efficient than AMLS-SIMa with less execution time.

Automated multi-level sub-structuring for fluid-solid interaction problems

The Automated Multi-Level Sub-structuring (AMLS) method is a powerful technique to determine a large number of eigenpairs with moderate accuracy of huge symmetric and definite eigenvalue problems in structural analysis. This paper is concerned with an adapted version of AMLS for eigenfrequency analysis of fluid-solid interaction systems. Although fluid-solid vibrations are governed by an unsymmetric eigenproblem the modified AMLS method needs approximately the same computational effort. An error bound related to the eigenvalue approximations is proved.

Automated multi-level substructuring (AMLS) for electromagnetics

We present an algebraic setting of an automated multi-level substructuring (AMLS) method for electromagnetic Maxwell eigenproblem solution. To our knowledge it is novel, and it involves a non-trivial generalisation of the corresponding method for elastodynamics and acoustics. In comparison to the AMLS method for elliptic problems the electromagnetic stiffness matrix is singular, it has a large kernel of gradients of H 1 -conforming shape functions. This difficulty was overcome by confining the method to solenoidal fields only. The goal of AMLS is to achieve a high level of dimensional reduction inexpensively, using multi-level modal substructuring, enabling reliable solution of huge response problems, without compromising the accuracy of the quantity of interest. As proof of concept we present simple numerical experiments on rectangular domains verifying a correct implementation of a truly multi-level version of the generalised algorithm. For real life applications the algorithm need to be equipped with, automated nested mesh dissection and an algorithm for automatic subdomain eigenspace truncation. Finally, the algorithm needs to be tuned and parallelised.

An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems

We describe an efficient implementation and present a performance study of an automated multi-level substructuring (AMLS) method for sparse eigenvalue problems. We assess the time and memory requirements associated with the key steps of the algorithm, and compare it with the shift-and-invert Lanczos algorithm. Our eigenvalue problems come from two very different application areas: accelerator cavity design and normal-mode vibrational analysis of polyethylene particles. We show that the AMLS method, when implemented carefully, outperforms the traditional method in broad application areas when large numbers of eigenvalues are sought, with relatively low accuracy.

Superelement, Component Mode Synthesis, and Automated Multilevel Substructuring for Rapid Vehicle Development

Superelement, Component Mode Synthesis, and Automated Multilevel Substructuring for Rapid Vehicle Development

High‐frequency response analysis via algebraic substructuring

AbstractHigh‐frequency response analysis (Hi‐FRA) of large‐scale dynamical systems is critical to predict the resonant behavior of modern micro‐devices and systems operated over MHz or GHz frequency range. Algebraic substructuring (AS) is a powerful technique to extract a large number of natural frequencies. In this work, we extend the AS technique for FRA between two specified cutoff frequencies ωmin and ωmax. The technique is referred to as ASFRA. ASFRA can be efficiently applied to Hi‐FRA, as demonstrated by two examples of microelectromechanical sensors operated at 1–2 and 200–250 MHz ranges. To some extent ASFRA generalizes the underlying numerical algorithm and functionality of commercially viable automated multi‐level substructuring (AMLS) technique. AMLS is designed for FRA up to a specific frequency ωmax, starting from the lowest, and is inefficient for Hi‐FRA. Copyright © 2008 John Wiley & Sons, Ltd.

Reducing Huge Gyroscopic Eigenproblems by Automated Multi-level Substructuring

Simulating numerically the sound radiation of a rolling tire requires the solution of a very large and sparse gyroscopic eigenvalue problem. Taking advantage of the automated multi-level substructuring (AMLS) method it can be projected to a much smaller gyroscopic problem, the solution of which however is still quite costly since the eigenmodes are non-real and complex arithmetic is necessary. This paper discusses the application of AMLS to huge gyroscopic problems and the numerical solution of the AMLS reduction. A numerical example demonstrates the efficiency of AMLS.

An A Priori Bound for Automated Multilevel Substructuring

The automated multilevel substructuring (AMLS) method has been developed to reduce the computational demands of frequency response analysis and has recently been proposed as an alternative to iterative projection methods like those of Lanczos or Jacobi--Davidson for computing a large number of eigenvalues for matrices of very large dimension. Based on Schur complements and modal approximations of submatrices on several levels, AMLS constructs a projected eigenproblem which yields good approximations of eigenvalues at the lower end of the spectrum. Rewriting the original problem as a rational eigenproblem of the same dimension as the projected problem and taking advantage of a minmax characterization for the rational eigenproblem, we derive an a priori bound for the AMLS approximation of eigenvalues.

An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics

We present an automated multilevel substructuring (AMLS) method for eigenvalue computations in linear elastodynamics in a variational and algebraic setting. AMLS first recursively partitions the domain of the PDE into a hierarchy of subdomains. Then AMLS recursively generates a subspace for approximating the eigenvectors associated with the smallest eigenvalues by computing partial eigensolutions associated with the subdomains and the interfaces between them. We remark that although we present AMLS for linear elastodynamics, our formulation is abstract and applies to generic H1 -elliptic bilinear forms. In the variational formulation, we define an interface mass operator that is consistent with the treatment of elastic properties by the familiar Steklov--Poincaré operator. With this interface mass operator, all of the subdomain and interface eigenvalue problems in AMLS become orthogonal projections of the global eigenvalue problem onto a hierarchy of subspaces. Convergence of AMLS is determined in the continuous setting by the truncation of these eigenspaces, independent of other discretization schemes. The goal of AMLS, in the algebraic setting, is to achieve a high level of dimensional reduction, locally and inexpensively, while balancing the errors associated with truncation and the finite element discretization. This is accomplished by matching the mesh-independent AMLS truncation error with the finite element discretization error. Our report ends with numerical experiments that demonstrate the effectiveness of AMLS on a model problem and an industrial problem.

Efficient Broadband Vibro-Acoustic Analysis of Passenger Car Bodies Using an FE-Based Component Mode Synthesis Approach

The costliest part of computationally optimizing automobiles for Noise, Vibration and Harshness (NVH) is the frequency response calculation of very large finite element models. In general, NVH is treated as a fluid-structure interaction between the body and the car interior fluid volume. In this paper, some features of the current approach to NVH computations at BMW are elucidated. First, the equations of motion for a typical NVH problem will be derived using three different formulations, differential, algebraic and modal. Next, three numerical approaches for solving the large systems of equations (direct, modal and Automated Multi-Level Substructuring (AMLS)) will be shown. Finally, the numerical precision and the efficiency of these approaches will be assessed.