Multi-level Substructuring Research Articles

A load balancing algorithm for the parallel automated multilevel substructuring method

The objective of this paper is to present a load balancing algorithm for the parallel automated multilevel substructuring (PAMLS) method. In the PAMLS method, load balancing is highly dependent on the computation time for the transformation and back transformation procedures corresponding to substructures. To balance the workload among threads, the proposed algorithm consists of two types of granularity: coarse-grained and fine-grained parallel algorithms. According to the level of substructures, the coarse-grained parallel algorithm splits both the transformation and back transformation procedures and assigns them to threads. Through fine-grained parallelism, more threads are exploited for the transformation of each substructure compared to threads used in the original PAMLS method. Without repartitioning, the proposed algorithm significantly improves the efficiency of the PAMLS method.

A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures

Based on the fixed interface component mode synthesis, a multiple and multi-level substructure method for the modeling of complex structures is proposed in this paper. Firstly, the residual structure is selected according to the structural characteristics of the assembled complex structure. Secondly, according to the assembly relationship, the parts assembled with the residual structure are divided into a group of substructures, which are named the first-level substructure, the parts assembled with the first-level substructure are divided into a second-level substructure, and consequently the multi-level substructure model is established. Next, the substructures are dynamically condensed and assembled on the boundary of the residual structure. Finally, the substructure system matrix, which is replicated from the matrix of repeated physical geometry, is obtained by preserving the main modes and the constrained modes and the system matrix of the last level of the substructure is assembled to the upper level of the substructure, one level up, until it is assembled in the residual structure. In this paper, an assembly structure with three panels and a gear box is adopted to verify the method by simulation and a rotor is used to experimentally verify the method. The results show that the proposed multiple and multi-level substructure modeling method is not unique to the selection of residual structures, and different classification methods do not affect the calculation accuracy. The selection of 50% external nodes can further improve the analysis efficiency while ensuring the calculation accuracy.

Effect of mounting bracket structure and material on automotive air conditioning compressor

ABSTRACT This paper proposes an analysis of Mounting Bracket Structure and Material effects on fixed-displacement swash-plate-type compressor. To decrease the interior sound pressure of a vehicle, both finite-element analysis (FEA) and vibration experiment of the automotive air conditioning compressor with the mounting bracket were conducted, and the effect of the material and structure of the mounting bracket on the noise and vibration of the compressor was analysed. The finite-element model of the fixed-displacement swash-plate-type compressor with a mounting bracket is established. For the mounting bracket with different materials and structures, vibration acceleration level (VAL) of the finite-element model was analysed. Results show that the mounting bracket MB3, which is made of QT400-10, provides the best noise and vibration performance, which is also verified through interior sound pressure level experiment. Abbreviations: FEA: finite-element analysis; VAL: vibration acceleration level; FRA: frequency response analysis; EMA: experimental modal analysis; LTI: Linear Time-Invariant; FEM: finite-element model; AMS: automatic multi-level substructuring; DOF: degree of freedom

A Domain Decomposition Rayleigh--Ritz Algorithm for Symmetric Generalized Eigenvalue Problems

This paper proposes a parallel domain decomposition Rayleigh--Ritz projection scheme to compute a selected number of eigenvalues (and, optionally, associated eigenvectors) of large and sparse symmetric pencils. The projection subspace associated with interface variables is built by computing a few of the eigenvectors and associated leading derivatives of a zeroth-order approximation of the nonlinear matrix-valued interface operator. On the other hand, the projection subspace associated with interior variables is built independently in each subdomain by exploiting local eigenmodes and matrix resolvent approximations. The sought eigenpairs are then approximated by a Rayleigh--Ritz projection onto the subspace formed by the union of these two subspaces. Several theoretical and practical details are discussed, and upper bounds of the approximation errors are provided. Our numerical experiments demonstrate the efficiency of the proposed technique on sequential/distributed memory architectures as well as its competitiveness against schemes such as shift-and-invert Lanczos and automated multilevel substructuring combined with $p$-way vertex-based partitionings.

Improving the computational efficiency of the enhanced AMLS method

The objective of this study is to improve the computational efficiency of the enhanced automated multilevel substructuring (EAMLS) method (Kim et al., 2015) for dealing with large finite element models in structural dynamics. In the EAMLS method, the compensation procedure of the residual mode effect is a bottleneck resulting in a large computational cost. The original EAMLS method indeed cannot be used for finite element models with more than one million degrees of freedom in personal computers. An improvement in the computational efficiency is achieved by an interface subspace reduction and a new compensation procedure derived by residual flexibility matrices for only bottom-level substructures. Through this approach with submatrix level computations, the computation time and memory requirements are significantly reduced. Several numerical examples are presented to show the effectiveness of the proposed method.

Structural and Phase Transfromations in Superlocalization Band in Ni3Ge Single Crystal

The paper studies the high-temperature plastic deformation in Ni3Ge alloy single crystals localizing in a band a few tens of microns wide that reaches the shear strain of thousands of percent. The electron backscatter diffraction and synchrotron radiation techniques allow investigating the structure of the single crystal in the band of super-localized plastic deformation. It is shown that super-localization of plastic deformation occurs due to the formation of polycrystalline multi-level substructure in the initial single crystal. The grain size and grain misorientation bimodal distributions are determined in the polycrystalline substructure. Also, regions of amorphous state are observed in the super-localization band at high homologous temperature Т > 0.5Tm. A partial destruction of the far atomic order is detected.

Structure, residual stresses and mechanical properties of LaB6–TiB2 ceramic composites

Bulk and powdered LaB6-TiB2 composites obtained by the methods of the floating zone melting (FZM) and atomizing of heated ceramic material were investigated. Microscopic and X-ray diffraction methods were used to study and compare the phase and structural as-grown states and their changes after annealing. It is shown, that the different values of thermal expansion coefficients of the matrix and fibers form a complex stress state and a multilevel substructure in the matrix phase of the composites, which reduces their hardness and fracture toughness. In bulk composites, the structure at the macrolevel is disoriented by 1.5–2° disclinational boundaries into fragments, which in turn, contain less disoriented dislocation ensembles. In powdered composites, a structure of the matrix phase is less disoriented and is formed by subgrains with 0.5° boundaries and relatively uniformly distributed dislocations. The annealing at 1400–1600°С allows achieving an equilibrium structural state in bulk and powdered composites. An increase of the fracture toughness (1.6 times) and Vickers hardness (1.2 times), that happens due to a decrease in the dislocation density, is shown.

Automated static condensation method for local analysis of large finite element models

In this paper, we introduce an efficient new model reduction method, named the automated static condensation method, which is developed for the local analysis of large finite element models. The algebraic multilevel substructuring procedure is modified appropriately, and then applied to the original static condensation method. The retained substructure, which is the local finite element model to be analyzed, is defined, and then the remaining part of the global model is automatically partitioned into many omitted substructures in an algebraic perspective. For an efficient condensation procedure, a substructural tree diagram and substructural sets are established. Using these, the omitted substructures are sequentially condensed into the retained substructure to construct the reduced model. Using several large practical engineering problems, the performance of the proposed method is demonstrated in terms of its solution accuracy and computational efficiency, compared to the original static condensation method and the superelement technique.

An atomistic Green’s function method hybrid with a substructure technique for coherent phonon analysis

ABSTRACTBy virtue of substructure technique, the atomistic Green’s function method can be applied to efficiently evaluate the coherent phonon transport in a large device. Comprised of several substructures without inner atoms, the device can be described by a small system of algebraic equations, which can give a small but crucial submatrix of the retarded Green’s function of the open device without loss in accuracy. While avoiding solving the entire device problem, the method can basically give all the information to describe the quantum transport within the ballistic limit, such as the transmission function and local density of states. The validity and efficiency of the proposed method is demonstrated by the one- and three-dimensional cases. The multilevel substructure technique is also feasible.

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.

Band Structure Characteristics of Nacreous Composite Materials with Various Defects

Abstract Nacreous composite materials have excellent mechanical properties, such as high strength, high toughness, and wide phononic band gap. In order to research band structure characteristics of nacreous composite materials with various defects, supercell models with the Brick-and-Mortar microstructure are considered. An efficient multi-level substructure algorithm is employed to discuss the band structure. Furthermore, two common systems with point and line defects and varied material parameters are discussed. In addition, band structures concerning straight and deflected crack defects are calculated by changing the shear modulus of the mortar. Finally, the sensitivity of band structures to the random material distribution is presented by considering different volume ratios of the brick. The results reveal that the first band gap of a nacreous composite material is insensitive to defects under certain conditions. It will be of great value to the design and synthesis of new nacreous composite materials for better dynamic properties.

Multi-objective optimization of two-dimensional phoxonic crystals with multi-level substructure scheme

Phoxonic crystal (PXC) is a promising artificial periodic material for optomechanical systems and acousto-optical devices. The multi-objective topology optimization of dual phononic and photonic max relative bandgaps in a kind of two-dimensional (2D) PXC is investigated to find the regular pattern of topological configurations. In order to improve the efficiency, a multi-level substructure scheme is proposed to analyze phononic and photonic band structures, which is stable, efficient and less memory-consuming. The efficient and reliable numerical algorithm provides a powerful tool to optimize and design crystal devices. The results show that with the reduction of the relative phononic bandgap (PTBG), the central dielectric scatterer becomes smaller and the dielectric veins of cross-connections between different dielectric scatterers turn into the horizontal and vertical shape gradually. These characteristics can be of great value to the design and synthesis of new materials with different topological configurations for applications of the PXC.

Effect of damaged circular flange-bolted connections on behaviour of tall towers, modelled by multilevel substructuring

The paper deals with non-linear analysis of a telecommunication tower with circular flange-bolted connections (CFBCs). They are composed of two flanges, welded to the structural tubes, and then connected together with pre-tensioned bolts. A rigorous FEM analysis is performed for finding the connection stiffness in two cases. One deals with all bolts undamaged and the second one with one or more bolts broken.The analysis, which includes contact and friction forces, shows that when joints are under tension, the bolts are not only subjected to axial forces, but also to bending moments due the prying effect. The value of stresses caused by bending depends strongly on the bolt pre-tension and flange thickness. Removing one of the six connection bolts significantly increases stresses in the remaining bolts. Knowing the behaviour of the connection, it is possible to study the behaviour of the whole structure. This is achieved by applying the multilevel substructuring approach. The first levels is related to the flanges and bolts, whereby the connection model is simplified, and compared with the rigorous one, the second level is related to the assembly of the whole tower.The paper is illustrated with several examples of connections of different thicknesses, and different bolt pre-tensions. The considered tower comes from a real design.

An efficient Laplace transform-wave packet method hybrid with substructure technique

By virtue of substructure technique, the wave packet method can be applied to efficiently evaluate the thermal conduction of each phonon mode in the large periodic harmonic lattice. The substructures are connected by the inter-belts due to the long-range interatomic interaction. A parallel dynamic substructure method is presented in the Laplace transform domain, without any approximation in the condensation process. The validity and efficiency of the proposed method is demonstrated by the one- and three-dimensional cases. By launching a wide band wave packet, the whole given phonon branch can be studied by a one-time calculation. Defined as the ratio between square of Fourier transform of the transmitted and incident wave packets, the resulted transmission coefficient agrees with that obtained by the atomistic Green’s function method. Moreover the time evolution of the wave field can be given by the method, and the multilevel substructure technique can also be used.

Design of nacreous composite material for vibration isolation based on band gap manipulation

Inspired by natural nacreous materials with the excellent performance, two kinds of two-dimensional (2D) phononic crystals with the Brick-and-Mortar pattern are compared. To discuss the band structure, a multi-level substructure algorithm is employed which is efficient. Additionally, the influences on the band gap with different material and geometric parameters of the model are examined, and the range of the band gap is turned out to be adjustable by changing these parameters. Furthermore, a kind of 3D nacreous composite material is designed based on the thought of staggered and combined soft and hard materials. In the 3D band structure analysis, the designed material generates an ultrawide low frequency band gap, similar to the results of the 2D case. The further numerical tests for the transmission characteristics and random vibration response reveal the significant vibration attenuation effect of the nacreous material compared with the reference materials. The results of the tests fit remarkably well with the band gap. Thus the devices for vibration reduction and isolation with various working frequencies can be designed by means of manipulating the band gap in the low frequency regime.

Band structure and transmission characteristics of complex phononic crystals by multi-level substructure scheme

A fast scheme based on the multi-level substructure technique is proposed for the band structure and transmission characteristics calculation of phononic crystals uniformly. The main idea is that finite element models of phononic crystals are divided into several domains by a special multi-level decomposition. For the band structure calculation, the upscaling calculation is employed to condense the internal stiffness matrix of the unit cell into the Bloch boundary. Due to the internal stiffness matrix does not change along with reduced wave vectors in an iteration process, the scheme can reduce the computational scale and improve the efficiency greatly, meanwhile it does not introduce approximation into the traditional finite element model. For the transmission characteristics calculation, the unit cell of the phononic crystal is periodic which is taken as a substructure with the same coefficient matrix. Moreover, the downscaling calculation of internal displacements can be selected flexibly. Some closely watched examples of the three-dimensional locally resonant, defect state of Lamb wave and Bragg waveguide are analyzed. Numerical results indicate that the proposed scheme is efficient and accurate, which may widely be applicable and suitable for complex phononic crystal problems, and provides a reliable numerical tool to optimize and design crystal devices.

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.

An efficient transient analysis method for linear time-varying structures based on multi-level substructuring method

In this paper, the Newmark-beta method for linear time-varying (LTV) structures is firstly derived, then a transient analysis method for LTV structures based on multi-level substructuring method is presented. In theory the proposed method could achieve the same precision as the overall structure analysis method, the computational accuracy is not influenced by substructure partition strategy. The proposed method has higher computational efficiency than overall structure analysis method. Furthermore, the partial substructure tree traversal strategy can further improve the computational efficiency without any loss of accuracy. Some numerical examples are given to illustrate the effectiveness and feasibility of the proposed methods.

BAND STRUCTURE AND TRANSMISSION CHARACTERISTICS CALCULATION OF PHONONIC CRYSTALS BASED ON MULTI-LEVEL SUBSTRUCTURE TECHNIQUE

A fast scheme based on multi-level substructure technique is proposed to analyze band structure and transmission characteristics of phononic crystals. The main idea is that the finite element model of phononic crystal is divided into several substructures by a special multi-level decomposition. For the calculation of band structure, tree travelling technique and static condensation method are used to convert the internal stiffness matrix into the Bloch boundary of unit cell. Because the internal stiffness matrix does not change along with reduced wave vector, the scheme can reduce computation budget and improve efficiency greatly, while it does not introduce approximation into traditional finite element model. For the calculation of transmission characteristics, the proposed scheme can be also used to reduce computation, because unit cell of phononic crystal is periodic which has the same coefficient matrix. Moreover, the back substitution of internal displacements of unit cell can be selected flexibly. Some examples of three-dimensional locally resonant system and two-dimensional Bragg scattering phononic crystal are analyzed. Numerical results indicate that the proposed scheme is efficient and accurate, and suitable for complex phononic crystal problems.

Ultrawide low frequency band gap of phononic crystal in nacreous composite material

Abstract The band structure of a nacreous composite material is studied by two proposed models, where an ultrawide low frequency band gap is observed. The first model (tension-shear chain model) with two phases including brick and mortar is investigated to describe the wave propagation in the nacreous composite material, and the dispersion relation is calculated by transfer matrix method and Bloch theorem. The results show that the frequency ranges of the pass bands are quite narrow, because a special tension-shear chain motion in the nacreous composite material is formed by some very slow modes. Furthermore, the second model (two-dimensional finite element model) is presented to investigate its band gap by a multi-level substructure scheme. Our findings will be of great value to the design and synthesis of vibration isolation materials in a wide and low frequency range. Finally, the transmission characteristics are calculated to verify the results.