Substructuring Scheme Research Articles

Dynamic Condensation for Efficient Band-Structure Calculations of 2D Periodic Structures

Efficient band-structure calculations are essential for understanding the mechanical behaviors of periodic materials, with significant implications in material design and phononic engineering. This paper introduces the application of the improved reduced system (IRS) technique to expedite elastic band-structure calculations. The IRS, a dynamic condensation method, partitions the unit cell degrees of freedom (DOFs) into primary and secondary sets. A strategic selection of primary DOFs retains a subset of interior DOFs alongside all exterior DOFs while truncating the remaining interior DOFs. The integration of IRS with the Craig–Bampton method for additional reduction and the imposition of Bloch boundary conditions yields a notable decrease in computational overhead. Additionally, for structures with a high number of interior DOFs, a substructuring scheme can be implemented to further enhance efficiency. This approach offers a compelling combination of accuracy and expedited computation, making it applicable across diverse periodic materials.

Modified Iterative-Order-Reduction Substructuring Method with Interface Boundary Reduction

The iterative-order-reduction (IOR) substructuring method is an efficient model order reduction (MOR) technique for estimating eigenvalues and eigenvectors in large-scale structural systems. However, it can be computationally expensive when applied to models with highly refined meshes and a substantial number of substructures because it retains all the interface degrees of freedom (DOF), which increase the size of the reduced structural matrices. Moreover, the reduced structural matrices may suffer from ill-conditioning after a certain number of iterations, potentially yielding inaccurate results. This paper presents a modified IOR substructuring method that integrates interface boundary reduction to reduce the size of the original IOR substructuring model by decreasing the number of interface DOF. The performance of the proposed method is demonstrated through a comprehensive numerical comparison with the original IOR substructuring and other widely used MOR techniques, including the Craig–Bampton, enhanced Craig–Bampton, and iterated improved reduced system with a substructuring scheme. The results show that the proposed method achieves comparable accuracy while significantly reducing the computational time and memory usage.

A probabilistic study on impedances and kinematic response factors of square pile groups in homogeneous soils

The complexity of the Soil-Structure Interaction (SSI) problem is mainly due to difficulties related to the modelling of the soil-foundation behaviour, which is not only frequency-dependent, but is also largely influenced by uncertainties related to soil properties and stratigraphic conditions. According to a sub-structuring scheme for the analysis of SSI problems, a deterministic approach is usually adopted for the analysis of the soil-foundation system assuming properties and geotechnical models based on the expert judgment, despite the fact that uncertainties related to the intrinsic variability of soil parameters are widely recognised and confirmed by experimental campaigns and laboratory tests.This paper presents a probabilistic study on the dynamic behaviour of square pile group foundations in homogeneous soils, focusing on the effects of the uncertainties related to (i) the frequency-dependent impedance functions and (ii) the kinematic response factors. Above quantities are required to define compliant restraints for the modelling of the soil-foundation system in performing inertial soil-structure interaction analysis, and for the definition of the foundation input motion starting from the free-field motion. Square pile groups are considered, and uncertainties are described through probabilistic distributions of parameters governing the soil-foundation dynamic response. The probabilistic analyses are performed through a numerical model developed by the Authors and some results are presented to show and discuss the variability of the results. In addition, a sensitivity analysis is performed to assess the influence of each variable uncertainty on the system response. Overall, response quantities are found to be very sensitive to shear wave velocity, although soil density and pile elastic modulus may often play a significant role.

An improved iterative model reduction technique to estimate the unknown responses using limited available responses

A major challenge in structural health monitoring (SHM) is the availability of responses at limited degrees of freedom (DOFs). This requires the determination of the transformation parameter to expand the available DOFs to the full size of a model. The present study proposes an improved iterative model reduction algorithm by eliminating the stiffness terms from the transformation equation. The modified equation is a function of measured modal responses and mass matrices. This enables obtaining the unknown responses without repeated evaluations in case of stiffness reductions typically involved in SHM problems. The transformation parameter is solved using an improved Levenberg-Marquardt (LM) based iterative least-squares optimization technique. Specifically, the LM algorithm is enhanced by introducing an adaptive damping term in the least-squares problem. The proposed approach is further integrated into the substructuring scheme so that it can be readily applied for large finite element models. The algorithm is numerically demonstrated by considering a beam and a ten-storey building model using modal data with Gaussian noise. The effectiveness of the proposed reduced-order model is studied for a gradually decreasing number of modes and available responses for various measurement configurations by comparing with the results of the existing model reduction techniques.

An iterative state-space component modal synthesis approach for damped systems

Efficiently calculating the high precision modal parameters and/or reduce-order model of large-scale and/or complicated damped structures are important in engineering applications such as optimal design, model modification and updating tasks. In this paper, an iterative procedure based state-space component modal synthesis method is developed to address the problem. The Kron’s substructuring method is modified into a state-space form first to include damping effects and the defectiveness of components and global system. Then an iterative scheme is developed to seek all desired complex eigen-pairs simultaneously in one round of iterations without complex operations. The consistency of the proposed method is proved and the convergency is briefly discussed for integrity. A detailed implementation is given for reference. Compared with other methods for solving modal parameters and/or model order reduction, the proposed method has such merits as high computational efficiency and still in a substructuring scheme. Numerical examples show that the proposed method can provide highly accurate results with low computational cost.

Basic examination of two substructuring schemes for shake table tests

This study examines two basic substructuring schemes for shake table tests where the dynamics of the table is significantly affected by a specimen. The hybrid simulation (HS) scheme, commonly adopted in substructuring experiments, directly uses the output of a numerical substructure as an input signal to a physical substructure. The dynamical substructuring system (DSS) scheme, developed long after HS, uses feedforward and feedback controllers to minimise the control error produced by the outputs of numerical and physical substructures. Before examining these two schemes, this study introduces a systematic formulation for dividing a multi-degree-of-freedom emulate system into a numerical substructure and a physical substructure with a shake table. Then, controller designs for HS and DSS are discussed for the substructures. The two schemes with basic control approaches were numerically examined through substructuring tests for a linear 3DOF emulate system. DSS with stability was powerful even under a control condition with a pure time delay and inaccurate estimation of the table dynamics, whereas these factors degraded the HS performance. In additional simulations in which nonlinear characteristics were considered, the performance of DSS (HS) was degraded mainly by the nonlinear (inaccurate estimation of the table dynamics). It was found that the stability of substructures with nonlinear characteristics could be roughly assessed by the Nyquist stability criterion. These simulations with/without nonlinear characteristics clarified the performances of the HS and DSS schemes through basic control approaches and stability analysis under practical conditions.

Frequency Based Substructuring for Structure with Double Bolted Joints: A Case Study

Adopting dynamic substructuring schemes is a common practice in the field of structural dynamics, where the dynamic behaviour of a structure is predicted by combining the multiple subsystems that are analysed individually. This paper investigated the dynamic behaviour of a structure with doubled bolted joints using the frequency based substructuring (FBS) method. In the attempt, the substructures were combined with the frequency domain that can be numerically derived or experimentally measured. However, the applicability of this method suffered from several issues where most of them were related to the frequency response function (FRF) of the rotational degree of freedom (DOF) at the subsystem’s interface. In some cases, the system’s interface vibrated in the rotary motion of certain modes, for instance, a car side rear view mirror. Therefore, excluding the rotational FRF during the coupling process could lead to a completely different result. This paper presents the use of the FBS method for a structure with double bolted joints by using the equivalent multipoint constraint (EMPC) method through which rotational DOFs can be completely neglected. The actual tested structure for this study was an assembled structure consisting of two substructures: a simple beam and an irregular plate steel structure. The FRFs of both substructures were derived from using the FRF synthesis from finite element models and combined together via the FBS method. This study reveals that the use of the FBS method with the EMPC method for a structure double bolted joints has led to very promising results.

An interpolation-based parametric reduced order model combined with component mode synthesis

This paper presents an efficient interpolation-based parametric reduced order model with component mode synthesis. The off-line sampling of a large-scale structural dynamic system containing many parameters requires many computations to investigate the parameter-dependency of a dynamical system. Therefore, we introduce a dynamic substructuring scheme to execute off-line sampling in the subdomain level. In addition, we suggest discriminating the interpolation of the subsystem depending on the characteristics of each substructural mode. We synthesize the substructures, and we then construct a two-level, semi-parametrized ROM by reducing the degrees of freedom of the interface in the on-line stage. We then demonstrate the efficiency and accuracy of the present method by computing the relative eigenvalue errors and the transient responses of the system with random values for the parameters, and we conduct the design optimization of large-scale systems under dynamic loading conditions. A comparison of the present method with the full order model and a conventional reduced order model indicates that the present method is useful in carrying out an efficient analysis and design optimization for various large-scale structural dynamic systems.

Substructuring tools for probabilistic analysis of instrumented nonlinear moving oscillator–beam systems

The study considers the application of finite element modeling, combined with numerical solutions of governing stochastic differential equations, to analyze instrumented nonlinear moving vehicle–structure systems. The focus of the study is on achieving computational efficiency by deploying, within a single modeling framework, three substructuring schemes with different methodological moorings. The schemes considered include spatial substructuring schemes (involving free-interface coupling methods), a spatial mesh partitioning scheme for governing stochastic differential equations (involving the use of a predictor corrector method with implicit integration schemes for linear regions and explicit schemes for local nonlinear regions), and application of the Rao–Blackwellization scheme (which permits the use of Kalman's filtering for linear substructures and Monte Carlo filters for nonlinear substructures). The main effort in this work is expended on combining these schemes with provisions for interfacing of the substructures by taking into account the relative motion of the vehicle and the supporting structure. The problem is formulated with reference to an archetypal beam and multi-degrees of freedom moving oscillator with spatially localized nonlinear characteristics. The study takes into account imperfections in mathematical modeling, guide way unevenness, and measurement noise. The numerical results demonstrate notable reduction in computational effort achieved on account of introduction of the substructuring schemes.

An improved fast multipole method for electrostatic potential calculations in a class of coarse-grained molecular simulations

This paper presents a novel algorithm to approximate the long-range electrostatic potential field in the Cartesian coordinates applicable to 3D coarse-grained simulations of biopolymers. In such models, coarse-grained clusters are formed via treating groups of atoms as rigid and/or flexible bodies connected together via kinematic joints. Therefore, multibody dynamic techniques are used to form and solve the equations of motion of such coarse-grained systems. In this article, the approximations for the potential fields due to the interaction between a highly negatively/positively charged pseudo-atom and charged particles, as well as the interaction between clusters of charged particles, are presented. These approximations are expressed in terms of physical and geometrical properties of the bodies such as the entire charge, the location of the center of charge, and the pseudo-inertia tensor about the center of charge of the clusters. Further, a novel substructuring scheme is introduced to implement the presented far-field potential evaluations in a binary tree framework as opposed to the existing quadtree and octree strategies of implementing fast multipole method. Using the presented Lagrangian grids, the electrostatic potential is recursively calculated via sweeping two passes: assembly and disassembly. In the assembly pass, adjacent charged bodies are combined together to form new clusters. Then, the potential field of each cluster due to its interaction with faraway resulting clusters is recursively calculated in the disassembly pass. The method is highly compatible with multibody dynamic schemes to model coarse-grained biopolymers. Since the proposed method takes advantage of constant physical and geometrical properties of rigid clusters, improvement in the overall computational cost is observed comparing to the tradition application of fast multipole method.

Nonlinear dynamic state estimation in instrumented structures with conditionally linear Gaussian substructures

Many problems of state estimation in structural dynamics permit a partitioning of system states into nonlinear and conditionally linear substructures. This enables a part of the problem to be solved exactly, using the Kalman filter, and the remainder using Monte Carlo simulations. The present study develops an algorithm that combines sequential importance sampling based particle filtering with Kalman filtering to a fairly general form of process equations and demonstrates the application of a substructuring scheme to problems of hidden state estimation in structures with local nonlinearities, response sensitivity model updating in nonlinear systems, and characterization of residual displacements in instrumented inelastic structures. The paper also theoretically demonstrates that the sampling variance associated with the substructuring scheme used does not exceed the sampling variance corresponding to the Monte Carlo filtering without substructuring.

The transient and frequency response analysis using the multi-level system condensation in the large-scaled structural dynamic problem

In large-scale problem, a huge size of computational resources is needed for a reliable solution which represents the detailed description of dynamic behavior. Recently, eigenvalue reduction schemes have been considered as important technique to resolve computational resource problems. In addition, the efforts to advance an efficiency of reduction scheme leads to the development of the multi-level system condensation (MLSC) which is initially based on the two-level condensation scheme (TLCS). This scheme was proposed for approximating the lower eigenmodes which represent the global behavior of the structures through the element-level energy estimation. The MLSC combines the multi-level sub-structuring scheme with the previous TLCS for enhancement of efficiency which is related to computer memory and computing time. The present study focuses on the implementation of the MLSC on the direct time response analysis and the frequency response analysis of structural dynamic problems. For the transient time response analysis, the MLSC is combined with the Newmark's time integration scheme. Numerical examples demonstrate the efficiency of the proposed method.

Experimental investigation of the behavior of RC bridge piers subjected to horizontal and vertical earthquake motion

This paper presents an experimental program aimed at assessing the effect of vertical earthquake ground motion on reinforced concrete bridge piers. The program comprised testing of four large scale piers by utilizing the Multi-Axial Full-Scale Sub-Structured Testing and Simulation (MUST-SIM) facility at the University of Illinois. Two hybrid analytical-experimental simulations were used to experimentally investigate the effects of vertical ground motion on pier loading and behavior. Two additional cyclic tests were conducted to allow direct comparison of pier response under tension to pier response under compression. Detail aspects of the experimental framework including a substructuring scheme, analytical representation of structure, test specimen design and advanced measurement system are provided. The observations and interpretation of experimental results, including the impact on both global and local behavior, are discussed. It is concluded that vertical ground motion can significantly impact Reinforced Concrete (RC) pier behavior and failure mode.

Iterative-Order-Reduction Substructuring Method for Dynamic Condensation of Finite Element Models

animprovediterativeorder-reductionmethodbasedonthestate-spacemodelisproposedfornonclassicallydamped systems by fully taking into account the effect of damping on the transformation matrix. By combining the original and improved iterative order-reduction schemes with the substructuring scheme, a new iterative order-reduction substructuring method is also proposed to overcome the problem of requiring more computational resources to constructthetransformationmatrixoflarge-scale finiteelementmodels.Theproposedmethodmaybeappliedtothe reduction problems of large-scale undamped and damped finite element models with limited computer storage and high computational efficiency. Two numerical examples are provided to demonstrate the effectiveness of the proposed method.

A three-dimensional Wiener-Hopf technique for general bodies of revolution, Part II: Substructuring and implementation.

This paper begins with a brief review of Part I’s main ideas and conclusions regarding: (a) Operator-induced symmetry and reciprocity leading to compatibility of the halfplanes of analyticity between the integral equation’s non-translational kernel and those of the Fourier-transformed displacements and pressures on the surface of a scattering or vibrating body of revolution; and (b) subsequent spectral factorizations in juxtaposition to those of the classical Wiener–Hopf case for the planar geometry. The spatial and spectral domains in question refer to the arclength variable along the body’s generator and to its transform wavenumber, respectively. The analysis described in this Part-II paper brings out some of the additional positive consequences of the pre-symmetrization of the integral equation that governs the response of the coupled fluid and structure. It then applies standard substructuring schemes to extend Part I’s formulation to include departures from axisymmetry resulting from generic appendages as discontinuities along the shape’s circumferential coordinate. The presentation ends with sample applications of its new 3-D Wiener–Hopf technique to canonical shapes of revolution, and with a summary of the physical insights made possible by the new approach relative to numerical results by boundary elements.

Iterative method for dynamic condensation combined with substructuring scheme

An iterated improved reduced system (IIRS) procedure combined with substructuring scheme for both undamped and nonclassically damped structures is presented. Iterated IIRS method is an efficient reduction technique because the highly accurate eigenproperties from the repeatedly updated condensed matrices can be obtained without consuming expensive computational cost. However, single domain direct approach of this method to large structures requires much computational resources and even makes analysis intractable in the case only limited computer storage is available. These problems can be overcome by combining the substructuring scheme with IIRS procedure. The newly developed IIRS method combined with a substructuring scheme can provide an efficient methodology for large-scale eigenvalue problems. The validation of the present method and the evaluation of computational efficiency are demonstrated through the numerical examples.

Improvement of substructuring reduction technique for large eigenproblems using an efficient dynamic condensation method

An accelerated substructuring reduction procedure for the iterated improved reduced system (IIRS) method is proposed. The iterated IIRS method can be combined with a substructuring scheme to provide an efficient methodology for large-scale eigenvalue problems. Not only can it reduce eigenvalue analysis errors through successive iterations, but the accuracy of the eigenanalysis is not sensitive to the selected master degrees of freedom. In practical structural eigenproblems, reducing the number of iterations can save a great deal of computation cost. The present substructuring technique modifies the iterative form of the transformation matrices in each substructure to achieve faster convergence. Applications of the present method to two numerical examples demonstrate that the proposed method can obtain lower eigensolutions of structures more accurately and efficiently, as compared with those of the current substructuring technique.

부구조화 기법을 연동한 반복적인 동적 축소법 (II) - 비비례 감쇠 구조 시스템 -

An iterated improved reduced system (IIRS) procedure combined with sub-structuring scheme for nonclassically damped structural systems is presented. For dynamic analysis of such systems, complex eigenproperties are required to incorporate properly the nonclassical damping effect. In complex structural systems, the equations of motion are written in the state space from. Thus, the number of degrees of freedom of the new equations of motion and the size of the associated eigenvalue problem required to obtain the complex eigenvalues and eigenvectors are doubled. Iterated IRS method is an efficient reduction technique because the eigenproperties obtained in each iteration step improve the condensation matrix in the next iteration step. However, although this reduction technique reduces the size of problem drastically, it is not efficient to apply this technique to a single domain finite element model with degrees of freedom over several thousands. Therefore, for a practical application of the reduction method, accompanying sub-structuring scheme is necessary. In the present study, iterated IRS method combined with sub-structuring scheme for nonclssically damped structures is developed. Numerical examples demonstrate the convergence and the efficiency of a newly developed scheme.

부구조화 기법을 연동한 반복적인 동적 축소법 (I) - 비감쇠 구조 시스템 -

부구조화 기법을 연동한 반복적인 동적 축소법 (I) - 비감쇠 구조 시스템 -

Zeroth-order coding of complex amplitude in two dimensions

Diffractive elements that modulate both the amplitude and the phase of an optical field are designed with use of an iterative Fourier-transform algorithm with gradual amplitude clipping at the plane of the element. Unlike phase-only diffractive elements, these complex-amplitude modulating elements do not generate noise outside the signal window W, and their diffraction efficiencies are comparable to the efficiencies of phase-only elements with a noiseless frame around W. Encoding techniques of the complex amplitude into variations of the diffraction efficiency and phase of the zeroth order of a carrier grating are introduced: the shape and the depth of a substructure within each rectangular pixel are modulated. Approximate and rigorous diffraction calculations are performed to compare different substructuring schemes, which all deflect the noise far away from W but differ, for example, in the manner in which this noise is distributed in the signal plane.