Dynamic Condensation Method 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.

Improvement to dynamic condensation of structural systems using an innovative virtual model reduction

Model reduction methods have primarily been used in the field of the structural dynamic analysis to reduce the computational time. However, it has progressively been extending to other engineering practices like the damage detection to extract the structural response by means of limited monitored degrees-of-freedom. This paper aims to improve the iterative dynamic condensation method, as a convenient method of the model reduction, for better assessment of the structural dynamic properties. The improvement consists of: (1) Capturing different modal frequencies by applying an iterative perturbation near the frequency obtained from the static condensation. The perturbation is evaluated in a manner that there is no need to solve eigenvalue problem successively; (2) Developing virtual dynamic condensation (VDC) procedure to generalize unique reduced stiffness and mass matrices, reflecting the influence of all modes of vibration, needed for a complete dynamic analysis. Finally, the accuracy of the proposed VDC method was proved by solving three examples.

The Effect of Nonlinear Sloshing Response of Water on Seismic Behavior of Reinforced Concrete Elevated Water Tanks

In this study, the fluid-structure interactive behavior by using the Westergaard approach and the Smoothed Particle Hydrodynamics (SPH) method of a 1000 m3 reinforced concrete (RC) elevated water tank under different seismic activities i.e., Kocaeli, Van, Kahramanmaraş and Kobe earthquakes was investigated. In the SPH method, the resulting hydrodynamic pressures were applied to the interior of the wall of the structure as an external force which is made as mass additions to the tank walls in Westergaard approach. With the help of software developed, the load carrying system of the elevated water tank was modeled using the MATLAB Partial Differential Equation Toolbox (PDE) software, based on the finite element method, and its verification was made by comparing with ANSYS software model. Dynamic condensation method was used to perform time history analyses for the cited earthquakes. The time-dependent solutions of partial differential equations are solved with the help of ODE45 functions based on Runge-Kutta method, by arranging the equation of motion in state-space format. The cited tank was analyzed linearly under different seismic activities for empty, half-filled (50%) and full-filled (100%) cases and the obtained results were compared with each other. It has been concluded that considering nonlinear behavior of the fluid during the earthquake with the SPH method provided consistency with the real behavior of RC elevated water tanks and gave more realistic results than the traditional Westergaard approach.

Review on the Theories and Applications of Dynamic Condensation and Component Mode Synthesis Methods in Solving FEM-based Structural Dynamics

Review on the Theories and Applications of Dynamic Condensation and Component Mode Synthesis Methods in Solving FEM-based Structural Dynamics

Numerical modeling of natural vibrations of coaxial shells partially filled with fluid, taking into account the effects on the free surface

This work is devoted to the numerical analysis of vertical elastic coaxial cylindrical shells completely or partially filled with a quiescent compressible fluid with account for sloshing of its free surface. The problem is solved in the axisymmetric formulation using a semi-analytical version of the finite element method. The fluid medium is described by the wave equation, which together with the conditions prescribed at its boundaries is reduced to the weak form by the Bubnov - Galerkin method. A mathematical formulation of the dynamic problem for thin-walled structures is developed using the variational principle of virtual displacements and linear theory of thin shells based on Kirchhoff - Love hypotheses. The fluid pressure on the walls of the structure is calculated according to Bernoulli's equation. Sloshing modes caused by gravitational effects on free surface of liquid medium are excluded from the resulting system of equations through the use of the iteration dynamic condensation method. The numerical model has been verified by comparing it with the known data for the case of a single shell partially filled with fluid. The influence of the fluid level on the lowest natural frequencies of the system vibrations at different variants of kinematic boundary conditions for shells (rigid clamping at both edges, cantilever support) and different widths of the annular gap between them has been evaluated. It has been found that for the examined configurations, the level of the fluid in the annular channel has a stronger influence on the frequency spectrum compared to the level of fluid in the cavity of the inner shell due to the fact that vibration frequencies change in a wider range.

On the Forced Vibration of Bending-Torsional-Warping Coupled Thin-Walled Beams Carrying Arbitrary Number of 3-DoF Spring-Damper-Mass Subsystems

This paper presents an analytical transfer matrix modeling framework for the forced vibration of a bending-torsional-warping coupling Euler-Bernoulli thin-walled beam carrying an arbitrary number of three degree-of-freedom (DOF) spring-damper-mass (SDM) subsystems. The thin-walled beam is divided into a series of distinct sub-beams whose ends are connected to the SDM subsystems. The transfer matrix for each sub-beam is developed based on the exact shape functions of the bending-torsional-warping coupling Euler-Bernoulli theory. Each SDM system is modelled by a set of effective springs based on the dynamic condensation method. The governing matrix equation is formulated based on the compatibility conditions of the placement and the force at the common interfaces of two adjacent sub-beams. Then, a closed-form expression for the frequency response function of the thin-walled beam system is proposed. The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method, which shows the accuracy and reliability of the proposed method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied. The presented method can simultaneously consider arbitrary number of SDM subsystems and boundary conditions. Furthermore, none of the associated matrices are larger than 12 × 12, which provides a significant computational advantage.

3D-based equivalent model of SMART control rod drive mechanism using dynamic condensation method

The SMART (System-integrated Modular Advanced ReacTor) is an integral-type small modular reactor developed by KAERI (Korea Atomic Energy Research Institute). This paper discusses the feasibility and applicability of a 3D-based equivalent model using dynamic condensation method for seismic analysis of a SMART control rod drive mechanism. The equivalent model is utilized for complicated seismic analysis during the design of the SMART. While the 1D-based beam–mass equivalent model is widely used in the nuclear industry for its calculation efficiency, the 3D-based equivalent model is suggested for the seismic analysis of SMART to enhance the analysis accuracy of the 1D-based equivalent model while maintaining its analysis efficiency. To verify the suggested model, acceleration response spectra from seismic analysis based on the 3D-based equivalent model are compared to those from the 1D-based beam–mass equivalent model and experiments. The accuracy and efficiency of the dynamic condensation method are investigated by comparison to analysis results based on the conventional modeling methodology used for seismic analysis.

Substructure Damage Identification Based on Model Updating of Frequency Response Function

Model updating of large-scale structures is difficult to carry out when using a frequency response function (FRF) for damage identification, as the solutions for the global system matrices with too many degrees of freedom are required in each iteration. In this paper, a substructure damage identification method is proposed based on the model updating of the acceleration FRF. The original finite element model is divided into several substructures using the improved reduced system (IRS) by the dynamic condensation method, resulting in the simplified substructure model. The final simplified model is composed of the simplified mass matrix and stiffness matrix of the substructure considered. The damage acceleration FRF to be identified is used to iteratively update the simplified model. The locations and extents of the damage elements are obtained by updating the results, which reduces the number of uncertain parameters to be updated and leads to the rapid convergence of the optimization process. In the iteration, L1 norm regularization is introduced to solve the ill-posed problem, which improves the stability of the identification results. A numerical simulation of a six-story steel frame structure under various working conditions was carried out to verify the effectiveness of the proposed method, which was also validated by the experiments. The robustness and performance of the proposed damage identification method based on substructures have been demonstrated.

A comparative study of different dynamic condensation techniques applied to multi-damage identification of FGM and FG-CNTRC plates

The dynamic condensation method has been recognized as an effective alternative for structural damage identification using spatially-incomplete modal measurements. However, comparative studies of different dynamic condensation techniques applied to the subject of structural damage identification have been scarcely found, especially for composite structures. In this regard, we conduct a comparative study of six typical dynamic condensation techniques utilized for addressing damage identification problems of composite plates made of functionally graded materials (FGM) and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials. Firstly, the six techniques consisting of Guyan’s method, Kidder’s method, Neumann series expansion-based second-order model reduction (NSEMR-II) method, improved reduced system (IRS) method, iterated IRS (IIRS) method, and iterative order reduction (IOR) method are reviewed. Then, their performance for reduced Eigen and optimization-damage identification problems are evaluated by studying two numerical examples of FGM plate and FG-CNTRC plate. For solving the optimization-damage identification problem of plate structures, the article proposes to use a hybrid global–local algorithm, Manta Ray Foraging Optimization—Sequential Quadratic Programming (MRFO-SQP), where the MRFO algorithm is utilized for global exploration and the SQP algorithm is used for the local searching process. The comparative study indicates that the IOR technique is the best dynamic condensation technique and is effective for addressing the structural damage identification problems when comparing with the other five techniques. It is also found that the damage identification approach based on the hybrid MRFO–SQP algorithm combined with the IOR technique can archive the high accuracy and low computational cost for damage localization and quantification.

Simulation of heavy-duty engine based on flexible multi-body dynamics

For heavy-duty diesel engine, the structural flexibility of the crankshaft and the engine block has an important influence on the dynamic characteristics of the engine. The deformation and modal resonance generated at work must be considered. Therefore, the multi-body dynamic analysis of the crankshaft and the body structure is based on the flexible body model of the crankshaft and the body. In this paper diesel engine dynamics behaviour is analysed based on flexible multi-body method. The dynamic substructure condensation method is used to reduce the he shaft and the bearing block, obtain the flexible body model of the crankshaft and the body, reduce the computational solution scale, and improve the computational efficiency.

A dynamic condensation method with free interface substructuring

Abstract In this paper, a novel dynamic condensation method is proposed with free interface substructuring, in which the substructures are completely decoupled. The substructures are independently defined and the substructural finite element (FE) models are individually reduced by adopting the improved reduced system (IRS) method. The reduced mass and stiffness matrices of substructures are assembled using a Lagrange multiplier vector, leading to the final reduced model. Using the proposed method, each of the substructural FE models can be reduced in parallel without coupling among the neighboring substructures. Thus, the method can be applied with ease to substructures connected through non-matching meshes. Moreover, complicated substructural interfaces can be accommodated when employing this method. The formulation of the proposed method is presented in detail, and its accuracy, computational efficiency, and modeling ability are investigated using several demonstrative engineering problems.

The mathematical modeling of the incomplete algebraic eigenvector and eigenvalue problem for obtaining the reduced frequency equation and its solution

This Paper studies the application of the Sequential Frequency–Dynamic Condensation Method developed by the authors for reducing the frequency equations obtained on the basis of the Finite Element Method (FEM) in the form of a Classical Mixed Method

Simulation of EHD lubrication for heavy-duty journal bearing based on flexible multi-body dynamics

For heavy-duty journal bearing, the structural flexibility of the shaft and the bearing block has an important influence on the dynamic characteristics of bearing behaviour. Therefore, the deformation and modal resonance generated at work must be considered. In this paper a computational method for solving Elastohydrodynamic lubrication based on flexible multi-body dynamics is introduced. The dynamic substructure condensation method is used to reduce the he shaft and the bearing block, obtain the flexible body model of the crankshaft and the body, reduce the computational solution scale, and improve the computational efficiency.

An order reduction method for single-walled carbon nanotubes with multi-vacancy defects

Presented here is a novel method of reducing the computational cost of analyzing the vibrational properties of a single-walled carbon nanotube (CNT) containing multi-vacancy defects (MVDs), while maintaining a high degree of accuracy. This reduced-order model method is described as modal domain reduction (MDR). In the MDR method, a linearized model of a perturbed CNT is first transformed into modal coordinates derived from the pristine system, taking into account the differences in the numbers of degrees of freedom for the pristine CNT and the CNT with MVDs. Next, the system is divided into primary modes, on the basis of the frequency range of interest, and remaining or secondary modes. The impact of secondary modes on primary modes is captured by the dynamic condensation method. The accuracy of MDR is demonstrated on linearized models of nanotubes of differing diameter and chirality and with different sized MVDs. The natural frequencies and normal modes of these reduced-order models are calculated and compared to the results for the full-order systems.

Dynamic condensation of non-classically damped structures using the method of Maclaurin expansion of the frequency response function in Laplace domain

As the number of degrees of freedom (DOFs) in structural dynamic problems becomes larger, the analyzing complexity and CPU usage of computers increase drastically. Condensation (or reduction) method is an efficient technique to reduce the size of the full model or the dimension of the structural matrices by eliminating the unimportant DOFs. After the first presentation of condensation method by Guyan in 1965 for undamped structures, which ignores the dynamic effects of the mass term, various forms of dynamic condensation methods were presented to overcome this issue. Moreover, researchers have tried to expand the dynamic condensation method to non-classically damped structures. Dynamic reduction of such systems is far more complicated than undamped systems. The proposed non-iterative method in this paper is introduced as 'Maclaurin Expansion of the frequency response function in Laplace Domain' (MELD) applied for dynamic reduction of non-classically damped structures. The present approach is implemented in four numerical examples of 2D bending-shear-axial frames with various numbers of stories and spans and also a floating raft isolation system. The results of natural frequencies and dynamic responses of models are compared with each other before and after the dynamic reduction. It is shown that the result accuracy has acceptable convergence in both cases. In addition, it is indicated that the result of the proposed method is more accurate than the results of some other existing condensation methods.

An iterative algebraic dynamic condensation method and its performance

A novel iterative reduced-order modeling method is proposed, which is based on the recently developed algebraic dynamic condensation method. The algebraic substructuring technique is employed to improve the reduction efficiency, and the initial reduced model is calculated using the substructural stiffness condensation and the interface boundary reduction procedures. Then, the initial reduced model is iteratively updated using the iterative substructural inertial effect condensation procedure until the solutions converge. The iterative formulation of the reduced model is represented simply with small submatrix operations to avoid huge computational cost induced by the iterative procedure resulting from the very large global transformation matrix. To verify the performance of the proposed method, we consider several large structural problems, and compare the numerical results to those of the iterated improved reduced system (IIRS) method, a widely used reduced-order modeling method.

A hierarchical non-iterative extension of the Guyan condensation method for damped structures

Reduction methods are commonly employed for solving the eigenvalue problems of systems with a large number of degrees of freedom. These methods are based upon dividing the system׳s degrees of freedom into masters and slaves, and obtaining a reduced system which is in terms of the masters only. Since 1965 when the Guyan condensation method for undamped structures was presented, which neglects the dynamic effects of the slaves entirely, there have been many efforts to overcome this by proposing various forms of dynamic condensation methods. These methods take into account the dynamics of the slaves through an iterative procedure. In this paper, a hierarchical, non-iterative reduction method has been proposed for damped dynamic systems. The method results in explicit forms of the effective stiffness, viscosity and mass, and also introduces higher order properties when third and higher order approximations are used. Furthermore, a procedure for the automatic selection of master degrees of freedom has been proposed which assures the convergence and increases the efficiency of the method. Application of the method for obtaining low-frequency eigenvalues of two example structures, with and without damping, reveals that results with good accuracy are obtained by using higher order approximations, as they consider the dynamics of the slaves properly.

A Simplified Numerical Method for Blast Induced Structural Response Analysis

Efficiently and accurately predicting structural dynamic response and damage to external blast loading is a big challenge to both structural engineers and researchers. The conventional numerical treatment to this problem is proved being able to give reliable predictions, however at the cost of enormous computational time and resource. Simplified SDOF approach is popularly used in design as it is straightforward to use and also gives good structural response predictions if the response is governed by a global response mode (shear or bending) and the accurate dynamic deflection curve is available, but it cannot predict the detailed local structural damage. In this study, a new numerical approach that combines the recently proposed two-step method and the static condensation method is proposed to analyze structure response and collapse to blast loads. The two-step method divides the structural response into two phases, i.e. forced vibration phase (blast loading duration) and free vibration phase. Single- Degree-of-Freedom system approach is adopted to solve the structural element responses at the end of the forced vibration phase, and the structural free vibration simulation is carried out using the hydro-code LS-DYNA to calculate the detailed structural response and damage. The static condensation technique is utilized to condense structural components that are relatively away from the explosion center to further reduce the computational effort. To demonstrate the proposed method, the structural responses of a three story RC frame to blast loads are calculated by four approaches, i.e. the traditional detailed FE simulation, the two-step method, the model condensation method, and the new combined two-step and dynamic condensation method. Through the results comparison, the efficiency and accuracy of the proposed combined approach are demonstrated.

Structural topology optimization based on system condensation

Topology optimization problem requires repeated evaluations of the objective function and design sensitivity in the design domain with various density distributions. A repeated computations in the optimization process, requires a large amount of computing time and resources. These issues have inspired the development of optimization techniques combined with a system reduction. In order to reduce the system, this study employs a system dynamic condensation method based on selected primary degrees of freedom. Based on a system reduction, this study performs a topology optimization to maximize the eigenvalue and linear summation of each eigenvalue. In the optimization procedure, mode tracking method, called MAC, is used to pursue target modes, and the design sensitivity is calculated by a method of the rigid body mode separation assuring the reliability of sensitivity regardless of the design variable perturbation size. Each result of the numerical examples based on the reduction system is compared to that of the full system. Through a few numerical examples, it is demonstrated that the proposed method can provide efficient and reliable results in topology optimization.

Damage Detection for Continuous Bridge Based on Static-Dynamic Condensation and Extended Kalman Filtering

As an effective and classical method about physical parameter identification, extended Kalman filtering (EKF) algorithm is widely used in structural damage identification, but the equations and solutions for the structure with bending deformation are not established based on EKF. The degrees of freedom about rotation can be eliminated by the static condensation method, and the dynamic condensation method considering Rayleigh damping is proposed in order to establish the equivalent and simplified modal based on complex finite element model such as continuous girder bridge. According to the requirement of bridge inspection and health monitoring, the online and convenient damage detection method based on EKF is presented. The impact excitation can be generated only on one location by one hammer actuator, and the signal in free vibration is analyzed. The deficiency that the complex excitation information is needed based on the traditional method is overcome. As a numerical example, a three-span continuous girder bridge is simulated, and the corresponding stiffness, the damage location and degree, and the damping parameter are identified accurately. It is verified that the method is suitable for the dynamic signal with high noise-signal ratio; the convergence speed is fast and this method is feasible for application.