Mode Acceleration Method Research Articles

A Harmonic Response Topology Optimization Method Based on Hybrid Cellular Automata

Abstract Hybrid cellular automaton (HCA) method is a topology optimization method with high convergence efficiency. Harmonic response topology optimization problem has been widely researched and common frequency-domain methods have been applied to solve this problem. Despite this, little work has so far been undertaken to utilize the HCA method to improve the calculation efficiency of harmonic response topology optimization. In this article, we present a methodology that applies the HCA method to solve harmonic response topology optimization problem. Mode displacement method (MDM), mode acceleration method (MAM), and full method (FM) are common frequency-domain methods that are utilized as harmonic response analysis methods in this study. The examples under rotating loads demonstrate that feasibility of this methodology at one specific frequency excitation and multiple frequencies excitation. This article presents and implements a high convergence efficiency method for harmonic response topology optimization, and this method can converge in 20–30 iterations. This method can extend the application range of the HCA method and improve the optimization efficiency of harmonic response topology optimization.

An Improved Correction Technique for the Prediction of the Dynamic Response of a Beam under a Moving Vehicle

This study presents a correction approach that can capture the discontinuities in the bending moment and shear force in the dynamic analysis of beam-like structures traveled by a moving vehicle. The proposed approach was based on the Dynamic Modal Acceleration Method (DyMAM) to correct the dynamic response of the supporting structure with a reduced number of vibration modes. The use of a two-axle vehicle model was adopted to consider the pitching effect in the presence of surface irregularity and damping. The interacting forces between the beam and vehicle were filtered to avoid undesirable high-frequency contributions. Subsequently, a new formulation for the entire vehicle-beam system was obtained. The corresponding equation was solved using the Newmark numerical scheme to obtain the system responses in each time step. A numerical example was illustrated, showing that the proposed method was in close agreement with previous correction solutions in the vehicle-beam interaction analysis.

Improved mode acceleration-based vibroacoustic coupling analysis of functionally graded shell under random excitation

This paper focuses on the vibroacoustic response of functionally graded material (FGM) shell structures under stationary random excitations. An eight-node finite element model of FGM shells is developed, which is valid for both flat and curved shells of FGM. Then, the finite element formulation of the acoustic-structural coupled system for FGM shells under random excitation is established. It is shown that the commonly used combined method of the pseudo excitation method (PEM) and mode displacement method (MDM) in previous work is highly efficient for stochastic dynamics problems, but its accuracy is reduced by the modal truncation error. To remedy this, the combined method of PEM and the conventional mode acceleration method (MAM) is an effective alternative, but not applicable to the coupled acoustic-structural system due to its singular stiffness matrix. To circumvent this, an efficient and accurate method integrating the PEM and the improved MAM (IMAM) is proposed for the vibroacoustic coupling analysis of FGM shell structures under stationary random excitation. Numerical examples are given to verify the effectiveness of the proposed method and to explore the random vibroacoustic properties of the FGM shells with different power-law distributions.

Topology optimization of laminated composite structures under harmonic force excitations

In this paper, a topology optimization approach for the design of laminated composite structures under harmonic force excitations is proposed. A novel method is developed to calculate the harmonic response for composite laminates, which consists of two steps: firstly, based on the strain energy approach, the damping matrix model of composite laminates is established with the proportional damping assumption; then, the displacement response is calculated by the mode acceleration method The design objective of topology optimization is to minimize the displacement amplitude at the concerning point with an excitation frequency or a frequency band. An extended polynomial interpolation scheme is introduced to penalize the stiffness, damped stiffness and mass of elements. The analytical sensitivities of the objective and constraint functions to the density variables are derived in detail, and the globally convergent method of moving asymptotes is used to solve the optimization problem. Numerical examples are performed to demonstrate the effectiveness and feasibility of the proposed topology optimization method in improving the dynamic performance of laminated composite structures. The influence rules of excitation frequency and layer sequence on topologic shape are also discussed.

Concurrent design of the free damping structure for minimizing the frequency response in a broad frequency band

This article develops a multiscale topology optimization model for the free damping structure to optimize the microstructure of the damping material and its macro layout. The objective is to minimize structural response in a broad frequency gap subject to volume constraints on both macro and micro levels. To overcome the convergence difficulty caused by the sharp jumping of the harmonic response around structural eigenfrequencies and maintain high-precision integral, the Gauss Legendre integration method is adopted to calculate the integration of objective function. Based on the density-based method, the sensitivity analysis of the objective function and the constraint is deduced. The mode acceleration method is used to reduce computational cost and a modified threshold Heaviside projection method is used to obtain the clear black-and-white designs. Several numerical examples are investigated to demonstrate the validity of the proposed model.

A multi-scale discrete material optimization model for optimization of structural topology and material orientations to minimize dynamic compliance

For fiber-reinforced composites, the combination of structural topology optimization and fiber angle design is an excellent way to suppress structural vibration more efficiently and acquire a lighter structure. This paper proposes a concurrent optimization model for fiber-reinforced composite structures, where the mathematical model is built by combining the polynomial interpolation scheme (PIS) and the Heaviside penalization of discrete material optimization (HPDMO). Therein, the former is adopted to avoid the localized mode phenomena due to the mismatch between element stiffness and mass. And the latter is used for multi-candidate fiber orientation design. The objective function is to minimize the integration of the dynamic compliance in a given frequency band subject to macro-volume constraint. In order to ease the heavy computational burden caused by hundreds of frequency steps in each iteration, too many freedom degrees, and discrete design variables in HPDMO, the modal acceleration method is used to obtain a high-precision estimation of displacement response, and a decoupled method suitable for multiphase fiber material sensitivity analysis is also developed. Finally, a modified threshold Heaviside projection is used to obtain a black-and-white design. Several numerical examples are presented to verify the validity and robustness of the proposed model.

Modal reduction procedures for flexible multibody dynamics

Through a critical review of the various component mode synthesis techniques developed in the past, it is shown that both Craig–Bampton’s and Herting’s methods are particular cases of the mode-acceleration method and furthermore, Rubin’s method is equivalent to Herting’s method. Consequently, the mode-acceleration method is the approach of choice due to its simplicity and because unlike the other methods, it imposes no restriction on the selection of the modes. Next, a general approach to the modal reduction of geometrically nonlinear structures is developed within the framework of the motion formalism, based on the small deformation assumption. The floating frame of reference is defined unequivocally by imposing six linear constraints on the deformation measures, which are defined as the vectorial parameterization of the relative motion tensor that brings the fictitious rigid-body configuration to its deformed counterpart. This approach yields deformation measures that are both objective and tensorial, unlike their classical counterparts that share the first property only. Derivatives are expressed in the material frame, leading to computationally advantageous properties: tangent matrices are functions of the deformation measures only and become nearly constant during the simulation. Numerical examples demonstrate the accuracy, robustness, and numerical efficiency of the proposed approach. With a small number of modal elements, the formulation is able to capture geometrically nonlinear effects accurately, even in the presence of inherently nonlinear phenomena such as buckling.

Concurrent topology optimization with uniform microstructure for minimizing dynamic response in the time domain

This paper aims to develop an efficient concurrent topology optimization approach for minimizing the maximum dynamic response of two-scale hierarchical structures in the time domain. Compared with statics problems, the dynamic response problems usually involve many time steps, which may lead to intensive computational burdens in both dynamic response and sensitivity analyses. This study thus proposes an enhanced decoupled sensitivity analysis method for concurrent topology optimization of the time-domain dynamic response problems. The mode acceleration method is incorporated for efficient dynamic response analysis. The three-field density-based approach is employed for topology optimization of macrostructure and microstructure. A previously proposed aggregation functional is employed to approximate the maximum dynamic response of the structure. The method of moving asymptotes (MMA) is employed to update the design variables. Three numerical examples are presented to demonstrate the effectiveness of the proposed approach. Some discoveries regarding the concurrent topology optimization for dynamics problems are presented and discussed. Furthermore, the potential of the concurrent topology optimization formulation for designing lightweight structures under dynamic loads is also demonstrated.

Estimation of seismic demand in tall buildings using mode-acceleration method

The current study presented a method to estimate seismic response of inter-story drift in tall buildings. Inter-story drift is obtained using combined mode-acceleration equations via modeling tall buildings through flexural and shear cantilever beams’ equations, and estimating dynamic components of tall buildings. In the provided equation to calculate seismic response of inter-story drift, it is necessary to know details of the building, which makes it easy and expedites the analysis. Inter-story drift of 10-, 15-, and 50-story buildings for three earthquake accelerograms was estimated using the provided equation, and via finite element software OPENSEES, the equation was verified. Results comparison showed that the equation provided a good approximation, which can be used instead of modeling, to calculate inter-story drift. In addition, the comparison between the calculated inter-story drift with those of the approximate method and finite element method showed that the mode-acceleration method was close to the accurate solution even using few modes.

Estimating Inter-Story Drift in High Rise Buildings with the Flexural and Shear Cantilever Beam and Mode-Acceleration Method

In this study, the seismic inter-story drift of structures is estimated by a combination of mode-acceleration equations with the modelling of high-rise buildings with flexural and shear cantilever beams. In the equation presented for calculating the inter-story drift, having less knowledge of the building is adequate and this issue is of significance in estimating the nonstructural component forces, especially in high-rise buildings and also in the initial design of structures. Also, a comparison of inter-story drift estimated by the approximation method with an exact method indicates that the application of the mode-acceleration method compared to mode-displacement with a fewer number of modes comes close to the exact calculation, which facilitates and expedites the analysis. In order to carry out an exact evaluation of the presented equation, inter-story drift is calculated and compared in 10, 15 and 50 story buildings during three seismic records using approximate relations. Exact analysis of those structures is done in finite element Opensees software. The results of comparisons show that the presented equation provides an adequate estimation without the need for modelling and lengthy software analysis.

Structural topology optimization under stationary random base acceleration excitations

Structural topology optimization subjected to stationary random base acceleration excitations is investigated in this paper. In the random response analysis, the Large Mass Method (LMM) which attributes artificial large mass values at each driven nodal Degree Of Freedom (DOF) to transforming the base acceleration excitations into force excitations is proposed. The Complete Quadratic Combination (CQC) which is commonly used to calculate the random responses in previous optimization has been proven to be computationally expensive especially for large-scale problems. In order to conquer this difficulty, the Pseudo Excitation Method (PEM) and the combined method of PEM and Mode Acceleration Method (MAM) are adopted into the dynamic topology optimization, and random responses are calculated using these two methods to ascertain a high efficiency over the CQC. A density-based topology optimization method minimizing dynamic responses is then formulated based on the integration of LMM and PEM or the combined method of PEM and MAM. Numerical examples are presented to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performance.

Integrated layout and topology optimization design of multi-component systems under harmonic base acceleration excitations

The integrated optimization of component layout and structural topology is studied in this paper to improve the dynamic performance of the multi-component structure systems under harmonic base acceleration excitations. Considering linear systems, including multi-point constraints interconnecting the components and structures as an integrity, the dynamic responses and the corresponding design sensitivities are analytically derived based on the mode acceleration method. To obtain precise dynamic response, structural real damping characteristics are measured using vibration experiments, which are relevant to the structural dynamic response, especially when the excitation frequencies fall into the resonant frequency band. Further verifications are done by the comparison of Rayleigh damping, constant damping ratio, and hysteretic damping model with experimental results of structures achieved by resin-based additive manufacturing. In this way, structural real damping characteristics are taken into account in the integrated optimization. Numerical examples and vibration testing results are presented to show the validity of the optimization procedure and its potential application in engineering.

Structural topology optimization under harmonic base acceleration excitations

This work is focused on the structural topology optimization methods related to dynamic responses under harmonic base acceleration excitations. The uniform acceleration input model is chosen to be the input form of base excitations. In the dynamic response analysis, we propose using the large mass method (LMM) in which artificial large mass values are attributed to each driven nodal degree of freedom (DOF), which can thus transform the base acceleration excitations into force excitations. Mode displacement method (MDM) and mode acceleration method (MAM) are then used to calculate the harmonic responses and the design sensitivities due to their balances between computing efficiency and accuracy especially when frequency bands are taken into account. A density based topology optimization method of minimizing dynamic responses is then formulated based on the integration of LMM and MDM or MAM. Moreover, some particular appearances such as the precision of response analysis using MDM or MAM, and the duplicated frequencies are briefly discussed. Numerical examples are finally tested to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performances.

The approximate reanalysis method for topology optimization under harmonic force excitations with multiple frequencies

In mode acceleration method for topology optimization related harmonic response with multiple frequencies, most of the computation effort is invested in the solution of the eigen-problem. This paper is focused on reduction of the computational effort in repeated solution of the eigen-problem involved in mode acceleration method. The block combined approximation with shifting method is adopted for eigen-problem reanalysis, which simultaneously calculates some eigenpairs of modified structures. The triangular factorizations of shifted stiffness matrices generated within a certain number of design iterations are utilized to calculate the modes. For improving computational efficiency, Basic Linear Algebra Subprograms (BLAS) are utilized. The reanalysis method is based on matrix-matrix operations with Level-3 BLAS and can provide very fast development of approximate solutions of high quality for frequencies and associated mode shapes of the modified structure. Numerical examples are given to demonstrate the efficiency of the proposed topology optimization procedure and the accuracy of the approximate solutions.

Dynamic response topology optimization in the time domain using model reduction method

The dynamic response topology optimization problems are usually computationally expensive, so it is necessary to employ the model reduction methods to reduce computational cost. This work will investigate the effectiveness of the mode displacement method(MDM) and mode acceleration method(MAM) for time-domain response problems within the framework of density-based topology optimization. Three objective functions, the mean dynamic compliance, mean strain energy and mean squared displacement are considered. It is found that, in general cases, MDM is not suitable for time-domain response topology optimization problems due to its low accuracy of approximation, while MAM works well for problems of a wide range, and when there are many time steps, the MAM based topology optimization approach is more efficient than the direct integration based approach. So for practical applications, when the problem needs many time steps, the MAM based approach is preferred and otherwise, the direct integration based approach is suggested.

Topology optimization of large-scale structures subjected to stationary random excitation: An efficient optimization procedure integrating pseudo excitation method and mode acceleration method

Structural topology optimization related to dynamic responses under stationary random force excitation is investigated in this paper. It is shown that the commonly used Complete Quadratic Combination method (CQC) in previous optimization work is not only computationally expensive but also results in non-convergent design pattern due to the low computing accuracy of random responses for large-scale problems. To circumvent these difficulties, an efficient and accurate optimization procedure integrating the Pseudo Excitation Method (PEM) and Mode Acceleration Method (MAM) is introduced into the dynamic topology optimization. In this framework, random responses are calculated using the PEM to ascertain a high efficiency over the CQC. More importantly, the accuracy of random responses is improved indirectly by solving the pseudo harmonic responses involved in the PEM with the help of the MAM. Numerical examples fully demonstrate the validity of the developed optimization procedure and its potential applications in practical designs.

High order modal approximation techniques in the frequency domain

This paper presents high order modal methods that increase the accuracy of the standard modal truncation scheme with an emphasis on frequency-domain accuracy. The existing Mode Acceleration and Modal Truncation Augmentation methods are compared alongside a new interpretation utilizing the interpolating polynomial. This technique is motivated by taking limits of the modal sum, introducing a residual that appears as a Laurent polynomial. The results presented focus on the case when the frequency range of interest is in the middle of the system’s spectrum of eigenvalues, meaning that at least some of the system’s natural frequencies lie below the lower bound of the frequency range. This case allows for a possible error in the Modal Truncation Augmentation method that is demonstrated here. Also, convergence properties of the methods are discussed and demonstrated.

A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations

This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account.

Analysis of three dimensional equivalent static wind loads of symmetric high-rise buildings based on wind tunnel tests

Using synchronous surface pressures from the wind tunnel test, the three dimensional wind load models of high-rise buildings are established. Furthermore, the internal force responses of symmetric high-rise buildings in along-wind, across-wind and torsional directions are evaluated based on mode acceleration method, which expresses the restoring force as the summation of quasi-static force and inertia force components. Accordingly the calculation methods of equivalent static wind loads, in which the contributions of the higher modes can be considered, of symmetric high-rise buildings in along-wind, across-wind and torsional directions are deduced based on internal forces equivalence. Finally the equivalent static wind loads of an actual symmetric high-rise building are obtained by this method, and compared with the along-wind equivalent static wind loads obtained by China National Standard.

Generalized mode acceleration and modal truncation augmentation methods for the harmonic response analysis of nonviscously damped systems

The modal truncation problem is frequently encountered in nonviscously (viscoelastically) damped systems since only the modes of interest are usually considered in the dynamic analysis of engineering problems. This study aims at accurately calculating the steady-state responses of nonviscously damped systems by only considering the modes of interest. Based on the Neumann expansion theorem and the frequency shifting technique, a property obtained from the first-order terms of the Neumann expansion of the frequency response function (FRF) matrix of nonviscously damped systems is given. However, this procedure cannot be extended to consider the further higher-order terms. It means a truncation expansion problem exists for nonviscously damped systems. By considering the first-order terms of the Neumann expansion, a generalized mode acceleration method (GMAM) is presented to handle the modal truncation problem. The GMAM can overcome the singular problem of the stiffness matrix. The modal truncation augmentation method (MTAM) is also presented to handle the modal truncation problem by making the equilibrium equations into a subspace equation spanned in terms of the columns of a projection basis given in the GMAM. Several conclusions concerning the implementation of the presented methods are formulated on the basis of the results of three examples.