Nonlinear dynamic response structural optimization using equivalent static loads

joined-wing configuration that was published by Wolkovich in 1986 has been studied by many researchers (Wolkovich, J., The Joined-Wing: An Overview, Journal of Aircraft, Vol. 23, No. 3, 1986, pp. 161―178. doi: 10.2514/3.45285). Thejoined-wing airplane is defined as an airplane that incorporates tandem wings arranged to form diamond shapes from both the top and front views. joined wing can lead to increased aerodynamic performances as well as a reduction in the structural weight. However, the joined wing has high geometric nonlinearity under the gust load. gust load acts as a dynamic load. Therefore, nonlinear dynamic (transient) behavior of the joined wing should be considered in structural optimization. In previous research, linear dynamic response optimization and nonlinear static response optimization were performed. It is well known that conventional nonlinear dynamic response optimization is extremely expensive. Therefore, in this research, nonlinear dynamic response optimization of a joined wing is carried out by using equivalent static loads. concept of equivalent static loads is expanded and newly proposed for nonlinear dynamic response optimization. Equivalent static loads are the load sets that generate the same response field in linear static analysis as that in nonlinear dynamic analysis. Therefore, nonlinear dynamic response optimization can be conducted by repeated use of linear response optimization. For the verification of efficiency of the proposed method, a simple nonlinear dynamic response optimization problem is introduced. problem is solved by using both the equivalent static loads method and the conventional method with sensitivity analysis using the finite difference method. procedure for nonlinear dynamic response optimization of a joined wing using equivalent static loads is explained, and the optimum results are discussed.

Vehicle crash optimization is a representative non-linear dynamic response structural optimization that utilizes highly non-linear vehicle crash analysis in the time domain. In the automobile industries, crash optimization is employed to enhance the crashworthiness characteristics. The equivalent-static-loads method has been developed for such non-linear dynamic response structural optimization. The equivalent static loads are the static loads that generate the same displacement field in linear static analysis as those of non-linear dynamic analysis at a certain time step, and the equivalent static loads are imposed as external loads in linear static structural optimization. In this research, the conventional equivalent-static-loads method is expanded to the crash management system with regard to the frontal-impact test and a full-scale vehicle for a side-impact crash test. Crash analysis frequently considers unsupported systems which do not have boundary conditions and where adjacent structures do not penetrate owing to contact. Since the equivalent-static-loads method uses linear static response structural optimization, boundary conditions are required, and the impenetrability condition cannot be directly considered. To overcome the difficulties, a problem without boundary conditions is solved by using the inertia relief method. Thus, relative displacements with respect to a certain reference point are used in linear static response optimization. The impenetrability condition in non-linear analysis is transformed to the impenetrability constraints in linear static response optimization.

The equivalent static loads method (ESLM) is a structural optimization method that can consider an analysis method other than linear static analysis. This method defines two separate domains: the analysis domain and design domain. Analysis is performed in the analysis domain, equivalent static loads (ESLs) sets are generated, linear static response optimization is carried out in the design domain using the ESLs and the process iterates until the stopping criteria are satisfied. This method is quite popular and some commercial systems have installed the method. Theoretical foundation of ESLM was validated for linear dynamic response optimization by Park and Kang (J Optim Theory Appl 18:191‑200, 2003). They claimed that when the ESLM process terminates, the optimum solution satisfies the Karush-Kuhn-Tucker (KKT) necessary condition. Some critical issues were raised by Stolpe (Struct Multidiscip Optim 50:921‑926, 2014). He showed that the theoretical results in Park and Kang are not valid. In this paper, the validation process of Park and Kang is amended according to the Stolpe’s corrections. It is shown that the original claim for the KKT condition is valid by adding some mathematical aspects.

An optimization method is proposed for the simultaneous design of structural and control systems using the equivalent static loads. The two structural and control systems are not completely independent and need to be considered in a unified fashion. Furthermore, an integrated system design is unavoidable to exhibit a good performance in the time domain. The analysis for the integrated system is conducted for the transient-state in a dynamic manner. The constraints for the structural and control systems are defined in the time domain as well. Therefore, a physically small scale problem in structural analysis easily becomes quite a large scale in an optimization problem. A new equivalent static loads (ESLs) method, which deals with the structural design variables as well as the control design variables, is proposed to solve physically large scale problems. A finite element dynamic equation is defined with control forces and a dynamic response optimization problem is formulated. Linear static response optimization is carried out with the ESLs. The control forces for the linear static response optimization are considered as design variables. Shape variables are utilized to handle the design variables for the control forces. Several examples are solved to validate the proposed method.

The forging process is the shaping of a workpiece using dynamic loads and typically consists of the multi-step process with the preforming process. Defects such as flash and unfilling occur and the distribution of effective strains is not even in the workpiece after the forging process. An optimized preform is necessary for reduction of the defects in the desired final forging. Optimization of the forging process is nonlinear dynamic response optimization because nonlinearities are involved in the analysis. When the conventional method is utilized in optimization of the forging process, the cost is extremely high due to repeated nonlinear analyses for function and sensitivity calculation. In this paper, the equivalent static loads method for non linear static response structural optimization (ESLSO) is employed to determine the preform shape which leads to reduction of the unfilled area and even distribution of the effective strain in the final forging shape. ESLSO is a structural optimization method where nonlinear dynamic loads are transformed to equivalent static loads (ESLs). ESLs are defined as the loads for linear analysis, which generate the same response field as that of nonlinear analysis. Two kinds of ESLs are proposed and they are the ESLs for the displacements and the ESLs for the effective strains. Examples of the forging process are solved using these ESLs and the results are discussed.

In this study, an efficient topology optimization method under crash loads is proposed by combining the equivalent static loads with a model order reduction method, which is referred as the reduced model–based equivalent static loads method for nonlinear dynamic response topology optimization method. Considering that some parts of the vehicle experience large nonlinear deformations, whereas others exhibit only small linear deformations in a vehicle crash scenario, the linear and nonlinear behavior parts are identified and the whole model of the complete structure is divided into nonlinear and linear sub-models. At each cycle, the model order reduction method is used in the linear sub-model during crash analysis to solve the low-density-elements-induced mesh distortion problem and accelerate this process. In the linear static topology optimization, the nonlinear sub-model that was initially used to describe the nonlinear behavior part is linearized by the equivalent static loads method and then reduced by the Guyan reduction method. Then, the reduced equivalent static load model is assembled into the linear sub-model that is defined as the design space to formulate a reduced topology optimization model of the complete structure and the reduced equivalent static loads that only act on master degrees of freedom are calculated. Finally, the linear static topology optimization is performed based on the reduced topology optimization model with the reduced equivalent static loads to enhance the efficiency and improve the numerical stability. The process is repeated until the convergence criterion is satisfied. The effectiveness of the proposed method is demonstrated by investing a numerical example. The results show that the proposed method provides a feasible strategy for the topology optimization under crash loads, which can effectively improve the numerical stability and convergence.

An approach for reliability-based topology optimization of interval parameters structures under dynamic loads is proposed. We modify the equivalent static loads method for non linear static response structural optimization (ESLSO) to solve the dynamic reliability optimization problem. In our modified ESLSO, the equivalent static loads (ESLs) are redefined to consider the uncertainties. The new ESLs including all the uncertainties from geometric dimensions, material properties and loading conditions generate the same interval response field as dynamic loads. Based on the definition of the interval non-probabilistic reliability index, we construct the static reliability topology optimization model using ESLs. The method of moving asymptotes (MMA) is employed as the optimization problem solver. The applicability and validity of the proposed model and numerical techniques are demonstrated with three numerical examples.

The success that has been achieved in applying optimization techniques to linear structural analyses has led to the desire to extend these techniques to nonlinear structural analyses. Studies have been done to apply both gradient and non-gradient based approaches to the nonlinear structural analysis problem. This paper presents a hybrid approach that enables the application of gradient based methods without incurring the expensive and difficult calculations required to provide sensitivities based on the nonlinear analysis. Specifically, it discusses the commercial implementation of the ESL (Equivalent Static Loads) method developed by Prof. G. J. Park and his research team into the MD Nastran computer code available from MSC Software. A high-level overview of the methods currently available to perform nonlinear structural optimization is followed by a brief review of the ESL method. Particular emphasis is placed on the special details required to place the prototype concepts into a commercial product. The implementation has emphasized placing minimal burden on the user to extend a file used as input to a nonlinear structural analysis task to one that can perform nonlinear structural optimization. Two examples are then provided to further illustrate the methods and demonstrate that it is capable of solving “real world” problems. Concluding remarks summarize the contributions of the paper and indicate further work that is being done at MSC to extend this initial implementation of nonlinear structural optimization.

Importance of ‘DAF’ in evaluating structural adequacy of gravity-load designed RC buildings

While significant advances have been made on the blast analysis, applications of nonlinear analysis of structures under explosion have been limited to the frames and structural behavior of bridges subjected to blast is an issue that has attracted attention of researchers in recent years. Nonlinear dynamic analysis can be lead to the most accurate responses of the structure under blast but the use of it is not justified in many cases because of its complexity and excessive computing times. Pushover method is often seen to be a viable and attractive alternative to nonlinear analysis because of its simplicity and ability to estimate deformation demands with acceptable accuracy without the intensive computational and modeling effort of a nonlinear dynamic analysis. In this study, a rectangular load pattern is developed to estimate nonlinear responses of composite bridges subjected to blast using pushover method. For this purpose, first, an efficient method is proposed to determine equivalent static blast loading and then the accuracy of pushover analysis based on the developed load pattern is evaluated. It is shown that the proposed method can be easily used to conservatively estimate maximum displacement of composite steel girder bridges under explosion.

Accurate prediction of the vibration response of aircraft engine assemblies is of great importance when estimating both the performance and the lifetime of its individual components. In the case of underplatform dampers, for example, the motion at the frictional interfaces can lead to a highly nonlinear dynamic response and cause fretting wear at the contact. The latter will change the contact conditions of the interface and consequently impact the nonlinear dynamic response of the entire assembly. Accurate prediction of the nonlinear dynamic response over the lifetime of the assembly must include the impact of fretting wear. A multi-scale approach that incorporates wear into the nonlinear dynamic analysis is proposed, and its viability is demonstrated for an underplatform damper system. The nonlinear dynamic response is calculated with a multiharmonic balance approach, and a newly developed semi-analytical contact solver is used to obtain the contact conditions at the blade-damper interface with high accuracy and low computational cost. The calculated contact conditions are used in combination with the energy wear approach to compute the fretting wear at the contact interface. The nonlinear dynamic model of the blade-damper system is then updated with the worn profile and its dynamic response is recomputed. A significant impact of fretting wear on the nonlinear dynamic behaviour of the blade-damper system was observed, highlighting the sensitivity of the nonlinear dynamic response to changes at the contact interface. The computational speed and robustness of the adopted multi-scale approach are demonstrated.

Given the vast advancements in computing power in the last several decades, nonlinear dynamic analysis has gained wide acceptance by practicing engineers as a useful way of assessing and improving the seismic performance of structures. Nonlinear structural analysis software packages give engineers the ability to directly model nonlinear component behavior in detail, resulting in improved understanding of how a building will respond under strong earthquake shaking. One key component, in particular, for understanding the behavior of moment-resisting frames is the plastic rotation of the flexural hinges. Performance-based standards typically use plastic rotation as the primary parameter for defining the acceptance criteria in moment-resisting frames. Since plastic rotation is a key parameter in the seismic damage assessment, the concept as well as the method to calculate this quantity must be understood completely. Though engineers rely on the plastic rotation output from seismic structural analysis software packages to determine acceptable performance, the actual calculation methods used in achieving such plastic rotation quantities usually lay within a so-called “black box”. Based on the outputs obtained from most structural analysis software packages, it can be shown that running an algorithm considering material nonlinearity by itself will produce reasonably accurate results. Moreover, separately running an algorithm considering geometric nonlinearity also can produce accurate results. However, when material nonlinearity is combined with geometric nonlinearity in an analysis, obtaining accurate results or even stable solutions is more difficult. The coupling effect between the two nonlinearities can significantly affect the global response and the local plastic rotation obtained from the analysis and therefore needs to be verified through some analytical means. Yet, the verification process is difficult because a robust analytical framework for calculating plastic rotation is currently unavailable and urgently needed. In view of this gap, an improved analytical approach based on small displacement theory is derived to calculate the plastic rotations of plastic hinges at various locations of moment-resisting frames. Both static and dynamic analysis with nonlinear geometric effects will be incorporated in the derivation. Here the element stiffness matrices are first rigorously derived using a member with plastic hinges in compression, and therefore the coupling of geometric and material nonlinearity effects is included from the beginning of the derivation. Additionally, plastic rotation is handled explicitly by considering this rotation as an additional nonlinear degree-of-freedom. Numerical simulation is performed to calculate the nonlinear static and dynamic responses of simple benchmark models subjected to seismic excitations. Results are compared with various software packages to demonstrate the feasibility of the proposed method in light of the output results among software packages in calculating plastic rotations.

This research investigates numerically the potential failure of different framed structures after removal of a corner column. This study is directed towards the role of the joint above the removed column in terms of framed structures stability and internal forces redistribution utilizing linear static, nonlinear static and nonlinear dynamic analysis procedures. First, the preliminary linear static analysis as a part of more detailed nonlinear static and dynamic analysis to model reinforced concrete frames is performed. For nonlinear numerical models, three different moment-rotation capacity curves are considered for the joint above the removed column. These curves are taken according to FEMA 356 regulation according to the reinforcement condition at the joint location to represent normal joint, anchorage deficient joint and shear deficient joint. In linear static models, the results confirmed that, the more moment the joint can resist at the removed column location, the less moment is generated at the beam’s other end. In nonlinear static analysis, the model with normal plastic hinge properties recorded a higher carrying capacity than the model with both shear and anchorage plastic hinges. Nonlinear dynamic analysis with less assumptions offer the more representative modeling for the considered case study which is obvious from its closer perdition of internal forces and deflection compared to the reference findings.

Accurate prediction of the vibration response of aircraft engine assemblies is of great importance when estimating both the performance and the lifetime of their individual components. In the case of underplatform dampers, for example, the motion at the frictional interfaces can lead to a highly nonlinear dynamic response and cause fretting wear at the contact. The latter will change the contact conditions of the interface and consequently impact the nonlinear dynamic response of the entire assembly. Accurate prediction of the nonlinear dynamic response over the lifetime of the assembly must include the impact of fretting wear. A multiscale approach that incorporates wear into the nonlinear dynamic analysis is proposed, and its viability is demonstrated for an underplatform damper system. The nonlinear dynamic response is calculated with a multiharmonic balance approach, and a newly developed semi-analytical contact solver is used to obtain the contact conditions at the blade–damper interface with high accuracy and low computational cost. The calculated contact conditions are used in combination with the energy wear approach to compute the fretting wear at the contact interface. The nonlinear dynamic model of the blade–damper system is then updated with the worn profile and its dynamic response is recomputed. A significant impact of fretting wear on the nonlinear dynamic behavior of the blade–damper system was observed, highlighting the sensitivity of the nonlinear dynamic response to changes at the contact interface. The computational speed and robustness of the adopted multiscale approach are demonstrated.

Aiming at the problems such as low computational efficiency and long process in dynamic topology optimization, combined with bi-directional evolutional structure optimization, the dynamic optimization problem is transformed into the static optimization problem in multiple load cases. Newmark method is used to analyze the structure dynamics load. The topology optimization model based on equivalent static load is established. On the basis of the equivalent static load, multi-objective dynamics evolutionary structure optimization is studied. And the mathematical model of the dynamic stiffness and natural frequency about the multi-objective optimization is established. The mathematical formula of the sensitivity is deduced. Then a complete optimization iterative process is given. The numerical example illustrates that it is feasible to solve the multi-objective dynamic topology optimization with equivalent static loads. Modal tracking technique is introduced to the multi-objective dynamic topology optimization. The numerical example shows that modal tracking technology can track the required modal in the optimization process.