Optimization of Nonlinear Structural Responses with MD Nastran

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.

Nonlinear dynamic response structural optimization using equivalent static loads

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 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.

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.

Structural optimization of crash-related problems usually involves nonlinearities in geometry, material, and contact. The Equivalent Static Load (ESL) method provides a method to solve such problems. It has previously been extended to employ an individual Finite Element model describing the deformed geometry at each considered time step under the name Difference-based Equivalent Static Load (DiESL) method. This paper demonstrates how an appropriate selection of the time steps in each cycle can further improve the convergence behavior of the DiESL method. It is shown that the adaptive selection of time steps leads to better objective values and more reliable convergence to the presumed global optimum. Furthermore, the DiESL extension enables the adaption of path-dependent structural properties of the original nonlinear problem like material stiffness in each linear auxiliary load case. In this paper, an adaption of the Young’s modulus on element level in the linear auxiliary problem corresponding to the local plasticization in the nonlinear dynamic problem is successfully implemented. Here, the test examples indicate that an observable improvement can only be obtained if neither the elements in the elastic nor in the plastic range are dominating the structure’s behavior.

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.

Equivalent static load and dynamic analyses methods are usually used for designing structures under and subjected to earthquake excitations. Estimation of site response from an earthquake is fundamental step to anticipate the possible damages and then to try to mitigate them. In this paper, the effect of nonlinearity on site response analyses summarized and evaluating ground surface response taking into account the local soil and subsurface soils properties for the proposed bridge over the river at Sirdjan Boulevard road subjected to earthquake vibration and provokes with assumption of rigid (viscoelastic) and elastic half space bedrock and quantify the site effect on the surface over a number of geotechnical areas has been notified. First, by field investigation, the required data were collected and by primary processing the acceptable data were selected. Then, in nonlinear analysis, for elastic and rigid half space bedrock, standard hyperbolic model was selected and executed, and then the results were compared to each other. The critical point of this work was to develop and use a computer code by the authors, named the “Abbas Converter”, with several advantages, such as work and quick installation, operating as a connecter function between the used softwares and generating the input data corresponding to defined format for them. Its output results can easily be exported to the other used softwares in this study. This code can make and render this study more easily than the previous softwares have done, and take over the encountered problem. This study clearly showed the applicability of the “Abbas Converter” for evaluation of site response with bedrock-type assumption on soil behavior under the earthquake excitations. The proposed scheme is used to analyze the ground motion data from the Bam earthquake in Kerman Province, Iran (2003, Mw 6.5).

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.

Convexity-oriented reliability-based topology optimization (CRBTO) in the time domain using the equivalent static loads method

To lightweight the engine connecting rod, and ensure it to meet the performance requirements, static equivalent load method was studied in the paper. Firstly, Connecting rod stress distribution was analyzed. Then, Multi-body dynamics model was established for the transient analysis, and connecting rod maximum stress was gotten when the engine working. Finally, Using equivalent static load method, the dynamic load was converted to an equivalent static load. And as the boundary conditions, it was used to the optimization process to optimize the shape of connecting rod. The weight of lightweight connecting rod was reduced by 9.5%. Under the condition of satisfying the stress, the purpose of the connecting rod optimization was reached.

In the proposed 2005 edition of the National Building Code of Canada (NBCC), the seismic hazard will be represented by uniform hazard spectra corresponding to a 2% probability of being exceeded in 50 years. The seismic design base shear for use in an equivalent static load method of design will be obtained from the uniform hazard spectrum for the site corresponding to the first mode period of the building. Because this procedure ignores the effect of higher modes, the base shear so derived must be suitably adjusted. A procedure for deriving the base shear adjustment factors for different types of structural systems is described and the adjustment factor values proposed for the 2005 NBCC are presented. The adjusted base shear will be distributed across the height of the building in accordance with the provisions in the current version of the code. Since the code-specified distribution is primarily based on the first mode vibration shape, it leads to an overestimation of the overturning moments, which should therefore be suitably adjusted. Adjustment factors that must be applied to the overturning moments at the base and across the height are derived for different structural shapes, and the empirical values for use in the 2005 NBCC are presented.Key words: uniform hazard spectrum, seismic design base shear, equivalent static load procedure, higher mode effects, base shear adjustment factors, distribution of base shear, overturning moment adjustment factors.

This paper proposes an equivalent static wind load calculation method based on response-related scale for the wind load calculation of large-span roof structures. The effectiveness of this method is verified by wind tunnel tests and finite element analysis of the roof of the east entrance of Hangzhou Convention and Exhibition Center. The research results show that the equivalent static wind load method based on response-related scale has high accuracy in reflecting the overall response characteristics of the structure, and its error in calculating the node displacement response is the smallest, only 4.6%, which is better than the traditional single-objective equivalent method and least squares method.

Practical nonlinear inelastic analysis method of composite steel-concrete beams with partial composite action