Equivalent Static Loads Method Research Articles

Practical Design Guidelines for Topology Optimization of Flexible Mechanisms: A Comparison between Weakly Coupled Methods

Industrial robots are complex systems, as they require the integration of several sub-assemblies to perform accurate operations. Moreover, they may experience remarkable dynamic actions due to high kinematic requirements, which are necessary to obtain reduced cycle times. The dynamic design of industrial robots can therefore be demanding, since the single structural component can induce an impact both in the design phase (development strategy and computational time) and at the machine level (global stiffness and natural frequencies). To this end, the present paper proposes first a topology optimization procedure based on the Equivalent Static Loads (ESL) method that integrates flexible multibody simulation outputs. The same procedure also foresees an intermediate static reduction to reduce and to precisely define the application points of the ESL. Secondly, an optimization procedure based on the Quasi-Static Loads (QSL) method integrating flexible multibody simulation outputs is proposed as well. The objective is to carry out a comparison between the two methods and consequently evaluate the benefits and drawbacks of the two. In the end, practical guidelines regarding the selection and application of the two methods are also provided to the reader.

An efficient online successive reanalysis method for dynamic topology optimization

In this study, an efficient reanalysis strategy for dynamic topology optimization is proposed. Compared with other related studies, an online successive dynamic reanalysis method and a Proper Orthogonal Decomposition (POD)-based approximate dynamic displacement strategy are integrated. In the dynamic reanalysis, the storage of the stiffness matrix decomposition can be avoided, and the reduced basis vectors should be successively updated according to the structural state in each iteration. Thus, the bottleneck of the combined approximation method for large-scale dynamic topology optimization can be overcome. Sequentially, POD is applied to obtain the approximate dynamic displacement, in which the Proper Orthogonal Mode (POM) of the displacement field is applied to obtain the approximate equivalent static loads of the Equivalent Static Load (ESL) method. Compared to the exact equivalent static loads at all the time intervals, the number of equivalent static loads is significantly reduced. Finally, the 2D and 3D test results indicate that the proposed method has remarkable speed-up effect under the premise of small relative error, support the strength of the proposed strategy.

Finite element analysis and optimization design of large vibrating screen based on equivalent static load method

Vibrating screens have wide applications in various industries, ranging from agriculture to coal mining. In recent years, improvement in production have put forward higher requirements on the efficiency of vibrating screens. However, these large screens are prone to fatigue damage, particularly crack formation, over time. The emergence of substantial dynamic inertial forces can be attributed to the considerable body mass of a large vibrating screen. This study analyzed the structural strength of the HZXZ200x300 large vibrating screen, employing finite element simulation to identify the maximum equivalent stress and primary stress distribution. The main frame model of the vibrating screen box was constructed by integrating the equivalent static load and submodule methods. The model transitioned from dynamic response to static optimization under equivalent static and dynamic loads, significantly reducing calculation scale and enhancing optimization efficiency. Equivalent static sub-models were employed for topology optimization, determining the optimal structure for material performance distribution. This process yielded an optimal conceptual model for reconstructing the actual model. The structural strength was further improved by comparing three-dimensional and dynamic local stress relationships post-topological optimization, followed by reinforcing the vibrating screen structure and introducing reinforced bars for increased stability. The study showed that lightweight topology optimization significantly decreased the stress levels and improved the fatigue durability of beams. Local strengthening, accomplished via topology optimization, effectively reduced the maximum equivalent stress to 76.487 MPa, a decrease of 46.8%. The vibrating screen mass also decreased by 225 kg, a reduction of 14.9%. In summary, this study employed structural analysis, topology optimization, and local reinforcement to mitigate stress levels, enhance the fatigue life of a vibrating screen, and reduce its weight. This study offers an effective solution to the dynamic optimization challenges of complex structures.

Kirigami pattern design for buckling-induced assembly 3D structures via topology optimization

Complex three-dimensional (3D) mesostructures are attracting an increasing interest due to their wide applications in microelectronics, biomedicine and other fields. A buckling-induced 3D structure can be assembled by buckling of a 2D plane precursor rapidly and precisely at low cost. However, the inverse design problem of finding 2D precursor configurations corresponding to the predetermined 3D mesostructures has not been well solved yet. In the present work, an inverse method is developed for buckling-induced assembly 3D structure design. Unlike the existing methods, the Moving Morphable Void (MMV) -based topology optimization approach is adopted for driving the inverse design. The voids in MMV are defined as kirigami cuts on the 2D precursor with explicit geometry parameters to generate complex buckling deformation. To cope with the costly non-linear optimization, the Equivalent Static Loads (ESL) method is incorporated, with which the non-linear optimization process can be transformed into a linear static optimization process. Therefore, the layout of the pattern of cuts is sought on the 2D precursor with linear sensitivity calculation. Some self-assembly numerical examples demonstrate the effectiveness and efficiency of the proposed approach.

A Footpad Structure with Reusable Energy Absorption Capability for Deep Space Exploration Lander: Design and Analysis

The footpad structure of a deep space exploration lander is a critical system that makes the initial contact with the ground, and thereby plays a crucial role in determining the stability and energy absorption characteristics during the impact process. The conventional footpad is typically designed with an aluminum honeycomb structure that dissipates energy through plastic deformation. Nevertheless, its effectiveness in providing cushioning and energy absorption becomes significantly compromised when the structure is crushed, rendering it unusable for reusable landers in the future. This study presents a methodology for designing and evaluating structural energy absorption systems incorporating recoverable strain constraints of shape memory alloys (SMA). The topological configuration of the energy absorbing structure is derived using an equivalent static load method (ESL), and three lightweight footpad designs featuring honeycomb-like Ni-Ti shape memory alloys structures and having variable stiffness skins are proposed. To verify the accuracy of the numerical modelling, a honeycomb-like structure subjected to compression load is modeled and then compared with experimental results. Moreover, the influence of the configurations and thickness distribution of the proposed structures on their energy absorption performance is comprehensively evaluated using finite element simulations. The results demonstrate that the proposed design approach effectively regulates the strain threshold to maintain the SMA within the constraint of maximum recoverable strain, resulting in a structural energy absorption capacity of 362 J/kg with a crushing force efficiency greater than 63%.

Nested loop approach for topology and shape optimization of crash-loaded deep-drawn components using contact forces for the inner loops

In this paper a new method for the optimization of deep-drawn shell components in crash by using substitute load cases in established sensitivity based static optimization is presented. The optimization process is split into an inner and an outer loop. In the outer loop the contact forces during crash are determined. These contact forces are used as substitute load cases in a static topology optimization, which builds the inner loop of the method. The presented method was inspired by the Equivalent Static Loads Method, but differs primarily in two aspects. These are the use of the contact forces as alternative to equivalent static loads in the linear static topology optimization and the extraction of a shell model that represents the voxel result obtained from topology optimization. A manufacturing constraint is applied in the topology optimization so that the results are shell components. This manufacturing constraint is enabling the mentioned extraction of a shell model. Two examples are presented where the method is applied to minimize the intrusion. The first example is a cantilever, the second example is a structure that is twisted. In both examples a rigid body impacts the structure. The method has shown to be numerical stable and suitable for the shown examples when the intrusion is the objective of the optimization. The intrusion was reduced significantly in both examples. The key novelty of the presented method is the ability to perform a static topology optimization of a voxel design space for a non-linear dynamic optimization model with the result being a shell model that can be analyzed for crash.

Optimization of framed structures subjected to blast loading using equivalent static loads method

In this study, the optimum design of a three-dimensional framed steel structure subjected to blast loading is considered. The main idea of this research is to develop a practical formulation for the design optimization problem and to study the effect of including blast loads in the design process. The optimization problem is formulated to minimize the total weight of the structure subjected to American Institution of Steel Construction (AISC) strength requirements and blast design displacement constraints. The design variables for beams and columns are the discrete values of the W-shapes selected from the AISC tables. A car carrying 250 lbs of trinitrotoluene with a 50 ft standoff distance from the front face is modeled as the source of the blast loading. Pressure–time histories are calculated on the front, sides, roof, and rear faces of the structure. Since the problem functions are not differentiable with respect to the design variables, the gradient-based optimization algorithms cannot be used to solve the problem. Therefore, metaheuristic algorithms are used to solve the optimization problem. Linear and nonlinear dynamic analyses are carried out in the optimization process. The problems are solved using metaheuristic optimization with the equivalent static loads method (MOESL). In MOESL, the dynamic load is transformed into equivalent static loads (ESLs) then the linear static analysis is carried out in the optimization process. The problems are 4-bay × 4-bay × 3-story frames under serviceability and blast loading. It is shown that a penalty on the optimum structural weight is substantial for designing structures to withstand blast loads.

Crash topology optimization for front-end safety parts of battery electric vehicle using an improved equivalent static loads method

Crash topology optimization is a typical nonlinear dynamic response structural topology optimization problem, which is one of the most difficult problem in the structural design field. The equivalent static load method (ESLM) provides a well-defined pattern to solve such difficult problems, which can convert a nonlinear dynamic response optimization into multi-load steps optimization problem with the equivalent static loads (ESLs). However, due to the large deformation in the crash condition and nodal characteristics of the ESLs, it is hard to solve the crash topology optimization directly using the standard ESLM. To expand the application scope of the ESLM, an improved ESLs calculation method is proposed by using the model order reduction method and energy principle, which only acting on some nodes and can be scaled adaptively. Correspondingly, to enrich the connotation of topology optimization, a crash topology optimization method is proposed by using the improved ESLs, which can solve the numerical problems in the design domain by guaranteeing the topology optimization with the improved ESLs perform in the linear rang. First, the principle of the standard ESLM is introduced, and corresponding problems and deficiencies in solving the crash topology optimization are summarized. Then, to solve the above problems, the improved ESLs calculation method is proposed. Meanwhile, the corresponding crash topology strategy is proposed based on the improved ESLs. Finally, to verify the effectiveness and engineering application value of the proposed crash topology optimization method, a test-verified frontal crash simulation model of the BEV front-end is established, and their safety parts are redesigned by using the proposed method. The results show that, the proposed method can effectively solve the crash topology optimization of thin-walled structures under large deformation crash condition. This method also provides a new idea and practical method for the crashworthiness and lightweight design of automobile structures.

A comparison of a level set method and the method of moving asymptotes for the topology optimization of flexible components in multibody systems

AbstractThe gradient‐based topology optimization of flexible multibody systems is considered, where the floating frame of reference method is utilized to model the flexible components with an appropriate efficiency. Thereby, the quality of the optimization results depends, among others, on the chosen gradient calculation strategy and the applied optimization algorithm. Here, both a fully‐coupled time‐continuous adjoint sensitivity analysis and a weakly‐coupled equivalent static load method are tested for gradient calculation. Moreover, both the method of moving asymptotes and a level set method are taken to solve the optimization problem. Different combinations of the mentioned gradient strategies and optimization algorithms are applied for the topology optimization of a flexible piston rod in a slider‐crank mechanism. The corresponding results and comparisons shall be used as quality benchmarks for further studies.

Multi-model optimization for various disciplines using the equivalent static loads method

Multi-model optimization for various disciplines using the equivalent static loads method

Blast responses of BFRP-bar reinforced AAC panels

Basalt fiber reinforced plastic (BFRP) bars can replace steel bars in concrete protective structures because of their advantages of corrosion resistance and high load-bearing capacity. In this paper, the BFRP-bar reinforced autoclaved aerated concrete (AAC) panels (BFRP-AACPs) were designed and prepared. Based on close-in explosion experiments, the dynamic responses and failure modes of the BFRP-AACPs were revealed. Through the quasi-static three-point bending experiment, the residual load capacity of each BFRP-AACP was evaluated, which can quantitatively reveal the damage degree of the BFRP-AACP after explosion. Calculated by the equivalent static load method, the critical scaled distance of the designed panel is 2.08 m/kg1/3, which provides a design basis for the application of the BFRP-AACP.

A Smooth Bidirectional Evolutionary Structural Optimization of Vibrational Structures for Natural Frequency and Dynamic Compliance

A smooth bidirectional evolutionary structural optimization (SBESO), as a bidirectional version of SESO is proposed to solve the topological optimization of vibrating continuum structures for natural frequencies and dynamic compliance under the transient load. A weighted function is introduced to regulate the mass and stiffness matrix of an element, which has the inefficient element gradually removed from the design domain as if it were undergoing damage. Aiming at maximizing the natural frequency of a structure, the frequency optimization formulation is proposed using the SBESO technique. The effects of various weight functions including constant, linear and sine functions on structural optimization are compared. With the equivalent static load (ESL) method, the dynamic stiffness optimization of a structure is formulated by the SBESO technique. Numerical examples show that compared with the classic BESO method, the SBESO method can efficiently suppress the excessive element deletion by adjusting the element deletion rate and weight function. It is also found that the proposed SBESO technique can obtain an efficient configuration and smooth boundary and demonstrate the advantages over the classic BESO technique.

Wind-induced response monitoring of large-span air-supported membrane structure coal-shed under the influence of typhoons

Air-supported membrane structures are being widely used for coal-sheds or the enclosed buildings of stockyards. With the emergence of new functional requirements for buildings, the demand for an increase in both the span and size of air-supported membrane structures has been rapidly increasing in China. Till now, the largest span and length of structures in China have reached 200 m and 1300 m respectively. For air-supported membrane structures, the load-bearing capacity comes from the difference in air pressure between the interior and exterior, which causes the structure to be sensitive to wind and snow loads. This paper studies the wind-induced response of a constructed air-supported membrane structure for a coal-shed cover building with a span of 115 m and width of 317 m, which is located in the southeast coastal city of Yueqing in Zhejiang Province. It is well known that the typhoon-induced response is closely related to the safety and stability of the structure. To ensure the safety of the structure, a newly improved Structure Health Monitoring System (SHMS) has been implemented on a membrane building which is one of the largest scale field monitoring projects anywhere in the world, from where the wind-induced response data was obtained for this paper. Based on the measured wind speeds, the wind pressure distribution coefficient of the membrane surface is derived by numerical simulation with Fluent software. Meanwhile, the displacement response is also calculated by ANSYS APDL. Detailed simulation results and field measurement data are compared for the displacement response caused by Typhoon “Chan-Thu”. It is shown that the former is larger than the latter with the treatment of equivalent wind load. However, the extreme value of the displacement response of the structure with consideration of fluctuating wind load is relatively much more consistent with the measured value. In conclusion, the wind-induced displacement response calculated by the equivalent static wind load method controlled by the principle of maximum displacement equivalence is suitable for engineering applications and wind resistance design, which will result in a slightly higher displacement response evaluation than the real cases.

Random Dynamic Analysis of Wind-Vehicle-Bridge System Based on ARMAX Surrogate Model and High-Order Differencing

To investigate the stochastic characteristics of vehicle-bridge (VB) system under crosswind, an efficient method which combines AutoRegressive Moving Average with eXogenous inputs (ARMAX) model, high-order differencing (HOD) and important sample was proposed in this paper. First, the wind turbulence spectra relative to a moving vehicle and equivalent static gust load method were adopted to simplify the turbulent wind field of VB system, and a wind-vehicle-bridge (WVB) model was established and verified. Then, an analysis framework for WVB system based on ARMAX model was proposed, and HOD method and important sample were used to improve the prediction performance of the surrogate model. Prediction accuracy and calculation efficiency of proposed AMRAX model were verified and compared by Monte Carlo simulation (MCS). Finally, the impacts of vehicle speed and wind velocity on the stochastic characteristics of train response were discussed. Results indicate that the HOD method has significantly improved the prediction performance of ARMAX model for lateral response of trains, and the train responses predicted by ARMAX model based on HOD and important sample show perfect agreement with target results. Compared with MCS, the calculation efficiencies of proposed ARMAX model are improved by about two orders of magnitude. The extreme values of the train response with different vehicle speed and wind velocity gradually obey right skewness distribution, especially the lateral acceleration.

Topology optimization using difference-based equivalent static loads

Topology optimization of crash-related problems usually involves a huge number of design variables as well as nonlinearities in geometry, material, and contact. The Equivalent Static Load (ESL) method provides an approach to solve such problems. This method has recently been extended under the name Difference-based Equivalent Static Load (DiESL) method to employ a set of Finite Element models, each describing the deformed geometry at an individual time step. Only sizing optimization problems were considered so far. In this paper, the DiESL method is extended to topology optimization utilizing a Solid Isotropic Material with Penalization approach (SIMP). The method is tested using an example of a rigid pole colliding with a simple beam, where the intrusion of the pole is minimized. The initial velocity of the pole is varied in order to examine the influence of inertia effects on the optimized structures. It is shown that the results differ significantly depending on the chosen initial velocity and, consequently, that they exhibit inertia effects. This cannot be seen in the results derived by the standard ESL method. Consequently, the results of the DiESL method’s show a considerable improvement compared to those of the standard ESL method.

Optimum design of sandwich composite structures under various dynamic loads using the equivalent static load method

Sandwich composite structures are high-performance materials with greater specific strength and specific stiffness properties, than widely used conventional metals. In this study, sandwich composite structures under dynamic loads were optimized using the equivalent static load method (ESLM). First, a finite element analysis was performed on the sandwich composite structure, using the high-order sandwich panel theory (HSAPT), considering the core material to be compressible. Further, it was optimized by applying the ESLM to three different sandwich composite structures under different types of dynamic loads. The results of the proposed method and conventional dynamic response optimization method were compared. The ESLM was observed to attain a convergence at a rate 1/24 times faster than the conventional method.

Kinematic Chain Equivalent Method for Tube Model and Elastodynamic Optimization for Parallel Mechanism Based on Matrix Structural Analysis

The platforms of parallel mechanisms usually suffer vibration loads. In these cases, structure elastodynamic analysis and elastodynamic optimization of parallel mechanisms are important. A tube structure is very common for parallel mechanisms. This work establishes the model of a tube structure based on matrix structural analysis. The kinematic pair equivalent method is used to simulate the surface contact between the inner and outer tubes. The corresponding mass and stiffness matrices are derived through the strain energy minimization method. The reconfigurable legged lunar lander has been used as an example to verify the effectiveness of this method. By adding the mechanism configuration to the optimization process, the equivalent static load method and the desirability approach are combined and modified. A procedure for the multi-objective elastodynamic optimization of parallel mechanisms is proposed. The optimization procedure is implemented on the lander and the results show a reduction in mass and an increase in natural frequency.

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

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

Owing to the wide existence of multi-source uncertainties in practical engineering, the reliability-based design optimization for structural dynamics is particularly important and meaningful. However, compared to the size and shape optimization, it has rarely been considered in topology optimization with the lack of reasonable uncertainty cognition and the difficulty in the sensitivity solution for reliability constraints calculated from dynamic characteristics. In this research, a new design strategy, which combines the equivalent static loads (ESLs) algorithm and the set-theoretical convex method, is proposed for dynamic response topology optimization under convex reliability constraints. Time-varying boundary rules of the layout response are determined using the state-space transformation and the set perturbation theory. As the safety measurement, a new non-probabilistic reliability index (related to the function of dynamic compliance near the peaks) is defined under the hypervolume-ratio principle and its design sensitivity is further analyzed by solving central finite difference (CFD) equations. Three typical optimization cases, including a rectangular plate, an equivalent control surface and a win rib, are presented to demonstrate the validity and effectiveness of the developed methodology.

Design optimization of thermally conductive support structure for laser powder-bed fusion process with part-scale thermal history

Although there has been a substantial volume of research for mitigating the thermally-induced residual stress with the heat accumulation in powder bed fusion (PBF) process, it has been challenging to quantify, access, and optimize the heat dissipation at part-scale. Furthermore, a practical method of designing three-dimensional support with a reasonable index for evaluating inhomogeneous cooling is also of high importance for facilitating the process. In this study, an optimization methodology for designing a thermally conductive lattice support structure is proposed considering the layerwise heating and cooling in the PBF process. A part-scale numerical model with homogenized properties of lattice material is constructed using the temperature-thread multiscale modeling approach to simulate the transient temperature field in PBF process. The transient analysis model is then integrated into the lattice structure topology optimization (LSTO) with the equivalent static loads method (ESLM) based sensitivity analysis to derive the optimal density profile of the support. In particular, a novel concept of the normalized liquid lifetime contour is proposed to analyze the inhomogeneous cooling in PBF process at part-scale. The normalized liquid lifetime is also utilized to build an effective part-scale thermal load to significantly reduce the overall computational cost for the optimization process. The proposed framework is also validated through thermo-mechanical process analysis and experiment for a benchmark case of manufacturing a twin cantilever beam.