Static Topology Optimization Research Articles

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.

Static and dynamic topology optimization: an innovative unifying approach

This paper presents a topology optimization approach that is innovative with respect to two distinct matters. First of all the proposed formulation is capable to handle static and dynamic topology optimization with virtually no modifications. Secondly, the approach is inherently a multi-input multi-output one, i.e., multiple objectives can be pursued in the presence of multiple loads. The input-to-output transfer matrix, say {{{varvec{G}}}}, is the key ingredient that governs the algebraic mapping between applied loads and structural response. In statics {{{varvec{G}}}} depends on the design variables only, whereas it depends on the frequency variable as well in the dynamic case. The Singular Value Decomposition (SVD) of {{{varvec{G}}}} represents then the core of the proposed approach. Singular values are shown to be the gains of the input/output mapping and are used to compute proper norms of {{varvec{G}}} that represent the goal functions to be minimized. Singular vectors provide at no extra cost the plant directions, i.e., the load combination factors that stress the structure the most. Numerical examples are discussed in much detail and open issues object of ongoing investigations are highlighted. A full Matlab code handling the static topology optimization problem is provided as an online Appendix to the manuscript. Its extension to the dynamic case may be gathered following the formulation proposed in Sect. 5.

Efficient structure crash topology optimization strategy using a model order reduction method combined with equivalent static loads

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.

Multi-objective optimization of crashworthiness of vehicle front longitudinal beam

Front longitudinal beam (FLB) is an important structure for the analysis of force transmission and energy absorption path in different crashing scenarios. In the present paper, the finite element (FE) model of a full-scale vehicle yun-100 Zotye is employed to explore the energy absorption of FLB, and aiming at its poor energy absorption problem, circular beam was used as the original topology optimization space of FLB. The advantages of both static and dynamic topology optimization methods are fully considered, and a variable density method is combined with a hybrid cellular automaton (HCA) method. The substructure dissipated energy and force for frontal crash in a full-scale vehicle are obtained at a speed of 13.8 m/s. The maximum stiffness in axial is obtained through the static topology optimization method and the maximum energy-absorbing structure of FLB which is satisfied with axial strength obtained through the dynamic topology optimization method. According to the results of topology optimization, a multi-cell thin-walled structure is interpreted as the novel FLB. A parametric study on geometric parameters is also performed to explore their effects on crashing characteristics of novel FLB, and it is found that they significantly influenced the crashworthiness of multi-cell thin-walled structure (especially the absorption of energy). Furthermore, in order to maximize the characteristics of novel FLB, a multi-objective optimization process is carried out using non-dominated sorting genetic algorithm (NSGA-II). The optimized FLB after multi-objective optimization is applied to Zotye of 100% vehicle frontal crash. The results showed that the peak value of acceleration decreased by 10.3% and the mass of the optimized FLB reduced to 0.59 kg. It indicated that the optimized FLB manifested excellent crashworthiness and better protective characteristics.

An Efficient Topology Optimization Strategy for Structural Crashworthiness Using Model and ESLs Reduction Method

This paper presents an efficient structural topology optimization strategy for crashworthiness. Although equivalent static loads (ESLs) provides a relatively effective way to solve nonlinear dynamic response topology optimization problems, efficiency is still required to improve. To overcome above disadvantage, the proposed method combines the ESLs with model reduction (MR) method. The ESLs method transforms crashworthiness topology optimization problems into a series of linear static topology problems. The model reduction scheme is applicable to reduce the linear matrix equation and ESLs to interface degrees of freedom (DOFs), by selecting interface DOFs between design and non-design space as master DOFs. Therefore, the linear static topology optimization is performed in a reduction manner and the efficiency is improved. The topology optimization results are filtered into a black-white design for the crash simulation to avoid the element distortion problem. The process is repeated until the convergence criteria is satisfied. A numerical example is used to verify the reliability of the proposed method. The results are compared with the dynamic response optimization method using ESLs (ESLSO). The results show that, the proposed method can efficiently solve structural topology optimization problems for crashworthiness.

Crashworthiness Optimization Design of Thin-Walled Tube Filled with Re-entrant Triangles Honeycombs

A novel re-entrant triangles-filled tube (RTT) is proposed through decoupling structural stiffness and energy absorption. Inner re-entrant triangles are employed to satisfy energy absorption, and outer thin wall is used to acquire high stiffness. This paper starts from establishment of theoretical models between geometric parameters of re-entrant triangles and relative density, equivalent elastic modulus and energy absorption characteristics, which are validated by experiments. On this basis, the optimal geometric parameters of unit cell are sought to maximize unit volume energy absorption and minimize relative density by adopting NSGA-II method. Subsequently, the cross-section of tube with optimal stiffness is obtained with targets for maximizing axial stiffness and lateral stiffness by employing static topology optimization method. To verify the proposed optimization method, RTT is analyzed and compared with positive Poisson’s ratio foam-filled tube (PFT), non-filled traditionally optimized tube (NTT) and pre-optimized square tube (PST). The results show that the novel RTT can improve stiffness and energy absorption performance simultaneously. Compared with the positive Poisson’s ratio material, re-entrant triangles honeycomb shows superior advantages in energy absorption. In comparison with the PFT, energy absorption of the RTT increases by 17.23%, and the peak crush force reduces by 5.04%. Therefore, the proposed decoupling design method demonstrates superiority in satisfying various performance requirements simultaneously.

An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads

The standard equivalent static loads (ESLs) method for stiffness topology optimization under crash condition may lead to exaggerated equivalent loads, which is not appropriate to be incorporated into the linear static topology optimization and whereby hinder the optimization process. To overcome this disadvantage, an improved ESLs algorithm based on energy-scaling ratio is proposed to guarantee the numerical stability, especially for the first several cycles with relatively larger differences of strain energy between the original crash simulation and equivalent static analysis. At each cycle, the equivalent external static forces are calculated by multiplying the stiffness matrix and the displacement vector at the time with maximal strain energy during the crash simulation. A further adaptive energy-scaling operation for those forces are performed by a weighting factor of square root of the energy ratio to the standard equivalent static loads for the crash problem based on the judging criterion. The newly equivalent loads are incorporated into the static topology optimization, and topology results are filtered into a black-white design for the crash simulation to avoid the numerical issues due to existing of low-density elements. The process is repeated until the convergence criteria is satisfied. The effectiveness of the proposed method is demonstrated by investing two crash design problems.

Topology optimization of a flexible multibody system with variable-length bodies described by ALE–ANCF

Recent years have witnessed the application of topology optimization to flexible multibody systems (FMBS) so as to enhance their dynamic performances. In this study, an explicit topology optimization approach is proposed for an FMBS with variable-length bodies via the moving morphable components (MMC). Using the arbitrary Lagrangian–Eulerian (ALE) formulation, the thin plate elements of the absolute nodal coordinate formulation (ANCF) are used to describe the platelike bodies with variable length. For the thin plate element of ALE–ANCF, the elastic force and additional inertial force, as well as their Jacobians, are analytically deduced. In order to account for the variable design domain, the sets of equivalent static loads are reanalyzed by introducing the actual and virtual design domains so as to transform the topology optimization problem of dynamic response into a static one. Finally, the novel MMC-based topology optimization method is employed to solve the corresponding static topology optimization problem by explicitly evolving the shapes and orientations of a set of structural components. The effect of the minimum feature size on the optimization of an FMBS is studied. Three numerical examples are presented to validate the accuracy of the thin plate element of ALE–ANCF and the efficiency of the proposed topology optimization approach, respectively.

On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation

This work deals with the topological design of vibrating continuum structures. The vibration of continuum structure is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. In comparison with well-known compliance minimization in static topology optimization, various objective functions are proposed in literature to minimize the response of vibrating structures, such as power flow, vibration transmission, and dynamic compliances, etc. Even for the dynamic compliance, different definitions are found in literature, which have quite different formulations and influences on the optimization results. The aim of this paper is to provide a comparison of these different objective functions and propose reference forms of objective functions for design optimization of vibration problems. Analytical solutions for two degrees of freedom system and topological design of plane structures in numerical examples are compared using different optimization formulations for given various excitation frequencies. The results are obtained by the finite element method and gradient based optimization using analytical sensitivity analysis. The optimized topologies and vibration response of the optimized structures are presented. The influence of excitation frequencies and the eigenfrequencies of the structure are discussed in the numerical examples.

Topology optimization based on level set for a flexible multibody system modeled via ANCF

A topology optimization methodology is proposed for the flexible multibody system undergoing both large overall motion and large deformation. The system of concern is modeled via the absolute nodal coordinate formulation. The equivalent static load method is employed to transform the topology optimization of the nonlinear dynamic response of the system into a static one, and evaluated to adapt to the absolute nodal coordinate formulation by splitting the elastic deformations of the flexible components from the overall motions of those components. During the static topology optimization, the material interface is implicitly described as the zero level set of a higher-dimensional scalar function. Then, the semi-implicit level set method with the additive operator splitting algorithm is employed to solve the corresponding Hamilton-Jacobi partial differential equation. In addition, the expert evaluation method of weights based on the grey theory is utilized to define the objective function, and a modified augmented Lagrange multiplier method is proposed to treat the inequality volume constraint so as to avoid the oscillation and drift of the volume. Finally, two numerical examples are provided to validate the proposed methodology.

Nonlinear dynamic response topology optimization using the equivalent static loads method

A novel method for nonlinear dynamic response topology optimization is proposed using the equivalent static loads (ESLs) method. The ESLs are the loads that generate the same response field of linear static analysis as that of nonlinear dynamic analysis at each time step. In the proposed procedure, nonlinear dynamic analysis is performed, ESLs are made and linear static topology optimization is carried out with the ESLs. The process cyclically proceeds until the convergence criterion, which is specifically defined for this problem, is satisfied. Since the density method for topology optimization is utilized, the low-density finite elements can cause mesh distortion in nonlinear dynamic analysis. Transformation variables are introduced for a new update method for the incorporating process of the topology results into nonlinear dynamic analysis. Also, a new objective function is proposed to minimize the peaks of the time dependent transient responses. A couple of standard problems and a practical problem are solved to validate the proposed method.

Application of the New Interpolation Functions on Topology Optimization of Continuum Structure for Static and Mode Analysis

To improve the convergence for topology optimization of continuum structure, a new material interpolation model is proposed. This function curve parameterized by certain penalty is similar to solid isotropic microstructures with penalization (SIMP) curve. In static topology optimization problem, the new interpolation function maintains the advantage of robustness and efficiency similar to the SIMP model. In low density, the ratio of density to interpolation function is kept in limited value. Localized mode problem is overcome naturally based on the new interpolation scheme in dynamic topology optimization problem. Two-dimensional numerical examples are used to test the proposed model and method.The results show that the proposed model and method are feasible and robust in topological optimization design of continuum structure.

Dynamic Response Topology Optimization in the Time Domain Using Equivalent Static Loads

Most topology optimization techniques find the optimal layout of a structure under static loads. Some studies are focused on dynamic response topology optimization because the dynamic forces act in the real world. Dynamic response topology optimization is solved in the time or frequency domain. A method for dynamic response topology optimization in the time domain is proposed using equivalent static loads. Equivalent static loads are static loads that generate the same displacement field as dynamic loads at each time step. The equivalent static loads are made by multiplying the linear stiffness matrix and the displacement field from dynamic analysis and used as multiple loading conditions for linear static topology optimization. The results of topology optimization are utilized in dynamic analysis again and a cyclic process is utilized until the convergence criterion is satisfied. The paradigm of the method was originally developed for size and shape optimizations. A new objective function is defined to minimize the peaks of the compliance in the time domain and a convergence criterion is newly defined considering that there are many design variables in topology optimization. The developed method is verified by solving some examples and the results are discussed.

등가정하중을 사용한 선형 동적반응 위상최적설계 기초연구

All the forces in the real world act dynamically on structures. Design and analysis should be performed based on the dynamic loads for the safety of structures. Dynamic (transient or vibrational) responses have many peaks in the time domain. Topology optimization, which gives an excellent conceptual design, mainly has been performed with static loads. In topology optimization, the number of design variables is quite large and considering the peaks is fairly costly. Topology optimization in the frequency domain has been performed to consider the dynamic effects; however, it is not sufficient to fully include the dynamic characteristics. In this research, linear dynamic response topology optimization is performed in the time domain. First, the necessity of topology optimization to directly consider the dynamic loads is verified by identifying the relationship between the natural frequency of a structure and the excitation frequency. When the natural frequency of a structure is low, the dynamic characteristics (inertia effect) should be considered. The equivalent static loads (ESLs) method is proposed for linear dynamic response topology optimization. ESLs are made to generate the same response field as that from dynamic loads at each time step of dynamic response analysis. The method was originally developed for size and shape optimizations. The original method is expanded to topology optimization under dynamic loads. At each time step of dynamic analysis, ESLs are calculated and ESLs are used as the external loads in static response topology optimization. The results of topology optimization are used to update the design variables (density of finite elements) and the updated design variables are used in dynamic analysis in a cyclic manner until the convergence criteria are satisfied. The updating rules and convergence criteria in the ESLs method are newly proposed for linear dynamic response topology optimization. The proposed updating rules are the artificial material method and the element elimination method. The artificial material method updates the material property for dynamic analysis at the next cycle using the results of topology optimization. The element elimination method is proposed to remove the element which has low density when static topology optimization is finished. These proposed methods are applied to some examples. The results are discussed in comparison with conventional linear static response topology optimization.

Evolutionary static topology optimization of cyclic-symmetry structures

The topology optimization of cyclic-symmetry structures is studied by means of bi-directional evolutionary structural optimization (BESO) method.Due to the retention of non-uniform finite element discretization relative to one such type of structures, it is found that the sensitivity result depends upon the volume of the concerned finite element.As a result,different finite clement discretizations will change the clement removal/growth sequence in the BESO procedure.For this reason,a new concept of element sensitivity density is proposed to avoid the element-volume-dependence of the optimal design.Meanwhile,a new volume-weighted filtering technique is developed to control the checkerboard for the corresponding mesh.To verify the proposed method,a variety of numerical tests related to the concentrated load are studied in detail.Discussions show that the size effects of basic cells are impor- tant.