Nonlinear response topology optimization using equivalent static loads—case studies

Nonlinear structural optimization is fairly expensive and difficult, because a large number of nonlinear analyses is required due to the large number of design variables involved in topology optimization. In element density based topology optimization, the low density elements create mesh distortion and the updating of finite element material with low density elements has a severe effect on the optimization results in the next cycles. In order to overcome these difficulties, the equivalent static loads method for nonlinear response structural optimization (ESLSO) primarily used for size and shape optimization has been applied to topology optimization. The nonlinear analysis is performed with the given loading conditions to calculate equivalent static loads (ESLs) and these ESLs are used to perform linear response optimization. In this paper, the authors have presented the results of five case studies with material, geometric and contact nonlinearities showing good agreement and providing justification of the proposed method.     

Shape optimization of the initial blank in the sheet metal forming process using equivalent static loads

In the sheet metal forming process, forming the final desired shape is difficult to obtain due to wrinkling, tearing, failure of material, etc. Various conditions of the forming process should be controlled for the desired shape. These conditions are the velocity of the punch, the friction factor, the blank holding force, the initial shape of the blank and others. Many researchers have conducted studies to predetermine the initial blank shape. The structural optimization technique is one of them. Non‐linear response structural optimization is required because non‐linearities are involved in the analysis of the metal forming process. When the conventional method is utilized, the cost is extremely high due to repeated non‐linear analysis for function and sensitivity calculation. In this paper, the equivalent static loads (ESLs) method is used to determine the blank shape which leads to the final desired shape and reduced wrinkling. The ESLs method is a structural optimization method where non‐linear dynamic loads are transformed into ESLs, and these ESLs are utilized as external loads in linear response optimization. The design is updated in linear response optimization. Non‐linear analysis is performed with the updated design and the process proceeds in a cyclic manner. An optimization formulation is defined for the examples, the formulated problems are solved to verify the proposed method and the results are discussed. Non‐linear analysis is performed using the commercial software LS‐DYNA, NASTRAN is used for calculating the ESLs and linear response optimization, and an interface program for LS‐DYNA and NASTRAN is developed. Copyright © 2010 John Wiley & Sons, Ltd.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

动载荷激励下板壳结构加筋自适应成长方法

自适应成长法是基于自然界分支系统形态形成机理的一种高效结构拓扑优化设计方法。通过引入等效静态载荷法理论,运用自适应成长法解决板壳加筋结构在承受动载荷激励下的动态响应拓扑优化设计问题。根据板壳结构所受的动载荷边界条件,构建以动柔度为目标的优化数学模型,推导迭代公式,使板壳结构的加强筋从"种子"开始,沿着使结构最佳力学性能方向成长,从而形成最优加强筋分布形态。研究在简谐载荷和冲击载荷作用下的板壳结构加强筋设计例,并与静态载荷作用下的设计结果进行比较。研究结果表明,板壳结构在动态载荷作用下,其主加强筋布局形态和在静态载荷作用下相同,但在靠近载荷作用点附近出现与主加强筋平行的截面积较小的加强筋,以增加抵抗动态载荷的作用;而冲击载荷作用下的加强筋与一般简谐载荷作用下的加强筋相比,多出一层较复杂的框型筋板抵抗瞬时冲击力。     

An efficient approach for reliability-based topology optimization

This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15–30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.     

On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented. The research is funded by the Danish Natural Science Research Council and the Danish Research Council for Technology and Production Sciences.     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

Topology optimization of energy absorbing structures with maximum damage constraint

A novel density‐based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed. This framework enables topologies to absorb large amount of energy via plastic work before failure occurs. To account for the plasticity and damage during the energy absorption, a coupled elastoplastic ductile damage model is incorporated with topology optimization. Appropriate material interpolation schemes are proposed to relax the damage in the low‐density regions while still ensuring the convergence of Newton‐Raphson solution process in the nonlinear finite element analyses. An effective method for obtaining path‐dependent sensitivities of the plastic work and maximum damage via adjoint method is presented, and the sensitivities are verified by the central difference method. The effectiveness of the proposed methodology is demonstrated through a series of numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

考虑设计相关载荷的稳态热传导拓扑优化设计

针对稳态热传导问题,以结构散热弱度最小为目标,建立了连续体传热结构的拓扑优化模型和方法,给出了相应的算例。优化方法中分别建立了设计相关载荷和非相关载荷的灵敏度列式,采用Rational Approximation of Material Properties (RAMP)方法对材料密度进行惩罚,利用优化准则法控制设计目标与材料分布,以敏度过滤技术抑制棋盘格效应。算例的结果直观显示了设计相关载荷和非设计相关载荷以及复合载荷对结构拓扑构型的影响规律,表明了该文考虑设计相关载荷的稳态热传导结构拓扑优化方法的合理性。     

A novel algorithm using an orthotropic material model for topology optimization

This article presents a novel algorithm for topology optimization using an orthotropic material model. Based on the virtual work principle, mathematical formulations for effective orthotropic material properties of an element containing two materials are derived. An algorithm is developed for structural topology optimization using four orthotropic material properties, instead of one density or area ratio, in each element as design variables. As an illustrative example, minimum compliance problems for linear and nonlinear structures are solved using the present algorithm in conjunction with the moving iso-surface threshold method. The present numerical results reveal that: (1) chequerboards and single-node connections are not present even without filtering; (2) final topologies do not contain large grey areas even using a unity penalty factor; and (3) the well-known numerical issues caused by low-density material when considering geometric nonlinearity are resolved by eliminating low-density elements in finite element analyses.     

A simple method for first dynamic mode of plastic deformation for suddenly loaded beams

This paper presents a simple method for determining the position of plastic hinges under dynamic loading of constant magnitude. The essence of the method is to convert a dynamic problem into a static one by means of D'Alembert's principle and then to perform equilibrium analyses. It provides a way of studying and understanding dynamic problems in some cases, and can also be regarded as an attempt to use the so called “equivalent structure technique” for dynamic loading. Four cases are studied and they show the merit of this method. For curved structures, to ensure a linear distribution of inertia forces for easy sketch, the mass is assumed to be uniformly distributed along the chord instead of the arc. Calculations of the cases indicate that this assumption is reasonable and the approximated solutions are in very good agreement with those from “exact” analyses. It is hoped that the method will be complementary to the existing one in better understanding the physics involved and in providing a quick, though maybe approximate, solution on the mode of plastic deformation of structures subject to sudden loading.     

Topology optimization for nonlinear dynamic problems: Considerations for automotive crashworthiness

Crashworthiness of automotive structures is most often engineered after an optimal topology has been arrived at using other design considerations. This study is an attempt to incorporate crashworthiness requirements upfront in the topology synthesis process using a mathematically consistent framework. It proposes the use of equivalent linear systems from the nonlinear dynamic simulation in conjunction with a discrete-material topology optimizer. Velocity and acceleration constraints are consistently incorporated in the optimization set-up. Issues specific to crash problems due to the explicit solution methodology employed, nature of the boundary conditions imposed on the structure, etc. are discussed and possible resolutions are proposed. A demonstration of the methodology on two-dimensional problems that address some of the structural requirements and the types of loading typical of frontal and side impact is provided in order to show that this methodology has the potential for topology synthesis incorporating crashworthiness requirements.     

Topology optimization of continuum structures for natural frequencies considering casting constraints

A method for the topology optimization on the natural frequency of continuum structures with casting constraints is proposed. The objective is to maximize the natural frequency of vibrating continuum structures subject to casting constraints. When the natural frequencies of the considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this article, the topology optimization is performed by the bi-directional evolutionary structural optimization (BESO) method. The effects of different locations of concentrated lump mass, different volume fractions and meshing sizes on the final topologies are compared. Both two and four parting directions are investigated. Several two- and three-dimensional numerical examples show that the proposed BESO method is effective in achieving convergent solid–void optimal solutions for a variety of frequency optimization problems of continuum structures.     

The elastic–plastic analysis of frames by the theorems of structural variation

The theorems of structural variation for rigidly jointed frames are extended to cover the nonlinear elastic–plastic analysis of frames to collapse. Strain hardening and other material changes are included in the analysis. It is shown that the entire load–deflection history of a frame, following the successive formation of plastic hinges up to and including collapse, is obtained from a single initial elastic analysis. The manner in which these theorems are used to insert real (or plastic) hinges to a frame is presented and details of the manual calculations involved in the elastic–plastic analysis of a portal is given to demonstrate the simplicity of the method. This is followed by a comparison of the results with those obtained experimentaly by Baker and Charlton. The versatility of the method is then demonstrated by altering the shape of a frame before its elastic–plastic analysis.     

Shifted integration technique in one-dimensional plastic collapse analysis using linear and cubic finite elements

The present study is concerned with the physical explanations of the linear and the cubic finite elements for beams and axisymmetric shells through comparisons of their strain energy approximations with those of the Rigid Bodies-Spring Models which are discrete elements suitable for plastic collapse analysis using the concepts of plastic hinges and hinge lines. The established conditions for the equivalence between these two modellings, which are given as the relations between the locations of the numerical integration points and those of the occurrence of plastic hinges, can be conveniently used in the economical plastic collapse analysis of framed structures and axisymmetric shells where the locations of plastic hinge formations are controlled by the movement of numerical integration points. Some numerical results are shown in order to prove numerically the obtained relations and to verify the validity of the proposed shifting technique of numerical integration points, which is identified as ‘the shifted integration technique’ in the present paper.     

巷道吸能支护的减震防冲效应分析

冲击地压是煤矿开采的主要动力灾害，巷道吸能支护是防御冲击地压灾害的新型支护方式和有效手段。吸能支护是在刚性支护基础上附加阻尼耗能构件形成的巷道支护，基于巷道顶板与支护相互作用的动力学模型，分析了在巷道刚性支护与吸能支护作用下的顶板-支护系统动力响应，同时就阻尼构件在吸能支护上的分布特征对减震防冲效应的影响进行了分析，研究了阻尼构件在支护中的串联、并联、混联3种分布特征下的支护吸能减震防冲效应。结果表明：相比于刚性支护，吸能支护不仅能有效抑制顶板的冲击响应，还对支护体的冲击响应具有自保护能力；串联吸能支护模式与混联吸能支护模式对顶板冲击位移的控制及支护体加速度的抑制作用相当，且均优于并联吸能支护模式，其中，在串联吸能支护模式下，顶板冲击位移可下降约89%，支护体加速度可下降约55%。进一步优化串联吸能支护模式可知，当采用支护体上端串联布置吸能构件时，构件吸能效果发挥的最好，并且支护体的变形、应力、等效塑性应变变化平稳且幅值较小，同时相比于在下端以及两端串联吸能构件时支护等效塑性应变分别下降约77%和96%。该研究为冲击地压动力灾害的防冲吸能支护动力可靠性设计提供思路。     

基于塑性铰模型的三塔斜拉桥抗震能力时程分析

摘要：采用非线性塑性铰考虑结构构件的混凝土开裂、钢筋屈服和材料滞回效应,利用APDL语言编制二分法程序使计算机进行自动计算和减少计算量,从而建立通过不断增大地震时程作用来计算斜拉桥结构抗震能力的方法,并以该方法分析研究了一座在建的三塔结合梁斜拉桥的抗震能力和地震反应特性。研究结果表明：在抗震能力状态,三塔斜拉桥中塔底内力响应要远大于边塔,但由地震引起的塔顶位移却相差很小,所以提高中塔抗震能力可以较快地提高整体结构的抗震能力;有塑性铰模型要比无塑性铰模型具有更高的抗震能力;采用二分法循环计算程序,不仅可以降低人工试算的烦琐而且能大幅减少循环计算次数,使大型结构的抗震能力分析较为简便。     

Structural shape optimization using equivalent static loads transformed from dynamic loads

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered only in small‐scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. This algorithm can be applied to large‐scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain. The optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large‐scale shape optimization problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Material nonlinear topology optimization considering the von Mises criterion through an asymptotic approach: Max strain energy and max load factor formulations

This paper addresses material nonlinear topology optimization considering the von Mises criterion by means of an asymptotic analysis using a fictitious nonlinear elastic model. In this context, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. Two nested formulations are considered. In the first, the objective is to maximize the strain energy of the system in equilibrium, and in the second, the objective is to maximize the load factor. In both cases, a volume constraint is imposed. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved using direct minimization of the structural strain energy using Newton's method with an inexact line-search strategy. Four numerical examples demonstrate the features of the proposed material nonlinear topology optimization framework for approximating standard von Mises plasticity.