Optimal topology design of structures under dynamic loads

When elastic structures are subjected to dynamic loads, a propagation problem is considered to predict structural transient response. To achieve better dynamic performance, it is important to establish an optimum structural design method. Previous work focused on minimizing the structural weight subject to dynamic constraints on displacement, stress, frequency, and member size. Even though these methods made it possible to obtain the optimal size and shape of a structure, it is necessary to obtain an optimal topology for a truly optimal design. In this paper, the homogenization design method is utilized to generate the optimal topology for structures and an explicit direct integration scheme is employed to solve the linear transient problems. The optimization problem is formulated to find the best configuration of structures that minimizes the dynamic compliance within a specified time interval. Examples demonstrate that the homogenization design method can be extended to the optimal topology design method of structures under impact loads.Presented at WCSMO-2, held in Zakopane, Poland, 1997     

Mixed-integer second-order cone programming for global optimization of compliance of frame structure with discrete design variables

Frame structures are extensively used in mechanical, civil, and aerospace engineering. Besides generating reasonable designs of frame structures themselves, frame topology optimization may serve as a tool providing us with conceptual designs of diverse engineering structures. Due to its nonconvexity, however, most of existing approaches to frame topology optimization are local optimization methods based on nonlinear programming with continuous design variables or (meta)heuristics allowing some discrete design variables. Presented in this paper is a new global optimization approach to the frame topology optimization with discrete design variables. It is shown that the compliance minimization problem with predetermined candidate cross-sections can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution is then computed with an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to examine computational efficiency of the proposed approach.     

Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.     

Topology optimization of frame structures with flexible joints

A method for structural topology optimization of frame structures with flexible joints is presented. A typical frame structure is a set of beams and joints assembled to carry an applied load. The problem considered in this paper is to find the stiffest frame for a given mass. By introducing design variables for beams and joints, a mass distribution for optimal structural stiffness can be found. Each beam can have several design variables connected to its cross section. One of these is an area-type design variable which is used to represent the global size of the beam. The other design variables are of length ratio type, controlling the cross section of the beam. Joints are flexible elements connecting the beams in the structure. Each joint has stiffness properties and a mass. A framework for modelling these stiffnesses is presented and design variables for joints are introduced. We prove a theorem which can be interpreted as the fact that the removal of structural elements, e.g. joints or beams, can be modelled by a small strictly positive material amount assigned to the element. This is needed for the computations of sensitivities used in the applied gradient based iterative method. Both two and three dimensional problems, as well as multiple load cases and multiple mass constraints, are treated.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

基于KRTG的动态拓扑结构的粒子群算法研究

标准的粒子群优化算法作为一种随机全局搜索算法,因其在种群中传播速度过快,易陷入局部最优解。基于KRTG的动态拓扑结构的粒子群算法（KRTGPSO）,从粒子间的拓扑结构出发,动态地调整种群的拓扑结构,增加种群的多样性,使算法收敛于全局最优解。通过测试函数以及与其他算法的比较,并通过实验表明,该算法在收敛速度与数据精度上收到了满意的效果。     

基于拓扑优化的中央空调模块机机架结构设计

为提高中央空调模块机的整体性能和品质,对模块机机架的结构框架进行最优化设计.基于拓扑优化方法,利用HyperWorks建立有限元模型并施加载荷和边界条件;以节点位移为约束条件,以模型体积最小化为目标函数,对模块机机架进行分析和结构优化.结果表明,拓扑优化方法可以获得较优的结构模型,可以改善结构的力学性能,减轻产品质量和降低制造成本.     

Component allocation and supporting frame topology optimization using global search algorithm and morphing mesh

This paper proposes a stepwise structural design methodology where the component layout and the supporting frame structure is sequentially found using global search algorithm and topology optimization. In the component layout design step, the genetic algorithm is used to handle system level multiobjective problem where the optimal locations of multiple components are searched. Based on the layout design searched, a new Topology Optimization method based on Morphing Mesh technique (TOMM) is applied to obtain the frame structure topology while adjusting the component locations simultaneously. TOMM is based on the SIMP method with morphable FE mesh, and component relocation and frame design is simultaneously done using two kinds of design variables: topology design variables and morphing design variables. Two examples are studied in this paper. First, TOMM method is applied to a simple cantilever beam problem to validate the proposed design methodology and justify inclusion of morphing design variables. Then the stepwise design methodology is applied to the commercial Boeing 757 aircraft wing design problem for the optimal placement of multiple components (subsystems) and the optimal supporting frame structure around them. Additional constraint on the weight balance is included and the corresponding design sensitivity is formulated. The benefit of using the global search algorithm (genetic algorithm) is discussed in terms of finding the global optimum and independency of initial design guess. It has been proved that the proposed stepwise method can provide innovative design insight for complex modern engineering systems with multi-component structures.     

Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.     

The exact weight of discretized Michell trusses for a central point load

A discretized optimal structure is derived in a closed analytical form based on Michell truss. The result shows that the discretized optimal structure is most similar to Michell truss in topology and shape. The difference in volume, displacement and strain energy between the discretized optimal structure and Michell truss decreased sharply as the number of members increased in discretized structure. A discretized optimal structure may be obtained from Michell truss by using finite members. This work is meaningful for studying discretized optimal topology based on Michell truss. This result is useful for engineering structural design.     

Topology optimization of frame structures—joint penalty and material selection

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.     

Analytical optimal designs for long and short statically determinate beam structures

Analytical solutions for optimal beam design may serve as benchmarks for numerical studies and for basic understanding. The influence of load case, of boundary conditions and of cross sectional type is severe, so many cases are studied. Short beams based on Timoshenko theory are included in the present energy approach. With a given amount of material/volume the objective is to minimize compliance, and the necessary optimality criterion is to obtain the same gradient of elastic energy along the beam axis x, i.e., for all volumes A(x) d x, where the design function is the area A(x). The beams considered in the paper are geometrically unconstrained, i.e., no minimum and/or maximum constraints are specified for the variable cross sectional area function A(x) other than the volume constraint. The obtained explicitly described designs may be used for comparison with obtained distribution of volume densities for two and three dimensional numerical models, or in the sense of topology optimization for the distribution of the number of active design pixels/voxels. Also the presented decrease in compliance, relative to compliance for uniform beam design, show how much it is possible to obtain for these optimal compliance designs of statically determinate cases.     

Adaptive volume constraint algorithm for stress limit-based topology optimization

Homogenization or density-based topology optimization methods work by distributing a fixed amount of material to the most effective areas of the design domain so as to create an optimal structural configuration that meets the minimum compliance criteria. These topology optimization methods generally cannot control the maximum stress levels of the structure; therefore, the smoothened optimum structure is not guaranteed to be ready for immediate use. This can be because it is either unsafe if the maximum stress at this structure exceeds the strength limit, or over designed if the maximum stress is far below the stress limit. Difficult and complex shape optimization must then be done to obtain a minimum-weight structure that meets the maximum stress constraint. This paper proposes an adaptive volume constraint (AVC) algorithm, a heuristic approach, in place of traditional topology optimization methods so that the maximum stress in the optimal structural configuration will be below the predefined stress limit and the smoothened structure will be directly applicable. In order to test the applicability and robustness of the AVC algorithm, topology optimization using both a traditional fixed volume constraint and an AVC are tested on a number of configuration design problems. To further illustrate the usefulness of the AVC algorithm, shape optimizations at the maximum stress constraint are also conducted on the smooth structural models by both optimization approaches on an identical problem set.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

An alternative interpolation scheme for minimum compliance topology optimization

We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material used. We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q. This increases the probability to obtain a zero-one solution of the relaxed problem. Received July 20, 2000     

Generalized shape optimization using stress constraints under multiple load cases

Optimal shape design using numerical techniques is an increasingly useful engineering tool. Generalized or layout optimal design where the topology of the object is not fixed is one of the emerging applications. These problems are numerically difficult to solve due to the large number of design variables and equality/inequality constraints. Solutions have focused primarily on compliance based minimization under a fixed volume. A more usual engineering approach would be one of minimizing the volume under a stress or deflection constraint. This, however, can lead to problems as stress is a local quantity and volume minimization of multiple load cases under stress constraints may not result in the stiffest design for the remaining material. The approach adopted here is based on a differential rate equation governed by a local operator that defines the state of each element at each time step. This algorithm forms the optimality criteria for the problem. To satisfy the global stress constraints, a feedback derivative is used, analogous to a Lagrange multiplier. The original method for a single load case developed by these authors is extended to deal with multiple load cases. Additionally, a discussion of the global behaviour is included.     

Multi-material structural topology optimization with decision making of stiffness design criteria

A new methodology for making design decisions of structures using multi-material optimum topology information is presented. Multi-material analysis contributes significant applications to enhance the bearing capacity and performance of structures. A method that chooses an appropriate material combination satisfying design stiffness requirement economically is currently needed. An alternative method of making design-decision is to utilize a multi-material topology optimization (MMTO) approach. This study provides a new computational design optimization procedure as a guideline to find the optimal multi-material design by considering structure strain energy and material cost. The MMTO problem is analyzed using an alternative active-phase approach. The procedure consists of three design steps. First, steel grid configurations and composite with material properties are defined as a given structure for automatic design decision-making (DDM). And then design criteria of the steel composites structure is given to be limited strain energy by designers and engineers. Second, topology changes in the automatic distribution of multi-steel materials combination and volume control of each material during optimization procedures are achieved and at the same time, their converged minimal strain energy is produced for each material combination. And third, the strain energy and material cost which is computed based on the material ratio in the combinations are used as design decision parameters. A study in constructional steel composites to produce optimal and economical multi-material designs demonstrates the efficiency of the present DDM methodology.     

Integrated layout and topology optimization design of multi-frame and multi-component fuselage structure systems

The purpose of this paper is to present an extended integrated layout and topology optimization method dealing with the multi-frame and multi-component fuselage structure systems design. Considering an aircraft or aerospace fuselage system including main structure, numbers of frames and featured components located on the frames, a simultaneous optimization procedure is proposed here including geometrical design variables of components and frames as well as topological design variables of main structure and frame structures. The multi-point constraints (MPC) scheme is used to simulate the rivets or bolts connecting the components, frames and structures. The finite circle method (FCM) is implemented to avoid the overlaps among different components and frames. Furthermore, to deal with the difficulties of large numbers of non-overlapping constraints, a penalty method is used here to compose the global strain energy and non-overlapping constraints into a single objective function. To guarantee the fuselage system’s balance, the constraint on the system centroid is also introduced into the optimization. Different numerical examples are tested and the optimized solutions have demonstrated the validity and effectiveness of the proposed formulation.     

Optimal structural topologies with transmissible loads

In optimal topological design of structures one obtains the configuration of optimal structures when the design domain, the displacement boundary conditions and the applied loads are specified. In the optimal structure one often notices a marked difference between the main bearing structure and the load transfer zones. The latter are composed of relatively light elements the exact nature of which is not always very distinct. The main purpose of this paper is to allow the main bearing part of the structure to emerge. Moreover the actual location of the load along its line of action is not always a design requirement. In order to include this relaxed condition regarding the loading position the concept of transmissible or sliding forces is introduced in topological design of structures. A transmissible force is a force of given magnitude and direction which can be applied at any point along the line of action of the force. The optimization formulation is similar to standard topological design procedure in addition to the condition of transmissability of the forces. It is shown that this condition reduces to an equal displacement constraint along the line of action of the forces. The method is illustrated by typical structural examples. It is observed that this numerical method produces indeed crisp images of the main structural components, unblurred by the secondary load transfer elements. It is also indicated that many results are often replicas of Prager structures which were previously obtained by analytical methods. Received March 3, 1999