Topology optimization of trusses with local stability constraints and multiple loading conditions?a heuristic approach

Dynamic fragmentation is one of the primary fracture types for rock mass in rock engineering. The dynamic fragmentation process and dynamic fragment size characteristics are important for the fragmentation optimization in field. Most researchers studied the fragmentation forming process via dynamic compression test using the traditional split Hopkinson pressure bar (SHPB) system, in which the multiple dynamic loading is generally applied on the rock specimen. However, the effect of the multiple loading on the dynamic rock fragmentation characteristics was rarely attempted. In this study, the momentum-trap system is used in SHPB to achieve the single dynamic compressive loading on the rock specimen. Westerly granite and Fangshan marble were chosen to investigate the effect of multiple dynamic compressive loading pulses on the fragmentation characteristics. During the dynamic loading, the fragmentation process was captured using the high-speed camera. The results show that under the specific loading rate, the multiple loading has no influence on measuring the dynamic compressive strength of rocks, but it was observed in the high-speed images that more fragments were produced in the multiple loading. Moreover, the failure mode of tested specimens under distinct compressive loading rates was examined. The results indicate that the fragmentized degree of rocks with single dynamic loading is less than that with multiple dynamic loading at a given loading rate. This trend is relatively obvious when the loading rate is lower. In addition, the D50 values of rocks with multiple dynamic loading are smaller than those with single dynamic loading. The fragment size distribution also indicates that the multiple loading pulses dominantly influence the produce and size distribution of the fine-grain fragment. Furthermore, the mean fragment size derived from Weibull distribution decreases as the loading rate increases. The multiple loading pulses could decrease the mean fragment size and generate more fine grains when comparing to the fragment size of rocks derived from the single loading pulse. Therefore, the multiple compressive loading may result in an inaccurate fragmentation size analysis, and the single dynamic compressive loading should be applied when evaluating the dynamic fragmentation efficiency of rocks.

A novel topology optimization model based on homogenization methods was developed for predicting bone density distribution and anisotropy, assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness. The feasibility and efficiency of this method were tested on a 2D model for a proximal femur under single and multiple loading conditions. The main aim was to compute homogenized optimal designs using an optimal laminated microstructure. The computational results showed that high bone density levels are distributed along the diaphysis and form arching struts within the femoral head. The pattern of bone density distribution and the anisotropic bone behavior predicted by the model in the multiple load case were both in good agreement with the structural architecture and bone density distribution occurring in natural femora. This approach provides a novel means of understanding the remodeling processes involved in fracture repair and the treatment of bone diseases.

This paper considers the mathematical properties of discrete or discretized mechanical structures under multiple loadings which are optimal w.r.t. maximal stiffness. We state a topology and/or sizing problem of maximum stiffness design in terms of element volumes and displacements. Multiple loads are handled by minimizing the maximum of compliance of all load cases, i.e., minimizing the maximal sum of displacements along an applied force. Generally, the problem considered may contain constraints on the design variables. This optimization problem is first reformulated in terms of only design variables. Elastic equilibrium is hidden in potential energy terms. It is shown that this transformed objective function is convex and continuous, including infinite values. We deduce that maximum stiffness structures are dependent continuously on the bounds of the element volumes as parameters. Consequently, solutions to sizing problems with small positive lower bounds on the design variables can be considered as good approximations of solutions to topology problems with zero lower bounds. This justifies heuristic approaches such as the well-known stress-rationing method for solving truss topology problems.

This paper presents an application of topology optimization and response surface method to optimize the geometry of a bicycle crank arm and the experimental validation of it. This is purposely to reduce the crank arm mass and create a preliminary design of a lightweight structure necessary for the high-performance bicycle development. A three-dimensional bike crank arm model was made in the SpaceClaim software followed by a static finite element analysis using ANSYS Workbench 2019 R1. A multiple cycling load was applied simultaneously in seven crank angles of 30, 45, 60, 90, 120, 135, and 150° relative to the horizontal position to create the multiple loads to the crank. From there, topology optimization was then conducted to investigate the effect of mass constraint, stress constraint, angle of cycling, and crank materials on the topological pattern result. To minimize stress concentration at corners, a shape optimization using the response surface method was conducted and obtained the final geometry. From the result, it is shown that both optimization methods not only successfully reduce the crank arm mass and provide several optimum design options but also are able to reduce the maximum stress in the crank arm up to 20% after the optimization process. The experimental validation using a newly developed wireless measurement system shows a considerable agreement to the numerical results.

A novel parameterization concept for structural truss topology optimization is presented in this article that enables the use of evolutionary algorithms in design of large-scale structures. The representational power of Boolean networks is used here to parameterize truss topology. A genetic algorithm then operates on parameters that govern the generation of truss topologies using this random network instead of operating directly on design variables. A genetic algorithm implementation is also presented that is congruent with the local rule application of the random network. The primary advantage of using a Boolean random network representation is that a relatively large number of ground structure nodes can be used, enabling successful exploration of a large-scale design space. In the classical binary representation of ground structures, the number of optimization variables increases quadratically with the number of nodes, restricting the maximum number of nodes that can be considered using a ground structure approach. The Boolean random network representation proposed here allows for the exploration of the entire topology space in a systematic way using only a linear number of variables. The number of nodes in the design domain, therefore, can be increased significantly. Truss member geometry and size optimization is performed here in a nested manner where an inner loop size optimization problem is solved for every candidate topology using sequential linear programming with move-limits. The Boolean random network and nested inner-loop optimization allows for the concurrent optimization of truss topology, geometry, and size. The effectiveness of this method is demonstrated using a planar truss design optimization benchmark problem.

Topology optimization (TO) is currently a focal point for researchers in the field of structural optimization, with most studies concentrating on single-loading conditions. However, real engineering structures often have to work under various loading conditions. Approaches addressing multiple-loading conditions often necessitate subjective input in order to determine the importance of each loading condition, aiming for a compromise between them. This paper proposes a so-called bisection constraint method (BCM), offering a unique, user-preference-independent solution for TO problems amidst multiple-loading conditions. It is well-known that minimizing the system’s compliance is commonly used in TO as the objective. Generally, compliance is not as sufficient as stress to be used as a response to evaluate the performance of structures. However, formulations focusing on minimizing stress levels usually pose significant difficulties and instabilities. On the other hand, the compliance approach is generally simpler and more capable of providing relatively sturdy designs. Hence, the formulation of min–max compliance is used as the target problem formulation of the proposed method. This method attempts to minimize compliance under only one loading condition while compliances under the remaining loading conditions are constrained. During the optimization process, the optimization problem is automatically reformulated with a new objective function and a new set of constraint functions. The role of compliance under different loading conditions, i.e., whether it is to be treated as an objective or constraint function, might be changed throughout the optimization process until convergence. Several examples based on the solid isotropic material with penalization (SIMP) approach were conducted to illustrate the validity of the proposed method. Furthermore, the general effectiveness of the compliance approach in terms of stress levels is also discussed. The calculation results demonstrated that while the compliance approach is effective in several cases, it proves ineffective in certain scenarios.

Topology optimization considering multiple loading

In this paper, topology optimization under multiple independent loadings with uncertainty is presented. In engineering practice, load uncertainty can be found in many applications. From the literature, researchers have focused mainly on problems containing only a single uncertain external load. However, such idealistic problems may not be very useful in engineering practice. Problems involving multi-loadings with uncertainty are more commonly found in engineering applications. This paper presents a method to solve a system which contains multiple independent loadings with load uncertainty. First, a two-level optimization problem is formulated. The upper level problem is a typical topology optimization problem to minimize the mean compliance in the design using the worst case conditions. The lower level optimization problem is to solve for the worst loadings corresponding to the critical structure response. At the lower level formulation, an unknown-but-bounded model is used to define uncertain loadings. There are two challenges in finding the worst loading case: non-convexity and multi-loadings. The non-convexity problem is addressed by reformulating the problem as an inhomogeneous eigenvalue problem by applying the KKT optimality conditions and the multi-uncertain loadings problem is solved by an iterative method. After the worst loadings are generated, the upper level problem can be solved by a general topology optimization method. The effectiveness of the proposed method is demonstrated by numerical examples.

Computer Aided Engineering (CAE) including the topology optimization technique has been successfully used in many mechanical industries such as automotive industries. However, most mechanical designers, in fact, cannot utilize CAE applications by themselves because of the sophisticated operations and a huge amount of time for the analysis. Recently, the concept of a new CAE, First Order Analysis (FOA), has been proposed in order to overcome these problems. In this paper, we present the topology optimization method using beam elements corresponding to the concept of FOA. First, the mutual stiffness is defined using a mutual energy concept. Next, a multi objective optimization problem to deal with the multiple loading problem is formulated, and the optimization procedure is developed based on the ground structure approach and sequential convex programming. Finally, some examples are provided to confirm the optimization method proposed here.

The design obtained from a topology optimization problem can be sensitive to the type and details of the ground structure used. A new type of ground structure containing hinged beam elements is described that increases the types of elements in the ground structure available to the optimizer. In addition to the pin-ended truss and rigidly-connected beam elements that have traditionally been used, two new types of elements are introduced: (1) a beam with a hinge on one end and a solid connection on the other end, (2) a beam with hinges on both ends. These elements have different deflection characteristics than those of the typical truss or beam elements, and can be used in ground structure-based compliant mechanism design. Given a reduced ground structure, these new elements effectively increase the design space. Pin-ended members with lengths larger than those of the ground structure cell size are permitted to develop, reducing the sensitivity of solutions to the cell density of the reduced ground structure. Furthermore, because these longer members do not require intermediate lateral support to provide stability, as series connections of shorter pin-ended members do, fewer ancillary members are required. Two compliant mechanism problems are solved to demonstrate the effectiveness of these new elements.

The methods of optimizing structure's topology include two classes, analytical method andnumerical method. The Michell theory is an analytical method,developed early and having a great influence on structural topologyoptimization study. Thispaper mainly focuses on the numerical methods of optimizing truss, frame and continuumstructures. The ground structure approach is usually used to optimize trusses and frames. Theoptimum topology is achieved by deleting parts of members in trusses or frames. To optimizecontinuum structures, the design region is divided into finite elements. The optimum topologyis achieved by deleting parts of elements to form continuum structures with holes. Three kindsof material models are generally used, isotropic, anisotropic and thatwith microstructures. The numerical instabilities in (0-1) topology optimization ofcontinuum structures are analyzed. The methods in common use to overcomethis difficulty include perimeter control, local gradient constraint, mesh-independence filter methods and topologyanalysis. The singular optimum in topology optimization of structures with stressconstraints is introduced briefly. The main mathematical methods to solve topologyoptimization are discussed.

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.

A kinematic stability repair algorithm for planar truss topology via geometric decomposition

Some design considerations related to the optimization of structural topologies are discussed. The basic difficulties involved in the solution process are first reviewed and the ground structure approach is discussed. Formulating the problem as a nonlinear programming problem, the computational effort involved in the solution process might become prohibitive and solutions with a singular optimal topology might be encountered. Neglecting the implicit compatibility equations, an explicit approximate problem is obtained. However, the approximate displacement constraints might be meaningless in cases where members are eliminated from the structure. If the latter constraints are neglected, the resulting problem can be stated as linear programming under certain assumptions. The difficulties involved in topological optimization suggest that a two-stage design procedure might be useful. Several such procedures, where an approximate solution is evaluated at the first stage and modified at the second stage, are presented. It is shown that the first-stage solution can be viewed as the final optimum if a set of control forces or displacements is applied. The latter can readily be determined at the second stage by solving a linear programming problem.

Topology optimization of truss structures is considered in this paper. Trusses are widely used in various constructions: bridges, towers, roof supporting structures. Topology optimization of trusses requires large amount of computing resources. Therefore distributed computer networks are used to solve this kind of problems. In this paper a distributed branch-and-bound combinatorial algorithm for topology optimization of trusses is investigated in computational grid infrastructure and clusters of personal computers.