Structural Topology Optimization for Multiple Load Cases While Avoiding Local Minima

Topology optimization considering multiple loading

An evolutionary shape optimization for elastic contact problems subject to multiple load cases

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.

The algorithms for multi-material topology optimization were developed to solve compliance-minimization problems and applied to engineering problems in automotive concepts and lightweight design. Two small-scale problems of a long cantilever and a control arm were studied initially to verify the effectiveness of the developed algorithms and in-house program. Optimal solutions achieved by the multi-material topology optimization method developed were compared to their counterparts obtained by standard single-material topology optimization. To efficiently solve real-world engineering problems, the algorithms were further advanced to incorporate extrusion constraints and to handle multiple load cases. The effectiveness and the efficiency of the proposed method were demonstrated by the study of two real-world engineering problems: (a) the conceptual design of a cross-member for a chassis frame; and (b) the conceptual design of an automotive engine cradle. The two optimization design problems both involved complex geometries, design and non-design domains, prescribed regions with specific material allocations, multiple load cases, and manufacturing extrusion constraints. It was explicitly demonstrated that, for the same weight, the optimum designs achieved by the multi-material topology optimization method were stiffer than those achieved by standard single-material topology optimization.

Multicriteria optimization that minimizes maximum stress and maximizes stiffness

This paper presents an iterative automated method for optimizing structures with multiple deflection criteria and load cases. The method is based on the principle of virtual work. Discrete sections are selected for structures with fixed geometries. An optimal structure is one which meets all strength and deflection criteria using minimal material. Four case studies are considered in this paper. A simple portal frame is presented to show how the method works. A 60-story frame is optimized to demonstrate the effectiveness of the method for large structures. A warehouse designed by professional engineers is presented to show how the method can be used for structures subjected to complex loading conditions and deflection criteria. The automated method’s solution is 4.5% lighter than the engineers’. Finally, a stepped cantilever is optimized and compared to results in literature. Material savings of up to 14.4% are realized.

Optimal designs of arches under multiple load cases using an automated design routine are presented. The design variables consist of geometric (arch shape) as well as member sizing variables (cross-sectional dimensions). The members are sized on the basis of working stress design procedure, and all the relevant provisions of the AISC specifications are met. The optimal arch shapes produced by the present method for a uniformly distributed load case show a good agreement with the theoretical results. The optimal designs under two and three load cases are also obtained. Under two load cases, the optimal shapes are not very different than a parabolic curve passing through the supports and crown of the arch.

Aiming at the problems such as low computational efficiency and long process in dynamic topology optimization, combined with bi-directional evolutional structure optimization, the dynamic optimization problem is transformed into the static optimization problem in multiple load cases. Newmark method is used to analyze the structure dynamics load. The topology optimization model based on equivalent static load is established. On the basis of the equivalent static load, multi-objective dynamics evolutionary structure optimization is studied. And the mathematical model of the dynamic stiffness and natural frequency about the multi-objective optimization is established. The mathematical formula of the sensitivity is deduced. Then a complete optimization iterative process is given. The numerical example illustrates that it is feasible to solve the multi-objective dynamic topology optimization with equivalent static loads. Modal tracking technique is introduced to the multi-objective dynamic topology optimization. The numerical example shows that modal tracking technology can track the required modal in the optimization process.

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.

Topology optimization of truss lattices, using the ground structure method, is a practical engineering tool that allows for improved structural designs. However, in general, the final topology consists of a large number of undesirable thin bars that may add artificial stiffness and degenerate the condition of the system of equations, sometimes even leading to an invalid structural system. Moreover, most work in this field has been restricted to linear material behavior, yet real materials generally display nonlinear behavior. To address these issues, we present an efficient filtering scheme, with reduced-order modeling, and demonstrate its application to two- and three-dimensional topology optimization of truss networks considering multiple load cases and nonlinear constitutive behavior. The proposed scheme accounts for proper load levels during the optimization process, yielding the displacement field without artificial stiffness by simply using the truss members that actually exist in the structure (spurious members are removed), and improving convergence performance. The nonlinear solution scheme is based on a Newton-Raphson approach with line search, which is essential for convergence. In addition, the use of reduced-order information significantly reduces the size of the structural and optimization problems within a few iterations, leading to drastically improved computational performance. For instance, the application of our method to a problem with approximately 1 million design variables shows that the proposed filter algorithm, while offering almost the same optimized structure, is more than 40 times faster than the standard ground structure method.

High-speed forward flight in helicopters causes high vibratory loads at the rotor hub, which are transmitted into the fuselage. This results in pilot fatigue and high maintenance requirements. Anti-vibration devices or vibration absorbers may be placed near the hub to reduce vibration transmission, while active vibration control systems may typically sense and reduce forces in specific areas of the fuselage. This paper focuses on developing strategies for optimal deployment (sizing, placement and control) of Circular Force Generators (CFGs) for vibration cancellation. Particle Swarm Optimization was used to carry out the optimization for two different load cases corresponding to different flight conditions. Converged solutions for CFG placement were obtained for: different individual load cases; multiple load cases; constrained and unconstrained actuator locations; and using different numbers of actuators. Two CFGs were found to be sufficient to cancel hub loads for a single load case. When two load cases are considered in multi-objective optimization, three actuators can substantially cancel the hub loads (to within 1%). And using four actuators can cancel the hub loads to within 0.1%. Finally, performance under actuator failure was considered as a way to distinguish among otherwise similarly-performing placement solutions.

Ill-loaded layout optimization of bi-modulus material

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.

In truss topology optimization against buckling constraints, the extension from considering a single load case to include multiple loading conditions remains an unsolved problem in the ground structure approach. The present paper suggests a heuristic method attempting to take the multiple load situation into account. A method by Pedersen (1993, 1994) considering only single loading conditions is generalized to include multiple load cases. Based on the ground structure approach the algorithm allows for variable ground structures allowing for, for instance, geometrical restrictions such as concave or even disconnected design domains (Smith 1995b).

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size