Some key issues in topology optimization
Zhang Weihong
Northwestern Polytechnical University, 710072 Xi’an, China
tel.(fax): 029-88495774
Abstract: In this paper, four key issues related to topology optimization of materials and structures are discussed: size effect in the integrated topology design of materials & structures, topology optimization of large-scale repetitive structures, design-dependent load problems and topology optimization of multi-component mechanical systems. These problems not only exist for static design problems, but also for dynamic and other loading and regime conditions. Some numerical results are given to illustrate recent advancements of theChino-French Laboratory of Concurrent Engineering (LACFIS) research group ofNorthwesternPolytechnicalUniversity in the mathematical modeling and developments of efficient optimization strategies in this field.
Key words: topology optimization, size effect, design dependent load, large-scale repetitive structures, multi-component layout
1. Introduction
Designs of efficient structures with lightweight and elegant mechanical properties, e.g.,energy absorption, thermal isolation, anti-impactconstitute always a challenging work for aerospace, aeronautical and automotive industries. In recent years, topology optimization becomes a powerful approach to fulfill this task. As illustrated by the survey work of Rozvany[1], Bendsøe and Sigmund [2], it was initially developed to perform overall structural layout design where the macroscopic solid-void solution pattern is obtained by means of either the homogenization method [3,4]，the SIMP (Solid Isotropic Material with Penalization) method [5, 6] or other methods like Voigt-Reuss mixing rule [7]. Now, it is expandedfor tailoring effective properties of cellular materials and the integrated design of materials and structures. The former deals with the microstructural configuration design of porous materials and multiphase composite materials. The latter deals with the design of the material microstructures involved in macrostructures for their optimal matching in local and global performances.
Although many results and successful applications have been achieved until now, some underlying issues are still unsolved perfectly. The motivation of the current wok is to deal with four key issues in this context: the problem of size effect indicates the influence of the representative cell size upon its optimal topology. This is because the size effect becomes often ignored when the material microstructure, e.g., the porous core design of sandwich panel, is replaced with an equivalent continuum by the homogenization method. Improving the efficiency of the design procedure is the basic concern for topology optimization of large-scale repetitive structures. With the structural periodicity feature, the substructure technique is adopted instead of using direct finite element analysis. Besides, as opposed to the fixed loads, design-dependent load problems concern with topology optimization under complicated loading conditions where loads vary with the iterative design solutions. For examples, thedistribution and magnitude of inertia loads, thermal loads, gravity loads depend on the spatial layout of materials when they areiteratively added or deleted in the specific design domain. As a result, the traditional concept “more materials, stiffer the structure” may become erroneous as the addition of materials will also introduce severe loads. Topology optimization of multi-component mechanical systems deals with the problem where the packaging and interconnections of a set of components of specific geometries have to be optimized systematically in a limited space to meet both the geometrical constraints and system performances.
2. Scale effect of the material microstructure[8-16]
A sandwich panel is illustrated in Fig. 1. Assume that the hexagonal microstructure of the core material can take different scales but with the same volume fraction of the solid material. As a result, the whole structure may exhibit different performances under the specific loading.This mechanism is the so-called scale effect. So, design optimization of the material microstructure requires that both the morphology and the scale of the unit cell have to be considered in a unified way.However, with the homogenization method written below,
(1)
the same effective elastic tensor will be obtainedin all cases of (a-d).
Figure 1. Scale effect of the cellular core with the same microstructure and volume fraction
The proposed procedure is to carry out the design in two steps: macroscale layout design and refined topology design of material microstructure. The first one is to find a preliminary material layout in the macroscale level. The second one is to find the refined configuration of material microstructure based on the first one. As the number of representative cellsincreases, the topology solution is found to converge asymptotically to the homogenized solution. In Fig. 2, the evolution of optimal configurations and values of structural compliance reveals clearly the size effect for the stiffness maximization design of a clamped bending beam structure.
C2×1＝165061 C4×2=168025.8 C8×4=176616.6 C∞=186282.8
Figure 2. Evolution of optimal topology solutions versus the cell size
3. Topology optimization of large-scale repetitive structures[17-20]
Arepetitive structure can be considered as a result of the translation or rotation of a representative substructure. As such a repetitive structure often corresponds to a large-scale finite element model, it is important to increase the computing efficiency and meanwhile to ensure the structural periodicity in the design procedure. Instead of using direct FE analysis (DFE) of the whole discretized structure, the super-element technique (SE) of substructure partition is applied for each reanalysis. The SE is defined by the representative substructureto condense internal d.o.fsand involved finite element densities are identified as design variables in topology optimization. Meanwhile, the design variable linking strategy is used to ensure the periodicity of the optimized configuration from one substructure to another. A comparison of computing time for one static FE analysis of a thin-walled cylinder is given in Table 1 to show the benefice of using SE substructure.
Method / Element number / D.o.fs / TimeCase 1 / DFE
SE / 90,000 / 334746 / 7’18”
2’12”
Case 2 / DFE
SE / 150,000 / 559377 / 16’20”
8’48”
Tab. 1 Comparison of FE analysis time with and without SE substructure
Figure 3. Optimal topology of thin-walled cylinder with SE substructure
4. Design-dependent load problems[20]
Generally speaking, the study of design-dependent loads started with the structural topology design under pressure loads whose location and direction change with the loading surface to be designed.Thus, the influence of the design-dependent loads constitutes an important issue to be clarified. The main problem lies in that the optimal solution might be singular with full zero material density over the design domain whenever an upper bound imposed to the total volume of all solid elements is formulated as an inequality constraint, or oppositely the optimal solution might be non-singular but upper bound of the volume constraint will be never attained.
Physically, it is easy to clarify the problem from FE system equation and sensitivity analysis.
(2)
(3)
With the design-dependent load effect, the first term in Eq.(3) will be no longer zero so that the derivative may change the sign and the objective function associated with will becomenon-monotonous. Therefore, the volume constraint involved in the topology optimization model may be inactive at the optimum solution. Consider the stiffness maximization design of the following rotating disk of cyclic symmetry, loaded tangentially by P. A volume fraction of 40% is assumed. A variety of optimal configurations are obtained depending upon the relative magnitude of ex-centrifugal forceand P. Note that andrepresent the structural compliance related to and, respectively.
Problem model /
/
/
Figure 4. Influence of design-dependent load upon optimal configurations
5. Topology optimization of multi-component mechanical system [21]
This is a coupled design of the components packing and supporting structures. As shown in Fig. 5, most of aeronautic and aerospace structural systems belong to this kind of multi-component system made up of a container, i.e., a design domain and a number of components and structures to support and interconnect the container and components.
Figure 5. A rocket structure
We propose a Coupled Shape and Topology Optimization (CSTO) technique. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a Finite-Circle Method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. Since the FE mesh discretizing the packing space, i.e., design domain, has to be updated iteratively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain are introduced in this work instead of using traditional pseudo density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure and design sensitivities are calculated w.r.t both geometric variables and density variables.Consider the design domain shown in Fig. 6. The basic mesh is divided into 30×60 finite elements. Suppose the density points are defined at the centroids of the basic elements. Two identical solid components are presently taken into account. Each component will be approximated with two circles.
Figure 6. Design domain and basic mesh of the plate
Figure 7. The final configuration and the iteration history
Both componentsare initially placed at the center of the plate. By minimizing the strain energy as the design objective and constraining the volume of the supporting structures to 30% of the total volume, this problem is carried out with CSTO. The final configuration is shown in Fig. 7(1), where the componentsact as a part of the structure. The iteration history of the objective function is shown in Fig.7(2).
6. Conclusions
In this paper, four key issues related to topology optimization of materials, structures and multi-components system are discussed. New design procedures and methods are proposed and some representative examples are provided to verify their validities. The size effect analysis indicates that the homogenization method leads to the asymptotic solution when the size of the representative cell tends to be zero. The superelement technique allows reducing greatly the reanalysis time for the topology optimization of repetitive structures. Based on the sensitivity analysis, the nomonotonicity of the objective function associated with design-dependent load problems reveals that the optimal solution may never attain the upper bound of volume constraint. The packing and connection design of multi-components system extends the current concept of topology optimization and demonstrate the potential applications in aerospace and aeronautical industries.
Acknowledgements: This work is supported by the Natural Science Foundation of China (90405016, 10676028), 973 Program (2006CB601205), 863 Program (2006AA04Z122), Doctoral Research Foundation (20060699006), Aeronautical Science Foundation (04B53080, 2006ZA53006), Shaanxi Province Research Program (2006K05-G25) and 111 Project（B07050）.
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