Computational Algorithms for Generating Efficient and Innovative Load-carrying Structures

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1 Computational Algorithms for Generating Efficient and Innovative Load-carrying Structures XIAODONG HUANG, YI MIN XIE, KAZEM GHABRAIE School of Civil, Environmental and Chemical Engineering RMIT University GPO Box 2476V, Melbourne, Victoria 3001 AUSTRALIA Abstract: - The evolutionary structural optimization (ESO) method is based on the simple concept of gradually removing underutilized material and at the same time adding efficient material to the structure. As a result, the resulting structural shape evolves towards an optimum. This paper presents the latest developments of ESO method in generating efficient and innovative structures, including optimal design of bridges, optimization of periodic structures, optimal design of a structure with multiple materials and shape and reinforcement optimization of underground tunnels. Key-Words: - Structural Optimization, Computational Algorithm, Periodic Structure, Underground Tunnel 1 Introduction Topology optimization for continuum structures has attracted considerable attention in the last three decades and many optimization methods have been developed based on the finite element analysis (FEA) [1-2]. With the development of the computer technology, structural optimization has achieved more popularity in structural design to generate efficient and innovative load-carrying structures. Initially, the evolutionary structural optimization (ESO) method was based on an empirical concept that a structure evolves towards an optimum by slowly removing inefficient material [2]. More recently, this idea had been extended to allow for efficient material to be added to the structure at the same time as the inefficient one is removed [3, 4]. The ESO method proves to be capable of solving size, shape and topology optimization problems for static, dynamic, stability and heat transfer and so on [2]. The ESO method appeals to practicing engineers and architects particularly because of its simplicity and effectiveness. It can be easily implemented and linked to commercial FEA packages such as ABAQUS, NASTRAN and ANSYS. Anyone who has a basic knowledge of FEA can readily understand and apply the ESO method. To demonstrate the ESO concept, an example of using ESO for topology optimization problem of a classical Michell s structure [5] is given below, in which one seeks to find the stiffest design of a cantilever as shown in Fig.1. The final design with 50% volume fraction is shown in Fig. 2 which agrees well with solutions from other topology optimization methods [1]. This paper will present some interesting and innovative structural designs resulting from the recent developments of ESO [4, 6-8]. Fig. 1 Topology optimization problem for a Michell s structure. Fig. 2 Optimal design by the ESO method ISSN: ISBN:

2 2 Optimal Bridge Designs The ESO method can be applied to various engineering practices such as bridge design. The objective of the design is maximizing the stiffness of the design with a prescribed material volume. The following examples will demonstrate the optimal bridge designs under various design requirements and boundary conditions. The first example is a bridge shown in Fig.3(a) for support conditions and Fig.3(b) for the crosssection of the design domain where the deck on the top of the bridge is not allowed to be removed and is therefore defined as non-designable domain (0.6m deep). ESO finds the optimal solution shown in Fig.4 which looks similar to the Sydney Bridge. Fig. 4 Optimal design of the bridge. The second example shows another bridge design with different design domain and dimensions. Fig.5 shows the design domain which is a 3D beam with a narrow gap of 1m width in the middle of the beam beneath the deck. No material is allowed to fill the gap so as to leave an enough space for ship. The top deck is simulated with 1.5m plates defined as non-designable domain. By adding and removing material simultaneously, ESO finds the optimal design shown in Fig m 0.6m (a) Fig. 5 Design domain for a bridge-type structure. p Fig. 6 Optimal design for the bridge. Nondesign domain 0.6m (b) Fig. 3 Supports conditions and dimensions of design domain. 3. Optimal Design for Periodic Structures Periodic structures e.g. the honeycomb core of a sandwich plate are widely used in the structural designs due to their lightweight and ease of fabrication. The objective of the problem is to find the stiffest design of structure and simultaneously satisfy the geometrical periodic pattern. For example, a sandwich cantilever undergoing four vertical concentrated loads is shown in Fig.7. The designable core of the size is divided into a mesh using eight-node cubic ISSN: ISBN:

3 elements and both non-designable skins with a unit thickness are divided into a mesh using four-node plate elements. Fig.8 and Fig.9 show the optimal designs from ESO program for the number of unit cells m=4 1 1 and m=4 2 1 respectively. 4. Optimal Design for Structures with Multiple Materials In practice, a structure may be designed with multiple materials e.g. a sandwich structure. For example, we design a cantilever with a concentrated load at the centre of the free end as shown in Fig.10. The design is made of two types of materials as Young s moduli E i and objective volume fractions V i given by E 1 =200GPa, E 2 =40GPa and V 1 =0.10, V 2 =0.15. The final optimal design with materials 1 and 2 is shown in Fig. 11 and design for material 1 in Fig. 12. Fig.7 The optimization problem for a sandwich structure. Fig.10 The optimization problem for a structure with multiple materials. Fig.8 Optimal design for a sandwich structure with m=4 1 1 Fig.11 Optimal solution for materials 1 and 2. Fig.9 Optimal design for a sandwich structure with m=4 2 1 Fig.12 Optimal solution for material 1. ISSN: ISBN:

4 5. Shape and Reinforcement Optimization of Underground Tunnels Topology and shape optimization techniques e.g. ESO have proved to be useful in solving the shape and reinforcement optimization of underground tunnels [].The tunnel is considered under biaxial insitu stress state with the horizontal to vertical stress ratio of λ=σ2/σ1=1.2. To model the in-situ stress conditions normal distributed loads with consistent magnitudes have been applied on top, right and left sides and the bottom is restrained against vertical displacement as shown in Fig.13. To prevent reinforcing far fields a non-designable rock element layer has been defined on the outer boundary of design domain. The minimum dimension of the opening is also restricted to height of 1.6m and width of 2.4m by a setting a rectangular area of nondesignable voids. Using the ESO technique, the final solution is shown Fig.14. Fig. 14 The final shape and reinforcement of the tunnel. Fig. 15 The final shape and reinforcement of the tunnel with the traffic load. Fig. 13 An initial guess design illustrating the design domain, non-designable elements, in-situ loading, and restraints. If a traffic load is considered by applying a distributed load on the tunnel floor over the width of non-designable void elements (2.4m). The magnitude of 0.25σ1 has been considered for this load. The final solution is illustrated in Fig.15. As expected the area under the traffic load attracts more reinforcement in compare to results obtained in previous example. It can also be seen that the optimum shape of the hole has also changed. When two identical and parallel tunnels are considered in the optimization problem, the optimal solution is shown in Fig.16. It can be seen that the shape and reinforcement of the tunnels are different from the above cases due to the interaction of two tunnels. Fig.16 The final shape and reinforcement of two parallel tunnels ISSN: ISBN:

5 6. Conclusion This paper has described recent developments in evolutionary structural optimization and given a variety of illustrative examples. The ESO algorithm is capable of solving a wide range of structural optimization problems and generating efficient and innovative load-carry structures. With the rapid development of computer technology, the ESO method has great potential to become a powerful design tool for structural designers. References: [1] Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comp. Meth. Appl. Mech. Engrng. Vol. 71, 1988, pp [2] Xie YM, Steven GP. Evolutionary Structural Optimization. London: Springer, [3] Yang XY, Xie YM, Steven GP, Querin OM. Bidirectional evolutionary method for stiffness optimization. AIAA Journal Vol. 37, No.11, 1999, pp [4] Huang X, Xie YM. Convergent and meshindependent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design, Vol.43, 2007, pp [5] Michell AGM. The limits of economy of material in frame structures. Phil. Mag. Vol.8, pp [6] Huang X, Xie YM. Optimal design of periodic structures using evolutionary topology optimization. Struct. Multidisc. Optim. DOI /s [7] Huang X, Xie YM. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics, (Accepted). [8] Ghabraie K, Xie YM and Huang X. Shape and reinforcement optimization of underground tunnels. Computers and Geotechnics (Submitted). ISSN: ISBN: