Simultaneous layout design of supports and structures systems

Transcription

1 8 th World Congress on Structural and Multidisciplinary Optimization June 1-5, 2009, Lisbon, Portugal Simultaneous layout design of supports and structures systems J.H. Zhu 12, P. Beckers 2, W.H. Zhang 1, D.H.Bassir 3 1 The Key Laboratory of Contemporary Design & Integrated Manufacturing Technology, Northwestern Polytechnical University, Xi an, Shaanxi , China 2 LTAS-Infographie, University of Liège, Liège 4000, Belgium 3 Faculty of Aerospace Engineering, TU Delft, the Netherlands 1. Abstract This paper is to demonstrate the further developments and applications of the related techniques in the integrated layout design of supports and structures based on the previously proposed techniques for the integrated layout optimization of multi-component system. The design procedure mainly consists of two parts. Firstly, the layout of the supports that is described with the positions of movable support components on the boundary of the design domain. These components are partially embedded into the design domain and subjected to the applied boundary conditions. Meanwhile, geometrical constraints are imposed to avoid the overlap of multiple components. Secondly, the layout optimization of the support components and the structure for which locations of the components and the pseudo-densities defined on the density points are assumed as the geometrical and topological design variables, respectively. The technique of embedded meshing is employed to adapt the topology optimization to the variation of the finite element mesh caused by the component movement. Varieties of numerical examples are tested and solved to validate the proposed method. Both surface load and self-weight load are taken into account. More complexities of partially supported components are introduced in the presented examples. 2. Keywords: Layout optimization; Support components; Density points; Embedded meshing 3. Introduction The supports layout design of a structural system plays a crucial role in improving the mechanical performances. As a descriptor of boundary conditions, a slight change of the support positions may lead to a significant variation of the structural responses. The optimal design of support positions is thus of great importance in most structural engineering fields, especially in building constructions, workpiece machining fixture, welding or rivets joints of marine and aircraft structures. In the earlier researches, the support design was formulated as a quasi-shape optimization problem where a few support positions were optimized for some beam or plate structures. Within this scope, optimal support locations were found to improve elastic and plastic responses of beams [1-4]. Likewise, the column support optimization was discussed for buckling load [5-7] or natural frequency maximization [8-13]. For example, different techniques like the minimum stiffness design of the accessional supports [8] and the concept of material derivative [9] were investigated to derive the design sensitivities of the natural frequency with respect to the support positions and to maximize the fundamental natural frequency. Later, the frequency sensitivity of closed-form to a support position was derived by means of the classical normal modal method for an Euler-Bernoulli beam [10] or by using the Rayleigh quotient in conjunction with the Lagrange multiplier [12]. The genetic algorithm was also employed to determine optimal support positions of beam structures for a wide variety of boundary conditions [11]. Normally, these structural supports are simulated with some spring elements. Ideas based on the element shape function were also applied to formulate the global stiffness matrix of the beam including a support located on the beam element [13]. Then, it was found that the optimal support position relies greatly on the support stiffness. And in many cases, the optimal support position is not unique for plate structures when supports are stiff enough [14-15]. The introduction of topology optimization method made it possible to carry out the layout design of the supports. The structural connections that are modeled with spring elements were firstly designed with the technique of topology optimization [16]. The concept of the pseudo-densities in topology optimization was introduced to describe the stiffness factors of these spring elements. Then an optimization procedure was further provided to deal with the number, position and stiffness of the supports [17]. Similar methods with the spring element simulating the structural supports were adopted to find the optimal support layout maximizing the natural frequencies of beams and plates [18]. Furthermore, simultaneous topology designs of the structural layout and their boundary conditions or interconnections were also addressed with more complexities. The pseudo-densities of both the spring elements and the structural elements are imposed as the design variables in topology optimization. Topology optimization can be carried out for 3D truss structures with support variables involved in the optimization problems in the 1

2 meantime [19]. In more recent works [20], the layout of the continuous structure and supports are designed simultaneously to find the efficient compliant mechanism or to maximize the global stiffness. Even numbers of independent solid components are treated as sub-domains with different volume fractions in topology optimization and their interconnections are modeled by spring elements [21-23]. The concept of integrated layout design was later proposed and implemented [22] for the component positions and the structures. The movements of rigid components were described using a predefined material interpolation model, i.e., the geometrical movement of a component was simulated as a physical variation of the material properties. Besides, intermediate stiffness was attributed to the elements located on the boundary of the components to interpolate the variation of the material properties. Recently, the technique of density points and embedded meshing [24] were proposed as alternative methods dealing with the topology optimization with movable components in the design domain. With the extension of this work, the integrated layout design of structures and supports is studied in this paper. Compared with existing models presented above, the supports are considered here as structural components of specific configurations and rigidities that act as joints between the structure and the boundary fixation condition. Hence, some movable supports are defined and embedded into the design domain to model the attached fixations. Positions of these components and pseudo-densities of elements in the design domain are assigned as geometrical and topological design variables, respectively. During the optimization process, these design variables will be updated simultaneously toward a design pattern of 0-1 material layout and a proper layout of supports. (a) (b) Fig. 1. Schematic of integrated layout design of supports and structure 2. Formulation of integrated layout optimization 2.1 Description of the problem As shown in Fig. 1, the integrated layout design of supports and structures is actually a structural topology design problem with the design-dependent boundary conditions. The layout of fixations is designable depending upon the locations of concerned support components. For a meshed design domain with specific loads, several support components of different shapes will be embedded along the design domain boundary and meshed into finite elements as integrated parts. The overall structural responses can be obtained by finite element analysis. With the designable boundary conditions associated with the support components, design optimization can be performed iteratively to optimize such a lightweight structure and the component positions as illustrated in Fig. 1(b). Thus, the support components may have a variety of prescribed forms and a portion of the support component is embedded into the design domain while the fixations are applied on the rest part of the component. Instead, in the previous works [20-23], fixations are directly imposed on the boundary nodes of the structure without support components. 2.2 Density points and embedded meshing techniques In the framework of traditional SIMP model-based topology optimization, it is known that the density variables used to describe the material layout are always defined with respect to a fixed finite element mesh, i.e., one density variable is assigned to one element. But for the current design problem, positions of the support components and the material layout of the structure will be modified simultaneously at the design stage. The finite element mesh of the design domain has to be updated iteratively to follow the variations of component positions in a consistent way. Consequently, the traditional topology optimization method cannot be directly applied for the integrated layout design. Here, the techniques of density points and embedded meshing are employed to take into account the mesh variation 2

3 caused by the movable locations of support components along the design domain boundary. As shown in Fig. 2, the support component and the design domain are meshed independently with quadrangular elements. Here, the initial mesh of the design domain is referred to as the basic mesh where the density points are defined at the centroids of the corresponding elements. Fig. 2. Basic mesh with density points and the meshed support component When the support component is embedded in the design domain as shown in Fig. 3(a), Boolean operations are carried out to subtract the overlapping elements between the basic mesh and the support component. Affected elements of both the basic mesh and the support component indicated with gray color will be locally modified by adding transition elements to ensure the consistent connection between them while the rest of the basic mesh elements are kept to be unchanged. Note that the transition elements of the design domain are locally restricted within each corresponding element of the original basic mesh as shown in Fig. 3(b). Hence, density values of the added transition elements will still be dominated by the same density points of the basic mesh and a remesh of the whole structure is avoided. In this way, when the support component changes its location as shown in Fig. 3(c), the basic mesh will be simply restored and only the Boolean operation and addition of the transition elements will be locally repeated. As shown in Fig. 3(d), one density point can dominate several finite elements in the transition area between the component and the structure. But in the rest part of the design domain, one density point corresponds to only one quadrangular element. (a) (b) (c) (d) Fig. 3. Schematic of density points and the process of embedded meshing 3

4 3. Mathematical formulations Based on the above presentation, the integrated layout design is a simultaneous optimization of structural topology and support positions. If the global compliance of the structural system is chosen as the objective function to be minimized, this problem can be mathematically stated as i i n s s ns find: η 1,2,..., ; 1,2,..., 1 min: C 2 T f G U (1) ( L) ( U) s.t.: 0 i 1, i 1,2,..., n ; s s s, 1,2,..., ns; ( L) ( U ) ( L) V V V ; g g, 1,2,..., ng where η is the vector of design variables defined by the pseudo-density values of density points. n η is the total number of pseudo-density variables, i.e. the number of density points. s is the vector of geometrical design variables describing the locations of the support components. f is the vector of external forces. G is the vector of self-weight load of the total structural system. C is the global strain energy of the structural system which is computed by the load vector times the nodal displacement vector U. V is the total volume of the consumed material for the structure. An equality constraint is assigned for the total volume of the materials used for the structure. For the easy processing, this constraint is relaxed to an inequality constraint with tighten bounds of V (L) and V (U). g ζ represents the geometrical constraint function used to avoid the overlap between the support components. It limits the distance between the components explicitly. 3.1 Sensitivity analysis In this work, the gradient-based optimization algorithm is used to solve problem (1). To do this, sensitivity analysis is needed as the basic information. Firstly, design sensitivities of the global strain energy are derived with respect to the pseudo-density variables. Suppose following relations hold for the material density and elastic modulus in terms of pseudo density variables, respectively. i ii0 (2) E P E i i i0 where E i0 and ρ i0 are the elastic modulus and density of solid material attributed to the elements controlled by the ith density point. P(η i ) is the penalization function whose expression will be detailed later. The derivative of the strain energy can be thus written as T C 1 T U G f G U (3) i 2 i i Then, based on the finite element equation of the system f G KU (4) the differentiation of both sides results in G K U K U (5) i i i 1 U G K K U (6) i i i The substitution of (6) into (3) produces C 1 T 1 G K T G f G K U U i 2 i i i (7) T G 1 T K U U U i 2 i At the element level, one can calculate the derivative of the self-weight load vector of the jth element dominated by the ith density point Gij i0g V ij (8) i where g is the gravity acceleration vector. V ij is the volume of the jth element dominated by the ith density point. Because the global load vector of self-weight G is an assemblage of all element gravity vectors, the derivative of G can be easily calculated as an assemblage of Eq. (8). 4

5 Note that in the right member of Eq. (7), the presence of first term implies the compliance function is a non-monotonic function of pseudo density variables. Similarly, at the element level, one has Kij P( i ) K (9) ij0 i where K ij is the stiffness matrix of the jth element dominated by the ith density point. The derivative of K with respect to η i can be evaluated by summing derivatives of the element stiffness matrices from all elements dominated by the ith density point. In the same way, one can evaluate sensitivities with respect to geometrical parameters noted with s ε. C T G 1 T K U U U (10) s s 2 s Theoretically, the semi-analytical sensitivity scheme [25] can be adopted so that derivatives of G and K can be approximated by finite difference * G G G G (11) s s s * K K K K (12) s s s where G * and K * are the global self-weight loading vector and global stiffness matrix after the mesh perturbation, respectively. In practical implementations, Eq. (10) can be directly approximated by finite difference due to the complexity of mesh perturbation. 3.2 Consistent interpolation model As is well known, the popular SIMP model takes the exponential form. Consider here different penalties for material density and Young s modulus introduced [26]. i ii0 (13) p Ei P( i ) Ei 0 i Ei 0 with p to be the penalty factor. The penalization function, P(η i ), and the ratio function, η i /P(η i ) are shown in Fig. 4. Obviously, the latter tends to be infinitely large when η i varies towards zero. This indicates that the mass and stiffness are relatively inconsistent [27]. Physically, this means that the element of weak density value is too compliant to support the self-weight. Such an inconsistence will further lead to the problem of localized deformation. Fig. 4. Standard SIMP interpolation model Fig. 5. Improved imterpolation model of Eq. (14) Therefore, consistent material interpolation models have to be used to penalize the element stiffness and the self-weight. To avoid the large ratio value, an alternative interpolation model is used in this paper. i ii0 1 p i (14) Ei P( i ) Ei 0 i Ei 0 Besides the penalty factor p, another factor α is introduced to control the maximum value of the ratio η i /P(η i ). The new penalization function and the corresponding ratio η i /P(η i ) are also plotted in Fig. 5 with α=16 and p=3. It is 5

6 seen that the large ratio value is well controlled. 4. Numerical examples In this section, two numerical problems with numbers of support components of different forms are tested to verify the proposed techniques. The GCM algorithm [29] implemented in BOSS-Quattro TM optimization module is used to solve the optimization problems. Two loading conditions associated with the surface load and the self-weight are taken into account in both examples. Material properties are as follows, Support components: E 0 = pa, ρ 0 = 7800kg/m 3 ; Structure: E 0 = pa, ρ 0 = 2700kg/m Bridge problem 1 The design domain is shown in Fig. 6. The basic mesh is composed of quadrangular finite elements. Suppose the road on the bridge is the non-designable area. As the movable fixations, four support components with an identical size of 0.6m 1m will be located symmetrically on the bottom boundary of the design domain. Due to the symmetry, only two position variables are involved as geometrical variables for the four components. Firstly, a uniform surface load of 10000N/m is assumed on the top of the road and the self-weight of the structural system is ignored. The volume fraction is constraint to be 40% of the design domain for the topology optimization of the structure. During the optimization, the material layout and the locations of the components are updated simultaneously. Intermediate design configurations are presented in Fig. 7. Fig. 6. The design domain and the support components The convergence history of the objective function is shown in Fig. 8. At the beginning of the optimization process, the support components may be located away from the main loading members. As a result, some instability occurs in the convergence curve. After 25 iterations, significant reductions of the global strain energy are obtained and a bridge-like structure is achieved with the satisfaction of volume fraction constraint. In the continuing iterations, the support components are still moving slightly to find their proper positions. In the second case, suppose the self-weight of the structural system is taken into account together with the uniform surface pressure. A gravity acceleration of 10m/s 2 is applied to the global structural system. The optimization starts from the same initial design with the same assignment of volume fraction. Several intermediate layout patterns of the iteration process are shown in Fig. 9. The evolution of the structural layout and the support components is similar to the first case. Although similar bridge-like structure and the support layout are obtained, one can observe that most of the structural materials are distributed near the support components against the self-weight. The convergence history of the objective function is shown in Fig. 10. Although the value of the objective function tends to be stable after about 40 iterations, the support components are still moving toward the final positions until the convergence. (a) 5 th iteration 6

7 (b) 10 th iteration (c) 25 th iteration (d) The final design, C =0.1298J Fig. 7. Iteration history of the design patterns Fig. 8. The convergence history of the global strain energy (a) 5 th iteration 7

8 (b) 10 th iteration (c) 40 th iteration (d) The final design, C =8.005 J Fig. 9. Iteration history of the design patterns Fig. 10. The convergence history of the global strain energy (a) 5 th iteration 8

9 (b) 10 th iteration (c) 25 th iteration (d) The final design Fig. 11. Iteration history of the design patterns Now, the example is further tested with only two allowed to move along the bottom edge, while the rest two ones have to be located on the left and right sides of the design domain. With the volume fraction of 40%, several structural patterns are shown in Fig. 11. The structural layout becomes clear very quickly. But it costs much more iterations to find proper positions for the support components. The convergence history of the objective function is shown in Fig. 12. Finally, the problem is tested by supposing that two support components are located on the left and right sides of the design domain, respectively. With the same definition as the previous one, some of the iterative solutions are shown in Fig. 13. Two support components are always located at the two bottom corners, which provide strong supports for the global structure from the beginning to the end. The structure pattern evolves quickly but the locations of the support components move rather slowly during the design process. The convergence curve is shown in Fig. 14. No instability is found during the iteration history. Fig. 12. The convergence history of the global strain energy 9

10 (a) 5 th iteration (b) 10 th iteration (c) 25 th iteration (d) The final design Fig. 13. Iteration history of the design patterns Fig. 14. The convergence history of the global strain energy 10

11 Fig. 15. Definition of the support component 4.2 Bridge problem 2 This test consists of support components with more complex shapes. As shown in Fig. 15, the solid parts with dashed boundary lines are not allowed to connect with the base structure of the design domain. The solid material layout is only attainable outside the void area. As illustrated in Fig. 16, four identical support components will be located symmetrically along the span of the design domain and the horizontal positions are assigned as the geometrical design variables. The basic mesh consists of quadrangular finite elements and two top layers of the elements are assigned as the non-designable area loaded with a uniform pressure 10000N/m and fixations are defined at the two ends. The volume fraction is still set to be 40% in the following optimization. Firstly, consider the case without gravity. The integrated layout optimization is carried out. Several intermediate layout configurations are presented in Fig. 17. In the first 10 iterations, support components vary significantly to find the proper locations whereas no clear structure pattern is generated in the design domain. Thereafter, the structural layout is figured out even some slight movements still take place in the following iterations. The optimal solution indicates that the final structure has a symmetric configuration and is strongly supported by the components. As expected, only connectible sections of the support components join together with the obtained structure. The convergence history of the objective function is shown in Figure 18. Fig. 16. Definition of the design domain and the initial configuration (a) 5 th iteration (b) 10 th iteration 11

12 (c) 25 th iteration (d) The final design Fig. 17. Iteration history of the design patterns Fig. 18. The convergence history of the global strain energy Secondly, consider the case with gravity. The acceleration rate is set to be 10m/s 2. The optimization starts from the same configuration as the previous one. Due to the effect of the self-weight, intermediate designs shown in Fig. 19 are obviously different from the previous ones. The components tend to be optimally placed after 10 iterations. The structure is also growing up from the connectible sections of the components to the loaded top surface. The final structural consists of some curved structural branches. Finally, the optimization gives rise to a clear configuration of the structural layout and component position with a relatively fast convergence. The convergence history is shown in Fig. 20. (a) 5 th iteration (b) 10 th iteration 12

13 (c) 25 th iteration (d) The final design Fig. 19. Iteration history of the design patterns Fig. 20. The convergence history of the global strain energy 5. Conclusions An integrated layout optimization method is proposed in this paper to deal with the simultaneous design of structure and support layout. Based on the previously proposed layout design techniques of multi-component system, the supports are considered as solid components that are partially embedded into the design domain. With the boundary conditions applied on these components, the support layout is thus described as the components layout. More limitations to the structural layout design is imposed with the definition of partially connectible components. Techniques like density points and embedded meshing are successfully implemented to avoid the conflict between the structural layout description and the mesh variation due to the movements of the components. Numerical examples with and without gravity are studied, respectively to show the effect of design-dependent load. To avoid the numerical singularity in the presence of gravity, an improved interpolation model is used instead of the SIMP model. Numerical iterations produce reasonable configurations that show evidently the optimal match between the structural layout and supports layout. 6. Acknowledgements This work is supported by the National Natural Science Foundation of China ( , ). 7. References [1] G. Rozvany, Optimization of unspecified generalized forces in structural design. Journal of Applied Mechanics-ASME. 41, , 1974 [2] Z. Mroz, G. Rozvany, Optimal design of structures with variable support positions. Journal of Optimization Theory and Applications. 15, , 1975 [3] W. Prager, G. Rozvany, Plastic design of beams: Optimal locations of supports and steps in yield moment. International Journal of Mechanical Sciences. 17, ,

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