Topology Optimization of Multicomponent Beam Structure via Decomposition-Based Assembly Synthesis

[+] Author and Article Information
Naesung Lyu, Kazuhiro Saitou

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125

J. Mech. Des 127(2), 170-183 (Mar 25, 2005) (14 pages) doi:10.1115/1.1814671 History: Received April 21, 2003; Revised April 19, 2004; Online March 25, 2005
Copyright © 2005 by ASME
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Outline of the approach (a) design domain, (b) ground structure consisting of basic members and potential joint locations, (c) the ground topology graph Gg, (d) optimization, (e) best product topology graph G (subgraphs representing components are annotated as C1–C3, and edges in joint set J is shown in dashed lines), and (f) optimal multicomponent structure
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Constraints for feasible topology: (a) feasible structure, (b) infeasible structure violating Connectivity Constraint 1, (c) points considered in Connectivity Constraint 2 (A: boundary condition, B: loading, and C: displacement), (d) infeasible topology violating Connectivity Constraint 2 (point A not connected), and (e) infeasible topology violating Connectivity Constraint 2 (point B–C not connected)
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Representative Pareto optimal designs: (a) R1 (3 components), (b) R2 (8 components), (c) R3 (14 components), (d) R4 (4 components), and (e) R5 (7 components). Thickness of components represents the size (width and height) of cross-sectional design.
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A sample topology defined by a topology vector t: (a) the ground structure, (b) Gg the ground topology graph, (c) a sample topology vector t and beam size vector w, (d) the original topology graph G0, defined by t and w, and (e) the corresponding topology. Note that only the black-colored elements of w are realized in (e).
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Components set defined by the joint library method: (a) topology defined by t and w, (b) a sample joint library vector JL. Note that the joint library JL1 at the location J1 does not exist (marked as NULL in JL), (c) corresponding subgraph, and (d) physical components set of (c) with two components C0 and C1.
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A sample joint model and modified joint assignment method used in this paper to assign joint attributes: (a) a sample structure with three components and (b) FE model for the J2 in (a), where each component is connected to the center node by using the torsional spring element (c) the modified individual joint attribute vector A _J2 that assigned to J2
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A spider diagram for the objective function values of the representative Pareto optimal designs (R1∼R5) in Case Study 2. Note that R5 shows a balanced performance in all six objective functions.
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Manufacturability calculation considering sheet metal working: (a) a beam component defined, (b) corresponding sheet metal components to be joined into the beam component defined in (a), and (c) die useable area Au calculated from the convex hull area and shearing parameter P for (b)
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Graph-based crossover operation: (a) parent structures P1 and P2 cut by crossover line L , (b) corresponding partitioning of P1 and P2 in graph representation, (c) assembly of offspring graphs C1 and C2 . Note that in C1 , edges e11 and e12 are copied from parent P1 because nodes n3,n4, and n5 are from P1 . Edges e16 and e13 are randomly assigned because n6 is from P2 while n4 and n5 are from P1 . (d) Offspring structures C1 and C2 constructed from their graphs. Both C1 and C2 have 2 components.
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Mutation operation (a) the original structure and graph (b) topology mutation (c) altering joint locations and (d) altering beam size
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Flowchart of multicomponent structure synthesis
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Case Study 1 model: (a) design domain, (b) ground structure with 15 beam elements, and (c) ground topology graph of 15 nodes (n0∼n14) and 44 edges (e0∼e43)
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Joint Library Type for a given topology: (a) Joint Location J0 and J2 have the same joint library Type 2 , and (b) Joint Library of the Type 3 Joint. Total 5 cases (C0∼C4) are in the Type 3 Joint Library.
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Typical convergence histories of GA runs with three different mutation probabilities for y (black line: 0.005, dark gray line: 0.05, and light gray line: 0.1)
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Distribution of designs at generation=100 for Case Study 1. In all plots, the utopia points are at the upper-right corner. Black-marked ones are designs in the Pareto Set with respect to all four objectives. Three representative Pareto optimal designs R1,R2, and R3 are shown in Fig. 14.
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Representative Pareto optimal designs for Case Study 1: (a) R1, (b) R2, and (c) R3.R1,R2, and R3 have 3, 3, and 4 components, respectively. Thickness of beams represents the size (width and height) of cross-sectional design.
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Case study 2 model: (a)–(c) Design domain with three loading and boundary conditions, (d) ground structure with 69 beam elements, and (e) ground topology with 69 nodes (n0∼n68) and 284 edges. Edge numbers (e0∼e283) are not shown in (e) due to the space limitation. Due to the symmetric design assumption, only the basic members in the right half plane in (d) contain the design variables. Left half plane has the symmetric design of the right half-plane design.
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Typical convergence histories of two GA runs
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Distribution of designs at generation=100 for Case Study 2. In all plots, the utopia points are at the upper-right corner. Black-marked ones are designs in the Pareto Set to all six objectives. Five representative Pareto optimal designs R1,R2,R3,R4, and R5 are shown in Fig. 18.



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