# Shape and topology optimization of plane frame

Geometry and topology optimization of truss-like structures is one of the standard topics in structural optimization, and has received considerable attention in the past[1][2]. Generally, optimization of truss-like structures can be classified into three categories; namely, sizing, geometry (or shape) and topology optimization. Among them, sizing optimization, which aims at finding the optimal cross-sectional properties of members, is commonly included in geometry and topology optimization, and the change of structural topology can be achieved in the standard framework of sizing optimization problem.

When geometry optimization is involved, one of the main challenges is the existence of melting nodes, or coalescent nodes, resulting in extremely short member in the structure and making the stiffness matrix singular[3]. In order to alleviate this diffculty, the force density method (FDM) is introduced for shape optimization which indirectly controls the member length[4]. A numerical example with initial structure shown in Fig.1 is investigated the effectiveness of the proposed method, and the optimization procedure shown in Fig.2 shows that a reasonble optimal result can be obtained by introducing FDM for determining the structural shape, where the red and blue lines indicate the positive and negative force density values, respectively.

Moreover, many researches have been proposed to consider uncertainty in structural optimization, such as robust optimization[5], reliability-based optimization[6] and the worst-case design[7], and I am also working on this research topic right now [8][9]. However, incorporating uncertainty into structural optimization is not a trival task expecially the diffculties in efficiency, accuracy and convergence when applied to large-scale problem, I would like to dig more in this research direction.

Reference

[1] Ohsaki M (2010) Optimization of finite dimensional structures. CRC Press, Boca Raton

[2] Lewiński T, Sokół T, Graczykowski C (2019) Michell structures. Springer, Berlin [3] Achtziger W (2007) On simultaneous optimization of truss geom- etry and topology. Struct Multidiscip Optim 33:285–304.

[4] Shen W, Ohsaki M (2021) Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model. Eng Comput 37:2029–2046.

[5] Kanno Y, Xu G (2010) A mixed integer programming for robust truss topology optimization with stress constraints. Int J Numer Methods Eng 83:1675-1699.

[6] Mogami K, Nishiwaki S, Izui K, Yoshimura M, Kogiso N (2006) Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques. Struct Multidiscip Optim 32(4):299-311.

[7] Ohsaki M, Yamakawa M, Fan W, Li Z (2019) An order statistics approach to multiobjective structural optimization considering robustness and confidence of responses. Mech Res Commun 97:33-38.

[8] Shen W, Ohsaki M, Yamakawa M (2020) Multiobjective robust shape and topology optimization of plane frames using order statistics. Struct Multidiscip Optim 75–94.

[9] Shen W, Ohsaki M, Yamakawa M (2021) Robust geometry and topology optimization of plane frames using order statistics and force density method with global stability constraint. Int J Numer Methods Eng 122:3653–3677.

**FIG.1:**Initial structure of numerical example

When geometry optimization is involved, one of the main challenges is the existence of melting nodes, or coalescent nodes, resulting in extremely short member in the structure and making the stiffness matrix singular[3]. In order to alleviate this diffculty, the force density method (FDM) is introduced for shape optimization which indirectly controls the member length[4]. A numerical example with initial structure shown in Fig.1 is investigated the effectiveness of the proposed method, and the optimization procedure shown in Fig.2 shows that a reasonble optimal result can be obtained by introducing FDM for determining the structural shape, where the red and blue lines indicate the positive and negative force density values, respectively.

**FIG.2:**Intermediate solutions of numerical example

Moreover, many researches have been proposed to consider uncertainty in structural optimization, such as robust optimization[5], reliability-based optimization[6] and the worst-case design[7], and I am also working on this research topic right now [8][9]. However, incorporating uncertainty into structural optimization is not a trival task expecially the diffculties in efficiency, accuracy and convergence when applied to large-scale problem, I would like to dig more in this research direction.

Reference

[1] Ohsaki M (2010) Optimization of finite dimensional structures. CRC Press, Boca Raton

[2] Lewiński T, Sokół T, Graczykowski C (2019) Michell structures. Springer, Berlin [3] Achtziger W (2007) On simultaneous optimization of truss geom- etry and topology. Struct Multidiscip Optim 33:285–304.

[4] Shen W, Ohsaki M (2021) Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model. Eng Comput 37:2029–2046.

[5] Kanno Y, Xu G (2010) A mixed integer programming for robust truss topology optimization with stress constraints. Int J Numer Methods Eng 83:1675-1699.

[6] Mogami K, Nishiwaki S, Izui K, Yoshimura M, Kogiso N (2006) Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques. Struct Multidiscip Optim 32(4):299-311.

[7] Ohsaki M, Yamakawa M, Fan W, Li Z (2019) An order statistics approach to multiobjective structural optimization considering robustness and confidence of responses. Mech Res Commun 97:33-38.

[8] Shen W, Ohsaki M, Yamakawa M (2020) Multiobjective robust shape and topology optimization of plane frames using order statistics. Struct Multidiscip Optim 75–94.

[9] Shen W, Ohsaki M, Yamakawa M (2021) Robust geometry and topology optimization of plane frames using order statistics and force density method with global stability constraint. Int J Numer Methods Eng 122:3653–3677.