Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model

Abstract

A new method is proposed for simultaneous optimization of shape, topology and crosssection of plane frames. Compliance against specified loads is minimized under constraint on structural volume. Difficulties caused by the melting nodes can be alleviated to some extent by introducing force density as design variables for defining the geometry, where the side constraints are assigned for force density to indirectly avoid the existence of extremely short members. Force density method is applied to an auxiliary cable net model with different boundary and loading conditions so that the regularity of force density matrix is ensured by positive force densities. Sensitivity coefficients of the objective and constraint functions with respect to the design variables are also explicitly calculated. After the optimal geometry of the frame is obtained, the topology is further improved by removing the thin members and combining closely spaced nodes. It is demonstrated in the numerical examples of three types of frames that rational geometry and topology can be achieved using the proposed method, and the effect of bending moment on the optimal solution is also discussed.

Robust geometry and topology optimization of plane frames using order statistics and force density method with global stability constraint

“…Let m and n represent the numbers of members and nodes of the auxiliary pin‐jointed truss structure, respectively. If member i connects nodes j and k , then the m ‐by‐ n connectivity matrix C is given by defining each entry as 11,13,44 where the subscript ( i , p ) indicates the entry of C at the i th row and p th column. The force density q i of the i th member is defined as where N i is the axial force and L i is the length of the i th member.…”

“…If the force densities are given for all members in the structure and the locations of fixed nodes are assigned, then the locations of free nodes can be obtained from Equation (3), that is It has been proved in Reference 45 that the matrix is nonsingular if at least one node is fixed and q is a non‐zero vector, resulting in the existence of a solution for x free and y free in Equation (4). Also, it is pointed out in Reference 13 that if the axial force N i in Equation (3) is non‐zero and the upper and lower bounds for q i have the same absolute value, then a lower bound for the member length L i can be indirectly assigned to prevent generating extremely short members.…”

“…By incorporating the FDM in Section 2, the locations of free nodes of a plane frame can be derived by solving Equation (4) of the corresponding auxiliary truss, and the optimization problem (10) is restated as where q is the force density vector. As stated in Reference 13, problems (10) and (12) are basically the same and will lead to the same solution if a set of q in problem (12) can define the optimal solution of problem (10), which means the optimal solution of problem (10) can be found by solving problem (1...…”

“…To alleviate the difficulty caused by melting nodes, Ohsaki and Hayashi 11 and Hayashi and Ohsaki 12 reformulated the objective and constraint functions in truss optimization problem using force density method (FDM). Shen and Ohsaki 13 extended this method to frame structures by introducing an auxiliary pin‐jointed truss or cable‐net structure.…”

Optimization of branching structures for free-form surfaces using force density method

“…Recently, Ohsaki and Hayashi [23] explored the merits of FDM in shape and topology optimization of trusses; the objective function, namely, the compliance, and the constraint function, namely, the structural volume, are expressed explicitly by the force densities based on the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members. Shen and Ohsaki [24] extended this method to simultaneous shape and topology optimization of planar frames. In their method, the nodal locations are expressed as functions of force densities of an auxiliary truss or a cable-net, and the cross-sectional areas of members of the primary frame are determined by solving a nonlinear programming problem.…”