Makoto Ohsaki

Genetic algorithm for topology optimization of trusses

A global search algorithm is presented for topology optimization of trusses based on the genetic algorithm. The nodal cost, as well as the member cost, is incorporated in the cost function. A topological bit is introduced to indicate the existence of each member. It is shown, based on the schemata theorem, that the use of the topological bit leads to rapid convergence of the solution to an optimal topology with small number of members. The efficiency of the proposed method is demonstrated in the examples of plane trusses and the effect of the nodal cost on optimal topologies is discussed.

Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints

Abstract Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as Semi-Definite Programming (SDP), and an algorithm is presented based on the Semi-Definite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed, the SDPA is especially useful for the case where the optimal design has multiple fundamental eigenvalues. Global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view. It is shown in the examples, that SDPA has advantage over existing methods in view of computational efficiency and accuracy of the solutions, and an optimal topology with five-fold fundamental eigenvalue is found without any difficulty.

Group Symmetry in Interior-Point Methods for Semidefinite Program

A class of group symmetric Semi-Definite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of primal-dual interior-point methods are group symmetric. Preservation of group symmetry along the search direction theoretically guarantees that the numerically obtained optimal solution is group symmetric. As an illustrative example, we show that the optimization problem of a symmetric truss under frequency constraints can be formulated as a group symmetric SDP. Numerical experiments using an interior-point algorithm demonstrate convergence to strictly group symmetric solutions.

A natural generator of optimum topology of plane trusses for specified fundamental frequency

Abstract A natural and practical method is developed for finding optimal topologies of large plane trusses for a specified fundamental natural frequency, where all the node locations are fixed and the set of members that may exist or may be adopted is prescribed. The method is based on the concept of an ordered set of optimal trusses introduced recently by the authors. The optimum designs are considered as functions of a problem parameter which specifies the set of minimum cross-sectional areas, and the optimal solutions are swept out with Taylor series expansions with respect to the parameter. The solution with vanishing minimum cross-sectional areas is considered to represent the truss with the optimal topology, and each solution throughout the process of decreasing the parameter is the global optimal solution corresponding to the set of minimum cross-sectional areas specified with the current value of the parameter. A method is presented of generating a truss with a practical optimal topology which contains long members. The efficiency of the proposed method is shown through the examples of large plane trusses, and the characteristics of the optimal topologies are discussed.

Sequential optimal truss generator for frequency ranges

Abstract The concept of an ordered set of optimum designs is introduced here and all the design variables and behavioral variables belonging to the optimum designs are represented by successive piecewise Taylor series. A most natural, direct, and efficient way of generating or sweeping out all the optimum designs sequentially in the set is devised. The procedure is started with the eigenvalue analysis on the truss defined by the set of all the minimum cross-sectional areas. Neither any further eigenvalue analysis nor application of any optimization technique is required in the proposed procedure. It is demonstrated that the proposed method is efficient not only for an ordered set of optimum designs of a large truss associated with the single lowest eigenvector but also for that associated with multiple lowest eigenvectors.

Optimum design with imperfection sensitivity coefficients for limit point loads

A computational method is presented for finding a sequence of optimum designs of a discrete system which exhibits limit point behaviour. Optimality conditions are derived in terms of the theory of imperfection sensitivity coefficients for the limit point load factor. Only those designs of the structures which exhibit limit point behaviour are considered as feasible designs, and the design change is conceived as generating a kind of imperfection. The efficiency of the proposed algorithm will be appreciated particularly for large structures, because incremental nonlinear analysis to find the limit point load factor needs to be carried out only once for the structure of trivial initial optimum design. The sequence of optimum designs is described by piecewise Taylor series expansions with respect to the specified limit point load factor. It is shown in the examples that the proposed method is efficient and of good accuracy for a large space truss.

Random search method based on exact reanalysis for topology optimization of trusses with discrete cross-sectional areas

Abstract A simple formulation applicable to topology modification is presented for exact reanalysis of static response of trusses based on general methods of inverting modified matrices. A random search algorithm is also presented for topology optimization of trusses with discrete list of cross-sectional areas. It is shown that the exact reanalysis is very effective for the case where the cross-sectional area of only one member is modified at each step for generating a neighboring solution. The difficulty due to discontinuity in the stress constraints is successfully avoided by considering the cross-sectional areas as discrete variables.

Optimization of imperfection-sensitive symmetric systems for specified maximum load factor

A method is presented for optimum design of a symmetric structure which reaches an unstable bifurcation point as the load factor is increased. Reduction of the maximum load level due to the antisymmetric imperfection is considered, and a straightforward algorithm is proposed for calculating the magnitude of reduction of the load factor corresponding to the most critical mode of antisymmetric imperfection. The sensitivity coefficients of the bifurcation load factor are calculated by using the interpolation method developed by the authors, and an approximate formulation is presented for sensitivity analysis of the maximum load factor. It is shown in the examples that the errors in the sensitivity coefficients do not lead to any significant difference in the optimum design.

Sequential generator of earthquake-response constrained trusses for design strain ranges

Abstract A new concept called “an ordered set of earthquake-response constrained designs” is introduced for trusses with respect to a parameter representing design strain levels. An efficient computational method is developed of sequentially generating such an ordered set with the use of successive piecewise Taylor series expansion of all the design variables and of formulas for estimating mean maximum responses. The numerical procedure is started with eigenvalue analysis of the truss defined by the set of all the minimum cross-sectional areas. No further eigenvalue analysis or any successive improvement is needed. The efficiency of the proposed method is demonstrated through the ordered sets of designs for a 480-bar truss and an 800-bar truss.

SENSITIVITY ANALYSIS OF BIFURCATION LOAD OF FINITE-DIMENSIONAL SYMMETRIC SYSTEMS

Three methods are presented for sensitivity analysis of bifurcation load factor of finite-dimensional conservative symmetric systems subjected to a set of symmetric proportional loads. In the first method, a conventional method with diagonalization is utilized to derive an explicit formula of sensitivity coefficients corresponding to a minor imperfection. Next, a new concept is introduced to find the sensitivity coefficients of the load factor, displacements and the eigenmodes under fixed lowest eigenvalue of the tangent stiffness matrix. Based on this concept, a method is presented for finding approximate sensitivity coefficients of the buckling load factor. Finally, a direct method is presented to find the accurate sensitivity coefficients of the bifurcation load factor, displacements at buckling and the buckling mode of a symmetric system. Note that different formula should be used for sensitivity analysis of a limit point load factor. In the examples, the proposed three methods are compared in view of accuracy of the results and simplicity in coding.