Yoshihiro Kanno

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.

Redundancy and Robustness, or When Is Redundancy Redundant?

The redundancy of a structure refers to the extent of degradation the structure can suffer without losing some specified elements of its functionality. However, because future structural degradation is unknown during design and analysis, it is evident that structural redundancy is related to robustness against uncertainty. This paper proposes a quantitative and widely applicable concept of strong redundancy and shows its relation to the info-gap robustness of the structure. In particular, one of this paper’s propositions establishes general conditions in which the strong redundancy is equivalent to the robustness. This paper also defines a concept of weak redundancy and presents propositions that relate it to the strong redundancy and the robustness. Results are illustrated with several heuristic and engineering examples.

Nonsmooth mechanics and convex optimization

Part I: Convex Optimization Over Symmetric Cone Cones, Complementarity, and Conic Optimization Proper Cones and Conic Inequalities Complementarity over Cones Positive-Semidefinite Cone Second-Order Cone Conic Constraints and Their Relationship Conic Optimization Optimality and Duality Fundamentals of Convex Analysis Optimality and Duality Application to Semidefinite Programming Applications in Structural Engineering Compliance Optimization Eigenvalue Optimization Set-Valued Constitutive Law Part II: Cable Networks: An Example in Nonsmooth Mechanics Principles of Potential Energy for Cable Networks Constitutive law Potential Energy Principles in Convex Optimization Forms More on Cable Networks: Nonlinear Material Law Duality in Cable Networks: Principles of Complementary Energy Duality in Cable Networks (1): Large Strain Duality in Cable Networks (2): Linear Strain Duality in Cable Networks (3): Green-Lagrange Strain Part III: Numerical Methods Algorithms for Conic Optimization Primal-Dual Interior-Point Method Reformulation and Smoothing Method Numerical Analysis of Cable Networks Cable Networks with Pin-Joint Cable Networks with Sliding Joints Form-Finding of Cable Networks Part IV: Problems in Nonsmooth Mechanics Masonry Structures Introduction Principle of Potential Energy for Masonry Structures Principle of Complementary Energy for Masonry Structures Numerical Aspects Planar Membranes Analysis in Small Deformation Principle of Potential Energy for Membranes Principle of Complementary Energy for Membranes Numerical Aspects Frictional Contact Problems Friction Law Incremental Problem Discussions on Various Complementarity Forms Plasticity Fundamentals of Plasticity Perfect Plasticity Plasticity with Isotropic Hardening Plasticity with Kinematic Hardening

SEQUENTIAL SEMIDEFINITE PROGRAMMING FOR OPTIMIZATION OF FRAMED STRUCTURES UNDER MULTIMODAL BUCKLING CONSTRAINTS

An algorithm based on Semi-Definite Programming (SDP) formulation is proposed for optimum design of structures for specified linear buckling load factors. Optimal trusses and frames are computed by using the primal-dual interior-point method based on SDP scheme. It is well known that optimizing structures under buckling constraints is difficult because of the non-differentiability of buckling load factors for the case of multimodal solutions. The examples studied indicate that optimum designs with multiple buckling load factors can be found with no difficulty by successively solving the SDP problems.

Contact Analysis of Cable Networks by Using Second-Order Cone Programming

A method based on mathematical programming is proposed for large deformation and contact analysis of cable networks. By explicitly considering these nonsmooth behaviors, we formulate the linear complementarity problems over symmetric cones under some practically acceptable assumptions. We also present the equivalent second-order cone programming (SOCP) problems, which can be regarded as the minimization problem of total potential energy and complementary energy with the subsidiary constraints on the displacements and contact forces, respectively. By solving the presented SOCP problems by using the primal-dual interior-point method, the equilibrium configurations and internal forces of several cable networks are obtained without any assumptions on stress states and contact conditions.

Global optimization of trusses with constraints on number of different cross-sections: a mixed-integer second-order cone programming approach

In design practice it is often that the structural components are selected from among easily available discrete candidates and a number of different candidates used in a structure is restricted to be small. Presented in this paper is a new modeling of the design constraints for obtaining the minimum compliance truss design in which only a limited number of different cross-section sizes are employed. The member cross-sectional areas are considered either discrete design variables that can take only predetermined values or continuous design variables. In both cases it is shown that the compliance minimization problem can be formulated as a mixed-integer second-order cone programming problem. The global optimal solution of this optimization problem is then computed by using an existing solver based on a branch-and-cut algorithm. Numerical experiments are performed to show that the proposed approach is applicable to moderately large-scale problems.

Alternating direction method of multipliers for truss topology optimization with limited number of nodes: a cardinality-constrained second-order cone programming approach

Abstract This paper addresses the compliance minimization of a truss, where the number of available nodes is limited. It is shown that this optimization problem can be recast as a second-order cone programming with a cardinality constraint. We propose a simple heuristic based on the alternative direction method of multipliers. The efficiency of the proposed method is compared with a global optimization approach based on mixed-integer second-order cone programming. Numerical experiments demonstrate that the proposed method often finds a solution having a good objective value with small computational cost.

Robustness Evaluation of Elastoplastic Base-Isolated High-Rise Buildings Subjected to Critical Double Impulse

A new method of robustness evaluation is proposed for an elastoplastic base-isolated high-rise building considering simultaneous uncertainties of structural parameters. Since it is difficult to evaluate the robustness of elastoplastic structures due to heavy computational load on the time-history response analysis including elastoplastic response, a double impulse input is used to provide a closed-form solution of the critical response of a single-degree-of-freedom (SDOF) elastic-perfectly plastic structure under a near-field ground motion. Introducing an equivalent elastoplastic SDOF model of a base-isolated high-rise building, the worst combination of uncertain structural parameters, i.e. the stiffness and yield deformation at the base-isolation story and the stiffness of the superstructure, can be derived which leads to the upper bound of the critical elastoplastic response. It is shown that, by using the derived upper bound of the critical response, the robustness function, a measure of the robustness, of elastoplastic structures can be evaluated efficiently. In numerical examples, the robustness of a 30-story base-isolated high-rise building is compared with those of other models with different yield deformations at the base-isolation story to find a preferable design with larger robustness.

A fast first-order optimization approach to elastoplastic analysis of skeletal structures

It is classical that, when the small deformation is assumed, the incremental analysis problem of an elastoplastic structure with a piecewise-linear yield condition and a linear strain hardening model can be formulated as a convex quadratic programming problem. Alternatively, this paper presents a different formulation, an unconstrained nonsmooth convex optimization problem, and proposes to solve it with an accelerated gradient-like method. Specifically, we adopt an accelerated proximal gradient method, that has been developed for a regularized least squares problem. Numerical experiments show that the presented algorithm is effective for large-scale elastoplastic analysis. Also, a simple warm-start strategy can speed up the algorithm when the path-dependent incremental analysis is carried out.