Ikuyo Kaneko

Optimum design of plastic structures under displacement constraints

Abstract This paper deals with the optimum design under given loads, of discrete (truss-like) linearly hardening or non-hardening plastic structures, subject to limitations on displacements and deformations and to linear technological constraints. Basic assumptions are: (i) the “cost” function is linear in the design variables; (ii) no local unstressing occurs under the given proportional loading, so that holonomic plastic laws can be adopted. Both elastic-plastic and rigid-hardening models are considered. A typical mathematical feature of the optimization problem is a (nonlinear, nonconvex) complementarity constraint. For situations where the local resistances, assumed to be design variables, do not affect the local stiffness, a branch-and-bound method is proposed and an alternative quadratic programming approach is envisaged. For situations where local strength and stiffness are coupled, a method is developed consisting basically of iterative applications of the procedure devised for uncoupled cases. The computational efficiency of the solution methods proposed is examined by means of numerical tests.

On the solution of some (parametric) linear complementarity problems with applications to portfolio selection, structural engineering and actuarial graduation

In this paper we discuss three applications of a class of (parametric) linear complementarity problems arising independently from such diverse areas as portfolio selection, structural engineering and actuarial graduation. After explaining how the complementarity problems emerge in these applications, we perform some analytical comparisons (based on operation counts and storage requirements) of several existing algorithms for solving this class of complementarity problems. We shall also present computational results to support the analytical comparisons. Finally, we deduce some conclusions about the general performance of these algorithms.

A parametric linear complementarity problem involving derivatives

The problem considered in this paper is given by the conditions:w = q + tp + Mz, w ≥ 0,ż ≥ 0,wTż = 0, where a dot denotes the derivative with respect to the scalar parametert ≥ 0. In this problem,q, p aren-vectors withq ≥ 0 andM is an byn P-matrix. This problem arises in a certain basic problem in the field of structural mechanics. The main result in this paper is the existence and uniqueness theorem of a solution to this problem. The existence proof is constructive providing a computational method of obtaining the solution asymptotically.

Complete solutions for a class of elastic-plastic structures

Abstract A mathematical programming method is proposed for determining the complete history of (generalized) stresses and strains during a loading process for a class of elastic-plastic structures. This method is based on a pivoting procedure similar to the simplex method for a linear program and is suited for computer implementation. The method is illustrated by a simple truss problem. Based on the method, some results on existence and uniqueness of the solution of the structural problem are established.

Linear complementarity problems and characterizations of Minkowski matrices

Abstract This paper unifies several recent characterizations of Minkowski matrices (nonsingular M-matrices) in terms of linear complementarity problems.

Isotone solutions of parametric linear complementarity problems

The parametric linear complementarity problem is given by the conditions:q + αp + Mz ⩾ 0,α ⩾ 0,z ⩾ 0,zT(q + αp + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz(α) of the above problem is montonically nondecreasing in the parameterα for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.

Some n by dn linear complementarity problems

Abstract The purpose of this paper is to study some recent applications of the n by dn LCP solvable by a parametric principal pivoting algorithm (PPP algorithm). Often, the LCPs arising from these applications give rise to large systems of linear equations which can be solved fairly efficiently by exploiting their special structures. First, it is shown that by analyzing the n by dn LCP we could study the problem of solving a system of equations and the (nonlinear) complementarity problem when the function involved is separable. Next, we examine conditions under which the PPP algorithm is applicable to a general LCP, and then present examples of LCPs arising from various applications satisfying the conditions; included among them is the n by dn LCP with a certain P -property. Finally we study a special class of n by dn LCPs which do not possess the P -property but to which the PPP algorithm is still applicable; a major application of this class of problems is a certain economic spatial equilibrium model with piecewise linear prices.

Methods for computing optimal bounds on deformation in the theory of workhardening adaptation

Abstract The paper is concerned with rigid-workhardening structures subjected to dynamic loadings. Within the theory of workhardening adaptation, a bounding technique is studied. Numerical methods for computing optimal bounds on deformation are given and illustrated by means of a simple frame structure. It is demonstrated that the computed optimal bounds are fairly accurate.

A reduction theorem for the linear complementarity problem with a certain patterned matrix

Abstract Matrices with a certain pattern are defined and their properties are studied. A reduction theorem is stated and proved which can be used to reduce the size of the linear complementarity problem defined by a matrix with the pattern. The identification of the pattern and the reduction theorem provide a mathematical model for a problem in structural mechanics when a certain symmetry prevails in the structural problem to be analyzed.