This paper presents several equivalent formulations for a structural design model where the load carrying capacity is maximized for a prescribed volume subject to bounds on complementary energy and stresses. This limit design model covers the full range of strictly elastic, elastic/plastic and strictly plastic designs and is based on the unified analysis model of Ben-Tal and Taylor [1]. While this design model is convex, it is of a very high dimension for topology design problems. Application of duality principles leads to several simpler but nonsmooth equivalent models. In particular, for the case when the design variables do not have explicit bounds, the dual models reduce to a minimization, subject to a single linear constraint, of a pointwise maximum of a finite number of convex functions. More importantly, these simpler design models are of greatly reduced size, since they contain only nodal variables. For these reduced models, the examples show that while the strictly plastic designs are effectively computed, this is not so yet for the case of limit design for strictly elastic and elastic/plastic responses.
In this paper, we study the rich class of formulations that arise in the limit analysis and design of elastic/plastic structures in the presence of contact constraints. It is well known that in the absence of contacts, both the limit analysis and limit design problems can be written as linear programs. However, when contact constraints are present, the structure effectively exhibits both softening and stiffening behavior under monotonic loading. The resulting limit analysis and limit design problems are nonconvex and are difficult to solve due to complementary type of equality constraints. We show that using a mixed form of the minimum principle, we can restate the limit analysis and limit design problems as two and three level formulations, respectively. Further, under a strong assumption on the problem and solution data, we can take advantage of the underlying convexity to reduce both these multilevel formulations to equivalent linear programs. While it may not be possible to always verify this assumption in practice, we show that a two-step iterative procedure is effective in reaching a solution to the limit design problem.
This paper presents the derivation of a broad variety of competitive learning vector quantization (LVQ) algorithms and investigates their relationship with fuzzy k-means algorithms and fuzzy learning vector quantization (FLVQ) algorithms. LVQ algorithms map a set of feature vectors into a set of prototypes by adapting the weight vectors of a feature map through an unsupervised learning process. The derivation of the proposed algorithms is accomplished by minimizing the average generalized mean between the input vectors and the prototypes using the gradient descent method. The close relationship between the resulting LVQ algorithms and fuzzy k-means is revealed by investigating the functionals that provided the basis for their derivation. In fact, fuzzy k-means algorithms result as a special case of the proposed algorithms if the learning rate is selected at each iteration to satisfy a certain condition. Moreover, the existing FLVQ algorithms are interpreted as one of the two possible implementations of the proposed LVQ algorithms. An alternative implementation of the LVQ algorithms developed in this paper is also proposed.