A Review of an Optimal Design Problem for a Plate of Variable Thickness

Abstract

Abstract. We revisit a classic design problem for a plate of variable thickness under the model of Kirchhoff. Our main contribution has two goals. One is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants. The other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold. In both situations, the cost functional can be pretty general.

On minimum compliance problems of thin elastic plates of varying thickness

“…This problem is not well posed. The hitherto known results on the relaxation of this problem are discussed in the review paper by Muñoz and Pedregal (2007).…”

“…27). In the paper by Muñoz and Pedregal (2007) a review of the relaxation methods of this problem can be found; the aim of the relaxation is to make the problem well posed, without losing its original setting.…”

Optimization of a plate with holes

“…It is to be noted that the literature on shape optimization problems, including unknown or variable domains, is mainly devoted to second order elliptic equations. Concerning fourth order boundary value problems, for instance plate models, there are papers Kawohl, Lang [8], Muñoz, Pedregal [11], Sprekels, Tiba [20], Arnautu, Langmach, Sprekels, Tiba [1] studying thickness optimization problems that may be reduced to optimal control problems by the coefficients. In Neittaanmäki, Sprekels, Tiba [14], Ch.…”

Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings

“…In recent years, variational inequality theory has been extended and generalized in several directions, using new and powerful methods, to study a wide class of unrelated problems in a unified and general framework. It turned out that odd-order and nonsymmetric obstacle, free, nonlinear equilibrium, dynamical network, optimal design, bifurcation and chaos, and moving boundary problems arising in various branches of pure and applied sciences can be studied via variational inequalities; see [1,2,[5][6][7][8][12][13]17]. One of the most important and difficult problems in this theory is the development of an efficient and implementable iterative algorithm for solving variational inequalities.…”

“…Variational inequalities introduced by Stampacchia [1] in the early sixties have witnessed an explosive growth in theoretical advances, algorithmic development, and applications across all the discipline of pure and applied sciences; see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and the references therein. In recent years, variational inequality theory has been extended and generalized in several directions, using new and powerful methods, to study a wide class of unrelated problems in a unified and general framework.…”