خواص موثر از یک بهینه سازی عددی پرفوره صفحه الاستیک توسط الگوریتم ژنتیک
|کد مقاله||سال انتشار||تعداد صفحات مقاله انگلیسی||ترجمه فارسی|
|10408||2001||24 صفحه PDF||سفارش دهید|
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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : International Journal of Solids and Structures, Volume 38, Issues 48–49, November–December 2001, Pages 8593–8616
We consider a perforated elastic plate in which identical traction-free holes form square or hexagonal periodic arrays. The optimization problem of finding the hole shape in this structure that maximizes either of its effective moduli is posed as a full scale inverse problem of elasticity. In this context, only limiting cases of a single hole and cellular solids with thin cell walls have been studied thus far. Here, we cover the gap between these cases numerically solving the problem by the genetic algorithm approach. A new time-saving scheme of the fitness evaluation provides reliable data even near the percolation limit. The presented numerical results comprehensively describe the optimal behavior of perforated plates in terms of equivalent homogeneous medium. Similar, though less sophisticated approach was used by the author Vigdergauz, (Int. J. Solids Struct. 38 (2001) 6851) to optimize an isolated inclusion in a plate.
Perforated thin plates find extensive applications in modern industry. Mathematically, they are modeled by a thin elastic porous plate, in which identical holes are placed on the nodes of a doubly periodic lattice. For this scheme to be applicable, one should know the effective moduli of the structure. These observable macroscopic constants describe the composite in the sense of an equivalent homogeneous and, generally speaking, anisotropic medium. Usually, they are derived from averaging the microscopic constitutive relations over a period. As a result, we obtain the equality of the elastic strain energy stored in the modeled heterogeneous medium to that of the hypothetical homogeneous material. This relates averages of an arbitrary stress macrofield to the induced average strains in the convenient form of the Hooke's law with a View the MathML source constant symmetric matrix of the effective elastic moduli. Since the equivalent composite properties generally vary with direction, this matrix may involve up to six independent entries. In this case the medium is said to be fully anisotropic on the macroscale. Of particular interest are less anisotropic structures with a square symmetry. Their comprehensive description requires only three moduli: the effective bulk modulus Ke, and two shear moduli μe, μe*, μe≠μe*. For higher symmetric honeycomb structures we further have: μe=μe*. Our concern here is only with these two composite types. Given the matrix elastic constants and the volume fraction c1<1 of the hole, its traction-free shape remains the only factor controlling the stress distribution in the plate and hence all the related quantities. In this context, we address the structural optimization problem of selecting the optimally shaped hole that maximizes one of the plate moduli over all such structures subject to the same trial loading. This corresponds individually to the maximal overall, shear or twisting stiffness of the plate, with deflections and stresses considerably reduced. At these settings a material with the maximal bulk modulus may be explicitly identified by application of the equi-stress concept (Cherepanov, 1974; Vigdergauz, 1994), when only uniform tangential stresses at the shape sought are allowed as a response to a hydrostatic external load. With this prerequisite, parametric equations of the optimal shape are derived analytically in terms of doubly periodic elliptic functions. In the opposite case of dominating shear stresses, the equi-stress principle is inapplicable due to the different analytic nature of the optimization problem for the moduli μe, μe*. Other analytical approaches are also yet unknown, so numerical methods become a natural choice. Computationally, any optimization process involves two main ingredients: the solution of given boundary value problem (a direct problem) which has to be repeated many times, and a minimization scheme (an inverse or shape optimization problem). Here, the direct problem is governed by the high-order equations of elasticity, which must be handled numerically in a special way to provide a compromise between accuracy and efficiency. The associated inverse problem is known to be ill-posed, namely large changes of the contour can correspond to small changes in the stored energy. By this reason, traditional gradient-base optimization methods would require enormous calculations of the local stresses at each control point in contrast to the objective energy function easily evaluated by averaging the stress field. This discrepancy is especially pronounced for angular points that may drastically improve the performance of a candidate to the μe or μe* optimum. Therefore, a more promising non-gradient alternative should be used when the search starts from different initial approximations, and proceeds according to some heuristic procedure. Here, we have applied a genetic algorithm (GA) advanced by Holland (1975). Till now, the GA has not been widely used in the full-scale theory of elasticity. One of the difficulties is that the energy assessment is computer time consuming, especially when more accurate models are used. Our contribution to remedy the situation is two-fold. First, the Kolosov–Muskhelishvili (KM) periodic potentials are combined in a fresh manner to perform an adequate timesaving procedure of fitness evaluation. Second, an effective self-adjusting scheme is proposed to encode randomly generated hole shapes with “automatic” retention of a given volume fraction c1<1. Together with some other numerical tricks this helps us to obtain a stable solution up to the values c1⩽0.85. The proposed technique has been successfully used in a less complicated shape optimization of a single hole View the MathML source in a plate (Vigdergauz, 2001). Here, this approach is applied to identify an optimal orthotropic material with a rectangular lattice, and an isotropic honeycomb structure. Our principal result is a numerical derivation of the optimal effective moduli in a wide range of admissible values of c1. The previous attempt to obtain them analytically (Vigdergauz, 1994) was unsuccessful as proven by Grabovsky and Kohn (1995). It should be noted, however, that no innovations are advanced in the GA itself. The ordinary approach with integer genes, tournament selection of random pairs for mating, one-point crossover, bit-wise mutation and elitism turned out to be sufficient for our purposes. The real novelty is a fresh combination of the GA with comprehensive analytical study of the problem. Because of this, we primarily address to the elasticity rather than to the GA community. Further, due to its heuristic nature, one cannot apply the GA to the considered problem automatically. A reliable assessment of the numerical results may be only provided by comprehensive theoretical analysis in combination with the common sense considerations and even with the programming experience. It is not surprising, then, that the required analytical manipulations take up a good portion of the paper. Together with this, the genetic configuration and application specifics are also covered in details. The paper is structured as follows: In Section 2, we display some basic formulae in the complex variable method of 2D elasticity. Section 3 states the optimization problem, while Section 4 describes the solution strategy based on the GA. Section 5 gives an encoding technique that automatically incorporates some geometrical constraint specifically pre-imposed on the moving interface to throw away most of the unpromising candidates. Proposed by the author Vigdergauz (2001) this not-too-standard scheme is reproduced here for making the paper more self-contained. In Section 6 we present a new numerical approach to solve stress–strain fields at given hole shape. On this basis, a highly performing scheme of the fitness evaluation is proposed as a principal part of GA. Though lengthy, 2, 3 and 6 provide a comprehensive theoretical basis required for reliable assessment of the numerical data obtained by the GA. In Section 7 we discuss some computational issues of the proposed scheme which shows good agreement with available experimental data. Detailed numerical results are given in Section 8. They validate the method and demonstrate its performance. Finally, some concluding remarks are made in Section 9.
نتیجه گیری انگلیسی
In our opinion, the GA is one of the most effective numerical methods yet devised to optimize the average characteristics of 2D composites. For the current application, the shape fitness is evaluated through numerical solution of the full-scale elasticity problem posed in terms of KM potentials. Two non-standard features are used in the approach to overcome enormous computational time necessary for GA. First, the traction-free boundary condition on a contour is reformulated in such a way as to exclude the doubly periodic part of the potential ψ(z) from further consideration. Second, a self-adjusting scheme of string encoding is implemented as an effective means against generating obviously unpromising candidates. As a result, this seemingly difficult optimization problem has been successfully solved in numerical way. The alternative possibility of optimizing non-convex contours is also worth consideration especially as the proposed encoding procedure may be easily adapted to the case. Omitted in the present study are equilateral triangular cells. Though more difficult technically, they attract considerable interest as possibly having extremal bulk and shear moduli that achieve the HSGC bounds. The arguments in favor of such conjecture are advanced by Torquato et al. (1998). This case is currently under development.