Shape optimization of two equal holes in an infinite elastic plate

Abstract

The optimal design of the stress state in elastic plate structures with openings is a problem of great significance in engineering practice. Achieving proper shape of hole can reduce stress concentration around the boundaries remarkably. The optimal shape of a single hole in an infinite plate under uniform stresses has been obtained by complex variable method based on different optimal criteria. The complex variable method is particularly suitable for the hole shape optimization in infinite plate, in which the continuous hole boundary can be represented by the mapping function. It can also be used to solve the shape optimization problems of two or more holes. However, because of the difficulty of finding the mapping function for multi connected domain, the holes are mapped onto slits or separately mapped onto a circle. In this article, the two symmetrical and identical holes are mapped onto an annulus simultaneously by the newly found mapping function, which has a general form. The maximum tangential stress around the boundaries is minimized to achieve the optimal hole shape. And the coefficients of mapping function which describe the boundary are calculated by differential-evolution algorithm.     

Surface tension-induced stress concentration around a nanosized hole of arbitrary shape in an elastic half-plane

This paper studies surface tension-induced stress concentration around a nanosized hole of arbitrary shape inside an elastic half-plane. Of particular interest is the maximum hoop stress on the hole’s boundary with relation to the point of maximum curvature and the distance between the hole and the free surface of the half-plane. The shape of the hole is characterized by a conformal mapping which maps the exterior of the hole onto the exterior of the unit circle in the image plane. On using the technique of conformal mapping and analytic continuation, the complex potentials of the half-plane are expressed in a series form with unknown coefficients to be determined by Fourier expansion method. Detailed numerical results are shown for elliptical, triangular, square and rectangular holes. Two basic conclusions are that the hoop stress increases with decreasing hole size and the maximum hoop stress generally appears nearby but not exactly at the point of maximum curvature. In addition, it is shown that the hoop stress nearby the point of maximum curvature on the hole’s boundary increases rapidly with decreasing distance between the hole and the free surface of the half-plane. On the other hand, if the distance between the hole and the free surface is more than three times the hole size, the effect of the free surface on the stress concentration around the hole is ignorable and the elastic half-plane can be treated approximately as an elastic whole plane.     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.     

基于表面弹性理论的开裂椭圆孔断裂力学研究

研究纳米尺度时开裂椭圆孔的III型断裂性能。基于表面弹性理论和保角映射技术,利用复势函数理论获得了缺陷(裂纹和椭圆孔)周围应力场和裂纹尖端应力强度因子的闭合解答。所得结果具有一般性,许多已有和新的解答可由本文退化的特殊情形得到。利用解析结果讨论了缺陷的绝对尺寸、椭圆孔的形状比以及裂纹的相对尺寸对应力强度因子的影响。结果表明：考虑表面效应且缺陷尺寸在纳米尺度时,应力强度因子具有显著的尺寸依赖效应;应力强度因子随椭圆孔形状比的变化规律受缺陷表面常数的影响;缺陷表面效应的影响取决于椭圆孔的形状比,非常大的形状比屏蔽了表面效应的影响;裂纹相对尺寸非常小时表面效应影响较弱,裂纹相对尺寸较大时表面效应较为明显。     

A semi-analytical elastic stress–displacement solution for notched circular openings in rocks

A semi-analytical plane elasticity solution of the circular hole with diametrically opposite notches in a homogeneous and isotropic geomaterial is presented. This solution is based on: (i) the evaluation of the conformal mapping function of a hole of prescribed shape by an appropriate numerical scheme and (ii) the closed-form solutions of the Kolosov–Muskhelishvili complex potentials. For the particular case of circular notches––which resemble to the circular cavity breakout in rocks––it is demonstrated that numerical results pertaining to boundary stresses and displacements predicted by the finite differences model FLAC^2D, as well as previous analytical results referring to the stress-concentration-factor, are in agreement with analytical results. It is also illustrated that the solution may be easily applied to non-rounded diametrically opposite notch geometries, such as “dog-eared” breakouts by properly selecting the respective conformal mapping function via the methodology presented herein. By employing a stress-mean-value brittle failure criterion that takes into account the stress-gradient effect in the vicinity of the curved surfaces in rock as well as the present semi-analytical solution, it is found that a notched hole, e.g. borehole or tunnel breakout, may exhibit stable propagation. The practical significance of the proposed solution lies in the fact that it can be used as a quick-solver for back-analysis of borehole breakout images obtained in situ via a televiewer for the estimation of the orientation and magnitude of in situ stresses and of strain–stress measurements in laboratory tests.     

On the Green's Functions and Boundary Integral Formulation of Elastic Planes with Cutouts

ABSTRACT

The problem of an infinite elastic plane that contains a hole of arbitrary shape and is subjected to a concentrated unit load is considered. The Green's function (influence function) for the problem is formulated by means of two complex potential functions. This is accomplished by mapping the region that is exterior to the hole onto a unit circle. A class of closed contour hole shapes is analyzed. Green's functions for an elliptical hole and a class of triangular holes are determined. Green's functions for a class of rectangular holes are also discussed. In order to determine stress and displacement fields for the finite plane problem, Green's function is employed and an indirect boundary integral equation is formulated, with the integrand of the integral equation incorporating the effect of the hole. The contour of the hole is no longer considered a part of the boundary and only the contour of the region that is exterior to the hole is subdivided into boundary elements. Examples for elliptical and triangular holes are solved.     

Contact problem for regular hexagon weakened with full-strength hole

A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six sides of the hexagon are subjected to uniform normal displacements via smooth rigid stamps, while the uniformly distributed normal stress is applied to the internal hole boundary. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials and the shape of the hole contour are determined from the condition that the circumferential normal stress is constant along the hole contour. Numerical results are given and shown in relevant graphs.     

Higher order topological derivatives in elasticity

The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain. In this work, we compute higher order topological derivatives for elasticity problems, so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain. A critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero; it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space. We present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion. We also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown.     

Surrogate based optimization of a laidback fan-shaped hole for film-cooling

The present work has been performed to evaluate the effects of geometric variables of a laidback fan-shaped hole on the film-cooling effectiveness using a Reynolds-averaged Navier-Stokes analysis, and to optimize the shape of the hole using the Kriging meta-modeling technique. The shape of the laidback fan-shaped hole is defined by four geometric design variables, namely, the injection angle of the hole, the lateral expansion angle of the diffuser, the forward expansion angle of the hole, and the ratio of the length to the diameter of the hole. From the results of a parametric study, effects of design variables on the film-cooling effectiveness are evaluated. The objective function, which is defined as the spatially averaged film-cooling effectiveness, is numerically evaluated through a RANS analysis at design points selected through Latin hypercube sampling. The Kriging model is used to approximate these objective function values at the design points, and sequential quadratic programming is used to search for the optimal point from the constructed Kriging model. The optimizations are carried out for two different blowing ratios, 0.5 and 2.5. The film-cooling effectiveness has been successfully improved with the optimization as compared to the reference geometry.     

Shape morphing and topology optimization of fluid channels by explicit boundary tracking

An integrated shape morphing and topology optimization approach based on the deformable simplicial complex methodology is developed to address Stokes and Navier‐Stokes flow problems. The optimized geometry is interpreted by a set of piecewise linear curves embedded in a well‐formed triangular mesh, resulting in a physically well‐defined interface between fluid and impermeable regions. The shape evolution is realized by deforming the curves while maintaining a high‐quality mesh through adaption of the mesh near the structural boundary, rather than performing global remeshing. Topological changes are allowed through hole merging or splitting of islands. The finite element discretization used provides smooth and stable optimized boundaries for simple energy dissipation objectives. However, for more advanced problems, boundary oscillations are observed due to conflicts between the objective function and the minimum length scale imposed by the meshing algorithm. A surface regularization scheme is introduced to circumvent this issue, which is specifically tailored for the deformable simplicial complex approach. In contrast to other filter‐based regularization techniques, the scheme does not introduce additional control variables, and at the same time, it is based on a rigorous sensitivity analysis. Several numerical examples are presented to demonstrate the applicability of the approach.     

On the optimum shape of apertures for a perforated plate subject to bending

The problem of finding the optimum shape of the holes in a perforated plate weakened by a triangular or square lattice of holes and subject to bending is considered by methods based on the theory of functions of a complex variable. The criterion determining the optimum shape of the hole is based on the condition that no stress concentration should occur on the hole contour or, alternatively, that a plastic region should be created around the whole contour of the hole at exactly the same instant.     

菱形开孔剪切钢板阻尼器的形状优化研究

矩形钢板阻尼器在剪力作用下耗能腹板各处受力不均匀,部分区域率先进入塑性而破坏,造成材料浪费甚至阻尼器延性欠佳。为使剪切钢板阻尼器充分发挥材料性能,可将矩形腹板开菱形孔,形成中部菱形孔钢板阻尼器,或两端去除半菱形,形成X型钢板阻尼器。然而,孔洞最优形状鲜有研究,而形状优化方法也很少提及。为了使菱形开孔剪切钢板阻尼器达到更好的滞回性能,对中部菱形孔和X型两种腹板开孔形式的阻尼器进行形状优化。在与试验结果验证的基础上,基于ABAQUS软件平台建立有限元分析模型,利用PYTHON语言开发了一种脚本优化方法,并与软件优化模块对比,分析优化前后的塑性分布和滞回耗能。结果表明,优化后最大塑性应变减小,塑性分布更均匀,利用脚本方法优化后的滞回曲线更饱满,耗能更好,为菱形开孔剪切型阻尼器的设计和优化提供了参考。     

A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature

Viscoplastic crack-tip deformation behaviour in a nickel-based superalloy at elevated temperature has been studied for both stationary and growing cracks in a compact tension (CT) specimen using the finite element method. The material behaviour was described by a unified viscoplastic constitutive model with non-linear kinematic and isotropic hardening rules, and implemented in the finite element software ABAQUS via a user-defined material subroutine (UMAT). Finite element analyses for stationary cracks showed distinctive strain ratchetting behaviour near the crack tip at selected load ratios, leading to progressive accumulation of tensile strain normal to the crack-growth plane. Results also showed that low frequencies and superimposed hold periods at peak loads significantly enhanced strain accumulation at crack tip. Finite element simulation of crack growth was carried out under a constant ΔK-controlled loading condition, again ratchetting was observed ahead of the crack tip, similar to that for stationary cracks.A crack-growth criterion based on strain accumulation is proposed where a crack is assumed to grow when the accumulated strain ahead of the crack tip reaches a critical value over a characteristic distance. The criterion has been utilized in the prediction of crack-growth rates in a CT specimen at selected loading ranges, frequencies and dwell periods, and the predictions were compared with the experimental results.     

THE STRESS ANALYSIS OF HOLE WALL OF FLOWER SHAPE OBTAINED BY INCLINATION RECTIFICATION DIGGING AND ENGINEERING APPLICATION

提出了一种带朵纠倾掏土孔的既有建筑纠倾新方法,给出了带朵纠倾孔孔壁最大应力的近似计算公式,用数值计算方法验证了近似公式的适用性。新方法相对于传统圆形掏土孔方法能更好地实现建筑纠倾工程的精确可控、快速纠倾、避免过倾。当大孔半径与小朵孔半径之比在1$\sim $6范围时,纠倾效果较好。     

以孔边绝对值最大的切向应力最小为优化准则的孔洞形状优化

提出了一个新的孔形优化准则－－孔边绝对值最大的切向应力最小，并给出了了基于这种优化准则寻找最优孔形的方法，在所得的孔洞周边，应力集中程度最小。     

马骨骼的承力机理研究

应用三维激光扫描仪、Pixform和ANSYS等软件建立了马掌骨和跖骨的三维有限元模型,分析了马掌骨的承力机理,研究证明生物体骨骼是一种等强度优化模型.在静态压力载荷下,斜椭圆孔模型更优,这正符合马跖骨代谢孔的形位特征,说明了生物体承力结构的进化合理性.     

Complex variable function method fob hole shape optimization in an elastic plane

In this paper, a complex variable function method for solving the hole shape optimization problem in an elastic plane is presented. In this method, the stresses in hole problems are analysed by taking advantage of the efficiency of the complex variable function method. To optimize the hole shape, the coeffecients in conformal mapping functions are taken as design variables, and the sensitivity analysis and gradient methods are used to reduce the largest circumferential stress in absolute value and at the same time to make the second largest circumferential stress in absolute value not to exceed the largest one (in fact, these two stresses are the stationary values of the circumferential stresses). The coefficients in conformal mapping function are revised by iteration step by step until the largest circumferential stress in absolute value is reduced to the second largest stress. This method guarantees the continuity, differentiability and accuracy of the stress solution along the boundary, and it is evident that this method is better than either the difference method or the finite element method.     

A fully automated Chimera methodology for multiple moving body problems

A fully automated Chimera methodology has been developed in this study to provide geometric or stencil information required to facilitate inter‐grid data communications. Chimera holes are cut automatically in each grid of an overset grid system based on whether the grid overlaps with non‐penetrable surfaces (NPS) and/or blocked regions. The efficiency of the hole‐cutting algorithm is boosted with search algorithms based on the state‐of‐the‐art alternating digital tree (ADT) data structures. The automated nature of the hole‐cutting algorithm is ideally suited for handling multiple moving body problems. Several cases, both steady and unsteady, are used to demonstrate the effectiveness of the methodology. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal Modification of Shape for Two-Dimensional Elastic Bodies

A variational formulation is presented for shape optimal design of two-dimensional elastic bodies. Optimal modification or remodel of a specified shape is determined for reinforcing (addition of material), lightening (removal of material), or redistribution of material by a specified amount. Necessary conditions are obtained in analytical form to provide the basis for an algorithm to solve the optimal remodel design problem computationally. The method is demonstrated for examples in both torsion and plane stress problems, including cases in which the optimal modification calls for introduction of a hole into an original shape.