Optimal Design of Hole Reinforcements for Composite Structures

In many technical applications like aerospace and automotive structures, holes in thin-walled composite components are necessary for some reason. It easily happens that the presence of a hole results in a detrimental stress concentration in the vicinity of the hole with a strength degradation and premature failure of the structure, respectively. In order to avoid the aforementioned overloading and to achieve a sufficient strength, in practice, a local reinforcement is employed. In the present study, reinforcements by elliptic doublers,as well as doublers adapted to reinforcement requirements in a layerwise manner, are considered. The increasing demands of a low weight and high strength for modern structures lead to the problem of an optimal reinforcement design. For this purpose, an appropriate optimization model is set up, a structural model is developed to describe the mechanical behavior (displacements, stresses, etc.) of such structures, and the techniques of mathematical structural optimization are used to find an optimal design in a systematic manner. In this study, the finite-element method is applied to the structural analysis. Eventually, an appropriate mathematical optimization algorithm is used to approach the desired design optimum in an iterative way. The implemented procedure works with a good reliability and efficiency and yields optimal reinforcement designs which are very useful for direct engineering applications.     

Use of the elastic-and-rigid-combined beam element for dynamic analysis of a two-dimensional frame with arbitrarily distributed rigid beam segments

This paper presents an elastic-and-rigid-combined beam element such that the dynamic characteristics of a hybrid beam and a two-dimensional frame composed of any number of elastic and rigid beam segments can be easily determined. First of all, the displacements for the two nodes of a rigid beam segment are determined in terms of the displacements of its centre of gravity (c.g.). Next, the mass and stiffness matrices for the elastic-and-rigid-combined beam element are derived using the above-mentioned nodal displacements of the rigid beam segment and those of the two adjacent elastic beam elements. Furthermore, for the transformation of state variables of last elastic-and-rigid-combined (or three-node) beam element between the local and global co-ordinate systems, a new transformation matrix is also presented. Finally, the overall property matrices of the entire vibrating system are determined with the conventional assembly technique of finite element method (FEM) and its natural frequencies and associated mode shapes are determined with the standard approach. Some important factors, such as length of rigid beam segment, position for the centre of gravity (c.g.) of rigid beam segment, and total number of rigid beam segments in the entire vibrating system, are investigated. Numerical results reveal that the above-mentioned parameters have significant influence on the dynamic characteristics of the structure with arbitrarily distributed rigid beam segments.     

Stochastic shape sensitivity in powder metallurgy processing

Stochastic shape sensitivity in forming process of powder metallurgy materials is analyzed. For this purpose the rigid-poroplastic material model has been assumed. The theoretical formulation for stochastic shape sensitivity is described which presents probabilistic distributions taking into account random initial and boundary conditions. The control volume approach is discussed. Stochastic finite element equations for rigid – poroplastic materials are solved for the first two probabilistic moments. Numerical simulations were performed to illustrate shape sensitivity problems in the process of compression of rigid-poroplastic cylinder. The differences in deterministic and stochastic sensitivities are presented. The results derived can be used for the subsequent quantitative stochastic shape design as well as stochastic shape optimization.     

On regularization in parameter-free shape optimization

This contribution is concerned with a parameter-free approach to computational shape optimization of mechanically-loaded structures. Thereby the term ’parameter-free’ refers to approaches in shape optimization in which the design variables are not derived from an existing CAD-parametrization of the model geometry but rather from its finite element discretization. One of the major challenges in using this type of approach is the avoidance of oscillating boundaries in the optimal design trials. This difficulty is mainly attributed to a lack of smoothness of the objective sensitivities and the relatively high number of design variables within the parameter-free regime. To compensate for these deficiencies, Azegami introduced the concept of the so-called traction method, in which the actual design update is deduced from the deformation of a fictitious continuum that is loaded in proportion to the negative shape gradient. We investigate a discrete variant of the traction method, in which the design sensitivities are computed with respect to variations of the design nodes for a given finite element mesh rather than on the abstract level by means of the speed method. Moreover, the design update process is accompanied by adaptive mesh refinement based on discrete material residual forces. Therein, we consider radaptive node relocation as well as hadaptive mesh refinement. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape sensitivity analysis of a quasi‐electrostatic piezoelectric system in multilayered media

The optimization of shape and topology of piezo‐patches or layered piezo‐electrical material attached to structural parts, such as elastic bodies, plates and shells, plays a major role in the design of smart structures, as piezo‐mechanic‐acoustic devices in loudspeakers or energy harvesters. While the design for time‐harmonic motions is genuinely frequency‐dependent, as has been reported in the literature in the context of density optimization with the SIMP‐method, time‐varying piezoelectric material has not been investigated with respect to the optimal design so far. Therefore, shape sensitivities for layered piezoelectric material and time‐varying loads and charges are derived in this paper. In particular, we provide the shape‐derivatives for nested piezo‐layers associated with a class of shape functional. More general layers can be dealt with similar approach. Copyright © 2010 John Wiley & Sons, Ltd.     

大跨度叠合梁斜拉桥多维多点非平稳随机地震响应分析

为研究地震动非平稳特性对大跨度叠合梁斜拉桥随机地震响应的影响,采用绝对位移求解的虚拟激励法在通用有限元软件中对某座叠合梁斜拉桥进行多维多点非平稳随机地震响应分析.研究结果表明:采用绝对位移求解的虚拟激励法,能在通用有限元软件中高效地实现对大跨度叠合梁斜拉桥多维多点非平稳随机地震响应分析;设计中假定地震动平稳性,得到的结果偏于保守;行波效应对大跨度叠合梁斜拉桥结构响应有显著影响,考虑行波效应对主塔塔顶位移和塔底内力是有利的,但应注意其对主梁跨中位移和内力的不利影响.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

Grid sensitivity analysis

This paper presents a method to compute the sensitivities of displacements and natural frequencies to the changes in grid locations. The sensitivities are computed within using the semi-analytical method via a program. New software was developed to assist in the selection of design variables and plot the spatial variation of the sensitivities on the finite element model. Two example problems demonstrate the technique.     

自然弯扭梁广义翘曲坐标的求解

提出了自然弯扭梁受复杂载荷作用时静力分析的一种理论方法,重点在于对控制方程的求解,其中考虑了与扭转有关的翘曲变形和横向剪切变形的影响．在特殊的情况下,可以比较容易地得到这些方程的解答,包括各种内力、应力、应变和位移的计算．算例给出了平面曲梁受水平和垂直分布载荷作用时广义翘曲坐标的求解方法．计算结果表明,求得的应力和位移的理论值和三维有限元结果非常接近．此外,该理论不限于具有双对称横截面的自然弯扭梁,同样可推广至具有一般横截面形状的情况．     

Free vibration analysis of a rigid bar supported by arbitrary elastic beams

For convenience, a two-node conventional elastic beam element (C beam element) with the displacements of its 2nd node replaced by those of center of gravity (c.g.) of the joined rigid bar is called the modified beam element (M beam element). The objective of this paper is to present a modified finite element method (modified FEM) such that the free vibration characteristics of a rigid bar supported by a number of elastic beams can be easily determined. First of all, the displacements for the 2nd node of a C beam element joined with the rigid bar are determined in terms of those for the c.g. of the joined rigid bar to establish the M beam element. Next, the mass and stiffness matrices for the M beam element are derived based on the displacements for the 1st node of the C beam element and those for the c.g. of the joined rigid bar. Then, the overall property matrices of the entire unconstrained vibrating system (i.e. a rigid bar supported by a number of elastic beams) can be determined by using the assembly technique of the conventional FEM and considering the effects of lumped mass and rotary inertia of the rigid bar. Finally, the boundary (supporting) conditions are imposed to produce the effective property matrices of the constrained vibrating system and then the free vibration characteristics are determined with the standard approach. In order to confirm the presented theory and the developed computer program, the rigid bar is modeled by a number of C beam elements with bigger Young’s modulus (E_R) and the conventional FEM is used to determine the natural frequencies and associated mode shapes of the vibrating system. It is found that the latter will converge to the corresponding ones obtained from the presented modified FEM when the magnitude of E_R increases to certain values.     

A General Hybrid Optimization Strategy for Curve Fitting in the Non-uniform Rational Basis Spline Framework

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem.     

Design Element Concept of squeeze casting process

Design sensitivity analysis and the application of Design Element Concept have been explored. Exploration has focused on expressed sensitivity with respect to material property and shape of the coolant channel. The Design Element Concept has been applied to the die domain, since the design elements can be considered as a direct mapping of the blocks that make up a die. Analytical methods such as Direct Differentiation Method (DDM) and Adjoint Variable Method (AVM) have been employed in calculating the design element sensitivities. All the calculated design element sensitivities were verified with the Finite Difference Method and the results showed close agreement. From the design element sensitivities distribution in the die, the results show that convergence can be observed as more design elements are employed.     

On curvature control in node–based shape optimization

We present and compare three approaches to control of the curvature of the domain boundary in two–dimensional shape optimization. Since the coordinates of the FE–nodes are used as design variables, objective and constraint functions have to be formulated in terms of the nodal coordinates. Therefore we investigate various formulations for the curvature of the domain boundary in terms of the coordinates of the boundary nodes. In addition, the sensitivities of these expressions with respect to the design nodes are required for the application of gradient–based optimization algorithms. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational sensitivity analysis and changes in the material configuration

This contribution is concerned with some aspects of variational design sensitivity analysis in the physical and material configuration. Sensitivity analysis is a branch of structural optimization, e.g. shape or topology optimization. In these disciplines we consider variations of the material configuration and we are interested in the change of the state variables and the objective functional due to these variations. In the context of structural optimization this is termed as design sensitivity analysis. The sensitivities are required in order to solve the corresponding Lagrangian equation within standard nonlinear programming algorithms. In many engineering applications, the energy functional of the problem is used as the objective functional. In this paper, we consider variations of the energy with respect to the state and the design and we investigate sensitivity relations for the physical and material problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element approach to the position problems in open-loop variable geometry trusses

The present paper looks at some kinematic and static-equilibrium problems that arise with variable-geometry trusses (VGTs). The first part of the paper looks at the use of active controls in the correction of static deformations, the second part at the position problems. The separation between deformable- and rigid-body displacements makes it possible to consider separately the corrections in each component of the structure. VGTs are considered as open-loop linkages with redundant rigid-body degrees of freedom. Owing to this redundancy, possible solutions to the inverse problem are in general infinite, for which reason it is necessary to use some optimization criteria. To tackle the problem an optimization procedure with constraints is developed for the purpose of minimizing the displacements of the actuators. Suitable use of the constraints allows us to solve the direct position problem using the same optimization procedure.     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

Vibrations of spatially suspended flexible pipe-beam with moving fluid

The nonlinear dynamic model of flexible pipe–beam suspended by spatial system of cables is proposed for vibration analysis of pipeline suspension bridges. The model, based on substructure technique, is considered as an assemblage of the following substructures: cables, hangers and pipe–beam. Equation of motion of pipe–beam is derived by Galerkin's FEM with the original finite element formulated in order to include moving mass of transported fluid. To describe cable vibrations, general continuum approach proposed in Ref.[1] is adopted with application into 3D model. Cable model takes into account initial sag, pre–tension force, large displacements and hangers' point reactions. Equation governing the motion of pipe–beam with cables and hangers is obtained regarding equilibrium conditions of interaction forces and compatibility of displacements at connection points between substructures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of fluid-structure systems by eigenvalues gap separation with sensitivity analysis

The design optimization process is frequently accompanied by repetitive time-consuming operations, through the design analysis with a few modifications in its parameters, until the required system is encountered. Whenever possible, it is more convenient to use the sensitivity analysis during the optimization, which permits a fast evaluation of the model performance without doing a completely new analysis. In this work, one methodology for obtaining dynamic optimal structural shape through parameters changing was developed for coupled fluid-structure systems, which took advantage of the modal sensitivity analysis and available nonlinear programming tools. The objective of this work is to maximize the gap between two adjacent eigenvalues in coupled fluid-structure systems, in order to avoid the resonance phenomena at a specific natural frequency interval, taking the height of the cross-section of the beam structural elements as design variables with some dimensional constraints and a constant structural volume. The sequential quadratic programming algorithm was used for solving the constrained nonlinear optimization problem, and the modal sensitivity analysis was considered for the modal prediction of the modified design, as well as for the gradient evaluation of the objective function and system's constraints, throughout the optimization. The study was verified in two-dimensional coupled fluid-structure systems, where the structural system is modeled as a beam. The implemented methodology can be applied to avoid resonant situations or to improve the vibroacoustic comfort, for example, in cabins and reservoirs.     

Large scale structural synthesis

A general purpose optimization program is coupled to a large scale finite element program to provide an efficient tool for structural synthesis. The resulting interface program may be used to design structures for minimum weight, subject to constraints on stress, displacement, and vibration frequencies. A variety of state-of-the-art techniques are employed, including design variable linking, constraint deletion, reciprocal variables, and formal approximations. The capability is demonstrated with the design of a gear housing using 30 design variables and over 5000 nonlinear inequality constraints. The finite element model consists of over 1600 elements and 7000 displacement degrees of freedom. The design required six detailed finite element analyses and approximately one hour on a Cray-1s supercomputer. It is concluded that structures of practical size and complexity can be efficiently designed using numerical optimization.