A Study of Two-Dimensional Plane Elasticity Finite Elements for Optimal Design

ABSTRACT

Design of structures using a variety of two-dimensional finite elements is considered in this paper. An efficient technique of computing first-order derivatives of pointwise stress constraints for simple and higher-order two-dimensional (membrane) finite elements is presented. Computational aspects of design sensitivity vector calculation, using a semi-analytical method versus traditional methods are presented. Implementation of a fully-stressed design approach to find a suitable initial estimate leads to increased computational efficiency. These aspects of the design procedure are illustrated through analysis of numerical examples. Experience indicates that a suitable mix of low and higher-order elements yields the most efficient and accurate design model.     

Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis

Abstract

Manifold theoretic ordinary differential equations of motion for holonomic mechanical systems that depend on problem data, or design variables, are shown to be well posed; i.e., they have a unique solution that depends continuously on problem data. It is proved that these differential equations are equivalent to the d’Alembert variational formulation and the index 3 Lagrange multiplier formulation of differential-algebraic equations of motion, which are also shown to be well posed. These results provide a foundation for dynamic system design sensitivity analysis, which requires differentiability of solutions of the equations of motion with respect to design variables.     

多自由度振动结构的随机优化方法

多自由度振动结构的随机优化问题的求解一般要比确定性优化问题的求解复杂得多，也没有行之有效的算法．本文利用概率原理，把随机优化问题转化为等价的确定性优化问题，优化参数取作随机设计变量的均值和标准差，并把该方法应用于多自由度线性扭振系统，在得设计参数的最优均值的同时，还得到了相应的最优标准差，为工程技术人员控制结构零部件的加工精度提供了依据．算例表明该方法是有效的，随机优化的结果比确定性优化的结果好得多．同时，在算例中还讨论了随机设计参数的标准差对优化结果的影响．     

小型飞行器空气动力学

对小型飞行器设计中涉及的空 气动力学问题进行了综述.描述了雷诺数和展弦比对固定翼飞行器的设计以及飞行 性能的影响.在低雷诺数飞行范围,翼型上边界层的特性对飞行器的设计尤为关键. 本文讨论了大量有关层流边界层(包括层流分离泡影响)的实验,作为例子,列举了几 个此飞行雷诺数范围的小型低空无人驾驶飞行器(UAVs).此外,对扑动翼推进的理论 模型进行了简述;其范围涵盖了早期的准定常附着流模型,以及后来计及非定常尾涡、 流动分离以及气动弹性等效应的模型.文中还介绍了那些与理论互补并最终导致扑 翼机设计成功的实验.     

Deployable lenticular stiffeners for origami-inspired mechanisms

Light-weight origami-inspired mechanisms can provide advantages in deployable space systems and other applications. However, a significant challenge in their design is ensuring that they are su?ciently stiff. Compliant, deployable stiffeners utilizing a profile that approximates the Euler Spiral are proposed as one possible solution. It is shown that a structure with this specific profile, called a lenticular stiffener, permits stiffeners to be flattened using a force applied at their edge. Formulas for calculating the increase in stiffness are developed. Relations needed to design the deployment behavior of the stiffeners are also derived. Finally, advantages of different configurations of stiffeners are evaluated. These results are verified through simulation and experiments.     

Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability

Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homogenization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing techniques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the microstructure according to the materials’ future application, a feature clearly demonstrated by the presented results. The methodology can be extended to incorporate other material properties of interest as well as the design of composite materials.     

基于模式运算的工程设计方法

研究了复杂的工程设计决策过程,讨论了设计问题的五个性质,提出了两类不同速度的设计过程:以模式识别为基础的快速设计过程和以模式加法、模式联想为基础的慢速设计过程。此外,还讨论了模蝴集合论和人工神经网络在实现模式运算中的应用。     

谐振腔参数对激光陀螺性能影响的分析

谐振腔是构成激光陀螺的主体，其参数直接影响陀螺的性能，从激光陀螺基本误差理论出发，分析了谐振腔腔长、光阑、毛细孔孔径一致性、毛细孔与光轴相对关系等与陀螺误差的关系。利用激光陀螺的尺度定律 分析了两种典型腔长的激光陀螺，给出了分析结果。根据激光陀螺放电引起的陀螺误差关系及谐振腔振荡模式特性，研究了谐振腔放电毛细孔的加工误差及谐振腔光阑对激光陀螺的影响，给出了毛细孔、光 阑的设计准则，并根据高斯光束特性得出光阑背向散射系数的表达式。这对激光陀螺的设计具有重要的参考价值。     

Computer simulation of the trajectories of large water jets

A three-dimensional computer simulation of the motion of a water jet is described which includes the effects of wind from any direction. The simulation is useful in the design of fire-fighting systems, particularly those used in offshore situations. The equations of motion are presented in vector form and the problem of the fluid dynamic drag variation is discussed. Semi-empirical approximations for the drag components along and across the jet are presented which involve four unknown constants. These are reduced to three by using previous data on the efficiency of vertical jets. To fix the remaining constants, information was available from a series of large jets tests carried out to prove an offshore fire fighting system. In these tests different nozzle shapes were tried and, using the best of these shapes, a large number of trajectories were measured photographically. These were used to fix the simulation drag constants and good agreement is shown between measurements and predictions. The simulation enables the effects of flow-rate, pressure, nozzle size, elevation and wind strength to be evaluated in the system design     

Design approach for cavitation tolerant hydrofoils and blades

Cavitation inception and growth on conventional shape hydrofoils and blades leads initially to a jump of their flow-induced noise, further to an amplification of flow-induced vibration with frequently assisted erosion and finally, to a lift/thrust decrease combined with the drag increase. These undesirable cavitation effects can be mitigated or even suppressed for stable partial cavities experiencing no tail pulsations. A design approach enhancing performance of cavitating hydrofoils/blades by maintaining stable partial cavities is described. Experimental data manifesting an increase of hydrofoil lift with reduction of its drag and of force pulsations by such design are provided. Application of this design approach to propeller/turbine blades and advantages of its employment for blades operating in non-uniform incoming flows are analyzed. The possibility of an increase of the lift to drag ratio and of a reduction of the cavity volume oscillation in gust flows for blade sections is numerically manifested.     

Minimum Weight Design for Structural Eigenvalue Problems by Optimal Control Theory*, †

The application of optimal control theory to minimum weight design of continuous one-dimensional structural elements subject to eigenvalue constraints is discussed. If not only the value of an eigenvalue is prescribed but also its position in the sequence of the ordered eigenvalues—for example, the critical buckling load of a column—the corresponding optimal control problem is shown to include necessarily all eigenvalues. Considering the unspecified eigenvalues as free parameters, necessary conditions for minimum weight design are derived. These conditions are compared with those obtained by use of variational methods. Attention is focused on the special case of multimodal solutions.     

相变传热问题的灵敏度分析与优化设计方法

研究了相变传热问题的优化设计及其灵敏度分析方法. 在有限元-时间差分和等效热容 法求解相变温度场的基础上,提出了相变温度场对设计变量一阶灵敏度的计算方法,给出直 接法和伴随法两种计算格式并分析了它们的特点,建立了相变温度场优化的模型和算法,在有限元分析与优化设计软件JIFEX中实现了该方法. 数值算例表明了灵敏度计算的精度和优 化方法的有效性.     

A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed ‘weak-contrast’ expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a ‘strong contrast’ expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution.

More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used.     

Analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems

This study is concerned with the analysis and design of the force and displacement transmissibility of nonlinear viscous damper based vibration isolation systems. Analytical algorithms are derived using the Ritz–Galerkin method to evaluate the transmissibility of SDOF displacement vibration isolation and force vibration isolation systems where a nonlinear viscous damper is used as an energy dissipating device. The results reveal that compared to linear dampers, nonlinear viscous dampers can more significantly improve the system vibration isolation performance in a wider frequency range. A procedure is then proposed based on the analysis results to facilitate the design of nonlinear viscous dampers for system vibration isolation purposes. These results have significant implications for the design of vibration isolation systems in many engineering applications.     

基于仿真和数据驱动的先进结构材料设计

先进结构材料近年来受到材料和结构设计领域的广泛关注,这些材料一般通过多个尺度的结构设计实现各种卓越的性能.在早期的材料设计中,有的基于设计者的丰富经验,从天然拓扑结构中抽象出合理的数学力学模型;有的基于生物系统的结构和功能特点提取出仿生力学模型.然而,仅依靠经验性的巧妙设计很难得到最优的设计方案,通过反复迭代设计和试验...     

Using geometric complexity to enhance the interfacial strength of heterogeneous structures fabricated in a multi-stage,multi-piece molding process

Interfaces in heterogeneous structures are typically engineered for optimal strength through the control of surface roughness and the choice of adhesives. Advances in manufacturing technologies are now making it possible to also tailor the geometries of interfaces from the nanoscale to the macroscale to create geometrically complex interfaces that exhibit enhanced performance characteristics. However, the impact of geometric complexity on the mechanical behavior of interfaces has not yet been ascertained. In this investigation, the first step is taken towards understanding the effects of geometric complexity on interfacial strength. A new multi-stage, multi-piece molding process is used to create heterogeneous polymer structures with geometrically complex interfaces consisting of rectangular and circular interlocking features. The structural integrity of these heterogeneous structures is characterized through interfacial tension testing. The full-field deformation measurement technique known as digital image correlation is also used during the testing to visualize the deformation fields around the geometrically complex features. Through this characterization, it is determined that the complex geometries increase the interfacial strength by approximately 20–25%, while reducing the statistical variation by 50%. These effects are attributed to a transition in the failure mechanism from interfacial fracture to homogeneous ligament failure. Results also indicate that geometrically complexity can be used on completely debonded interfaces to increase the strength to at least 25–35% of the bonded interface. Based on these results, some simple design rules have been proposed that enable geometrically complex interfaces to be engineered with enhanced strengths approaching the weaker of the two base materials. These design rules can also be used in the engineering of interfaces to facilitate the development of heterogeneous structures using new design paradigms, such as design for recyclability and the design of products based on bio-inspired concepts.     

Numerical modelling of wind flow over buildings in two dimensions

Numerical solutions to the Navier–Stokes equation may provide designers with predictions of the wind environment of buildings under design. To investigate this possibility, two complementary solution procedures are implemented for two-dimensional geometry: a random vortex method to depict the flow evolution, and a control volume method to depict the steady flow field. These are both illustrated by specific application to the case of a building form with a roof of arbitrary pitch.     

Shape optimization and its extension to topological design based on isogeometric analysis

In most of structural optimization approaches, finite element method (FEM) has been employed for structural response analysis and sensitivity calculation. However, the approaches generally suffer certain drawbacks. In shape optimization, cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary. In topology optimization, design results are generally restricted on the initial design space and additional post-processing is required for communication with CAD systems. These drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis: spline basis functions are used in design and geometric modeling whereas Lagrangian and Hermitian polynomials in analysis. Isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional FEM in structural optimization. In isogeometric analysis, the same spline information such as control points and spline basis functions which represent geometries in CAD systems are also used in numerical analysis. Such unification of the mathematical languages in CAD, analysis and design optimization can resolve the issues mentioned above. In this work, structural shape optimization using isogeometric analysis is studied on 2D and shell problems. The proposed framework is extended to topology optimization using trimming techniques. New inner fronts are introduced by trimming spline curves in topology optimization. Trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization. Some benchmarking problems in shape and topology optimization are treated using the proposed approach.     

Unkeyed hash function based on chaotic sponge construction and fixed-point arithmetic

Chaotic maps have various properties that mirror the security requirements of cryptographic algorithms. As such, researchers have utilized them in the design of algorithms such as hash functions. Although there exist a wide range of chaos-based hash functions in literature, most of them are designed in an ad hoc manner rather than relying on well-established design paradigms. In addition, they are commonly implemented using floating-point operations which are inefficient as compared to their bitwise counterparts. The combination of convoluted designs and floating-point representation also leads to hash functions that are difficult to analyze; therefore, claims of security cannot be verified easily. These issues are some of the reasons why chaos-based hash functions have not seen widespread use in practice. This paper proposes a new unkeyed hash function based on a chaotic sponge construction and fixed-point arithmetic to overcome the aforementioned problems. The use of a sponge construction provides provable security justifications, whereas the use of fixed-point arithmetic allows chaotic map operations to be implemented using bitwise operations. The combination of these design elements leads to a design that is both efficient and facilitates future cryptanalysis for security verification. Security and performance evaluations indicate that the proposed hash function has near-ideal diffusion, confusion, collision resistance, and distribution properties in addition to a hashing speed that is at least on par with the current state of the art in chaos-based hash functions.

     

A superposition principle in optimal plastic design for alternative loads

The paper is concerned with the optimal plastic design of sandwich beams, frames and trusses for alternative loading conditions. Upper and lower bounds for the optimal weight of a beam are derived, for single as well as for alternative loading conditions. These bounding theorems are used to establish a superposition principle. If no explicit bounds on the cross-sectional areas are prescribed, the optimal design for alternative loading conditions P₁ and P₂ can be obtained by superposition of the optimal designs for the single loading conditions and . If the cross-sections are to have at least given non-zero values, the principle furnishes upper and lower bounds to the optimal weight.The principle is illustrated by a simple example.