A general formulation of structural topology optimization for maximizing structural stiffness

This paper presents a general formulation of structural topology optimization for maximizing structure stiffness with mixed boundary conditions, i.e. with both external forces and prescribed non-zero displacement. In such formulation, the objective function is equal to work done by the given external forces minus work done by the reaction forces on prescribed non-zero displacement. When only one type of boundary condition is specified, it degenerates to the formulation of minimum structural compliance design (with external force) and maximum structural strain energy design (with prescribed non-zero displacement). However, regardless of boundary condition types, the sensitivity of such objective function with respect to artificial element density is always proportional to the negative of average strain energy density. We show that this formulation provides optimum design for both discrete and continuum structures.     

Determination of optimal actuator forces and positions in smart structures using adjoint method

The problem of optimal design of structures with active support is analyzed in the paper. The sensitivity expressions with respect to the generalized force and the position of actuator are derived by the adjoint structure approach. Next, the optimality conditions are formulated by means of an introduced Lagrangian function. The problem of introduction of a new actuator is also considered and the condition of modification is expressed by means of the topological derivative. The obtained sensitivity formula, optimality conditions and modification conditions are applied in the optimization algorithm with respect to the number, positions and generalized forces of the actuators. Numerical examples of optimal control of beams illustrate the procedure proposed in the paper.     

On optimal topology of grillage structures

Two problems of optimum topological design of grillages are discussed: (1) the Equilibrium Linear Programming (ELP), where the analysis model is based only on equilibrium conditions and (2) the Nonlinear Program (NLP), where the ELP formulation is extended to include compatibility conditions. The structural topology is optimized by allowing elimination of elements. Three different force method formulations are presented for each of the problems. It is shown that the optimal topology for the NLP problem might correspond to a singular point in the design space. The optimal topology for the ELP problem is obtained by solving a linear program (LP).

Conditions for selecting a geometry of Multiple Optimal Topologies (MOT) are derived. The objective function for the MOT geometry is shown to be independent of the redundant forces, and some of the optimal topologies are usually statically determinate structures. In such cases the lower bound on the optimal value obtained by the ELP solution is equal to the final global optimum. Examples are given to illustrate how the optimal topology and its corresponding load path change with the geometric parameters. Design procedures that combine automated optimization and CAD techniques are most suitable for solving the presented problems.

     

航空发动机叶片丢失整机响应及安全性分析

为了研究航空发动机结构叶片丢失后整机响应与连接结构安全性,在瞬态显式动力学软件ANSYS/LS-DYNA中建立了真实大涵道比涡扇发动机的全三维有限元模型,并开展了一个转动周期的仿真计算。结果表明:丢失叶片与包容机匣及追随叶片的撞击过程中,撞击力分别在叶尖首次撞击机匣、叶片根部受追随叶片撞击、叶根撞击包容机匣时出现峰值;风扇机匣与承力结构的位移和应力最大值区域随着丢失叶片运动而变化,且始终发生在丢失叶片同一方向上。前轴承和中轴承所受载荷较大,后轴承载荷较小,且随着发动机转动近似正弦曲线。前安装吊耳处位移最大,其次为推力杆前端,后吊耳处位移较小。     

On some simplified models for optimal design of structural systems

Some simplified models for optimal design of large structural systems are reviewed. The design variables represent the members' cross-sections and the constraints are related to stress, displacement and design considerations. Using displacement analysis formulation, several explicit behavior models which do not involve multiple implicit analyses are presented. It is shown that approximations of the displacements, often used in optimization of large systems, essentially lead to solutions which do not satisfy equilibrium.Approximations along a line in the design variables space are developed, and simplifications based on the virtual load method and inverse design variables formulations are presented. The relationships between the various models are derived; it is shown that, under the usual assumption of linear relationship between the stiffness matrix elements and the design variables, some of the approximations become equivalent.     

填埋场典型复合衬垫系统单剪试验数值模拟

为研究填埋场典型复合衬垫系统接触面间的滑动特性,建立典型复合衬垫系统单剪试验的有限元模型,用ANSYS对其进行数值仿真分析.研究复合衬垫不同界面在不同法向力下的滑动特征;绘制衬垫系统不同界面节点的应力 位移关系曲线,得到不同法向力作用下各个界面不同位置点沿剪切位移方向和垂直剪切位移方向的切应力变化规律;分析接触面内节点的剪切位移随载荷步的变化特征.结果表明：随法向力的增加,滑动界面将发生转移;在低法向力（56 kPa）条件下,土工网 土工膜界面发生滑动;在高法向压力条件下（压力超过560 kPa）,土工膜 黏土界面发生滑动.     

Topological structural analysis

Topological structural analysis is a terminology proposed for studying the behaviour of structures when the constitutive law is reduced to a least formulation. In addition to the equilibrium and deformation compatibility equations we only require that the extensional and contractile strains correspond, respectivley, to tensile and compressive stresses, without further specifying the nature of the constitutive law. The only analysis parameter in these equations is the equilibrium matrix which incorporates pure topological information such as node positions and bar connectivities. It is shown that in a topological context the internal forces, which can be realized by a structure, are bounded by a convex combination of the internal forces of its embedded statically determinate substructures. It is also shown that this structural equilibrium space is nonconvex. Consequently, the internal forces are bounded, component by component, by the internal forces in the statically determinate solutions. Having established that the structural equilibrium space is a small subset of the equilibrium space it is shown that pure equilibrium solutions, such as are obtained in plastic analysis and design, are not always feasible. It is conjectured that topological design of structures may benefit from using topological analysis rather than a pure equilibrium analysis.     

On Clamping Planning in Workpiece-Fixture Systems

Deformations of contacts between the workpiece and locators/clamps resulting from large contact forces cause overall workpiece displacement, and affect the localization accuracy of the workpiece. An important characteristic of a workpiece-fixture system is that locators are passive elements and can only react to clamping forces and external loads, whereas clamps are active elements and apply a predetermined normal load to the surface of workpiece to prevent it from losing contact with the locators. Clamping forces play an important role in determining the final workpiece quality. This paper presents a general method for determining the optimal clamping forces including their magnitudes and positions. First, we derive a set of “compatibility” equations that describe the relationship between the displacement of the workpiece and the deformations at contacts. Further, we develop a locally elastic contact model to characterize the nonlinear coupling between the contact force and elastic deformation at the individual contact. We define the minimum norm of the elastic deformations at contacts as the objective function, then formulate the problem of determining the optimal clamping forces as a constrained nonlinear programming problem which guarantees that the fixturing of the workpiece is force closure. Using the exterior penalty function method, we transform the constrained nonlinear programming into an unconstrained nonlinear programming which is, in fact, the nonlinear least square. Consequently, the optimal magnitudes and positions of clamping forces are obtained by using the Levenberg–Marquardt method which is globally convergent. The proposed planning method of optimal clamping forces, which may also have an application to other passive, indeterminate problems such as power grasps in robotics, is illustrated with numerical example.      

Anchorage plates for cable-stayed bridges

The design of Cable-Stayed bridges requires the engineer to examine in detail the local force distribution in the plates which transfer the cable force to the main structure. It is therefore the intention of this paper to present the response of these plate elements, when subjected to the cable forces, using the finite element method. The results will be presented such that rapid evaluation of their strength can be determined.     

Prager-structures: Archgrids and cable networks of optimal layout

This paper is concerned with a new class of optimal structures termed ‘Prager-structures’ which satisfy the following design criteria: all members have the same constant stress throughout the system and the total structural weight is minimized with respect to the member layout as well as the location of external loads along their line of action. In contrast to Michell frames, Prager structures (i) must have member forces of the same sign throughout; and (ii) have been found to constitute ‘surface structures’ whose middle surface contains the centroidal axis of all members. Using Prager's theories, global optimality is established for broad classes of loading and boundary conditions. The results are compared with those for Michell structures and optimal grillages.     

Constrained optimization of structures with displacement constraints under various loading conditions

Optimization problems often involve constraints and restrictions which must be considered in order to obtain an optimum result and the resultant solution should not deviate from any of the imposed constraints. These constraints and restrictions are imposed either on the design variables or on the algebraic relations between them. Constraints of allowable stress, minimum size and buckling of members in the absence of allowable displacement constraint are the most important factors in optimization of the cross-sectional area of structural elements. When the allowable displacement constraint is included in the problem as a determinant parameter, since the specifications of most of elements affect the displacement rate, the way of imposing and considering this constraint requires special care. In this research the way of simultaneous imposition of multi displacement constraints for optimum design of truss structures in several load cases is described. In this method various constraints for different load cases are divided into active and passive constraints. The mathematical formulation is based on the classical method of Lagrange Multipliers. Overall, this simple method can be employed along with other constraints such as buckling, allowable stress and minimum size of members for imposing the displacement constraint in various load cases.     

An investigation of the effect of counterweight configuration on main bearing load and crankshaft bending stress

In this study, effects of counterweight mass and position on main bearing load and crankshaft bending stress of an in-line six-cylinder diesel engine is investigated using Multibody System Simulation Program, ADAMS. In the analysis, rigid, beam and 3D solid crankshaft models are used. Main bearing load results of rigid, beam and 3D solid models are compared and beam model is used in counterweight configuration analyses. Twelve-counterweight configurations with a zero degree counterweight angle and eight-counterweight configurations with 30° counterweight angle, each for 0%, 50% and 100% counterweight balancing rates, are considered. It is found that maximum main bearing load and web bending stress increase with increasing balancing rate, and average main bearing load decreases with increasing balancing rate. Both configurations show the same trend. The load from gas pressure rather than inertia forces is the parameter with the most important influence on design of the crankshaft. Results of bearing loads and web bending stresses are tabulated.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

A minimal mass deployable structure for solar energy harvesting on water canals

This paper produces a design for a minimal mass, deployable support structure for a solar panel covering of water canals. The results are based upon the minimal mass properties of tensegrity structures. The efficient structure is a tensegrity system which has an optimal complexity (i.e. an optimal number of members) for minimal mass. This optimal complexity is derived in this paper, along with deployable schemes which are useful for construction, repairs, for Sun following, and for servicing. It is shown that the minimal structure naturally has deployable features so that extra mass is not needed to add the multifunctional features. The design of bridge structures with tensegrity architecture will show an optimal complexity depending only on material choices and external loads. The minimization problem considers a distributed load (from weight of solar panels and wind loads), subject to buckling and yielding constraints. The result is shown to be a Class 1 Tensegrity substructure (support structure only below the deck). These structures, composed of axially-loaded members (tension and compressive elements), can be easily deployable and have many port-able applications for small spans. The focus of this paper is an application of these minimal mass tensegrity concepts to design shading devices to prevent or reduce evaporation loss, while generating electric power with solar panels as the cover. While the economics of the proposed designs are far from finalized, this paper shows a technical solution that uses the smallest material resources, and shows the technical feasibility of the concept.     

On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

This paper presents a three-dimensional extension of graphic statics using polyhedral form and force diagrams for the design of compression-only and tension-only spatial structures with externally applied loads. It explains the concept of 3D structural reciprocity based on Rankine’s original proposition for the equilibrium of spatial frames. It provides a definition for polyhedral reciprocal form and force diagrams that allows including external forces and discusses their geometrical and topological characteristics. This paper furthermore provides a geometrical procedure for constructing a pair of reciprocal polyhedral diagrams from a given polyhedron representing either the form or force diagram of a structural system. Using this method, this paper furthermore suggests a design strategy for finding complex funicular spatial forms in pure compression (or tension), based on the construction of force diagrams through the aggregation of convex polyhedral cells. Finally, it discusses the effect of changes in the geometry of the force diagram on the geometry of the form diagram and the distribution of forces in it.     

The complex method applied to optimal truss configuration

The Complex Method of Box is applied to the determination of optimal member sizes and geometric configuration for minimum weight of 3-dimensional truss structures. Design constraints include bounds on member size, joint coordinates, member stresses, Euler buckling and joint displacements. The displacement method of structural analysis is used and the system is assumed to be linearly elastic. Statically indeterminate structures under multiple loading conditions can be optimized and the sizing as well as the geometric design variables may be linked. Limited topological changes are permitted for designs which have small member forces.The design spaces for sizing and geometric variables are separated. Geometry is modified by the Complex Method and member sizes by stress ratio and a scaling procedure for stiffness.The method is applied to two numerical examples from the literature. Results indicate favourable design improvements and rates of convergence. Substantial additional improvement resulting from member deletion is demonstrated.     

平行梁式电容传感器极板耦合角位移计算和分析

为了解决平行梁式电容称重传感器极板耦合角位移带来的输出电容值与输入载荷之间的非线性问题,运用力法对平行梁式弹性体超静定结构进行了计算分析,得出了与上下极板夹角关联的主要结构参数,通过内力对极板角度进行了解析,并在有限元软件中验证了计算的正确性。此结果为平行梁通过结构参数设计优化耦合角位移提供了理论方法,并根据非平行板电容计算公式,构建了更加精确的载荷与电容的数学模型,为传感器输出输入特性曲线的拟合提供了依据。     

大型通用静力试验平台研制与应用

为了验证大型结构体的设计合理性和产品可靠性,掌握结构体负荷承载能力,给出科学评价依据和指导意见,需要科学有效的试验手段为受试对象施加载荷,获取结构体应变、位移等信息,为力学分析提供必要数据支撑;分析了试验平台的主要功能需求,研究了由荷载测控系统、应变测量系统、位移观测系统、试验管理系统和图像监控系统组成的试验平台方案,重点论述了平台的核心关键技术,通过具体应用实例和试验数据,指出了试验平台的应用领域及所能解决的实际问题,最后总结了试验平台的特点和意义。     

Structural optimization in magnetic fields using the homogenization design method — Part II: Reduction of vibration caused by magnetic forces

Summary The homogenization design method has been expanded to obtain the optimal topology of a structure in magnetic fields to maximize the magnetic energy. In this study, the homogenization design method (HDM) is applied to a structure in magnetic fields to reduce the vibration level of a structure excited by the magnetic forces, especially by the magnetic harmonic forces. This is accomplished by obtaining the optimal material distribution of the structure to minimize the frequency response. The Maxwell stress method is used to compute the magnetic force and the HDM is applied for the optimization. It is verified that the HDM is useful to minimize the frequency response by some actual applications. The effects of mesh density of the design domain and the rotor-stator position are also examined.