基于无网格Galerkin法的灵敏度分析与形状优化

结构形状优化已经在工程应用中得到重视,将无网格法与形状优化相结合能够从根本上解决优化过程中出现的有限单元扭曲或畸变问题。为此在无网格Galerkin法的基础上,利用离散导数法,提出一种基于离散型的节点位移灵敏度分析方法,其中采用了拉格朗日乘子法来施加本质边界条件。该方法的最大优势是求解过程与无网格Galerkin法的求解过程相似,容易实现。另外对形状优化的数学模型和节点位移的设计速度域进行了讨论。采用具有解析解的实例,对所提出的灵敏度分析方法进行了验证,所得结果显示两者非常吻合。利用上述所建立的形状优化算法,完成了两个工程实例的形状优化设计。     

Analysis of rectangular stiffened plates under uniform lateral load based on FSDT and element-free Galerkin method

This paper presents an element-free Galerkin (EFG) method for the static analysis of concentrically and eccentrically stiffened plates based on first-order shear deformable theory (FSDT). The stiffened plates are regarded as composite structures of plates and beams. Imposing displacement compatible conditions between the plate and the stiffener, the displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate. The strain energy of the plate and stiffener can be superimposed to obtain the stiffness matrix of the stiffed plate. Because there are no elements used in the meshless model of the plate, the stiffeners need not to be placed along the meshes, as is done in the finite element methods. The stiffeners can be placed at any location, and will not lead to the re-meshing of the plate. The validity of the EFG method is demonstrated by considering several concentrically and eccentrically stiffened plate problems. The present results show good agreement with the existing analytical and finite element solutions. The influences of support size (denoted by a scaling factor d_max) and order of the complete basis functions (N_c) on the numerical accuracy are also investigated. It is found that larger support size and higher order of basis function will furnish better convergence results.     

用随机无网格法和免疫算法分析结构可靠性

本文首先用摄动随机无网格伽辽金法(PSEFGM)求解随机结构的响应。摄动随机无网格伽辽金法具有不需要划分单元和精度高等特点。然后应用人工免疫遗传算法对结构可靠性进行了分析。人工免疫遗传算法是一种新的综合人工免疫算法和遗传算法的智能优化算法,它避免了遗传算法易出现早熟。搜索效率低和不能很好地保持个体多样性等问题。数值实例表明在随机结构可靠性分析方面随机无网格迦辽金法与人工免疫遗传算法具有明显的优势和广泛的应用前景。     

无网格法在形状优化中的应用研究

建立了结构形状优化的数学模型，根据无网格法的离散策略定义了节点位移的设计速度域；引入Lagrange乘子法和罚函数来施加边界条件，借助直接微分法，分别建立了一种离散型的基于无网格Galerkin法的设计灵敏度分析算法；将建立的优化算法结合曲线描述方式对两个工程应用实例进行了形状优化研究，并与基于有限元优化的结果进行了比较，所得结果能够满足工程实际的需求。     

基于退货滞后惩罚的库存控制模型研究

随着竞争的日益激烈,销售商竞相延长产品允许退货时间。此时,如何设置合理的允许退货时间长度成为困扰销售商日常经营的难题。针对于此,本文提出了一种基于退货滞后惩罚的库存控制模型和求解算法,从而解决了上述问题。并采用数值实例分析了退货率和时间惩罚因子对最优允许退货时间长度、最佳订货量以及最大产品收益的影响。实例结果表明,选择合适的时间惩罚因子,并规定合理的允许退货时间长度,可以有效增加销售商的收益。最后,对销售商的经营提出了具有现实指导意义的建议。     

基于随机无网格伽辽金法的结构可靠性分析

基于摄动随机无网格伽辽金法（perturbation stochastic element-free Galerkin method，PSEFGM）进行随机结构的可靠性分析。摄动随机无网格伽辽金法具有不需要划分单元和精度高等特点。数值实例表明，在随机结构的可靠性分析方面随机无网格迦辽金法具有优良的特点和应用前景。     

A Petrov-Galerkin Natural Element Method Securing the Numerical Integration Accuracy

An improved meshfree method called the Petrov-Galerkin natural element (PG-NE) method is introduced in order to secure the numerical integration accuracy. As in the Bubnov-Galerkin natural element (BG-NE) method, we use Laplace interpolation function for the trial basis function and Delaunay triangles to define a regular integration background mesh. But, unlike the BG-NE method, the test basis function is differently chosen, based on the Petrov-Galerkin concept, such that its support coincides exactly with a regular integration region in background mesh. Illustrative numerical experiments verify that the present method successfully prevents the numerical accuracy deterioration stemming from the numerical integration error.     

基于无单元伽辽金法和Matlab遗传算法工具箱的结构形状优化研究

将无网格Galerkin法(element-free Galerkin method,EFGM)和改进的Matlab遗传算法工具箱(matlab genetic toolbox,MGT)相结合,提出了一种新的连续体结构形状优化设计方法。针对形状优化设计的特点,建立了一种规则网格与有限元网格混合的EFG积分背景网格法。详细介绍了M函数文件的修改方法,解决了现有Matlab遗传算法工具箱不能直接施加的非线性与隐性约束条件的难题。最后,完成了一个工程实例的结构形状优化,优化结果表明该方法是有效可行的。     

无网格伽辽金方法在线弹性断裂力学中的应用研究

通过对移动最小二乘形函数进行局部修正,将混合变换法应用于无网格伽辽金方法,给出分析线弹性断裂力学问题的有效的无网格伽辽金方法。这一方法克服了无网格伽辽金方法中常用的拉格朗日乘子法和罚函数法的缺点,实现了本质边界条件在节点处的精确施加。运用线弹性断裂力学理论,采用基于t-分布的新型权函数和部分扩展基函数,对有限板单边裂纹的应力强度因子和拉剪复合型裂纹的扩展进行分析。由于该方法仅需节点信息,而不需要节点的连接信息,从而避免了有限元方法中的网格重构,大大简化了裂纹扩展的分析过程。数值计算结果表明了方法的有效性。     

Dynamic analysis of laminated composite plates with piezoelectric sensor/actuator patches using the FSDT mesh-free method

This paper investigates the active control of laminated composite plates with piezoelectric sensor/actuator patches using an efficient mesh-free method, i.e. the element-free Galerkin (EFG) method. The formulation of the problem is based on the first-order shear deformation plate theory (FSDT) and the principle of virtual displacements. A simple control algorithm coupling the direct and converse piezoelectric effect is used to control the dynamic response of the laminate plate with distributed sensor/actuator patches through a closed control loop. Several example problems are studied to show the influence of stacking sequence and position of sensor/actuator patches on the dynamic responses of the laminate plate. These simulations provide us with the best location of the sensor/actuator patches for active control of the laminate plate.     

A new high-efficiency error compensation system for CNC multi-axis machine tools

To enhance the accuracy of CNC machines for the request of modern industry, an effective static/quasi-static error compensation system composed of an element-free interpolation algorithm based on the Galerkin method for error prediction, a recursive software compensation procedure, and an NC-code converting software, is developed. Through automatically analyzing the machining path, the new error prediction method takes into consideration the fact that the machine structure is non-rigid, and can efficiently determine the position errors of the cutter for compensation without computing a complex error model on-line. The predicted errors are then compensated based on a recursive compensation algorithm. Finally, a compensated NC program will be automatically generated by the NC-code converting software for the precision machining process. Because of the advantage of the element-free theory, the error prediction method can flexibly and irregularly distribute nodal points for accurate error prediction for a machine with complex error distribution characteristics throughout the workspace. To verify the algorithm and the developed system, cutting experiments were conducted in this study, and the results have shown the success of the proposed error compensation system.     

Modelling and simulation of the superelastic behaviour of shape memory alloys using the element-free Galerkin method

This paper is concerned with the application of the element-free Galerkin method to simulate the superelastic behaviour of shape memory alloys (SMA). The meshfree shape functions are derived from a moving least-squares interpolation scheme. A thermomechanical SMA constitutive law is used to describe the superelastic effect. The incremental displacement-based formulation for large deformation is developed by employing the meshfree shape functions and the continuum tangent stiffness tensor in the weak form of the equilibrium equations. By eliminating the unknown constrained nodal variables from the discrete equations, an effective approach is developed for the imposition of the essential boundary conditions. The numerical tests show that the proposed meshfree scheme can successfully reproduce the superelastic behaviour of shape memory alloys.     

用无网格法进行大变形柔性机构拓扑优化设

基于连续体结构的拓扑优化理论,将无网格伽辽金数值方法引入分布式大变形柔性机构拓扑优化设计,并解决了优化中的几何非线性问题。在优化问题中,基于各向同性固体材料惩罚模型（solidisotropic material with penalization,SIMP)和折衷规划法,同时考虑结构的柔性和刚度要求,建立了柔性机构拓扑优化的多准则优化模型,并利用优化准则法求解。采用无网格伽辽金法将求解域离散成节点,避免了有限元方法在处理大变形问题时由于使用网格而产生网格畸变等问题。求解经典算例,与基于线性理论的优化结果相比较分析,说明了该方法的正确性和有效性。     

谐振管内非线性驻波的间断Galerkin方法*

基于伪一维瞬态可压缩Navier-Stokes方程和理想气体状态方程,并结合保持强稳定性的三阶Runge-Kutta格式和斜率限制器技术,构建一种整体轴向谐振激励下求解变截面谐振管内非线性驻波的间断Galerkin方法。分别对直圆柱形、指数形以及圆锥形谐振管内的非线性驻波进行数值仿真计算,通过与现有数值仿真结果对比,验证了本方法的正确性,并重现了已有试验中非线性驻波压力波形中的第二峰值。在直圆柱形和指数形谐振管内研究间断Galerkin方法的两种网格加密方式对于不同形状下谐振管内非线性驻波仿真计算的数值精度、消除数值振荡和准确捕捉激波的不同作用,以及不同CFL数对于数值精度的影响。在指数形谐振管内,通过与有限体积法不同网格数目下的数值仿真结果和计算时间对比,验证了本方法具有耗时短、效率高以及计算精度高的优点。在圆锥形谐振管内,研究不同流速下谐振管内压力和速度波形的变化,并发现随着流速的增加,谐振管小端处的压力幅值增加,大端处速度幅值也随之增加且速度波形中会产生激波。通过间断Galerkin方法分析研究几种谐振管内非线性驻波的各种物理属性,为谐振管形状优化提供了重要参考,也为实现谐振管内非线性驻波的工程应用奠定了良好的基础。     

Behavior of a viscoelastic composite plates under transient load

An efficient numerical algorithm for analyzing dynamic response of orthotropic viscoelastic composite laminates has been developed in the time domain. The integral form of the constitutive laws is exploited. Generalized Wiechert model is adopted to simulate the viscoelasticity of the structure Mindlin-Reissner plate theory is utilized in finite element formulation employing the consistent mass matrix. The proposed algorithm is incorporated in the framework of the finite element based on the displacement model. Newmark Constant average acceleration method is used for integration equation of motion. The developed recurrence formula permits the new time solutions to be evaluated using only previous time values. The developed solution technique is applied to orthotropic plate under two types of force, which are the step-pulse and sin-pulse force.     

Analysis of stiffened corrugated plates based on the FSDT via the mesh-free method

The elastic bending of unstiffened and stiffened corrugated plates is studied in this paper, and a mesh-free Galerkin method is presented for the analyses. A corrugated plate is treated as an orthotropic plate that has different flexure properties in two perpendicular directions. The equivalent flexure properties are estimated by applying constant curvature conditions to the corrugated sheet. The stiffened corrugated plate is considered as a composite structure of an orthotropic plate with beams. By superimposing the strain energy of the orthotropic plate and the beams, and imposing the displacement compatibility conditions between the plate and the beams, the stiffness matrix of the structure is obtained. Because no mesh is needed in the proposed method, there is no limitation to the position of the stiffeners (beams). Changes in the positions of the stiffeners do not require the re-meshing of the plate. Several numerical examples are employed to show the accuracy and convergence of the proposed method. The computation results demonstrate good agreement with the solutions given by ANSYS, and different profiles of corrugated plates are considered.     

Numerical simulation of three-dimensional bulk forming processes by the element-free Galerkin method

A new approach for simulating three-dimensional (3D) bulk metal forming processes is proposed by combining element-free Galerkin method (EFGM) with the flow theory of 3D rigid-plastic/viscoplastic mechanics. Different from the conventional rigid-plastic FEM, the velocity field is constructed by the moving least-squares (MLS) approximation. Special emphasis is placed on the treatments of essential boundary conditions, incompressibility constraint and friction boundaries. The stiffness equation for the analysis of 3D bulk metal forming using EFGM is derived and its key algorithms are given. To test the validity of the proposed meshless approach, a typical 3D upsetting forming process is analyzed and the numerical results are compared with those obtained by commercialized rigid-plastic FEM software Deform^3D. As results show, when handling 3D plastic deformations, the proposed approach eliminates the need of expensive meshing and remeshing procedures unavoidable in conventional FEM, and still provides results that are in good agreement with finite element predictions.     

On a MEMS based dynamic remote temperature sensor using transverse vibration of a bi-layer micro-cantilever

In this paper a novel remote temperature sensor based on a bi-layer micro cantilever beam has been proposed, which can sense the temperature of a given heat source from a finite distance dynamically. Proposed sensor works based on radiation heat transfer and thermally induced vibration. In order to dispose a relationship between temperature of the heat source, and a sensible characteristic like as tunneling current or capacitance, thermo-mechanical behavior of such a structure has been investigated. Heat transfer equation of such a beam includes a nonlinear term due to the radiation heat transfer that has been solved numerically using Rung-Kutta explicit integration method. Galerkin based mode summation method has been utilized to solve the partial differential equation of the beam dynamics and the Houbolt implicit integrating method is used to solve obtained ordinary differential equations. Dynamic response to a step, and semi-harmonic excitations and frequency responses, for the proposed sensor with different geometrical or physical properties have been presented and discussed considering the temperature relaxation time of the sensor. In order to gain a semi explicit solution a perturbation method has been disposed and the results have been compared to the numerical direct integration ones, the accuracy and agreement are excellent.     

粗糙表面热弹塑性接触自适应无网格分析

为了在较真实地模拟接触状态的同时节省计算耗费,采用自适应无网格法求解粗糙表面热弹塑性接触问题。计算中考虑了屈服强度温度相关因素,将基于应变能梯度的自适应无网格法与线性规划-增量初应力法相结合,构建了热弹塑性接触自适应无网格分析模型,并给出相应的程序流程。通过粗糙表面与弹塑性平面热弹塑性接触算例进行验证,分别对两种不同工程材料考虑切向摩擦力、材料应变硬化和材料屈服强度温度相关等情况进行了讨论。结果表明,采用自适应无网格法能有效求解粗糙表面热弹塑性接触问题,在保证计算精度与整体加密相当的情况下,自适应加密的计算耗费约为整体加密计算耗费的10%。     

金属塑性成形过程无网格伽辽金法数值模拟技术研究

将无网格伽辽金法与刚(粘)塑性流动理论相结合,对无网格伽辽金法在金属塑性成形过程数值模拟中的应用技术进行了研究,分别采用混合变换法和罚函数法实现速度边界条件和体积不变条件,提出了基于刚(粘)塑性力学流动理论的无网格伽辽金法,给出了金属塑性成形无网格伽辽金法数值模拟的关键算法,拓展了无网格方法在金属成形领域的应用范围。典型算例的数值计算结果表明了方法的可行性。