基于反应扩散方程的水平集结构拓扑优化方法与实现

为实现更加先进的拓扑优化算法,研究采用反应扩散方程的水平集结构拓扑优化方法,通过理论推导给出算法中的参数选择建议.该方法允许在拓扑优化过程中生成新的孔洞,初始结构无须包含孔洞,不需要重新初始化步骤,从而可提高算法的收敛性.针对传统拓扑优化中主要采用体积约束、以柔度最小为目标和体积保留率设定存在一定主观性的问题,探究不同体积保留率下的结构应力水平的变化规律,结果显示可以依据结构最大应力水平与体积保留率的变化规律确定最优体积保留率.     

非结构网格下曲线变形的水平集方法

采用基于非结构网格的水平集方法对曲线变形问题进行了数值模拟。通过度量函数最优化过程得到曲线变形的驱动方程,驱动方程中增加曲线曲率项对度量函数进行优化,空间上采用有限体积法求解水平集方程,时间上采用Runge-kutta显式方法,时间、空间均达到2阶精度。实验结果得到了高质量的过渡曲线,显示复杂几何拓扑形变可以理想地实现。     

基于水平集的遗传算法优化的改进

现有的遗传算法大多数没有给出收敛性准则,且存在早熟收敛和收敛速度较慢的难题,为此提出一类新型遗传算法．该算法首先从被优化函数的因变量出发,引入了水平集的新概念,对每一代种群进行分类,把与目标相关的所有信息有机地结合在一起,从而提高了算法的优化速度;其次通过对变异算子进行改进,提高了种群的多样性,有效地避免了遗传算法的早熟收敛;同时还证明了变异算子能提高种群多样性以及新算法能收敛于全局最优解,最后给出了算法的收敛准则．实验表明,该算法正确有效,搜索效率与精度均优于其他方法．     

Fisher准则和正则化水平集方法分割噪声图像

随着活动轮廓模型的不断成熟和发展,模型的抗噪能力又成为重要的研究课题.为了精确地分割图像的同时去除图像的噪声,针对噪声图像用非负稳健函数作为边缘保持函数,从而保证图像在去噪的过程中边缘和纹理信息不被模糊.首先创造性地将分类器中的Fisher准则函数引入到图像分割中,从分类的角度对C-V模型给出了Fisher解释.把Fisher准则作为分割的标准来建立一个基于区域和边缘相结合的同时完成去噪和分割变分水平集分割模型.其次详细讨论了该模型的数值求解方法.最后实验验证了用Fisher值来衡量分割标准的理论的正确性和可靠性以及模型中正则项约束在去噪过程中的边缘保持功能.通过3组实验检验了提出的模型对噪声图像的去噪和分割比聚类算法、松弛迭代算法、Mean Shift算法有更好的效果.     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.     

联合紧支径向基函数和微分进化算法的电磁优化方法

针对工程电磁装置的优化设计问题,提出一种基于紧支径向基函数的近似模型和改进微分进化算法的组合优化方法。利用基于紧支径向基函数的近似模型得到多维设计空间中构造复杂目标函数和约束函数的显式函数关系。结合改进的微分进化算法获得精确度较高的全局最优解。实例分析和比较表明新方法的收敛速度和优化效率相对于其他的随机类优化算法和模型有明显的优势。     

测量点集的简化及其隐式曲面重建误差分析

基于测量点集的模型重建是逆向工程中的关键环节,为提高模型重建精度和重建效率、保证为模型重建提供必需的信息,简化测量点集、分析重建误差是十分必要的。首先实现了一种测量点集的快速简化算法,然后提出了采用紧支撑径向基函数建立简化后点集的隐式曲面方程,从而实现重建误差分析的方法。实例结果表明,本文简化算法效率较高、效果良好,运用隐式曲面实现的重建误差分析为简化测量点集提供了误差依据。     

保特征散乱数据的曲面重构——变分水平集方法

研究了保特征散乱数据的曲面重构问题。根据主曲率的差可以刻画图像的棱角特征这一特性,提出了一种新的能量模型。通过变分法,能量得到了新的微分方程,并利用有限元方法求解。试验结果表明,该方法有良好的重构效果,并很好地保持了棱角特征。     

用蛙跳算法优化RBF神经网络参数的研究

针对径向基函数（Radial Basis Functions,RBF）神经网络结构参数确定问题,提出了一种基于蛙跳算法优化RBF神经网络参数的新方法。将RBF神经网络参数组成一个多维向量,作为蛙跳算法中的参数进行优化。以适应度函数为标准,在可行解空间中搜索最优解,并对蛙跳算法进行了改进。非线性函数逼近实验结果表明,该优化算法相对标准遗传优化算法、粒子群优化算法有较小的均方误差,具有更好的逼近能力。     

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.     

一种基于轮廓的医学图像弹性配准方法

提出了一种基于轮廓的医学图像弹性配准方法。该方法先对医学图像进行轮廓提取,计算轮廓所围部分(目标)的质心。围绕质心每隔一定角度抽得轮廓点作为特征点。通过计算相似度确定两组特征点的对应关系。采用Wendland提出的ψ函数作为弹性变换的基函数。通过两组对应点确定变换系数。最后根据配准参数对形变图像进行变换,从而得到配准图像。实验表明,本方法对具有形变的医学图像的配准是快速的、有效的。     

A parallel updating scheme for approximating and optimizing high fidelity computer simulations

Approximation methods are often used to construct surrogate models, which can replace expensive computer simulations for the purposes of optimization. One of the most important aspects of such optimization techniques is the choice of model updating strategy. In this paper we employ parallel updates by searching an expected improvement surface generated from a radial basis function model. We look at optimization based on standard and gradient-enhanced models. Given N_p processors, the best N_p local maxima of the expected improvement surface are highlighted and further runs are performed on these designs. To test these ideas, simple analytic functions and a finite element model of a simple structure are analysed and various approaches compared.     

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

A Geometric Representation Scheme Suitable for Shape Optimization

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A computationally efficient numerical approach for multi-asset option pricing

Numerical solution of the multi-dimensional partial differential equations arising in the modelling of option pricing is a challenging problem. Mesh-free methods using global radial basis functions (RBFs) have been successfully applied to several types of such problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we propose a numerical scheme to solve the Black–Scholes equation for valuation of options prices on several underlying assets. We use the derivatives of linear combinations of multiquadric RBFs to approximate the spatial derivatives and a straightforward finite difference to approximate the time derivative. The advantages of the scheme are that it does not require solving a full matrix at each time step and the algorithm is easy to implement. The accuracy of our scheme is demonstrated on a test problem.     

A Modified Quasi-Newton Method for Optimization in Simulation

Optimization in Simulation is an important problem often encountered in system behavior investigation; however, the existing methods such as response surface methodology and stochastic approximation method are inefficient. This paper presents a modification of a quasi-Newton method, in which the parameters are determined from some numerical experiments. To demonstrate the validity of the devised method, two examples resembling the M/M/1 queueing problem are solved. The closeness of the converged solutions to the optimal solutions and a comparison with two stochastic approximation methods indicate that the modified quasi-Newton method as devised in this paper is a robust and efficient method for solving optimization problems in simulation.     

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.