耦合对流传热的Stokes流体形状优化设计

本文研究了耦合对流传热的Stokes流体中的形状优化问题.利用不可压缩的定常Stokes方程耦合对流传热的模型来描述流体的特性,运用形状导数方法分析依赖于区域的状态方程解的极小化问题.通过引入共轭状态方程,计算出目标函数的微分形式,并构造求解该形状优化问题的梯度型算法.数值实验的结果验证了所用方法的有效性和可行性.     

求解Stokes方程的一种自适应变分多尺度方法

针对Stokes方程,本文给出一种新的变分多尺度方法.此方法通过构造特殊投影并采用局部化技巧优化了经典变分多尺度方法.同时,本文对此方法进行了后验误差估计,并给出可靠的后验误差估计子及相应的自适应算法.此自适应变分多尺度方法计算简单且易于程序实现.L型区域问题和方腔流问题的数值结果验证了方法的有效性和稳定性.     

非结构子网格压力方法抑制爆轰模拟中的砂漏畸变

本文推广包括子网格压力和边粘性的相容流体算法到非结构网格,研制了包括子网格压力和边粘性的非结构网格相容流体程序,对Saltzman活塞问题和平面爆轰问题进行了数值模拟。从数值结果分析可知,包括子网格压力和边粘性的非结构网格相容流体算法能够有效捕捉流体的激波间断,成功抑制虚假涡漩,消除爆轰模拟中的砂漏畸变。     

基于径向基函数的自适应网格方法

本文给出了一种基于径向基函数的自适应网格方法．该方法利用网格依赖方法的解与径向基函数插值解的信息来细化或粗化网格，充分利用了径向基函数计算格式简单、节点配置灵活的优点与网格依赖方法的稳健性．提出的算法很容易编程实现．数值算例表明该算法可以在解变化剧烈的区域加密网格，在解变化平缓的地方粗化网格，从而在保证相同数值求解精度的情况下，能够极大地节省计算量．     

颗粒流体系统Euler/Lagrange模型中自适应三角网格的生成

针对二维颗粒流体系统Euler／Lagrange模型的有限元模拟,建立了三角网格生成的自适应算法。该算法能够根据颗粒分布与颗粒大小自适应地调整网格的疏密程度,使其网格密度在系统边界附近及颗粒边缘附近较大,而在其它地方较小。与此同时,网格的光滑化也提高了网格质量, 从而为颗粒流体系统介观尺度的有限元模拟奠定了基础。     

三维定常/非定常Stokes方程的维数分裂算法

本文针对三维柱形区域提出了定常/非定常Stokes方程基于一致分裂格式的维数分裂算法(DSA).文章推导了三维定常/非定常Stokes方程维数分裂方法的数值迭代格式.新算法的优势在于一系列的二维问题能够并行执行,而且数值计算中避免了三维网格的生成.大量的数值结果表明新算法既能获得最优收敛阶,而且能获得比采用四面体元求解更精确的逼近解.最后,通过采用并行求解新算法能够得到比较好的加速比和并行效率.     

三维定常/非定常Stokes方程的维数分裂算法（英文）

本文针对三维柱形区域提出了定常/非定常Stokes方程基于一致分裂格式的维数分裂算法(DSA).文章推导了三维定常/非定常Stokes方程维数分裂方法的数值迭代格式.新算法的优势在于一系列的二维问题能够并行执行,而且数值计算中避免了三维网格的生成.大量的数值结果表明新算法既能获得最优收敛阶,而且能获得比采用四面体元求解更精确的逼近解.最后,通过采用并行求解新算法能够得到比较好的加速比和并行效率.     

基于变域变分原理的一种有限元方法

本文从变域变分原理出发,提出了一种新的有限元方法,并将所提算法用于求解形状最优化问题。我们给出了最优化问题解的存在性及其离散形式。讨论了数值解的收敛性,给出了数值解与精确解之间的误差估计。最后给出了算法的具体实施步骤和一个具有代表性的数值算例,数值结果表明该算法是正确、高效的。     

形状误差数据处理的线性规划和单纯形解法

本文简述最优化方法在形状误差数据处理中的应用,并对线性规划和单纯形解法处理圆度误差数据的最优化方法进行实例计算和数学证明。指明了最优化方法是解决形状误差数据处理中减少评定方法误差和简化笔算的途径。     

形状梯度法求解不可压缩流的形状优化问题

流体中物体形状优化设计在实践中有重要的应用。对于区域内由Navier-Stokes方程描述的流体,本文研究以流体状态的泛涵为目标函数的优化问题。基于共轭方法与函数空间参数化方法,本文得到了问题的形状导数。在此基础上构造了一种共轭梯度算法。数值例子表明本文的方法是可行的和稳定的。     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

基于虚载荷变量的无网格法形状优化的研究

将选择施加在"虚结构"控制点上的虚载荷作为形状优化的设计变量,并将它与无网格Galerkin法相结合来开展结构形状优化研究,采用罚函数法来施加边界条件,通过直接微分法建立了结构形状优化的离散型灵敏度分析算法,利用无网格法研究了节点坐标关于设计变量导数的计算。所提出的算法简单明了,它不仅解决了网格的畸变问题,而且简化了优化模型和迭代流程,并可使结构的受力特性得到进一步的改善。最后用2个工程实例验证了所建立的算法,并得到了形状优化结果。     

Topology design with optimized,self-adaptive materials

Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topoiogy; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analytical-computational approach is proposed to solve the relaxed optimization problem. Both stress and displacement analysis methods are presented. Since rank-2 layered composites are known to achieve optimal energy bounds, we restrict the design space to this class of microstructures whose effective properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremization operators and analytically optimizing the microstructural design fields. This results in optimization problems involving the distribution of an adaptive material that continuously optimizes its microstructure in response to the current state of stress or strain. A further reduced problem, involving only the response field, can be obtained in the stress-based approach, but the requisite interchange of extremization operators is not valid in the case of the displacement-based model. Finite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. Care must be taken in selecting the discrete function spaces for the design density and displacement response, since the reduced problem is a two-field, mixed variational problem. An improper choice for the solution space leads to instabilities in the optimal design similar to those encountered in mixed formulations of the Stokes problem.     

Accurate computation of design sensitivities for structures under transient dynamic loads using time marching scheme

Design sensitivities for structures under transient dynamic loads with constraints on displacements and stresses are sensitive to proper space and time discretization. Accuracy within acceptable error limit is feasible when an appropriate time increment coupled with an optimal mesh is used. In this paper, we handle this problem by systematically achieving an adaptive mesh for a reasonably fine but constant time step. Design sensitivities calculated for a good number of examples demonstrate the behaviour of this integrated approach. Comparison is made in terms of total computational time between time-marching scheme and modal superposition method in the context of design sensitivity calculation. Optimal meshes are also obtained corresponding to adaptive time stepping and accurate values of design sensitivities are computed using the optimal mesh and the values of the time increment obtained adaptively. © 1998 John Wiley & Sons, Ltd.     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. A suitable and very general technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

In this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one‐way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Lagrangian finite element analysis of Newtonian fluid flows

A fully Lagrangian finite element method for the analysis of Newtonian flows is developed. The approach furnishes, in effect, a Lagrangian implementation of the compressible Navier–Stokes equations. As the flow proceeds, the mesh is maintained undistorted through continuous and adaptive remeshing of the fluid mass. The principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries, fluid/fluid or fluid/solid interfaces. In contrast to Eulerian approaches, boundary conditions are enforced at material surfaces ab initio and therefore require no special attention. Consistent tangents are obtained for Lagrangian implicit analysis of a Newtonian fluid flow which may exhibit compressibility effects. The accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking. The finite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing © 1998 John Wiley & Sons, Ltd.     

Designing polymer spin packs by tailored shape optimization techniques

We consider optimal shape design problems for polymer spin packs which are widely used in the production of synthetic fibers and nonwoven materials. The design goal is the minimization of the residence time of the polymer, which can be achieved by adjusting the wall shear stress along the boundary. Depending on the specific industrial setting we construct two tailored algorithms. First, we consider the design in three spatial dimensions based on a PDE constrained shape optimization problem. Here, the constraint is given by the Stokes flow. Second, we change the design goal and want to construct shapes in two spatial dimensions which allow for a lower bound on the wall shear stress. This can be incorporated as an additional state constraint. By relaxing this condition and employing the method of mapping we can pull-back the problem onto a fixed reference domain. We get an elegant formulation of this state constrained optimization problem, in which geometric constraints on the boundary can also be included. After discretization we end up with a large-scale NLP which can be handled by existing solvers. Finally, we present numerical results underlining the feasibility of our approach.