基于优化原理的单元尺寸场光滑化：模型和应用

发布时间：2018-07-05 02:34

本文选题：网格生成 + 单元尺寸场　；参考：《温州大学》2017年硕士论文

【摘要】：网格生成是有限元分析的前置处理环节,也是其主要性能瓶颈。为控制网格规模的同时保证后续分析的精度,网格生成算法需借助单元尺寸函数,在问题域的不同位置布置不同尺寸的单元,以适应几何特征和物理量梯度在问题域的变化。给定的初始单元尺寸场通常不满足光滑过渡的要求,距离靠得很近的点其单元尺寸值相差很大,其间不可避免会产生质量很差的网格单元。因此,单元尺寸场光滑化是有限元网格生成研究的重要课题。已有方法主要利用直角背景网格定义尺寸场,存在背景网格规模过于庞大、相应算法时空效率不佳等缺陷。基于非结构网格自动以及自适应生成能力高这一特点,本文选用非结构背景网格定义尺寸场,研究了相应的尺寸场光滑化方法。主要工作包含两个方面:(1)基于合理的基本假设,将单元尺寸场在直线域、三角形域过渡受控的数学条件拓展到四面体域。继而以最小化尺寸场的变化为目标,以尺寸场在三维体域过渡受控为约束,建立了适用于尺寸场光滑化问题的优化模型。证明了模型的凸性和唯一最优解的存在性。(2)基于内点法开发了优化模型的数值求解算法,并应用算法于几何自适应尺寸场的光滑化。结合初始几何自适应尺寸场计算、Delaunay网格生成等算法,实现了一套几何自适应四面体网格全自动生成软件流程。数值实验表明,和已有也基于非结构背景网格的经典方法相比,新方法成功解决了 H-Correction方法无法保证单元尺寸场在背景单元内光滑过渡(导致最终网格包含一定数量的低质量过渡单元),以及GradH-Correction算法改动初始尺寸场过于剧烈(导致最终网格单元大范围“过度细化”)等缺陷。
[Abstract]:Mesh generation is not only the preprocessing of finite element analysis, but also the main bottleneck of its performance. In order to control the size of the grid and ensure the accuracy of the subsequent analysis, the mesh generation algorithm needs to use the element size function to arrange different size elements in different positions of the problem domain, so as to adapt to the variation of the geometric characteristics and the gradient of physical quantities in the problem domain. The given initial element size field usually does not satisfy the requirement of smooth transition, and the size of the element is very different from the point close to it, so it is inevitable that the quality of the mesh element will be very poor. Therefore, the smoothing of element size field is an important topic in the research of finite element mesh generation. The existing methods mainly use right-angle background mesh to define the size field, which has some defects such as the large scale of background grid and the inefficient spatio-temporal efficiency of the corresponding algorithm. Based on the characteristics of automatic and adaptive generation of unstructured meshes, this paper uses unstructured background meshes to define dimension fields, and studies the corresponding smoothing methods of dimension fields. The main work includes two aspects: (1) based on the reasonable basic assumptions, the mathematical conditions of the transition control of the element size field in the straight line domain and the triangular domain are extended to the tetrahedron domain. With the objective of minimizing the change of the size field and the constraint of the transition of the size field in the three-dimensional volume domain, an optimization model for the smoothing of the size field is established. The convexity of the model and the existence of the unique optimal solution are proved. (2) based on the interior point method, the numerical solution algorithm of the optimization model is developed, and the algorithm is applied to the smoothing of the geometric adaptive size field. Based on the algorithm of Delaunay mesh generation based on the initial geometric adaptive dimension field, a set of automatic generation software flow of geometric adaptive tetrahedron mesh is implemented. Numerical experiments show that, compared with the classical methods based on unstructured background meshes, The new method successfully solves the problem that the H-Correction method can not guarantee the smooth transition of the element size field in the background element (resulting in the final grid containing a certain number of low-quality transition units), and the GradH-Correction algorithm changes the initial size field too violently. Resulting in the ultimate grid cells in a wide range of "excessive refinement") and other defects.
【学位授予单位】：温州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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