无网格自然邻接点Petrov-Galerkin法及其在结构分析和拓扑优化中的应用
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摘要
拓扑优化是一个相对较新但急速发展的研究领域,目前,在连续体结构拓扑优化中主要的分析方法是有限元法(FEM:Finite Element Method),而且主要集中在静力柔度的拓扑优化,涉及动力问题的拓扑优化研究相对较少。然而,有限元法有其固有的缺点,有限元法在处理诸如大变形和移动边界问题时需要不断重新划分和重构网格以解决网格畸变和网格移动等问题。无网格方法克服了对网格的依赖性,彻底或部分消除了网格的划分,因此无网格方法在处理裂纹扩展问题,超大变形问题,高速冲击等问题中具有明显的优势。本文基于自然邻接点Petrov-Galerkin (NNPG)法主要研究了连续体结构的静力、动力响应及其拓扑优化设计问题。
首先综述了拓扑优化和无网格法的发展历史和国内外研究现状,对各种典型的拓扑优化方法进行了回顾。基于SIMP(Solid Isotropic Material with Penalization)密度-刚度插值模型和优化准则法,以最小柔度的连续体拓扑优化设计为例,介绍了优化模型的建立、求解方法、计算步骤等基本理论；讨论了数值计算的稳定性问题和目前存在的解决方法,提出了一种连续的密度插值方法,有效消除了棋盘格现象；为了获得网格独立的拓扑优化设计,在连续密度场的基础上引入密度和刚度之间的“非局部关系”,构造修正的连续密度场一并解决了传统基于FEM的拓扑优化数值方法普遍存在的棋盘格现象和网格依赖性问题。
对于连续体结构的拓扑优化问题,精确的结构响应分析是至关重要的。在将自然邻接点Petrov-Galerkin法应用于连续体结构的拓扑优化设计之前,有必要对NNPG法的求解精度和收敛性问题进行验证。采用自然邻接点Petrov-Galerkin法相继分析平面问题、Mindlin板问题的静力响应和动力响应问题。在求解过程中,利用成熟的Delaunay三角形剖分技术,采用FEM三节点三角形单元的形函数作为权函数,可以减少域积分中被积函数的阶次,提高了计算效率。基于自然邻接点插值构造的试函数,具有Kronecker delta函数性质,能够方便地施加本质边界条件和处理不连续面问题。数值算例表明：对于平面问题、Mindlin板问题的静力、动力问题,NNPG法具有精度高、稳定性好和实现简单的特点。
利用无网格NNPG法和所提出的连续密度场方法,基于SIMP拓扑优化模型和优化准则法研究了平面问题、Mindlin板问题的最小柔度拓扑优化设计。采用自然邻接点插值形函数构造连续的名义密度场,有效消除了棋盘格现象,获得无棋盘格的高分辨率解的设计。在连续密度场的基础上,考虑非局部效应,通过引入半径为Rmin的影响域对原密度场进行重构,以约束设计变量空间(这对于确保解的存在性是十分必要的)。通过影响域半径Rmin控制优化结构的复杂性,从而一并解决了传统基于FEM的拓扑优化数值方法普遍存在的棋盘格现象和网格依赖性问题。
利用无网格NNPG法,研究了平面问题、Mindlin板问题的动力拓扑优化设计,其中包括自由振动结构的特征值优化和受迫振动结构的频率响应优化问题。对动力拓扑优化模型的建立、低密度区域可能存在的局部模态、特征值优化目标函数的振荡问题以及非结构质量对拓扑优化设计的影响等问题进行了讨论和研究。对于特征值优化存在的目标函数振荡的问题,提出了修改灵敏度信息,控制优化路径的方法,一定程度上可以解决目标函数的振荡问题,获得与最优解比较接近的拓扑优化设计。
对于连续体结构的最小柔度拓扑优化设计、自由振动结构的特征值优化和受迫振动结构的频率响应拓扑优化设计,数值算例表明本文方法均能获得合理的、无棋盘格的、节点分布独立的、黑白的拓扑优化设计,证明了本文方法的有效性和可行性。
Topology optimization is a relatively new but extremely rapidly expanding research field. Up to now, the prevailing analysis method in the topology optimization is the finite element method (FEM), and the bulk part of the work focus on static compliance optimization, while limited efforts were put into the dynamic topological optimization. However, for the FEM, there are some shortcomings such as mesh distortion, frequent remeshing when dealing with large deformation or moving boundary problems, etc. The meshless method avoids the dependence of meshes, and thoroughly or partly eliminates meshing. It makes this kind of methods possess great advantages when dealing with crack propagation problems, super-large deformation problems and high velocity impact problems, etc. In this dissertation, the natural neighbor Petrov-Galerkin method (NNPG) is used to study the elasto-static, dynamic problems and the topology optimization of the continuum structures.
At the beginning of the dissertation, the history and recent developments of the topology optimization and meshless method are overviewed, and several typical topology optimization methods are introduced particularly. The fundamental theory of topology optimization of continuum structure is discussed, including the formulation and solution of the optimization model, implementation and computational procedure, which based on the SIMP (Solid Isotropic Material with Penalization) density-stiffness interpolation model and the optimality criteria method. Numerical instabilities such as checkerboard and mesh-dependence as well as the available ways to circumvent them are also discussed. A continuous bulk density fields is proposed to eliminate the checkerboard patterns of the material distribution. In order to achieve mesh-independent designs by the topology optimization, a non-local relationship between the density and stiffness is introduced, based on the proposed modified continuous bulk density fields, both of the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are circumvented simultaneously.
When dealing with topology optimization problems, it is vital to obtain accurate structural responses. It's very important to testify the accuracy and convergence of the NNPG method before applying the method to study the topology optimization of the continuum structure. To this end, both of the elasto-static and dynamic problems in2D plane problems as well as the Mindlin plate problems are studied by the natural neighbor Petrov-Galerkin method. In the implementation, based on the well established Delaunay triangulation, the FEM shape functions of three nodes triangular are taken as test functions, which reduce the order of integrands involved in domain integrals and improves the computational efficiency of the method. The trial functions which are constructed based on the natural neighbor interpolations have the Kronecker Delta function property, which facilitate imposition of essential boundary conditions and dealing with discontinuity problems. Numerical examples show that the present method possesses high accuracy and good performance of stability, and is easy to implement.
The topology optimization for plane and Mindlin plate problems based on the meshless NNPG method and the proposed continuous bulk density fields, where the objective is to minimize compliance, is investigated. The numerical approach presented here is based on the SIMP approach and the optimality criteria method. A continuous bulk density fields constructed by the natural neighbor interpolation shape functions is proposed to eliminate the checkerboard pattern. This approach is used to get higher resolution solutions with mesh refinement and the checkerboard control. Based on the continuous bulk density fields, the non-local effect is considered. To this end, the original bulk density fields is reconstructed by introducing a influence domain whose radius is Rmin, in this way, the design space is restricted, which is very necessary to ensure the existence of solution. By adjusting Rmin, the complication of the optimized structures is controlled by the proposed method. In this way, the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are all circumvented.
The dynamic topological optimization for plane and Mindlin plate problems based on the meshless NNPG method is also studied, which include the eigenvalue optimization problems for free vibration of structures and the frequency response optimization problems for forced vibration of structures. The formulation of the dynamic topological optimization and the possibility of localized modes in low density areas are discussed. While the oscillation of the objective function during the eigenvalue optimization process and the effect of the non-structural mass to optimized structures by the topology optimization are investigated particularly. A heuristic approach, which modifies the sensitivity information to control the optimization path, is presented to avoid the oscillation of the objective function in eigenvalue optimization. By this way, some "optimized" design which closing to the global optimum of the original problem can be obtained.
Numerical results show that the present method obtains reasonable, checkerboard controlled, node independent and black-white optimization results for the topology optimization of the continuum structures including the minimum compliance topology optimization, the eigenvalue optimization for the free vibration of structures and the frequency response optimization for the forced vibration of structures, which demonstrate the feasibility and validity of the present method for these problems.
首先综述了拓扑优化和无网格法的发展历史和国内外研究现状,对各种典型的拓扑优化方法进行了回顾。基于SIMP(Solid Isotropic Material with Penalization)密度-刚度插值模型和优化准则法,以最小柔度的连续体拓扑优化设计为例,介绍了优化模型的建立、求解方法、计算步骤等基本理论；讨论了数值计算的稳定性问题和目前存在的解决方法,提出了一种连续的密度插值方法,有效消除了棋盘格现象；为了获得网格独立的拓扑优化设计,在连续密度场的基础上引入密度和刚度之间的“非局部关系”,构造修正的连续密度场一并解决了传统基于FEM的拓扑优化数值方法普遍存在的棋盘格现象和网格依赖性问题。
对于连续体结构的拓扑优化问题,精确的结构响应分析是至关重要的。在将自然邻接点Petrov-Galerkin法应用于连续体结构的拓扑优化设计之前,有必要对NNPG法的求解精度和收敛性问题进行验证。采用自然邻接点Petrov-Galerkin法相继分析平面问题、Mindlin板问题的静力响应和动力响应问题。在求解过程中,利用成熟的Delaunay三角形剖分技术,采用FEM三节点三角形单元的形函数作为权函数,可以减少域积分中被积函数的阶次,提高了计算效率。基于自然邻接点插值构造的试函数,具有Kronecker delta函数性质,能够方便地施加本质边界条件和处理不连续面问题。数值算例表明：对于平面问题、Mindlin板问题的静力、动力问题,NNPG法具有精度高、稳定性好和实现简单的特点。
利用无网格NNPG法和所提出的连续密度场方法,基于SIMP拓扑优化模型和优化准则法研究了平面问题、Mindlin板问题的最小柔度拓扑优化设计。采用自然邻接点插值形函数构造连续的名义密度场,有效消除了棋盘格现象,获得无棋盘格的高分辨率解的设计。在连续密度场的基础上,考虑非局部效应,通过引入半径为Rmin的影响域对原密度场进行重构,以约束设计变量空间(这对于确保解的存在性是十分必要的)。通过影响域半径Rmin控制优化结构的复杂性,从而一并解决了传统基于FEM的拓扑优化数值方法普遍存在的棋盘格现象和网格依赖性问题。
利用无网格NNPG法,研究了平面问题、Mindlin板问题的动力拓扑优化设计,其中包括自由振动结构的特征值优化和受迫振动结构的频率响应优化问题。对动力拓扑优化模型的建立、低密度区域可能存在的局部模态、特征值优化目标函数的振荡问题以及非结构质量对拓扑优化设计的影响等问题进行了讨论和研究。对于特征值优化存在的目标函数振荡的问题,提出了修改灵敏度信息,控制优化路径的方法,一定程度上可以解决目标函数的振荡问题,获得与最优解比较接近的拓扑优化设计。
对于连续体结构的最小柔度拓扑优化设计、自由振动结构的特征值优化和受迫振动结构的频率响应拓扑优化设计,数值算例表明本文方法均能获得合理的、无棋盘格的、节点分布独立的、黑白的拓扑优化设计,证明了本文方法的有效性和可行性。
Topology optimization is a relatively new but extremely rapidly expanding research field. Up to now, the prevailing analysis method in the topology optimization is the finite element method (FEM), and the bulk part of the work focus on static compliance optimization, while limited efforts were put into the dynamic topological optimization. However, for the FEM, there are some shortcomings such as mesh distortion, frequent remeshing when dealing with large deformation or moving boundary problems, etc. The meshless method avoids the dependence of meshes, and thoroughly or partly eliminates meshing. It makes this kind of methods possess great advantages when dealing with crack propagation problems, super-large deformation problems and high velocity impact problems, etc. In this dissertation, the natural neighbor Petrov-Galerkin method (NNPG) is used to study the elasto-static, dynamic problems and the topology optimization of the continuum structures.
At the beginning of the dissertation, the history and recent developments of the topology optimization and meshless method are overviewed, and several typical topology optimization methods are introduced particularly. The fundamental theory of topology optimization of continuum structure is discussed, including the formulation and solution of the optimization model, implementation and computational procedure, which based on the SIMP (Solid Isotropic Material with Penalization) density-stiffness interpolation model and the optimality criteria method. Numerical instabilities such as checkerboard and mesh-dependence as well as the available ways to circumvent them are also discussed. A continuous bulk density fields is proposed to eliminate the checkerboard patterns of the material distribution. In order to achieve mesh-independent designs by the topology optimization, a non-local relationship between the density and stiffness is introduced, based on the proposed modified continuous bulk density fields, both of the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are circumvented simultaneously.
When dealing with topology optimization problems, it is vital to obtain accurate structural responses. It's very important to testify the accuracy and convergence of the NNPG method before applying the method to study the topology optimization of the continuum structure. To this end, both of the elasto-static and dynamic problems in2D plane problems as well as the Mindlin plate problems are studied by the natural neighbor Petrov-Galerkin method. In the implementation, based on the well established Delaunay triangulation, the FEM shape functions of three nodes triangular are taken as test functions, which reduce the order of integrands involved in domain integrals and improves the computational efficiency of the method. The trial functions which are constructed based on the natural neighbor interpolations have the Kronecker Delta function property, which facilitate imposition of essential boundary conditions and dealing with discontinuity problems. Numerical examples show that the present method possesses high accuracy and good performance of stability, and is easy to implement.
The topology optimization for plane and Mindlin plate problems based on the meshless NNPG method and the proposed continuous bulk density fields, where the objective is to minimize compliance, is investigated. The numerical approach presented here is based on the SIMP approach and the optimality criteria method. A continuous bulk density fields constructed by the natural neighbor interpolation shape functions is proposed to eliminate the checkerboard pattern. This approach is used to get higher resolution solutions with mesh refinement and the checkerboard control. Based on the continuous bulk density fields, the non-local effect is considered. To this end, the original bulk density fields is reconstructed by introducing a influence domain whose radius is Rmin, in this way, the design space is restricted, which is very necessary to ensure the existence of solution. By adjusting Rmin, the complication of the optimized structures is controlled by the proposed method. In this way, the checkerboard and mesh-dependence problems which pertaining to the conventional numerical FEM-based topology optimization methods are all circumvented.
The dynamic topological optimization for plane and Mindlin plate problems based on the meshless NNPG method is also studied, which include the eigenvalue optimization problems for free vibration of structures and the frequency response optimization problems for forced vibration of structures. The formulation of the dynamic topological optimization and the possibility of localized modes in low density areas are discussed. While the oscillation of the objective function during the eigenvalue optimization process and the effect of the non-structural mass to optimized structures by the topology optimization are investigated particularly. A heuristic approach, which modifies the sensitivity information to control the optimization path, is presented to avoid the oscillation of the objective function in eigenvalue optimization. By this way, some "optimized" design which closing to the global optimum of the original problem can be obtained.
Numerical results show that the present method obtains reasonable, checkerboard controlled, node independent and black-white optimization results for the topology optimization of the continuum structures including the minimum compliance topology optimization, the eigenvalue optimization for the free vibration of structures and the frequency response optimization for the forced vibration of structures, which demonstrate the feasibility and validity of the present method for these problems.
引文
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