约束非结构网格生成方法研究
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摘要
在科学和工程计算领域中,网格生成作为数值模拟与分析的载体,是其中必不可少的技术步骤,并且网格质量的好坏直接影响数值计算的精度,关系数值模拟的成败。非结构网格没有网格节点的结构性限制,易于控制网格单元的大小、形状及网格点的位置,具有很强的灵活性,可以根据计算区域特征和尺度的要求科学合理地分布网格的疏密,方便进行自适应计算,进而提高计算精度,对模拟真实复杂外形和解决间断计算问题的适应能力非常强。
本文从非结构网格实际应用的角度出发,致力解决网格生成过程中的约束边界剖分和各种预设约束条件的完全复原问题。本文的创新点是:
(1)提出了三种二维约束非结构网格生成方法,它们均在无需加入Steiner点的前提下,对二维任意区域均实现了带约束的非结构网格生成;
(2)利用单侧Steiner点扰动和矢量边界推进三角化方法得到了一个保持边界剖分完整性的三维网格边界还原方法,利用双侧Steiner点扰动和矢量边界推进三角化方法得到了一个三维约束非结构网格生成方法;
(3)结合阵面推进法和Conforming方法的优势得到了一个保边界非结构网格生成方法;
(4)在二维计算区域,通过分析边界曲率、计算区域近似中轴线以及相邻节点之间的光滑效应,对预先指定的尺度函数作了优化,使之更趋光滑合理;
(5)采用弱区域指示函数描述计算区域形状,并基于该函数得到了一个简单高效的非结构网格生成方法。
本文具体工作如下:
(1)提出了三种二维约束非结构网格生成方法,分别是:
1)矢量边界推进三角化;
2)基于局部重构的约束Delaunay三角化;
3)基于边交换的约束Delaunay三角化。其中,方法1)在不作任何坐标变换的前提下,能够对空间任意封闭平面区域实现约束非结构三角网格生成,方法2)和3)对离散的物面边界没有方向要求,具有很强的普适性。它们的收敛性和稳定性本文均给出了相关定理和证明。
(2)分三个步骤得到一种三维约束非结构网格生成方法:
1)在Delaunay非结构四面体网格中,利用矢量边界推进三角化思想通过在约束条件中加入Steiner点,得到一种能够恢复计算区域形状的Conforming网格生成方法;
2)在Conforming网格边界上,利用单侧点扰动技术迫使位于边界约束面上的Steiner点逐个转移,并利用矢量边界推进三角化方法重构约束面网格剖分,进而得到一个保持边界剖分完整性的三维网格边界还原方法,其优点是在此过程中不需添加Steiner点;
3)在Conforming网格中,利用双侧点扰动技术迫使位于网格约束面上的Steiner点逐个转移,并利用矢量边界推进三角化方法重构约束面网格剖分,从而得到一个三维约束非结构网格生成方法。其优点是能够对所有约束边和约束面进行完全恢复,而且对物面边界剖分没有方向要求,普适性广。
关于这三种方法的收敛性和稳定性本文中也给出了相关定理和证明。
(3)结合阵面推进法和Conforming方法的优势得到了一个保边界非结构网格生成方法。该方法充分结合了阵面推进法能够保持物面边界剖分的优势和Conforming网格对任何复杂区域均能生成网格结构并恢复边界形状的特点。首先采用water-tight层次阵面推进技术,由计算区域表面向内部推进生成几层网格单元,然后利用Conforming技术生成剩余区域的网格,并在两种网格分界处对Steiner点进行“缝合”,从而在保证边界完整性的同时形成整个计算区域的网格结构。
(4)通过求解常微分方程组优化了二维尺度函数,使其能够充分体现计算区域边界曲率、近似中轴线以及相邻节点之间的光滑效应。
(5)得到一个隐方式下的网格生成方法。采用弱区域指示函数描述计算区域形状,并基于该函数得到了一个简单高效的非结构网格生成方法,同时证明了对于任意由显方式或隐方式描述的封闭区域均可以由多个不同的弱区域指示函数描述,并给出了构造方法。最后将该方法推广应用至自适应网格。
在上述网格生成过程中,本文一律采用概率筛选布点方法对计算区域进行布点,分别通过弹簧振子和边交换(边/面交换)技术优化网格节点位置和结构,最后利用边吞噬技术剔除所有形状较差的网格单元。
As a carrier of numerical simulation and analysis, mesh generation is one of the essential technical steps in the field of scientific and engineering computing. Mesh quality affects the numerical accuracy, and poor grid often leads to the failure of numerical simulation.
Without structural constraints of nodes, it’s easy and flexible to control the unit size, shape, and location of the points in the unstructured mesh. According to the scale function and the calculating requirements, we can get rational unit distributions for the field. It’s convenient to generate unstructured mesh on complex shape of computational field and solve discontinue problems adaptively, which can improve the numerical accuracy.
With practical application of unstructured grid in mind, we try to solve the constrained unstructured mesh generation in 2D and 3D. The innovations are:
(1) Three methods are presented to generate 2D constrained unstructured mesh without adding Steiner points, and these methods can generate constrained mesh for arbitrary field in 2D.
(2) A constrained boundary recovery method for 3D mesh is obtained through perturbing Steiner points unilaterally and triangulation method by vector boundaries advancing (TMVBA), and A 3D constrained unstructured mesh generation method is obtained through perturbing Steiner points bilaterally and TMVBA.
(3) A new method for constrained boundary recovery in mesh generation is studied, which combines the advantage of the advanced front method and conforming Delaunay method.
(4) The scale function is optimized by solving ODEs by analysizing the boundary curvature, axis and smooth between adjacent points.
(5) The unstructured mesh is generated by weak domain indicating function (WDIF).
The detailed works:
(1) The generation of 2D constrained unstructured mesh. In this thesis we present three methods to generate 2D constrained unstructured mesh:
a) triangulation method by vector boundaries advancing (TMVBA),
b) constrained Delaunay mesh generation based on local reconstruction,
c) constrained Delaunay mesh generation based on side-swapping. TMVBA could triangulate any closed domain without coordinate transformation, and the last two methods can triangulate plane domain without boundary orientation. The convergence and stability of these three methods are proved.
(2) The generation of 3D constrained unstructured mesh. We achieve this goal in three steps:
a) A conforming mesh generation method is obtained by combining TMVBA and adding Steiner points into the Delaunay mesh structure, which can preserve the shape of the field.
b) A constrained boundary recovery for three dimensional Delaunay triangulations is obtained through moving all points apart from constrained sides and faces by unilaterally perturbing Steiner points (UPSP) and retriangulating the constrained faces by TMVBA. The advantage of this approach is no Steiner point needed.
c) A constrained recovery for three dimensional Delaunay triangulations is obtained through moving all points apart from constrained sides and faces by bilaterally perturbing Steiner points (BPSP) and retriangulating the constrained faces by TMVBA. The advantages of this approach are that no boundary orientation is needed and that all constrained faces and sizes are recovered.
The convergence and stability of these methods are proved.
(3) A new method for constrained boundary recovery in mesh generation is studied. This method is a combination of advancing front method and conforming Delaunay method. This new methods exerts all the advantages of advancing front method and conforming Delaunay method, while avoiding the disadvantages of them. Here we triangulate the remanent domain on the advanced front of water-tight mesh by conforming Delaunay method, and then couple the two meshes at Steiner points.
(4) In 2D, we optimize scale function by analyze the boundary curvature, axis and smooth between adjacent points to make it much smoother.
(5) For the field described by implicit function, we first deduct WDIF to describe it, and prove the existence and multiplicity of WDIF. Then, we generate the unstructured mesh by it. Lastly, we generalize this method to adaptive mesh.
In the thesis, we employ the probability filer method to distribute mesh points, optimize the location of points and structure of mesh by Spring and side swapping (side-face flip) respectively. The side eating means could get rid of all the bad-shaped elements from mesh.
本文从非结构网格实际应用的角度出发,致力解决网格生成过程中的约束边界剖分和各种预设约束条件的完全复原问题。本文的创新点是:
(1)提出了三种二维约束非结构网格生成方法,它们均在无需加入Steiner点的前提下,对二维任意区域均实现了带约束的非结构网格生成;
(2)利用单侧Steiner点扰动和矢量边界推进三角化方法得到了一个保持边界剖分完整性的三维网格边界还原方法,利用双侧Steiner点扰动和矢量边界推进三角化方法得到了一个三维约束非结构网格生成方法;
(3)结合阵面推进法和Conforming方法的优势得到了一个保边界非结构网格生成方法;
(4)在二维计算区域,通过分析边界曲率、计算区域近似中轴线以及相邻节点之间的光滑效应,对预先指定的尺度函数作了优化,使之更趋光滑合理;
(5)采用弱区域指示函数描述计算区域形状,并基于该函数得到了一个简单高效的非结构网格生成方法。
本文具体工作如下:
(1)提出了三种二维约束非结构网格生成方法,分别是:
1)矢量边界推进三角化;
2)基于局部重构的约束Delaunay三角化;
3)基于边交换的约束Delaunay三角化。其中,方法1)在不作任何坐标变换的前提下,能够对空间任意封闭平面区域实现约束非结构三角网格生成,方法2)和3)对离散的物面边界没有方向要求,具有很强的普适性。它们的收敛性和稳定性本文均给出了相关定理和证明。
(2)分三个步骤得到一种三维约束非结构网格生成方法:
1)在Delaunay非结构四面体网格中,利用矢量边界推进三角化思想通过在约束条件中加入Steiner点,得到一种能够恢复计算区域形状的Conforming网格生成方法;
2)在Conforming网格边界上,利用单侧点扰动技术迫使位于边界约束面上的Steiner点逐个转移,并利用矢量边界推进三角化方法重构约束面网格剖分,进而得到一个保持边界剖分完整性的三维网格边界还原方法,其优点是在此过程中不需添加Steiner点;
3)在Conforming网格中,利用双侧点扰动技术迫使位于网格约束面上的Steiner点逐个转移,并利用矢量边界推进三角化方法重构约束面网格剖分,从而得到一个三维约束非结构网格生成方法。其优点是能够对所有约束边和约束面进行完全恢复,而且对物面边界剖分没有方向要求,普适性广。
关于这三种方法的收敛性和稳定性本文中也给出了相关定理和证明。
(3)结合阵面推进法和Conforming方法的优势得到了一个保边界非结构网格生成方法。该方法充分结合了阵面推进法能够保持物面边界剖分的优势和Conforming网格对任何复杂区域均能生成网格结构并恢复边界形状的特点。首先采用water-tight层次阵面推进技术,由计算区域表面向内部推进生成几层网格单元,然后利用Conforming技术生成剩余区域的网格,并在两种网格分界处对Steiner点进行“缝合”,从而在保证边界完整性的同时形成整个计算区域的网格结构。
(4)通过求解常微分方程组优化了二维尺度函数,使其能够充分体现计算区域边界曲率、近似中轴线以及相邻节点之间的光滑效应。
(5)得到一个隐方式下的网格生成方法。采用弱区域指示函数描述计算区域形状,并基于该函数得到了一个简单高效的非结构网格生成方法,同时证明了对于任意由显方式或隐方式描述的封闭区域均可以由多个不同的弱区域指示函数描述,并给出了构造方法。最后将该方法推广应用至自适应网格。
在上述网格生成过程中,本文一律采用概率筛选布点方法对计算区域进行布点,分别通过弹簧振子和边交换(边/面交换)技术优化网格节点位置和结构,最后利用边吞噬技术剔除所有形状较差的网格单元。
As a carrier of numerical simulation and analysis, mesh generation is one of the essential technical steps in the field of scientific and engineering computing. Mesh quality affects the numerical accuracy, and poor grid often leads to the failure of numerical simulation.
Without structural constraints of nodes, it’s easy and flexible to control the unit size, shape, and location of the points in the unstructured mesh. According to the scale function and the calculating requirements, we can get rational unit distributions for the field. It’s convenient to generate unstructured mesh on complex shape of computational field and solve discontinue problems adaptively, which can improve the numerical accuracy.
With practical application of unstructured grid in mind, we try to solve the constrained unstructured mesh generation in 2D and 3D. The innovations are:
(1) Three methods are presented to generate 2D constrained unstructured mesh without adding Steiner points, and these methods can generate constrained mesh for arbitrary field in 2D.
(2) A constrained boundary recovery method for 3D mesh is obtained through perturbing Steiner points unilaterally and triangulation method by vector boundaries advancing (TMVBA), and A 3D constrained unstructured mesh generation method is obtained through perturbing Steiner points bilaterally and TMVBA.
(3) A new method for constrained boundary recovery in mesh generation is studied, which combines the advantage of the advanced front method and conforming Delaunay method.
(4) The scale function is optimized by solving ODEs by analysizing the boundary curvature, axis and smooth between adjacent points.
(5) The unstructured mesh is generated by weak domain indicating function (WDIF).
The detailed works:
(1) The generation of 2D constrained unstructured mesh. In this thesis we present three methods to generate 2D constrained unstructured mesh:
a) triangulation method by vector boundaries advancing (TMVBA),
b) constrained Delaunay mesh generation based on local reconstruction,
c) constrained Delaunay mesh generation based on side-swapping. TMVBA could triangulate any closed domain without coordinate transformation, and the last two methods can triangulate plane domain without boundary orientation. The convergence and stability of these three methods are proved.
(2) The generation of 3D constrained unstructured mesh. We achieve this goal in three steps:
a) A conforming mesh generation method is obtained by combining TMVBA and adding Steiner points into the Delaunay mesh structure, which can preserve the shape of the field.
b) A constrained boundary recovery for three dimensional Delaunay triangulations is obtained through moving all points apart from constrained sides and faces by unilaterally perturbing Steiner points (UPSP) and retriangulating the constrained faces by TMVBA. The advantage of this approach is no Steiner point needed.
c) A constrained recovery for three dimensional Delaunay triangulations is obtained through moving all points apart from constrained sides and faces by bilaterally perturbing Steiner points (BPSP) and retriangulating the constrained faces by TMVBA. The advantages of this approach are that no boundary orientation is needed and that all constrained faces and sizes are recovered.
The convergence and stability of these methods are proved.
(3) A new method for constrained boundary recovery in mesh generation is studied. This method is a combination of advancing front method and conforming Delaunay method. This new methods exerts all the advantages of advancing front method and conforming Delaunay method, while avoiding the disadvantages of them. Here we triangulate the remanent domain on the advanced front of water-tight mesh by conforming Delaunay method, and then couple the two meshes at Steiner points.
(4) In 2D, we optimize scale function by analyze the boundary curvature, axis and smooth between adjacent points to make it much smoother.
(5) For the field described by implicit function, we first deduct WDIF to describe it, and prove the existence and multiplicity of WDIF. Then, we generate the unstructured mesh by it. Lastly, we generalize this method to adaptive mesh.
In the thesis, we employ the probability filer method to distribute mesh points, optimize the location of points and structure of mesh by Spring and side swapping (side-face flip) respectively. The side eating means could get rid of all the bad-shaped elements from mesh.
引文
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