The Mesh of Engineering Simulation 3: Element Quality

In simulation software such as Finite Element Analysis (FEA) tools, mesh quality is critical to the computation. High-quality elements improve computational speed, whereas low-quality elements compromise accuracy or may even terminate the computation. Common metrics for evaluating element quality include aspect ratio, Jacobian ratio, maximum corner angle, parallel deviation, skewness, characteristic length, orthogonal quality, and warping factor. This article discusses these common mesh (element) quality parameters from the perspectives of practical application and software development. Aspect Ratio The aspect ratio describes the elongation of an element. Defined as the ratio of the longest edge to the shortest edge, it indicates whether the element is excessively elongated. Elongated elements fail to accurately capture gradients along their short-edge direction, leading to significant errors in bending and boundary layer stress calculations. The ideal aspect ratio is 1, and higher values correspond to poorer element quality. Overall, aspect ratios are useful for checking meshes in boundary layers, thin-walled structures, and stretched regions. Press enter or click to view image in full size For general models, the overall aspect ratio should be less than 20; less than 10 in areas of stress concentration. When the aspect ratio exceeds 20, elements become severely stretched, resulting in poor FEA accuracy and convergence difficulties. In CFD boundary layer regions where the mesh is refined in the normal direction, aspect ratios between 30 and 100 are permissible. The numerical methods for calculating aspect ratios are straightforward. For triangles, the aspect ratio is the ratio of the longest edge to the diameter of the equivalent inscribed circle. For quadrilaterals, it is the ratio of the average lengths of the two pairs of opposite edges: max(L1, L2) / min(L1, L2). Tetrahedra are treated similarly to triangles, with the aspect ratio derived from the ratio of the circumscribed sphere radius (R) to the inscribed sphere radius (r). For hexahedra, the aspect ratio is the ratio of the maximum to the minimum length among the 12 edges. As these methods demonstrate, the aspect ratio is a purely geometric length-based calculation that does not utilize Gaussian points or Jacobian determinants. Jacobian Ratio The Jacobian ratio is the ratio of the minimum Jacobian determinant to the maximum Jacobian determinant within an element. The Jacobian uniformity metric reflects distortion gradients and measures how uniform the mapping is across the element. It is specifically designed to capture nodal distortion and local contraction or expansion in high-order elements, serving as a core quality control indicator for higher-order elements. The value ranges from -1 to 1. A ratio of 1 indicates a uniform, undistorted element, while values ​​closer to zero indicate significant distortion. The empirical standards are as follows: Jacobian Ratio | 0.3–1.0 | 0.1–0.3 | 0–0.1 | ≤ 0 Element Quality | High-quality element; stable solution | Distorted; reduced stress accuracy; prone to oscillation | Severely distorted; not recommended for high-precision calculations | Negative Jacobian / Degenerate; solver error There are two standards for calculating the Jacobian ratio that are based on Gaussian points and corner nodes, respectively. The Gaussian point method aligns with the integration locations of the solver’s stiffness matrix, making it a reliable mesh metric for the finite element method. The corner node method only checks geometric vertices; while computationally efficient, it may fail to detect cases where vertices are positive but integration points are negative, so it is typically used for a precursory geometric screening. Press enter or click to view image in full size For linear elements, the Jacobian determinant is constant throughout the element, so the Jacobian ratio remains 1; this fails to reflect the element’s quality. In this case, the element’s Jacobian determinant should be compared against that of an ideal element in order to yield a Jacobian ratio that is not always 1. Additionally, since the Jacobian determinant is constant within the element, the calculation only needs to be performed once per linear element, without iterating through all nodes or Gaussian points. For nonlinear elements, such as second-order elements with midside nodes or higher-order elements, shape functions are of higher degree. Consequently, the Jacobian matrix varies across different integration points within the element, requiring dynamic calculation by substituting coordinates at each integration point. Maximum Corner Angle The maximum corner angle is an intuitive metric for assessing element quality, capable of identifying elements with extremely obtuse angles or those that are nearly flat (degenerate). The maximum angle within the element is reported, typically in degrees. A larger value indicates more severe element distortion; values ​​closer to the ideal equiangular state are preferable. The optimal angle for a quadrilateral is 90° (a right angle) and a triangle is 60°. Angles exceeding 150° indicate a poor-quality element, and elements with angles exceeding 165° are not recommended for solver. Press enter or click to view image in full size Angles (measured in degrees) can be calculated using the formula: θ = arccos(u·v(|u|·|v|))·180/π, where u and v represent the vectors of the two sides forming the angle. For a triangular element, the angles θ0, θ1, and θ2 at the three vertices are calculated; the maximum value among them — max{θ0, θ1, θ2} — is defined as the maximum corner angle. Similarly, the maximum corner angle of a quadrilateral element is max{θ0, θ1, θ2, θ3}. Write on Medium For 3D elements, the maximum corner angle of the element is determined by identifying the maximum corner angle on each face and then comparing these values. For a tetrahedral element with four triangular faces (each with three interior angles), the maximum of all 12 interior angles is taken. A hexahedral element has six quadrilateral faces (each with four vertex angles), so all 24 angles are evaluated to obtain the maximum corner angle of the hexahedron. Skewness Skewness describes the tilt and distortion of an element, measuring the extent to which a quadrilateral or hexahedron deviates from a perfect rectangle or cube. It is a normalized, comprehensive metric that penalizes both excessively large and excessively small angles. As a key indicator of element quality for finite element solvers, it is one of the most commonly used general metrics. The value ranges from 0 to 1: a value of 0 indicates an ideal, orthogonal, and perfectly regular element, while a value of 1 indicates a completely skewed element unsuitable for finite element analysis. Generally, values ​​below 0.2 indicate high-quality elements; values ​​below 0.5 are acceptable. Values ​​above 0.8 indicate poor element quality, leading to reduced numerical accuracy, convergence difficulties in iterative calculations, and potential stress singularities. Press enter or click to view image in full size The general calculation method for skewness is as follows: S = max{ (θmax − θideal) / (180 − θideal), (θideal − θmin) / θideal } Here, θmax is the maximum corner angle, θmin is the minimum interior angle, and θideal is the ideal interior angle. The ideal internal angle for a triangle is 60° and for a quadrilateral is 90°; the ideal dihedral angle for a tetrahedron is approximately 70.5° and for a hexahedron is 90°. For 3D elements such as tetrahedra and hexahedra, skewness is calculated using the maximum or minimum dihedral angle. This method, also known as the equiangular method, supports all element types, including prisms and pyramids. For triangular and tetrahedral elements, skewness can also be calculated using the equi-volume method. The formula is: S = (V_ideal — V_actual) / V_ideal, where V_ideal is the volume of a standard equilateral element sharing the same circumsphere. For pyramid and wedge elements, equiangular skewness is calculated for all quadrilateral faces and equi-volume skewness for all triangular faces; the maximum value obtained is taken as the skewness. Element Quality Element quality is a comprehensive metric for evaluating mesh quality, with values ​​ranging from 0 to 1. A value of 1 represents a perfect square (2D) or a perfect cube (3D), while a value of 0 indicates zero volume or negative volume (indicating mesh distortion failure). Press enter or click to view image in full size The metric is determined as follows: For 2D quadrilateral or triangular elements, the calculation uses the ratio of the element area to the sum of the squares of the edge lengths: Q = C * (A / ΣL²). The area of ​​a triangular element can be obtained as half the magnitude of the vector cross product. Quadrilateral elements can be divided into two adjacent triangles to calculate their area. For 3D elements, the calculation uses the ratio of the element volume to the square root of the cube of the sum of the squares of the edge lengths: Q = C * (V / (ΣL²)^(3/2)). The volume of a 3D element can be obtained by calculating the determinant of the element’s Jacobian matrix. The value of constant C depends on the element type, as specified below: Element | Triangle | Quadrilateral | Tetrahedron | Hexahedron | Wedge | Pyramid Constant Value | 6.928 | 4.0 | 124.70 | 41.57 | 62.354 | 96 In element quality assessment, a quality value greater than 0.7 indicates an acceptable element. For elements in stress concentration zones, a value greater than 0.8 is desirable. Elements with values ​​below 0.05 are considered severely defective and must be repaired. Conclusion This article discusses the quality of common element types, as calculating and visualizing element quality is an essential feature of modern general-purpose finite element analysis software. These quality metrics are used not only for finite element meshing but also for interactive mesh inspection. By visualizing element quality via contour plots, users can quickly and intuitively assess mesh quality and decide whether to use the current mesh for subsequent calculations. Although this article does not discuss triangular prism elements or wedge elements in detail, their quality metrics are calculated similarly to those of tetrahedral and hexahedral elements and can be determined using the same methods. Other quality metrics — such as parallel deviation, orthogonal quality, and warping factor — are not addressed here.

工程仿真软件中的网格3：单元质量

有限元等仿真软件中网格的质量对于计算非常重要。高质量的单元可以提升计算速度，而低质量的单元会影响计算精度，甚至导致计算无法完成。常见的衡量单元质量的纲量有：纵横比，雅可比比值，最大内角，平行偏差，偏斜度，特征长度，正交质量，和翘曲因子等等。本文从实际应用与开发的角度，讨论常见的网格（单元）质量参数。 纵横比Aspect Ratio 纵横比描述单元的细长程度。用单元最长边/最短边的比值，表征单元是否过度细长；细长单元沿短边方向梯度捕捉会失效，使得弯曲、边界层应力误差巨大。理想值为1，数值越大表明单元质量越差。纵横比可以用于边界层、薄壁结构、拉伸网格的检查。 Image 对于通用的模型，整体的纵横比需要小于20，在应力集中区小于10。当纵横比大于20时，单元严重拉伸，有限元计算精度差、迭代难收敛。对于CFD中的边界层，法向加密区域允许 30~100的纵横比。 单元纵横比的数值方法直接。三角形通过计算最长边 / 等效内切圆直径的比值得到纵横比。四边形取两组对边平均长度比值max(L1, L2) / min(L1, L2)得到纵横比。四面体与三角形相似，基于内切球半径 r与外接球半径 R的比值得到纵横比。六面体取12条棱长度的最大 / 最小比值得到纵横比。通过计算方法可见，纵横比是一种纯几何长度计算，不使用高斯点、不使用雅可比。 雅可比比率Jacobian Ratio 雅可比比率是单元内最小与最大雅可比行列式的比值。雅可比变化的均匀性能反映扭曲梯度 ，可以衡量单元内部映射均匀性，专门捕捉高阶单元中节点扭曲、局部收缩膨胀，是二阶单元核心质控指标。值的范围是[-1, 1]，比值为1时，表明单元均匀无扭曲；比值越接近零，单元扭曲大。经验标准如下： 雅可比比值 0.3～1.0 0.1～0.3 0 ～0.1 小于等于0 单元质量 优质单元，求解稳定 畸变，应力精度下降，易震荡 严重畸变，不建议用于高精度计算 负雅可比 / 退化，求解报错、发散 雅可比比率的计算有两套标准，分别基于高斯积分点和顶点节点。由于高斯点的计算方式匹配求解器刚度矩阵积分位置，是有限元方法中可靠的网格指标。顶点节点计算方式仅几何顶点校验，计算效率高，但可能会出现 “顶点全正、积分点负” 漏判，常用于快速几何筛查。 Image 对于线性单元，雅可比行列式在单元内部为常数，雅可比比率恒为1，无法表现出线性单元的质量，此时可以对比理想单元的雅可比行列式，得到非恒为1的雅可比比率。同时由于单元内部雅可比行列式恒定，每个线性单元只需计算1 次，无需遍历所有节点或高斯点。 对于非线性单元，如二阶偏置或高阶单元，其形函数是多次的，此时单元内不同积分点的雅可比矩阵是不同的，需要在每个积分点上动态代入坐标计算。 最大内角度 Max Corner Angle 最大内角是一种直观的衡量单元质量的纲量，可以识别钝角极大、近乎扁平退化的单元。常用角度值作为单位，输出单元内最大角度。此数值越大，单元畸变越严重；越接近理想等角越好。四边形最优角度是90°，即直角；三角形最优角度是 60°；超过 150° 为劣质单元。超过165度的单元不建议使用。 Image 角度制的内角可以通过以下公式计算：θ=arccos(u⋅v /∥u∥⋅∥v∥ )⋅180/ π ，u 和v表示内角两条边的向量。三角形单元3 顶点的角度求得θ0, θ1, θ2，取其最大值 max { θ0, θ1, θ2 }，即为最大内角。同理，四边形单元的最大内角是max { θ0, θ1, θ2, θ3 }。 对于三维单元，取每一个面上的最大内角，再比较每个面上的最大内角得到单元最大内角。如四面体单元，4 个三角形面，每个面 3 个内角，取全部 12 个内角最大值。六面体有 6 个四边形面，每个面 4 个顶角，遍历所有24个面内角，即可得到该六面体的最大内角。 偏斜度 Skewness 偏斜度描述单元的倾斜扭曲，衡量四边形/六面体偏离矩形/立方体的程度。它是归一化的综合指标，同时惩罚过大角、过小角，是有限元求解器看重的单元质量，因此是最常用的通用单元指标之一。数值范围是[0, 1]，0表示单元理想正交，是标准规整单元，1表示单元完全歪斜，无法用于有限元计算。通常情况下，小于0.2表示优质单元；小于0.5的单元都可接受；大于0.8表示单元质量差，数值精度下降，迭代计算难以收敛、可能会导致应力奇异。 Image 偏斜度的通用计算方法如下， S=max{ (θmax−θideal)/(180−θideal), (θideal−θmin)/θideal } 其中，θmax是最大内角，θmax是最小内角。θideal是理想内角。三角形理想内角为60°；四边形是90°；四面体的理想二面角约 70.5°，六面体理想二面角是90°。对于三维单元如四面体和六面体单元，使用的是二面角的最大或最小值来计算偏斜度。此方法也叫等角法，支持所有的单元类型，包括棱柱与金字塔单元。 对于三角形和四面体单元，我们还可以使用体积法来计算偏斜度。公式如下：S= (Videal – Vactual)/ Videal，理想体积Videal是同外接球等边标准单元体积。 对于金字塔和楔形单元，可以分别计算所有四边形面的等角偏斜度，和所有三角形面的等体积偏斜度，取最大值即得到偏斜度。 单元综合质量 单元质量是一项综合网格质量评判指标，数值范围为 0～1。数值为 1 代表完美正方形（二维）或完美正方体（三维）；数值为 0 代表单元体积为零或出现负体积（网格畸变失效）。 Image 该指标的计算方式如下： 对于二维四边形 / 三角形单元，采用单元面积与各边长平方和的比值计算；Q = C* (A/∑L²)。三角形单元的面积可以通过向量叉乘模长的一半 得到。四边形单元可以划分为两个相邻的三角形单元，通过这个方法可以计算四边形单元的面积。 对于三维单元，则采用单元体积与各边长平方和立方的平方根的比值计算。Q = C (V/(∑L²)^1.5)。三维单元体积的可以通过计算单元的雅可比矩阵行列式实现。 常数C的大小取决于单元类型，具体数值如下： 单元 三角形 四边形 四面体 六面体 楔形 金字塔形 常数值 6.928 4.0 124.70 41.57 62.354 96 在单元质量评价中，质量值 >0.7为合格单元；应力集中区的单元值期望大于0.8。当单元值小于0.05时，存在严重坏单元，必须修复。 总结 本文讨论了常见单元类型的单元质量，计算并显示单元质量是现代通用有限元的一个必备功能。计算这些纲量不仅用于有限元网格划分，也用于网格的交互检查，云图显示单元质量，用户可以快速直观的了解网格的质量，并决定是否使用当前网格用于后续的计算。 本文未对三棱柱单元与楔形单元作详细讨论，但其单元质量的计算方式与四面体和六面体相似，可以用相同的方法计算得出。本文未对平行偏差，正交质量，翘曲因子等单元质量纲量做讨论。