球面有限元网格的最大角条件的精确表征
Exact characterisation of maximum-angle conditions for spherical finite element meshes
浏览论文内容
中文总结 AI 辅助
本研究精确表征球面有限元网格的本征球面最大角条件，推导其与弦半正则性参数的关系，为各向异性有限元分析提供无需最小角或形状正则性条件的几何控制。
中文摘要 AI 辅助
最大角条件是标准的有限元网格假设，其允许各向异性三角形，而这类三角形会被最小角条件或形状正则性假设排除。然而，对于精确球面三角形而言，弦仿射核心的角度与本征球面角未必一致，且径向几何会引入曲率尺度畸变。我们通过由三个顶点向量计算得到的无量纲量，给出了本征球面最大角条件的代数表征。该量的一致正下界等价于一致球面最大角界，且无需评估球面角或球面面积。我们确定了将弦外接圆半径与径向畸变关联的支撑平面几何，并推导了本征球面与弦半正则性参数之间的精确比较关系，其中存在与尖锐常数2/√3相关且与局部性无关的比较关系，以及经表征的等号情形。研究还表明，基于面积的参数小于球面半正则性参数，在局部平坦极限下其尖锐常数为1。对于球面有限元网格，顶点准则意味着弦半正则性具有显式常数，而相对加密使径向畸变因子一致收敛到1。因此，本征准则结合相对加密，可在不施加最小角或形状正则性条件的情况下，为各向异性有限元分析提供所需的几何控制。
英文摘要
Maximum-angle conditions are standard finite-element mesh hypotheses that permit anisotropic triangles excluded by minimum-angle or shape-regularity assumptions. For exact spherical triangles, however, the angles of the chordal affine core and the intrinsic spherical angles need not coincide, while radial geometry introduces curvature-scale distortion. We give an algebraic characterisation of the intrinsic spherical maximum-angle condition through a dimensionless quantity computed from the three vertex vectors. A uniform positive lower bound on this quantity is equivalent to a uniform spherical maximum-angle bound and requires neither spherical-angle nor spherical-area evaluation. We identify the support-plane geometry linking the chordal circumradius to radial distortion and derive sharp comparisons between the intrinsic spherical and chordal semi-regularity parameters. In particular, a locality-independent comparison holds with sharp constant $2/\sqrt3$ and a characterised equality case. An area-based parameter is shown to be smaller than the spherical semi-regularity parameter, with sharp constant one in the local flat limit. For spherical finite-element meshes, the vertex criterion implies uniform chordal semi-regularity with an explicit constant, while relative refinement makes the radial-distortion factors converge uniformly to one. Thus, the intrinsic criterion, together with relative refinement, provides the geometric controls used in anisotropic finite-element analysis without imposing a minimum-angle or shape-regularity condition.
