This study mathematically examines chemical and biomaterial models by employing the finite element method. Unshaped biomaterials’ complex structures have been numerically analyzed using Gaussian quadrature rules. It has been analyzed for commercial benefits of chemical engineering and biomaterials as well as biorefinery fields. For the computational work, the ellipsoid has been taken as a model, and it has been transformed by subdividing it into six tetrahedral elements with one curved face. Each curved tetrahedral element is considered a quadratic and cubic tetrahedral element and transformed into standard tetrahedral elements with straight faces. Each standard tetrahedral element is further decomposed into four hexahedral elements. Numerical tests are presented that verify the derived transformations and the quadrature rules. Convergence studies are performed for the integration of rational, weakly singular, and trigonometric test functions over an ellipsoid by using Gaussian quadrature rules and compared with the generalized Gaussian quadrature rules. The new transformations are derived to compute numerical integration over curved tetrahedral elements for all tests, and it has been observed that the integral outcomes converge to accurate values with lower computation duration.

中文翻译：

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通过离散化为生物材料研究的六面体单元对椭球体上的一些任意函数进行数值积分

本研究采用有限元方法对化学和生物材料模型进行数学检验。使用高斯求积规则对不定形生物材料的复杂结构进行了数值分析。它已被分析为化学工程和生物材料以及生物精炼领域的商业利益。在计算工作中，以椭球为模型，将其细分为六个具有一个曲面的四面体单元。每个弯曲四面体单元都被视为二次和三次四面体单元，并转换为具有直面的标准四面体单元。每个标准四面体单元进一步分解为四个六面体单元。提出了数值测试来验证导出的变换和求积规则。使用高斯求积规则对椭球上有理函数、弱奇异函数和三角测试函数的积分进行收敛性研究，并与广义高斯求积规则进行比较。新的变换是为了计算所有测试的弯曲四面体单元上的数值积分而导出的，并且观察到积分结果以较短的计算持续时间收敛到准确的值。