Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

中文翻译：

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与几何无关的样条有限元方法分析二维热传导和弹性问题


传统的等几何分析涉及冗余的几何计算，并且在施加边界条件时比有限元方法更复杂。针对这一问题，提出了一种与几何无关的样条有限元方法，该方法集成了等几何分析的几何构造技术和有限元方法的单元构造技术。首先，利用非均匀有理B样条和B样条构造几何精度元来描述几何和物理场，并引入变换矩阵构造具有插值性质的形函数。其次，通过参数映射推导出二维热传导和弹性问题的计算格式。最后通过数值算例验证了该方法的可行性和准确性。与等几何分析不同，在该方法中，几何和物理是分离的，并且形状函数具有插值属性，这减少了冗余的几何计算，并允许边界条件直接应用于节点。数值结果表明，与原始等几何分析相比，该方法具有更高的收敛速度。