非线性弹性复形与有限元图表追踪
Nonlinear elasticity complex and a finite element diagram chase
AI总结：
提出非线性弹性复形编码等距嵌入与曲率等几何信息，将其正合性对应刚性定理与黎曼几何基本定理，并推广BGG复形有限元构造的代数方法，通过单射连接映射约简自由度，导出二维应变复形。
摘要：
本文提出线性弹性（Calabi、Kröner，黎曼形变）复形的非线性版本，该复形编码等距嵌入、度量、曲率和Bianchi恒等式。我们将刚性定理与黎曼几何基本定理重新表述为该复形的正合性。随后推广一种构造Bernstein-Gelfand-Gelfand（BGG）复形有限元的代数方法，特别讨论在BGG图中利用单射连接映射进行自由度约简。通过图表追踪导出二维空间中的应变复形。
英文摘要：
In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kröner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.
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