基于离散外微积分的线弹性几何表述
A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus
AI总结：
本文提出了一种基于离散外微积分的线弹性直接表述方法，通过使用原始0-cochain表示位移，原始向量值1-cochain表示位移差，双值向量2-cochain表示内部力，建立了宏观本构关系，并通过音乐同构映射在原始0-cells上实现动量平衡，最终通过非局部非对角材料Hodge星运算求解Laplace方程，验证了该方法在经典问题中的有效性。
摘要：
本文提出了一种基于离散外微积分的线弹性直接表述方法。主要未知量是位移，用原始向量值0-cochain表示。位移差和内部力分别用原始向量值1-cochain和双值向量2-cochain表示。通过音乐同构映射将cochains映射到光滑场及其反向，从而在原始0-cells上强制宏观本构关系。线性动量平衡在原始0-cells上建立。 governing equations被作为具有非局部和非对角材料Hodge星的Laplace方程求解。几种经典问题的数值模拟和解析解被展示以验证该表述。与已知解的良好一致被获得。该表述提供了一种方法，用于计算任意晶格结构中位移差和内部力之间的关系，当结构需要遵循预设的宏观弹性行为时。这也是在开发细胞复形中耗散过程的表述方法中的第一步和关键步骤。
英文摘要：
A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector-valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.
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