超弹性梁后屈曲问题的规范对偶有限元法
Post-buckling Solutions of Hyper-elastic Beam by Canonical Dual Finite Element Method
AI总结：
采用规范对偶有限元法分析大变形梁的后屈曲问题，通过规范对偶应力重构非凸势能泛函，提出原对偶算法求解所有可能解，发现纯互补能量全局最大值对应稳定屈曲构型，局部极值对应不稳定状态且对单元数和外载敏感。
摘要：
采用规范对偶有限元法(CD-FEM)分析大变形梁的后屈曲问题。该方法的特点是正确选择规范对偶应力，使得原始非凸势能泛函被重新表述为同时包含位移场和应力场的混合互补能量形式，并在有限维空间中显式构造纯互补能量。基于规范对偶理论及相关的三重性定理，提出了一种原对偶算法，可用于寻找该非凸后屈曲问题的所有可能解。数值结果表明，纯互补能量的全局最大值导致梁的稳定屈曲构型，而纯互补能量的局部极值则呈现不稳定的变形状态。我们发现不稳定屈曲状态对总单元数和外载荷非常敏感。通过数值算例验证了理论结果，并观察到了该大变形梁后分叉中的一些有趣现象。
英文摘要：
Post buckling problem of a large deformed beam is analyzed using canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal-dual algorithm is proposed, which can be used to find all possible solutions of this nonconvex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam. While the local extrema of the pure-complementary energy present unstable deformation states, especially. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.
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