In this paper, we discuss reduced order modelling approaches to bifurcating systems arising from continuum mechanics benchmarks. The investigation of the beam's deflection is a relevant topic of investigation with fundamental implications on their design for structural analysis and health. When the beams are exposed to external forces, their equilibrium state can undergo to a sudden variation. This happens when a compression, acting along the axial boundaries, exceeds a certain critical value. Linear elasticity models are not complex enough to capture the so-called beam's buckling, and nonlinear constitutive relations, as the hyperelastic laws, are required to investigate this behavior, whose mathematical counterpart is represented by bifurcating phenomena. The numerical analysis of the bifurcating modes and the post-buckling behavior, is usually unaffordable by means of standard high-fidelity techniques such (as the Finite Element method) and the efficiency of Reduced Order Models (ROMs), e.g.\ based on Proper Orthogonal Decomposition (POD), are necessary to obtain consistent speed-up in the reconstruction of the bifurcation diagram. The aim of this work is to provide insights regarding the application of POD-based ROMs for buckling phenomena occurring for 2-D and 3-D beams governed by different constitutive relations. The benchmarks will involve multi-parametric settings with geometrically parametrized domains, where the buckling's location depends on the material and geometrical properties induced by the parameter. Finally, we exploit the acquired notions from these toy problems, to simulate a real case scenario coming from the Norwegian petroleum industry.

翻译：本文讨论连续介质力学基准问题中分叉系统的降阶建模方法。梁的挠度研究是一个具有基础意义的课题，对其结构分析和健康设计具有重要影响。当梁承受外力作用时，其平衡状态可能发生突变——当沿轴向边界作用的压缩力超过某一临界值时便会引发此现象。线性弹性模型不足以捕捉所谓的梁屈曲行为，需要使用超弹性定律等非线性本构关系来研究该现象，其数学表现即为分叉现象。对于分叉模态和屈曲后行为的数值分析，通常难以通过标准的高保真技术（如有限元法）实现，因此需要基于本征正交分解（POD）的降阶模型（ROM）效率，才能在重建分叉图时获得一致的加速效果。本研究旨在为基于POD的降阶模型在二维和三维梁屈曲现象中的应用提供见解，这些梁由不同本构关系控制。基准问题将涉及几何参数化域的多参数设置，其中屈曲位置取决于由参数引发的材料和几何特性。最后，我们利用从这些典型问题中获得的知识，模拟了来自挪威石油工业的实际案例。