Applications of Geometrical Singular Perturbation Theory in Hyperplasticity Accelerated Ratcheting Models

研究单桩横向荷载的一种方法是由Orsted和牛津大学合作开发的Hyperplastic Accelerated Racheting Model （HARM）框架[1-3]。利用HARM可以研究地基的荷载响应作为应力的函数。求解位移作为侧向荷载的函数是一个问题，其中在结构的生命周期中应用了许多不同振幅的循环（由于波浪等），并且由于主要风向的加载偏差，经常观察到棘轮。所提出的博士论文的目标是显著减少模拟循环加载的模拟时间，以便在桩设计的优化阶段引入一系列扩展的卸载/再加载循环。所使用的方法针对的是几何奇异摄动理论（GSPT）的某种变体[4-6]。在我的硕士论文中，我们利用GSPT框架证明了一个相关的慢速动力系统的存在，该系统相当于HARM模型的连续版本，其中关键流形的双曲度与每个循环的卸载和重新加载部分一致。这一结果可能直接解决当前长期棘轮和预期迟滞分析所面临的一些问题。对于HARM的0-D宏观建模方法，主要有两个方向受到[7]中发现的慢-快动力系统的推动：-首先，研究GSPT作为独立工具的性能，以获得沿桩深经验棘轮的可积（因此是解析的）和非保守上界。如果这些结果与收集的实验/模拟一致，它们可以用作直接的设计参数。-其次，研究如何准确利用HARM模型对当前数值求解器变得僵硬的知识。一个想法是将快速和慢速零件分开建模并将它们粘合在一起-避免机器编号精度的问题。这不仅可以带来更快的结果，而且更准确。在[7]中，我们只证明了具有塑性表面的0-D宏观模型的慢-快系统的存在。为了与目前使用的数值解决方案保持一致并提高复杂性，博士学位的一个主要部分还将是将理论结果扩展到更高的维度（或新的本构模型，如HySand）。由于土壤模型一般来说是复杂的，我们期望我们的理论工作达到一个边界点。“直到那时”达到的结果可能会激发新的数值方法，也有望在博士论文中涵盖。最后，理论和数值工作将通过对牛津大学在约克郡、肯特郡和敦刻尔克的试桩的实际数据进行综合研究来验证。投资和支持领域：该项目属于“EPSRC地面工程研究领域”。参考文献[10]G. Houlsby和A. Puzrin，“速率无关耗散材料本构模型的热力学框架”，《国际塑性杂志》，第16卷，第2期。9，第1017-1047页，2000。(在线)。T. Balaam，“粘土中单桩基础循环荷载模型的开发与校准”，博士论文，牛津大学，2020.[0C. Abadie，“无黏性土中单桩基础的循环侧向荷载”，博士论文，牛津大学，2015. 01 - 01K. U. Kristiansen，“多时间尺度动力学的回顾：基本现象和数学方法”，2013，在评论中C. Kuehn，“两个快变量和一个慢变量的多时间尺度动力学”，博士论文，康奈尔大学，2010年5月C. K. R. T. Jones，几何奇异摄动理论。

EPSRC Project Title: Applications of Geometrical Singular Perturbation Theory in Hyperplasticity Accelerated Ratcheting ModelsAn approach for studying lateral loading for monopile designs is the Hyperplastic Accelerated Racheting Model (HARM) framework developed in collaboration between Orsted and Oxford University [1-3]. Using HARM one can study the loading response of the foundation as a function of stresses. Solving for displacement as a function of lateral loading is a problem in which many cycles of different amplitude are applied during the lifetime of the structure (due to waves etc), and because of a bias in loading from predominant wind directions, ratcheting is often observed [1].The objective of the proposed PhD thesis is to significantly reduce simulation time for modelling cyclic loading such that extended series of unloading/reloading cycles can be introduced in the optimization phase of the pile design. The method used is targeted to be some variation of Geometrical Singular Perturbation Theory (GSPT) [4-6].In my Master Thesis, we utilize the GSPT framework to prove the existence of a relevant slow-fast dynamical system equivalent to a continuous version of the HARM model, where the hyperbolicity of the critical manifold align with unloading and reloading parts of each cycle. This result may address direct solutions to some of the currently faced issues with long-term ratcheting and expected hysteresis analysis [7]. For the 0-D macro modelling approach of HARM two main directions are heavily motivated by the found slow-fast dynamical system in [7]: - First, a study in the performance of GSPT as a standalone tool to obtain integrable (thereby analytical) and non-conservative upper bounds for experienced ratcheting along pile depth. If such results are in alignment with gathered experiments/simulations, they can be used as immediate design parameters. - Secondly, a study on utilizing the knowledge of exactly where the HARM model becomes stiff to current numerical solvers. An idea is to model the fast and slow parts separately & glue them together - avoiding issues with machine number precision. This could lead to not only faster results but also more accurate.In [7] we have only shown existence of a slow-fast system for the 0-D macro model with plasticity surfaces. To align with currently used numerical solutions and improve complexity, a major part of the PhD will also be to extend the theoretical results to higher dimensions (or to new constitutive models such as HySand). As soil modelling is in general complex, it is expected that we reach a boundary point for the theoretical work. The 'until then' reach results are likely to motivate new numerical methods, also expected to be covered within the PhD thesis. Lastly, both the theoretical and numerical work will be validated by comprehensive studies on real-world data from Oxford's test pilings at Yorkshire, Kent and Dunkirk.Areas of investment & support: This project falls within the 'EPSRC Ground Engineering research area'. References[1] G. Houlsby and A. Puzrin, "A thermomechanical framework for constitutive models for rate-independent dissipative materials," International Journal of Plasticity, vol. 16, no. 9, pp. 1017-1047, 2000. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S074964199900073X[2] T. Balaam, "Development and calibration of cyclic loading models for monopile foundations in clays," Ph.D. dissertation, Oxford University, 2020.[3] C. Abadie, "Cyclic lateral loading of monopile foundations in cohesionless soils," Ph.D. dissertation, Oxford University, 01 2015.[4] K. U. Kristiansen, "A review of multiple time scale dynamics: Fundamental phenomena and mathematical methods," 2023, in review.[5] C. Kuehn, "Multiple time scale dynamics with two fast variables and one slow variable," Ph.D. dissertation, Cornell University, 05 2010.[6] C. K. R. T. Jones, Geometric singular perturbation theory.