Nonlinear finite element technology for stable and locking-free analysis of large deformation problems

在所有行业分支中，虚拟原型取代物理原型的趋势正在持续，不仅可以提高效率和加速开发过程，而且还可以通过节省资源来提高可持续性，这对仿真工具的预测可靠性提出了越来越高的要求。有限元法是计算工程中的一种重要方法，但值得注意的是，在有限元技术中仍有一些基本问题没有得到解决。一个例子是一个普遍适用的有限元公式，是免费的锁定和免费的人工不稳定性，同时在涉及大变形的问题的持续追求。该建议建立在以前的研究项目自适应，变形相关的有限元公式稳定和锁定自由分析的大变形问题。两个主要主题是现有元素的分析，目的是更好地理解非线性区域中的元素行为，以及新的非线性有限元的开发，无论是关于非线性锁定现象和人工不稳定性（沙漏）。最重要的科学成果是检测到以前未知的明显的非线性锁定现象，不存在于线性分析中，包括解决它的方法，对人为不稳定现象的彻底分析，这导致了几何和材料沙漏的分类，并开发了一种新方法，以避免在大压缩应变存在下的人工沙漏。拟议的后续研究项目的第一个目标是回答问题，并解决在第一个资助期内出现的相关问题，特别是关于材料沙漏，包括材料各向异性，以及在不均匀变形状态下新发现的几何沙漏现象。此外，新开发的有限元公式，方法和算法将进行测试和扩展，适用于扭曲和非结构化网格和等几何公式。与第一个项目相比，将更加关注单元稳定性而不是非线性锁定。科学工作方案是沿着首要主题设计的，即在开发补救措施之前仔细分析寄生现象的起源。在这里，理论力学的分析结果将被重新审视，并投入到数值方法的前景。哲学是消除原因，而不是对抗症状。工作计划包括开发和记录一套全面的非线性基准设置，以测试非线性有限元的收敛性和稳定性。提供相应的原始数据以及公开数据库的开放源码软件将是支持和促进这一领域研究的一项重要产出。

The ongoing trend in all branches of industry to replace physical by virtual prototypes, not only to increase efficiency and accelerate development processes, but also to increase sustainability by saving resources, puts increasing demands on forecasting reliability of simulation tools. The finite element method is one of the work horses in computational engineering and thus it is remarkable that there are still some fundamental unsolved problems in finite element technology. An example is the persisting quest for a generally applicable finite element formulation that is free from locking and free from artificial instabilities at the same time in problems involving large deformations. This proposal builds upon previous research from the project Adaptive, deformation-dependent finite element formulations for stable and locking-free analysis of large deformation problems. The two main topics were the analysis of existing elements, with the goal to attain a better understanding of element behavior in nonlinear regimes, and the development of new nonlinear finite elements, both with respect to nonlinear locking phenomena and artificial instability (hourglassing). The most important scientific results were the detection of previously unknown distinctly nonlinear locking phenomena, not present in linear analysis, including a method to resolve it, a thorough analysis of artificial instability phenomena, which gave rise to a classification into geometric and material hourglassing and the development of a new method to avoid artificial hourglassing in the presence of large compressive strains. The first objective of the proposed follow-up research project is to answer questions and resolve related issues that have emerged during the first funding period, particularly concerning material hourglassing, including material anisotropy, as well as a newly detected geometrical hourglassing phenomenon occurring in inhomogeneous states of deformation. Moreover, the newly developed finite element formulations, methods and algorithms will be tested and extended for applicability to distorted and unstructured meshes and isogeometric formulations. Compared to the first project, there will be a stronger focus on element stability than on nonlinear locking. The scientific work program is devised along the overarching motif of carefully analyzing the origin of parasitic phenomena before developing remedies. Here, analytic results from theoretical mechanics will be revisited and put into the prespective of numerical methods. The philosophy is to remove causes instead of fighting symptoms. The work program includes development and documentation of a comprehensive set of nonlinear benchmark setups to test convergence and stability properties of nonlinear finite elements. Providing corresponding primary data as well as open-source software on a publicly available database will be an important output in order to support and foster research in this area.