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

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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.