Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics

本项目的研究目标是广义混合有限元法的数学基础和工程应用，以及固体力学中非线性问题的新的非标准方法的发展和分析。在计算工程中的实际应用将是工作组LUH在汉诺威与工作组HU在洪堡大学合作的重点在柏林的数学基础上的新的离散计划。联合目标是在非线性连续介质力学中进行有效和可靠的模拟，并开发基于超弱公式的自适应数值离散，混合和不连续Galerkin或Petrov-Galerkin有限元方法。在第一个资助期内，CNOLUH开发了不同的不连续离散化方法。一个有效的扩展/增强原来的不连续伽辽金有限元法（dG FEM）避免剪切锁定效应和体积锁定（几乎）不可压缩和弹塑性材料的行为。该研究所开发和分析了一个非线性模型问题的不连续彼得罗夫-伽辽金（dPG）有限元与LUH合作，并证明了自适应dPG和最小二乘方法的最佳收敛速度的线性弹性问题。进一步的研究课题是保证误差界逐点对称离散线弹性和分析的dPG有限元polyconvex material.The第二个资助期的重点将是进一步密切合作的两个工作组的dPG有限元扩展到非线性弹性材料的行为。因此，LUH工作组将对各种dPG公式进行研究，并就相关的机械问题进行练习。在AceGen中的实施有利于快速和有效的比较不同的离散化方面的收敛行为，这种新的有限元制剂。该协会将继续他们的非线性问题的分析，提高难度从Hencky材料到多凸材料和几何非线性配置。非线性问题dPG方法的最新突破激发了具有内置误差控制的自适应dPG方案在超弹性、障碍问题和时间演化弹塑性等进一步问题中的应用。将研究自适应非线性LS和dPG方法以及Arnold-Winther FEM的最优收敛速度，以及涉及显式常数和正确缩放的dPG方法的保证误差估计。外联活动，如共同组织minisymopsia或Oberwolfach研讨会（例如2015年，2018年的»计算工程«），促进了SPP内外的思想交流和富有成效的合作。

The research of this project aims at the mathematical foundation and the engineering application of generalized mixed FEM as well as the development and the analysis of new non-standard methods that yield guaranteed results for nonlinear problems in solid mechanics. The practical applications in computational engineering will be the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universität zu Berlin with focus on mathematical foundation of the novel discretization schemes. The joint target is the effective and reliable simulation in nonlinear continuum mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or Petrov-Galerkin Finite Element Methods. In the first funding period, the workgroup LUH developed different discontinuous discretization methods. An efficient extension/enhancement of the original discontinuous Galerkin Finite Element Method (dG FEM) avoids shear-locking effects and volumetric-locking for (nearly) incompressible and elasto-plastic material behaviour. The workgroup HU developed and analysed a discontinuous Petrov-Galerkin (dPG) FEM for a nonlinear model problem in collaboration with the workgroup LUH and proved optimal convergence rates of adaptive dPG and least-squares methods for linear elastic problems. Further topics of research were guaranteed error bounds for pointwise symmetric discretizations in linear elasticity and the analysis of nonconforming FEM for polyconvex materials.The focus of the second funding period will be a further close collaboration of both workgroups regarding the extension of the dPG FEM to nonlinear-elastic material behaviour. Therefore, various dPG formulations will be investigated and exercised on relevant mechanical problems by the Workgroup LUH. The implementation in AceGen facilitates expeditious and efficient comparison among different discretizations with respect to convergence behaviour of this novel finite element formulations. The workgroup HU will continue their analysis on nonlinear problems in raising difficulty from Hencky material to polyconvex material and geometric nonlinear configurations. Recent breakthroughs in the dPG methodology for nonlinear problems motivate the application of adaptive dPG schemes with built-in error control to further problems such as hyperelasticity, the obstacle problem and time-evolving elastoplasticity. Optimal convergence rates of adaptive nonlinear LS and dPG methods and Arnold-Winther FEM and guaranteed error estimation for dPG methods involving explicit constants and correct scaling will be investigated.Outreach activities such as coorganization of minisymposia or Oberwolfach workshops (e.g. »Computational Engineering« in 2015, 2018) have fostered the exchange of ideas and fruitful collaborations within the SPP and beyond.