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的数学基础和工程应用，以及对固体力学中非线性问题的新的非标准方法的发展和分析。计算工程中的实际应用将是汉诺威莱布尼茨大学LUH工作组与柏林洪堡大学HU工作组合作的重点，重点是新型离散化方案的数学基础。联合目标是随着基于非协调、混合和间断Galerkin或Petrov-Galerkin有限元的超弱格式的自适应数值离散的发展，在非线性连续介质力学中进行有效和可靠的模拟。在第一个供资期间，LUH工作组制定了不同的不连续离散化方法。对于(几乎)不可压缩和弹塑性材料的特性，对原不连续Galerkin有限元方法进行了有效的扩展/增强，避免了剪切锁定效应和体积锁定效应。HU工作组与LUH工作组合作，开发并分析了非线性模型问题的不连续Petrov-Galerkin(DPG)有限元，并证明了解线弹性问题的自适应DPG和最小二乘方法的最优收敛速度。进一步的研究主题是线弹性力学中逐点对称离散的保证误差界和多凸体材料的非协调有限元分析。第二个资助期的重点将是两个工作组进一步密切合作，将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配方进行调查，并就相关的机械问题进行练习。在AceGen中的实现方便了不同离散化之间关于这种新的有限元公式的收敛行为的快速和有效的比较。HU工作组将继续对非线性问题进行分析，将难度从Hencky材料提高到多凸材料和几何非线性构型。最近在非线性问题的DPG方法方面的突破促使具有内置误差控制的自适应DPG格式应用于进一步的问题，如超弹性、障碍问题和时间演化弹塑性。将研究自适应非线性LS和DPG方法以及Arnold-Winther有限元方法的最佳收敛速度和DPG方法的保证误差估计，包括显式常数和正确的尺度。共同组织微型符号或Oberwolfach研讨会(例如，2015、2018年的计算工程研讨会)等外联活动促进了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.