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

本项目旨在研究广义混合有限元法的数学基础和工程应用，以及开发和分析新的非标准方法，以保证固体力学非线性问题的结果。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Universität zu HU工作组合作的重点，重点是新离散化方案的数学基础。基于非协调、混合和不连续Galerkin或Petrov-Galerkin有限元方法之间的超弱公式，发展了自适应数值离散化，从而实现了非线性连续介质力学的有效、可靠的模拟。在第一个资助期，LUH工作组开发了不同的不连续离散化方法。对原始的不连续Galerkin有限元方法（dG FEM）进行了有效的扩展/增强，避免了（几乎）不可压缩和弹塑性材料行为的剪切锁定效应和体积锁定。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）有限元法，并证明了线性弹性问题的自适应dPG法和最小二乘法的最优收敛速度。进一步的研究课题是线弹性中点对称离散化的保证误差边界和多凸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在将dPG FEM扩展到非线性弹性材料行为方面的进一步密切合作。因此，LUH工作组将对各种dPG配方进行相关力学问题的研究和实践。在AceGen中的实现促进了不同离散化之间关于这种新颖有限元公式收敛行为的快速和有效的比较。HU工作组将继续分析从henky材料到多凸材料和几何非线性构型的非线性问题。非线性问题的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。研究了自适应非线性LS和dPG方法的最优收敛速率以及Arnold-Winther有限元法和包含显式常数和正确标度的dPG方法的保证误差估计。诸如共同组织小型研讨会或Oberwolfach研讨会（例如2015年和2018年的“计算工程”）等外展活动促进了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.