High-order immersed-boundary methods in solid mechanics for structures generated by additive processes

我们项目的中心研究课题是有限细胞法，一种高阶虚拟域方法。我们计划将该方法进一步扩展到非线性结构力学中，包括多物理场、多尺度和演化域问题。作为该方法的演示应用，我们将产品和过程模拟用于增材制造。在我们的项目中，最重要的是先进的机械建模，高效算法研究和严格的数学分析之间的紧密互动，目标是为一类难题开发预测和质量保证的模拟能力，其中现有方法的价值有限。第一个项目阶段侧重于基本的算法问题，如切割细胞的准确和有效集成，多物理场问题的局部自适应，以及线性问题的第一次误差估计，包括后验误差分析，第二阶段将侧重于扩展到更复杂的非线性问题。我们期望有限单元法在区域内部继承高阶p-FEM的众所周知的基本性质。然而，必须特别强调在嵌入域的边界处对切割细胞的适当处理。关于非线性问题，我们将集中在几乎不可压缩的材料上，包括大变形，弹塑性，以及在动态适应hp近似的情况下对历史变量的处理。我们对瞬态域耦合问题（增材制造工艺的一个核心方面）的研究重点将在于先前空洞零件的适当能量守恒初始化问题。此外，我们将研究的形状函数类扩展到样条，因为它们在等几何分析中非常成功地使用。因此，我们将能够研究元间连续性和可微性对耦合多物理场问题近似质量的影响。另一个重要的焦点是后验误差控制的推导和有限单元法的自适应方案的发展以及上述问题的类别，其中特别注意切割单元和相关的正交。最后但并非最不重要的是，我们将继续积极参与确定和计算在优先方案范围内制定的基准问题。

The central research topic of our project is the Finite Cell Method, a high-order fictitious domain approach. We plan to further extend this method to nonlinear structural mechanics including multi-physics, multi-scale and evolving domain problems. As a demonstrator application for this method, we use product and process simulation for additive manufacturing. Most important in our project is the tight interaction between advanced mechanical modeling, research on efficient algorithms, and rigorous mathematical analysis, with the goal of developing predictive and quality assured simulation capabilities for a class of hard problems, where existing approaches are only of limited value. Whereas the first project phase focused on essential algorithmic questions like accurate and efficient integration of cut cells, local adaptivity for multi-physics problems, and first error estimations including a posteriori error analysis for linear problems, the second phase will focus on an extension to more complex nonlinear problems. We expect that the Finite Cell Method inherits well-known basic properties of the high-order p-FEM in the interior of domains. Yet, special emphasis has to be laid on a proper treatment of cut cells at the boundary of the embedded domain. Concerning nonlinear problems we will concentrate on nearly incompressible material including large deformations, elastoplasticity, and the treatment of history variables in the case of dynamically adapted hp-approximations. A focal point of our research on coupled problems with transient domains (a central aspect of additive manufacturing processes) will lie on the question of a proper energy conservative initialization of previously void parts. Additionally, we will extend the class of investigated shape functions to splines as they are very successfully used in Isogeometric Analysis. Thereby, we will be able to study the influence of inter-element continuity and differentiability properties on the approximation quality of coupled multi-physics problems. A further important focus is on the derivation of a posteriori error controls and the development of adaptive schemes for the Finite Cell Method and the classes of problems specified above, where particular attention is paid to cut cells and related quadratures.Last but not least, we will continue to actively participate in the definition and calculation of benchmark problems that are developed within the priority programme.