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元方法，这是一种高阶虚拟区域方法。我们计划将这种方法进一步扩展到非线性结构力学，包括多物理、多尺度和演化领域的问题。作为该方法的一个示范应用，我们使用了加法制造的产品和过程模拟。在我们的项目中，最重要的是先进的机械建模、高效算法的研究和严格的数学分析之间的紧密互动，目标是为一类困难问题开发预测和质量保证的模拟能力，其中现有的方法只有有限的价值。第一个项目阶段集中于基本的算法问题，如准确和有效的切割单元积分，多物理问题的局部适应性，以及包括线性问题的后验误差分析在内的第一个误差估计，而第二阶段将专注于更复杂的非线性问题的扩展。我们期望有限元方法继承高阶p-有限元在区域内部的众所周知的基本性质。然而，必须特别强调对嵌入区域边界处的切割单元的适当处理。关于非线性问题，我们将集中在几乎不可压缩的材料，包括大变形，弹塑性，以及在动态适应hp近似的情况下历史变量的处理。我们对具有瞬变区域的耦合问题(附加制造工艺的一个中心方面)的研究重点将在于对先前无效的零件进行适当的能量守恒初始化的问题。此外，我们将把所研究的形状函数类扩展到样条线，因为它们在等距分析中被非常成功地使用。因此，我们将能够研究单元间连续性和可微性对耦合多物理问题的逼近质量的影响。另一个重要的重点是推导后验误差控制和为有限单元法和上述各类问题制定自适应方案，其中特别注意切割单元和相关的四次区域。最后但并非最不重要的是，我们将继续积极参与在优先方案范围内形成的基准问题的定义和计算。

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.