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

本计画的中心研究课题为有限单元法，一种高阶虚拟区域法。我们计划进一步将这种方法推广到非线性结构力学，包括多物理场，多尺度和发展领域的问题。作为这种方法的演示应用，我们使用产品和过程模拟进行增材制造。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，目标是为一类困难的问题开发预测和质量保证的仿真能力，现有的方法只有有限的价值。第一个项目阶段侧重于基本的算法问题，如切割单元的准确和有效集成，多物理问题的局部自适应性，以及包括线性问题后验误差分析在内的第一次误差估计，第二阶段将专注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了高阶p-FEM的基本性质。然而，特别强调必须放在一个适当的处理切割细胞的边界处的嵌入域。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，和处理的历史变量的情况下，动态适应hp-近似。 我们对瞬态域耦合问题（增材制造工艺的一个核心方面）的研究重点将在于对先前空部件进行适当的能量守恒初始化的问题。此外，我们将扩展类的调查形状函数样条，因为它们是非常成功地用于等几何分析。因此，我们将能够研究的影响，单元间的连续性和可微性的耦合多物理问题的近似质量。另一个重要的重点是推导后验误差控制和开发有限单元法和上述问题的自适应方案，特别注意切割单元和相关的求积。最后但并非最不重要的是，我们将继续积极参与优先计划中开发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.