Coordination Funds

协调基金

• 批准号: 431464129
• 负责人: Professor Dr. Axel Voigt
• 金额: --
• 依托单位: Professur für Wissenschaftliches Rechnen und Angewandte Mathematik
• 依托单位国家: 德国
• 项目类别: Research Units
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/431464129?language=en
• 关键词: Coordination; Funds

PDEs on surfaces remain an active field of research in applied mathematics. Due to their coupling with the geometry such surface PDEs are intrinsically nonlinear. This leads to new challenges in modeling and numerical analysis. Most of these challenges are addressed for scalar-valued surface PDEs. In the scalar case the coupling between surface geometry and the PDE is relatively weak, and thus numerical approaches established in flat spaces are applicable after small modifications. For vector- and tensor-valued surface PDEs these approaches are no longer sufficient. The surface vector- and tensor-fields often need to fulfill additional constraints. One example is the tangentiality of these fields, for which they need to be considered as elements of the tangent bundle, leading to a nonlinear coupling between the surface geometry and the PDE. To deal with these new challenges was the motivation for this research unit. Within the last three years we have seen a tremendous growth in research activities, with development of new models, new numerical methods, new numerical analysis results, new software tools, and new applications. Our research unit has significantly contributed to these developments and in several fields also initiated them. In the second funding period we will deepen this knowledge for a description and understanding that uncovers universal principles and makes similarities and differences between the considered applications transparent. Different from the first funding period a focus will be on surface dynamics, including deformable fluid surfaces, rate-independent evolution of prestrained plates, growth and swelling phenomena and the development of numerical methods and their analysis for these new challenges. Addressing these challenges can only be explored by combining various mathematical disciplines for which this research unit is perfectly suited. We strengthen our expertise in numerical analysis and mechanics and based on the success of the first funding period expect our results to further foster this fast growing research field within mathematics and other disciplines to enable breakthrough developments.

曲面上的偏微分方程组一直是应用数学中一个活跃的研究领域。由于它们与几何的耦合，这种表面偏微分方程组本质上是非线性的。这给建模和数值分析带来了新的挑战。这些挑战中的大多数都是针对标量曲面偏微分方程组的。在标量情况下，表面几何形状与偏微分方程组之间的耦合相对较弱，因此在平坦空间建立的数值方法在稍作修改后即可适用。对于向量值和张量值曲面偏微分方程组，这些方法不再是充分的。表面矢量场和张量场通常需要满足附加约束。一个例子是这些场的切性，对于这些场，它们需要被视为切线束的元素，从而导致曲面几何和PDE之间的非线性耦合。应对这些新的挑战是这个研究单位的动机。在过去的三年里，我们看到了研究活动的巨大增长，开发了新的模型、新的数值方法、新的数值分析结果、新的软件工具和新的应用。我们的研究单位对这些发展作出了重大贡献，在几个领域也发起了这些发展。在第二个资助期，我们将加深这方面的知识，以揭示普遍原则，并使所考虑的申请之间的异同变得透明。与第一个供资期间不同的是，重点将放在表面动力学上，包括可变形的流体表面、与速率无关的预应变板的演变、生长和膨胀现象以及针对这些新挑战的数值方法及其分析的发展。解决这些挑战只能通过将本研究单位非常适合的各种数学学科结合起来来探索。我们加强了我们在数值分析和力学方面的专业知识，并在第一个资助期取得成功的基础上，预计我们的成果将在数学和其他学科内进一步促进这一快速增长的研究领域，以实现突破性发展。