Simulation and Optimization of Rate-Independent Systems with Non-Convex Energies

该项目致力于对具有非凸能量的速率无关系统进行分析、仿真和优化。连续介质力学中的各种系统表现出近似的速率无关性。脆性材料的损伤演化就是一个突出的例子。基于正齐次凸耗散泛函和能量泛函，给出了相应的演化变分不等式形式。如果后者也是凸的，则解通常是唯一的和时间连续的。相比之下，具有非凸能量的速率无关系统，如损伤模型，导致了一些数学挑战：一般情况下，解在时间上不连续，并且存在多个解的概念，其在时间上的不连续性具有不同的特征。此外，解决方案往往不是唯一的，即使局限于特定的解决方案概念。因此，具有非凸能量的速率无关系统的数值模拟和优化是一项具有挑战性的任务。该项目将基于平衡粘性(BV)解的概念来面对这些问题，该概念最近被提出以使得能够更精确地刻画时间上的不连续性。BV解的存在性可以用粘性正则化来表示，并伴随着对消失粘性的极限分析。然而，到目前为止，出于这一目的，人们大多假设数据在时间上是平滑的。该项目分为三个分支：第一个分支包括将存在理论扩展到非光滑数据，以及对解集的分析性质进行精细表征，例如紧致性性质。这两个方面对其他两个分支机构来说都是非常重要的。第二个是关于高效和稳健的数值方法的开发、分析和实现，以近似BV解。此外，在第三个分支中，应该建立和分析合适的优化问题。对于BV解的数值逼近，我们使用时间增量局部极小化方法。虽然最初的数值结果看起来很有希望，但到目前为止，严格的收敛分析仅限于相当严格的假设，例如在损坏的情况下不能满足的假设。此外，对于这类问题，时间自适应方法到目前为止还没有被研究过。在优化方面，我们所知的更少。这特别涉及到通过粘性正则化来逼近最优解，这也提供了解决优化问题的机会。在提出的项目中，我们专注于这些开放的问题，并旨在为具有非凸能量的速率无关系统的模拟和优化建立一个分析上合理的框架。脆性材料中的损伤过程将作为它的典型应用。

The project is concerned with the analysis, simulation, and optimization of rate-independent systems with non-convex energies. Various systems in continuum mechanics behave approximately rate-independent. A prominent example is the damage evolution in brittle materials. The associated models are given in form of evolutionary variational inequalities based on a positive homogeneous and convex dissipation functional and an energy functional. If the latter is convex, too, then solutions are typically unique and time-continuous. In contrast to that, rate-independent systems with non-convex energies, such as damage models, lead to several mathematical challenges: in general, solutions are not continuous in time, and there are multiple solution concepts with different characterizations of the discontinuities in time. Moreover, solutions are often not unique, even when restricted to a specific notion of solution. Therefore, the numerical simulation and all the more the optimization of rate-independent systems with non-convex energies is a challenging task.The project will face these issues based on the notion of Balanced-Viscosity-(BV)-solutions, which was recently developed to enable a more precise characterization of the discontinuities in time. Existence of BV-solutions can be shown by viscous regularization accompanied by a limit analysis for vanishing viscosity. So far however, it is mostly assumed for this purpose that the data are smooth in time. The project is divided into three branches: The first branch includes an extension of the existence theory to non-smooth data and a refined characterization of the analytical properties of the solution set, like for instance compactness properties. Both aspects will be of major importance for the other two branches. The second one is concerned with the development, analysis, and implementation of efficient and robust numerical methods to approximate BV-solutions. Furthermore, in the third branch, suitable optimization problems shall be formulated and analyzed.For the numerical approximation of BV-solutions, we apply time-incremental local minimization methods. While first numerical results look promising, a rigorous convergence analysis is so far only known for rather restrictive assumptions that are for instance not fulfilled in case of damage. Moreover, time-adaptive methods have not been investigated so far for this type of problems. With regard to optimization, even less is known. This in particular concerns the approximation of optimal solutions via viscous regularization, which also offers an opportunity to solve the optimization problems. With the proposed project, we focus on these open questions and aim to establish an analytically sound framework for the simulation and optimization of rate-independent systems with non-convex energies. Damage processes in brittle materials will serve as prototypical application therefor.