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

该项目关注的是具有非凸能量的与速率无关的系统的分析、模拟和优化。在连续介质力学中，各种系统的行为近似与速率无关。一个突出的例子是脆性材料中的损伤演化。相关的模型给出了基于正齐次凸耗散泛函和能量泛函的演化变分不等式的形式。如果后者也是凸的，那么解通常是唯一的和时间连续的。与此相反，具有非凸能量的率无关系统，如损伤模型，导致了几个数学挑战：一般来说，解在时间上是不连续的，并且存在多个解概念，具有不同的时间不连续性特征。此外，解决方案往往不是唯一的，即使局限于解决方案的特定概念。因此，数值模拟和非凸能量的速率无关系统的优化是一项具有挑战性的任务。该项目将基于平衡粘度（BV）解的概念来面对这些问题，该解是最近开发的，可以更精确地表征时间上的不连续性。BV-解的存在性可以通过粘性正则化和粘性消失的极限分析来证明。然而，到目前为止，大多数情况下都假设数据在时间上是平滑的。该项目分为三个分支：第一个分支包括扩展的存在性理论的非光滑数据和一个精细的表征的分析性质的解决方案集，如例如紧凑性。这两个方面对其他两个部门都非常重要。第二个是有关的发展，分析和实施的有效和强大的数值方法来近似BV-解决方案。在第三个分支中，我们将构造和分析合适的优化问题，对于BV-解的数值逼近，我们采用时间增量局部极小化方法.虽然第一个数值结果看起来很有希望，但严格的收敛分析到目前为止只知道相当限制性的假设，例如在损坏的情况下不满足。此外，时间自适应方法还没有研究到目前为止，这类问题。关于优化，我们知道的更少。这特别涉及通过粘性正则化逼近最优解，这也提供了解决优化问题的机会。 通过该项目，我们专注于这些开放性问题，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合理的框架。脆性材料中的损伤过程将作为其原型应用。

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.