Numerical Relaxation for the Description of Microstructure Evolution in Functional Magnetic Materials

该项目在第一个供资期间的重点是开发本构模型，以根据弛豫能势预测性地描述非线性、滞后和各向异性磁致伸缩。项目延续的中心目标是将开发的建模概念从特殊情况解决方案扩展到广泛的适用性。本着这种概括的精神，应消除对磁性形状记忆合金、简单几何形状和二维设置的分析限制。因此，该项目集中在三个重点领域的理论模型开发和随后的算法实现：在第一步中，实现了能量弛豫模型到基于有限元的解决方案完全耦合的磁力边值问题。以这种方式，创建一个模拟工具，不仅捕捉能量最小化的微观结构的影响，磁致伸缩均匀的问题，但也可以用来模拟任意几何形状或多晶聚集体的单晶样品。还可以将宏观场施加到中尺度代表性体积元素（RVE）上--这为通过FE 2方法实现直接数值均匀化建立了算法基础和接口。第二步是对基于弛豫的材料模型进行扩展。这主要包括对三维效应的概括，同时考虑到微观结构演化机制的全谱，旨在提高材料模型的适用性。在这方面，仔细研究的效率和稳定性的基本数值计划是至关重要的。在第三步中，将执行建模和模拟概念的转移。为此，其他材料系统的应用-其宏观磁致伸缩行为可能是由不同的微观结构机制驱动-应予以解决，包括实验验证。另一个重要的方面是定量比较预测的扩展能量弛豫模型和模拟的基础上的替代概念（相场，基于层压的混合物理论），以允许一个有根据的评估的性能和适用性的不同建模方法。

The focus of the project during the first funding period was placed on the development of constitutive models for the predictive description of nonlinear, hysteretic and anisotropic magnetostriction on the basis of relaxed energy potentials. The central goal of the project continuation is to take the developed modeling concept from special case solutions to broad scale applicability.In the spirit of such a generalization, the restriction of the analysis to magnetic shape memory alloys, simple geometries, and two-dimensional settings shall be removed. The project thereby concentrates on the theoretical model development and subsequent algorithmic implementation in three focus areas: In a first step, the implementation of the energy relaxation models into finite element-based solution schemes for fully-coupled magnetomechanical boundary value problems is realized. In this manner, a simulation tool is created, that not only captures the influence of energy minimizing microstructures on magnetostriction in homogeneous problems, but that can also be employed to simulate single crystal samples of arbitrary geometry or polycrystalline aggregates. It shall also be possible to impose macroscopic fields onto meso-scale Representative Volume Elements (RVEs)---this establishes the algorithmic basis and an interface for the implementation into direct numerical homogenization via the FE2 method. The second step is focused on the extension of the relaxation based material model. This centrally comprises the generalization to three-dimensional effects while accounting for the full spectrum of microstructural evolution mechanisms and aims to improve the applicability of the material model. In this context, careful studies of the efficiency and stability of the underlying numerical schemes are of the essence. In a third step, a transfer of the modeling and simulation concepts will be performed. To this end, the application to other material systems---whose macroscopic magnetostriction behavior may be driven by differing microstructural mechanisms---shall be addressed, including experimental validation. Another important aspect is the quantitative comparison between predictions of the extended energy relaxation model and simulations on the basis of alternative concepts (phase field, laminate based mixture theory), to allow a well-founded assessment of the performance and applicability of the different modeling approaches.