Distance in the space of elasticity tensors: optimal choices of Hookean models

在我的研究中，我努力加强对地震现象和地球物质特性的定量描述。我的工作属于地球科学背景下的应用数学领域。为了定量地描述构成地球的材料，我使用了一个由胡克固体概念提供的数学模型。这种假想固体的性质，即描述它的数学表达式的性质，可以作为实际材料物理性质的类比。正如我在我的研究中所展示的那样，胡克固体展示了八种对称类。这在我的工作中是至关重要的，因为真实材料的行为可以通过考虑具有特定对称性的胡克固体来近似。然而，由于真实的材料不是胡克固体，我们不能说一个给定的材料属于一个特定的对称类。我们只能考虑这种材料与某一类别的接近程度。必须援引两个概念来检验这种接近性，这是我目前研究的重点。首先，我们必须在Hookean立体的抽象空间中引入“距离”的概念。其次，为了为现场数据提供实用的方法，我们必须在实验误差的背景下考虑距离；我们可能会问，数据分辨率是否允许我们区分两种不同对称的胡克固体的距离。为了评估这些问题答案的可靠性，我将使用不确定性分析，这需要先进的分析方法和高效的计算技术。这项研究的实际重要性是由于最近实验仪器的准确性提高，从而激发了定量公式的进一步发展。换句话说，所获取的数据中有比标准技术所能提取的更多的信息。就其直接应用而言，我的研究对关注全球和勘探问题的地球科学家都很重要。它提供有关材料特性的信息及其准确性的估计，以增加基于这些信息的科学和工业决策的信心。

In my research, I strive to enhance the quantitative description of seismic phenomena and material properties of the Earth. My work belongs to the realm of applied mathematics in the context of geosciences. To describe quantitatively materials that constitute the Earth, I use a mathematical model provided by the concept of a Hookean solid. Properties of such a hypothetical solid, which are properties of mathematical expressions that describe it, serve as analogies for physical properties of actual materials. As I have demonstrated through my research, Hookean solids exhibit eight symmetry classes. This is of paramount importance in my work, since the behaviour of a real material can be approximated by considering a Hookean solid of a particular symmetry. However, since real materials are not Hookean solids, we cannot say that a given material belongs to a particular symmetry class. We can consider only the proximity of this material to a particular class. Two concepts must be invoked to examine such a proximity, which is the focus of my present research. First, we must invoke a concept of "distance" within the abstract space of Hookean solids. Second, to provide a pragmatic methodology for field data, we have to consider distance in the context of experimental errors; we might ask whether or not the data resolution allows us to distinguish between distances to Hookean solids of two distinct symmetries. To evaluate the reliability of answers to such questions, I will use uncertainty analysis, which requires both advanced analytical methods and efficient computational techniques. The pragmatic importance of this research is motivated by recent increase of accuracy in experimental apparatus that inspires further advancements of quantitative formulations. In other words, there is more information in acquired data than can be extracted by standard techniques. In its immediate application, my research is important to geoscientists concerned with both global and exploration issues. It provides information on material properties with estimates of their accuracy to increase confidence in both scientific and industrial decisions based on such information.