Elegant, effective and exact: New mathematical models for materials science

The value of such fundamental research is also demonstrated by the fact that Carolin Kreisbeck has now received funding from the German Research Foundation (DFG) and its Spanish counterpart (AEI) for her current project in cooperation with her Spanish colleague Prof. Dr. Carlos Mora Corral (Autonomous University of Madrid). Under the title"Nonlocal gradients in variational analysis and materials modeling: limits, kernels, boundaries", the two scientists want to put a non-local deformation theory on a stable footing over the next three years. "We are working on the theoretical foundation with the aim of obtaining realistic descriptions of material behavior that offer advantages over previous approaches", explains Carolin Kreisbeck, who holds the Chair of Analysis at the KU. Among other things, they hope that various effects can be combined in a unified model: "The idea is to be able to simultaneously describe elastic and plastic deformations as well as fractures and other damage effects in a nonlocal model"

For the field of material science this could mean even more effective and efficient simulations. Although Kreisbeck admits that many engineers already use nonlocal models in their professional practice: "They often work quite well, but they lack a clear mathematical foundation, a mathematical proof." As an applied mathematician, she is driven by two motivations: "For me, the mathematical questions are thrilling and motivate me to formulate the most general concepts possible. But I also want what I'm working on to have a clear relation to reality." Kreisbeck and her colleague Mora Corral see potential applications for the nonlocal models not only in materials science, but also in areas such as image processing or machine learning.

Local and nonlocal gradients

The mathematical core objects of the project, the nonlocal gradients, fall under the concept of differentiation. The derivative of a function measures its local rate of change. If you plot a function of a single variable as a curve in a coordinate system, the derivative at a point indicates how steeply the curve rises or falls there. In mathematics, it does not stop at this simple form of functions with a single variable. "The concept also works in multiple dimensions, where, for instance, a pair of real numbers is assigned another real number ", explains mathematics professor Kreisbeck. "The graph of a function with two real variables looks a bit like a flying carpet." The gradient is designed precisely for such functions with multiple variables.  As a generalization of the derivative, it indicates the direction and extent of change of a function in multiple dimensions, for example, in space.

In practical applications, however, these "normal", local gradients sometimes reach their limits, as nonlocal interactions and global effects must also be taken into account in order to make more precise predictions. Carolin Kreisbeck compares a nonlocal gradient to a ball rather than a point: "Instead of looking at a single point, we consider changes averaged over the entire volume of the ball, which allows us to account for certain small-scale effects". This approach, for example, makes it possible to bridge different length scales.

Gradients in practical use

For the mathematical layperson, this becomes more tangible when looking at a field of application. In the automotive industry, for example, various composite materials are used, i.e. mixtures of different material components with a fine structure. Materials research is not so much concerned with  what exactly the composition is at a specific point, says Kreisbeck: "The important thing is rather: How does the material behave on a macroscopic level? What are the mechanical properties of the material?" The fine structures are crucial for material science, but are difficult to analyze. Nonlocal gradients can be used to incorporate certain averaging effects over fine scales into the models. "If I average out the fine scales in this step, I make life easier for myself and still represent the effective behavior", explains Kreisbeck. "This way, I can capture the entire effect at once in a meaningful way.”

Nonlocal gradients are also attractive and exciting objects purely from a mathematical perspective, emphasizes the professor: “These operators are more general than classical gradients; they also work for complex functions that, for example, have multiple jumps and don’t produce a smooth curve.” Thus, they can be meaningfully applied in practice to functions that describe fracture behavior, for instance. This is also the basis for the hope of creating a unified model to explain various mechanical properties. “With nonlocal gradients, we have a more general concept for describing change behavior at our disposal.”

The project places particular emphasis on asymptotic variational analysis, i.e. limit values and limit processes. "We are particularly interested in nonlocal-to-local convergence", says Carolin Kreisbeck. Related to the ball-and-point metaphor: If the ball as a symbol of the nonlocal gradient shrinks to a point, then you should end up with the local gradient again. "This is an important cross-check: Does the new, more general model make sense? Does it actually yield in the limit what people have done so far with the local approach?"  

Long-standing German-Spanish exchange as a basis

The fact that she is now working on such questions together with Prof. Dr. Carlos Mora Corral from the Autonomous University of Madrid, funded by the German Research Foundation DFG and the Spanish Research Association AEI, is both a coincidence and the result of many years of networking. The two project partners first met at a workshop more than ten years ago and came into contact again five years ago through a publication on the same topic. They maintained their contact and sent doctoral students to each other's groups for research visits. When the DFG and AEI published a call for proposals specifically for German-Spanish collaborations in the field of mathematics, it was a twist of fate for the two mathematicians – and there was no question that they would be submitting a joint project proposal.