A finite element approach for vector- and tensor-valued surface PDEs

We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on general manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided

Nematic liquid crystals on curved surfaces - a thin film limit

We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.Comment: 20 pages, 4 figure

Simulation of dendritic-eutectic growth with the phase-field method

Solidification is an important process in many alloy processing routes. The solidified microstructure of alloys is usually made up of dendrites, eutectics or a combination of both. The evolving morphologies are largely determined by the solidification process and thus many materials properties are dependent on the processing conditions. While the growth of either type of microstructure is well-investigated, there is little information on the coupled growth of both microstructures. This work aims to close this gap by formulating a phase-field model capable of reproducing dendritic, eutectic as well as dendritic-eutectic growth. Following this, two-dimensional simulations are conducted which show all three types of microstructures depending on the composition and processing conditions. The effect of the dendritic-eutectic growth on the microstructural lengths, which determine materials properties, is investigated and the morphological hysteresis between eutectic growth and dendritic-eutectic growth is studied by employing solidification velocity jumps. Further, the influence of primary crystallization is investigated in large-scale two-dimensional simulations. Finally, qualitative three-dimensional simulations are conducted to test for morphological changes in the eutectic.Comment: 51 pages, 19 figure

Properties of surface Landau-de Gennes Q-tensor models

Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase transition type for these models, and suggest experiments to identify proper model assumptions

Bayesian optimization framework for data-driven materials design

The improvement of experimental design and the optimization of materials’properties with complex and partially unknown behaviors are common problems in material science. In the context of aqueous foams, the microstructure has a major influence on the properties of the resulting foam. Multiple interlinked parameters yield a large design space that requires tuning to tailor the microstructure evolution and resulting physical qualities. Our goal is a data-driven framework that uses machine learning to guide both experiments and simulations in an autonomous closed-loop. This iterative approach presents a valuable opportunity to accelerate materials development processes. A design of experiments methodology utilizing Bayesian Optimization is used to efficiently explore and exploit the search space, while minimizing the number of required evaluations. This approach allows to select the next most informative evaluation to perform, autonomously and adaptively learning from the already acquired data. The designed workflow is implemented into the data platform Kadi4Mat1, which provides the possibility of storing heterogeneous provenance data, along with a common workspace to integrate analysis methods and visualization. Our contribution within Kadi4Mat strongly relies on the reuse of data, and it is an example of the close interoperability between experimental and simulation research that the platform supports, in full alignment with the FAIR principles. Acknowledgements: This work is funded by the Ministry of Science, Research and Art Baden-Württemberg (MWK-BW) in the project MoMaF–Science Data Center, with funds from the state digitization strategy digital@bw (project number 57)

Brittle anisotropic fracture propagation in quartz sandstone: insights from phase-field simulations

<jats:title>Abstract</jats:title><jats:p>We developed a generalized multiphase-field modeling framework for addressing the problem of brittle fracture propagation in quartz sandstones at microscopic length scale. Within this numerical approach, the grain boundaries and crack surfaces are modeled as diffuse interfaces. The two novel aspects of the model are the formulations of (I) anisotropic crack resistance in order to account for preferential cleavage planes within each randomly oriented quartz grain and (II) reduced interfacial crack resistance for incorporating lower fracture toughness along the grain boundaries that might result in intergranular crack propagation. The presented model is capable of simulating the competition between inter- and transgranular modes of fracturing based on the nature of grain boundaries, while exhibiting preferred fracturing directions within each grain. In the full parameter space, the model can serve as a powerful tool to investigate the complicated fracturing processes in heterogeneous polycrystalline rocks comprising of grains of distinct elastic properties, cleavage planes, and grain boundary attributes. We demonstrate the performance of the model through the representative numerical examples.</jats:p&gt