The Web-method is a meshless finite element technique which uses weighted extended B-splines (Web-splines) on a tensor product grid as basis functions. It combines the computational advantages of B-splines and standard mesh-based elements. In particular, degree and smoothness can be chosen arbitrarily without substantially increasing the dimension. Hence, accurate approximations are obtained with relatively few parameters. Moreover, the regular grid is well suited for hierarchical refinement and multigrid techniques. This article should serve as an introduction to finite element approximation with B-splines. We first review the construction of Web-bases and discuss their basic properties. Then we illustrate the performance of Ritz-Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects.
Most commercially used electrode materials contract and expand upon cycling. This change in volume influences the microstructure of the cell stack, which in turn impacts a range of performance parameters. Since direct observation of these microstructural changes with operando experiments is challenging and time intensive, a simulation tool that takes a real or artificially generated 3D microstructure and captures the volumetric changes in a cell during cycling would be valuable to enable rapid understanding of the impact of material choice, electrode and cell design, and operating conditions on the microstructural changes and identification of sources of mechanically-driven cell aging. Here, we report the development and verification of such a 3D electrochemical-mechanical tool, and provide an example use-case. We validate the tool by simulating the microstructural evolution of a graphite anode and a Li(Ni,Mn,Co)O2 cathode during cycling and comparing the results to x-ray tomography datasets of these electrodes taken during cycling. As an example use case for such a simulation tool, we explore how different volumetric expansion behaviors of the cathode material impact strain in the cell stack, illustrating how the material selection and its operation impact the mechanical behavior inside a cell.
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