An isogeometric finite element formulation for phase fields on deforming
  surfaces

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Abstract

This paper presents a general theory and isogeometric finite element implementation of phase fields on deforming surfaces. The problem is governed by two coupled fourth order partial differential equations (PDEs) that live on an evolving manifold. For the phase field, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance. For the surface deformation, the PDE is the thin shell equation following from Kirchhoff-Love kinematics. Both PDEs can be efficiently discretized using \(C^1\)-continous interpolation free of derivative dofs (degrees-of-freedom) such as rotations. Structured NURBS and unstructured spline spaces with pointwise \(C^1\)-continuity are considered for this. The resulting finite element formulation is discretized in time by the generalized-\(\alpha\) scheme with time-step size adaption, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples considering spheres, tori and double-tori.

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Visualization and quantification of electrochemical and mechanical degradation in Li ion batteries.

Martin Ebner, Federica Marone, Marco Stampanoni … (2013)

High-energy-density materials that undergo conversion and/or alloying reactions hold promise for next-generation lithium (Li) ion batteries. However, these materials experience substantial volume change during electrochemical operation, which causes mechanical fracture of the material and structural disintegration of the electrode, leading to capacity loss. In this work, we use x-ray tomography during battery operation to visualize and quantify the origins and evolution of electrochemical and mechanical degradation. Tomography provides the time-resolved, three-dimensional chemical composition and morphology within individual particles and throughout the electrode. In the model material tin(II) oxide, we witness distributions in onset and rate of core-shell lithiation, crack initiation and growth along preexisting defects, and irreversible distortion of the electrode, highlighting tomography as a tool to guide the development of durable materials and strain-tolerant electrodes.