A finite element approach for vector- and tensor-valued surface PDEs
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In this paper, a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on Riemannian manifolds is derived, which allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surfaces PDEs.About:
This article is published in Journal of Computational Physics.The article was published on 2019-07-15 and is currently open access. It has received 34 citations till now. The article focuses on the topics: Tensor field & Tensor.read more
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Hydrodynamic interactions in polar liquid crystals on evolving surfaces
TL;DR: In this paper, the effect of hydrodynamics on the interplay of topology, geometric properties, and defect dynamics on evolving surfaces is considered. But the model is derived as a thin-film limit, and a finite element method is used to study the effect.
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Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
TL;DR: The Local Monge Parametrization (LMP) method as discussed by the authors approximates tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations.
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A numerical approach for fluid deformable surfaces
TL;DR: In this article, the authors derived the governing equations as a thin film limit and provided a general numerical approach for their solution, which demonstrated the rich dynamics resulting from this interplay, where, in the presence of curvature, any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flows.
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Active morphogenesis of epithelial monolayers.
TL;DR: In this paper, an active-hydrodynamic theory of early-stage embryo development is presented, which is closed under shape-changing dynamics, i.e., the degrees of freedom that encode monolayer geometry appear properly as broken-symmetry variables.
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Liquid Crystals on Deformable Surfaces
TL;DR: A thermodynamically consistent Landau-de Gennes-Helfrich model is derived which considers the simultaneous relaxation of the Q-tensor field and the surface and the resulting system of tensor-valued surface partial differential equation and geometric evolution laws is numerically solved.
References
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Book
Manifolds, tensor analysis, and applications
TL;DR: In this paper, the authors provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists, including manifolds, dynamical systems, tensors, and differential forms.
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Discrete exterior calculus
TL;DR: In this article, a theory of discrete exterior calculus (DEC) is proposed, which includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects.
Journal ArticleDOI
CutFEM: Discretizing geometry and partial differential equations
TL;DR: Recent advances on robust unfitted finite element methods on cut meshes designed to facilitate computations on complex geometries obtained from computer‐aided design or image data from applied sciences are discussed and illustrated numerically.
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Finite element methods for surface PDEs
Gerhard Dziuk,Charles M. Elliott +1 more
TL;DR: Finite element methods for approximating the solution of partial differential equations on surfaces are considered, focusing on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods.
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Topology and dynamics of active nematic vesicles
Felix C. Keber,Etienne Loiseau,Tim Sanchez,Stephen J. DeCamp,Luca Giomi,Luca Giomi,Mark J. Bowick,M. Cristina Marchetti,Zvonimir Dogic,Zvonimir Dogic,Andreas R. Bausch +10 more
TL;DR: The spatiotemporal patterns that emerge when an active nematic film of microtubules and molecular motors is encapsulated within a shape-changing lipid vesicle are studied to demonstrate how biomimetic materials can be obtained when topological constraints are used to control the non-equilibrium dynamics of active matter.
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