Browsing by Author "Garcke, Harald"

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    An anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Garcke, Harald; Kraus, Christiane
    We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy
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    Interfaces, Free Boundaries and Geometric Partial Differential Equations
    (Zürich : EMS Publ. House, 2024) Elliott, Charles M.; Garcke, Harald; Niethammer, Barbara; Simonett, Gieri
    Partial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research. The workshop focused on new developments and emerging new themes. At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed. By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions.
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    Multi-material phase field approach to structural topology optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Blank, Luise; Farshbaf-Shaker, M. Hassan; Garcke, Harald; Rupprecht, Christoph; Styles, Vanessa
    Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an H1-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.
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    Relating phase field and sharp interface approaches to structural topology optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Blank, Luise; Farshbaf-Shaker, M. Hassan; Garcke, Harald; Styles, Vanessa
    A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.
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    Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, non-local and discrete structures
    (Zürich : EMS Publ. House, 2019) Garcke, Harald; Kornhuber, Ralf
    Partial differential equations in complex domains with free\linebreak boundaries and interfaces continue to be flourishing research areas at the interfaces between PDE theory, differential geometry, numerical analysis and applications. Main themes of the workshop have been PDEs on evolving domains, phase field approaches, interactions of bulk and surface PDEs, curvature driven evolution equations. Applications particular from biology, such as cell and cancer modelling and fluid as well solid mechanics have been subjects of the conference.