Computational Micromagnetism

1. Front Matter

   Pages I-XVIII

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2. Numerical Stationary Micromagnetism

   1. Front Matter

      Pages 1-13

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   2. Direct Minimization

      • Andreas Prohl

      Pages 15-41

   3. Convexified Micromagnetism

      • Andreas Prohl

      Pages 43-100

   4. Relaxed Micromagnetism using Young Measures

      • Andreas Prohl

      Pages 101-117

3. Numerical Nonstationary Micromagnetism

   1. Front Matter

      Pages 119-133

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   2. The Landau-Lifshitz-Gilbert Equation

      • Andreas Prohl

      Pages 135-196

   3. The Maxwell-Landau-Lifshitz-Gilbert Equations

      • Andreas Prohl

      Pages 197-232

   4. Nematic Liquid Crystals

      • Andreas Prohl

      Pages 233-285

   5. Summary and Outlook

      • Andreas Prohl

      Pages 287-289

4. Back Matter

   Pages 291-304

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In this work, we study numerical issues related to a common mathematical model which describes ferromagnetic materials, both in a stationary and non­ stationary context. Electromagnetic effects are accounted for in an extended model to study nonstationary magneto-electronics. The last part deals with the numerical analysis of the commonly used Ericksen-Leslie model to study the fluid flow of nematic liquid crystals which find applications in display technologies, for example. All these mathematical models to describe different microstructural phe­ nomena share common features like (i) strong nonlinearities, and (ii) non­ convex side constraints (i.e., I m I = 1, almost everywhere in w C JRd, for the order parameter m : w -+ JRd). One key issue in numerical modeling of such problems is to make sure that the non-convex constraint is fulfilled for computed solutions. We present and analyze different solution strategies to deal with the variational problem of stationary micromagnetism, which builds part I of the book: direct minimization, convexification, and relaxation using Young measure-valued solutions. In particular, we address the following points: • Direct minimization: A spatial triangulation 'generates an artificial exchange energy contribution' in the discretized minimizing problem which may pollute physically relevant exchange energy contributions; its minimizers exhibit multiple scales (with branching structures near the boundary of the ferromagnet) and are difficult to be computed efficiently. We exploit this observation to construct an adaptive scheme which better resolves multiple scale structures (cubic ferromagnets).

• Direct Minimization
• Micromagnetism
• Nematic Liquid Crystals
• Numerical Nonstationary
• Numerical Stationary
• Relaxed Micromagnetism
• finite element method
• magnetic material
• magnetism
• model

Dr. Andreas Prohl, Universität Kiel

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