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Investigations on enhanced Fischer–Burmeister NCP functions: application to a rate-dependent model for ferroelectrics

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Abstract

This contribution deals with investigations on enhanced Fischer–Burmeister nonlinear complementarity problem (NCP) functions applied to a rate-dependent laminate-based material model for ferroelectrics. The framework is based on the modelling and parametrisation of the material’s microstructure via laminates together with the respective volume fractions. These volume fractions are treated as internal-state variables and are subject to several inequality constraints which can be treated in terms of Karush–Kuhn–Tucker conditions. The Fischer–Burmeister NCP function provides a sophisticated scheme to incorporate Karush–Kuhn–Tucker-type conditions into calculations of internal-state variables. However, these functions are prone to numerical instabilities in their original form. Therefore, some enhanced formulations of the Fischer–Burmeister ansatz are discussed and compared to each other in this contribution.

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Acknowledgements

The financial support of the German Research Foundation (DFG) of the research group FOR 1509, “Ferroic Functional Materials: Multi-scale Modeling and Experimental Characterization” in projects P6 “Microstructural Interactions and Switching in Ferroelectrics” and P7 “Numerical Relaxation Techniques for the Modeling of Microstructure Evolution in Multifunctional Magnetic Materials” is gratefully acknowledged.

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Authors and Affiliations

  1. Institue of Mechanics, TU Dortmund, Leonhard-Euler-Strasse 5, 44227, Dortmund, Germany

    T. Bartel, R. Schulte & A. Menzel

  2. Division of Solid Mechanics, Lund University, P.O. Box 118, 22100, Lund, Sweden

    A. Menzel

  3. Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg, Lampadiusstr. 4, 09599, Freiberg, Germany

    B. Kiefer

  4. Chair of Material Mechanics, RWTH Aachen, Schinkelstr. 2, 52062, Aachen, Germany

    B. Svendsen

  5. Microstructure Physics and Alloy Design, Max-Planck-Institut fuer Eisenforschung GmbH, Max-Planck Str. 1, 40237, Duesseldorf, Germany

    B. Svendsen

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  1. T. Bartel
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  3. A. Menzel
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  5. B. Svendsen
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Correspondence to T. Bartel.

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Derivation of the smoothed Fischer–Burmeister NCP function

Derivation of the smoothed Fischer–Burmeister NCP function

The roots of the original NCP function need to be solved in an approximative manner for the smoothed Fischer–Burmeister NCP approach. Thus, the perturbation parameter \(\delta \) controls the accuracy and the equality constraint to be solved. The corresponding derivation of the smoothed Fischer–Burmeister function provided that \(r\ge 0\) and \(\varGamma \ge 0\) reads as follows

$$\begin{aligned} r\,\varGamma= & {} \delta ^2 \end{aligned}$$
(50)
$$\begin{aligned} 2\,r\,\varGamma= & {} 2\,\delta ^2 \end{aligned}$$
(51)
$$\begin{aligned} r^2+\varGamma ^2+2\,r\,\varGamma= & {} r^2+\varGamma ^2+2\,\delta ^2 \end{aligned}$$
(52)
$$\begin{aligned} \left[ r+\varGamma \right] ^2= & {} r^2+\varGamma ^2+2\,\delta ^2 \end{aligned}$$
(53)
$$\begin{aligned} r+\varGamma= & {} \sqrt{r^2+\varGamma ^2+2\,\delta ^2} \end{aligned}$$
(54)
$$\begin{aligned} 0= & {} \sqrt{r^2+\varGamma ^2+2\,\delta ^2} - \left[ r+\varGamma \right] \end{aligned}$$
(55)

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Bartel, T., Schulte, R., Menzel, A. et al. Investigations on enhanced Fischer–Burmeister NCP functions: application to a rate-dependent model for ferroelectrics. Arch Appl Mech 89, 995–1010 (2019). https://doi.org/10.1007/s00419-018-1466-7

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