Numerical and experimental analysis of diffusioninduced aging in engineering solid mixture components
Applied Mechanics, Statics and Dynamics
Final Report Abstract
Within the funding period of the project a simulation tool for the numerical treatment of diffusion controlled phase decomposition and coarsening problems in multi-component systems has been developed. The FEA tool is based on the concept of isogeometric analysis (IGA) and includes refinement schemes in space and time for higher-order continuous phase-field computations in two and three dimensions. Since in the context of IGA non uniform rational B-splines (NURBS) are used as ansatzfunction a subdivision method had to be developed for the spatial refinement. To maintain the controllability of the continuity of the basis functions at the boundary of the computational domain within the subdivision framework, the scheme has been extended as part of this project. Depending on the number of repeated knots and hence, the reduced continuity, a recursive definition of adapted subdivision matrices has been developed. The proposed approach maintains the partition of unity on a global level for the refined basis without changing the predefined continuity. The extension to the multivariate case follows in a straightforward manner by taking the tensor product structure of the NURBS basis functions into account. In order to deal with the peculiarity of Galerkin type transport analysis to show artificial oscillations at the boundary a newly methodology for the diffusion equation, using a C n -continuous outflow condition at the Neumann boundaries has been introduced. This technique stabilizes the Galerkin approach for arbitrary geometries, maintaining the high efficiency of the global discretization approach for the multi-field higher-order phasefield problem at hand. After the investigation of C n -continuous boundary conditions, application to higher-order domain decomposition methods maintaining the required and predefined continuity across the volumetric interface could be introduced as well. This side benefit of the previous development allows for more complex geometries and, more importantly, in the long term the application of higher-order phase-field models to advanced multi-grid methods. After the numerical extension of our FEA code, multi-field problems have been addressed. This refers to multi-component materials as well as to different loadings. Specifically we studied thermal diffusion, i.e., a phase coarsening driven by a non-uniform temperature field. For the coupling of temperature and diffusion additional fields of unknowns needed to be incorporated into the finite-element code. In a first approach, a thermo-mechanical model for a mixture of two very similar polymeric components has been employed. The difference of both polymeric components lays in their thermal diffusity, specific heat capacity and thermal conductivity; density and thermal expansion coefficient are similar. Concluding computational studies of thermomigration events within this polymeric mixture and of various benchmark problems corroborated finally the robustness and stability of the simulation tool.
Publications
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Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int. J. Numer. Meth. Engng., 99:906–924, 2014
C. Hesch and K. Weinberg
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A high-order finite-deformation phasefield approach to fracture. C. Continuum Mech. Thermodyn., 2015
C. Hesch and K. Weinberg
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Innovative numerical approaches for multi-field and multi-scale problems. Springer-Verlag, 2016
K. Weinberg, S. Schuß, and D. Anders
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Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Meth. Appl. Mech. Eng., 303:185–207, 2016
C. Hesch, S. Schuß, M. Dittmann, M. Franke, and K. Weinberg
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Modeling and numerical simulation of crack growth and damage with a phase field approach. GAMM-Mitteilungen, 39:55–77, 2016
K. Weinberg, T. Dally, S. Schuß, M. Werner, and C. Bilgen
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Thermomigration in Sn-Pb solder bumps: Modelling and simulation. Proc. Appl. Math. Mech., 16:483–484, 2016
S. Schuß, C. Hesch and K. Weinberg
