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Phase field modeling of fracture with isogeometric analysis and machine learning methods

  • This thesis presents the advances and applications of phase field modeling in fracture analysis. In this approach, the sharp crack surface topology in a solid is approximated by a diffusive crack zone governed by a scalar auxiliary variable. The uniqueness of phase field modeling is that the crack paths are automatically determined as part of the solution and no interface tracking is required. TheThis thesis presents the advances and applications of phase field modeling in fracture analysis. In this approach, the sharp crack surface topology in a solid is approximated by a diffusive crack zone governed by a scalar auxiliary variable. The uniqueness of phase field modeling is that the crack paths are automatically determined as part of the solution and no interface tracking is required. The damage parameter varies continuously over the domain. But this flexibility comes with associated difficulties: (1) a very fine spatial discretization is required to represent sharp local gradients correctly; (2) fine discretization results in high computational cost; (3) computation of higher-order derivatives for improved convergence rates and (4) curse of dimensionality in conventional numerical integration techniques. As a consequence, the practical applicability of phase field models is severely limited. The research presented in this thesis addresses the difficulties of the conventional numerical integration techniques for phase field modeling in quasi-static brittle fracture analysis. The first method relies on polynomial splines over hierarchical T-meshes (PHT-splines) in the framework of isogeometric analysis (IGA). An adaptive h-refinement scheme is developed based on the variational energy formulation of phase field modeling. The fourth-order phase field model provides increased regularity in the exact solution of the phase field equation and improved convergence rates for numerical solutions on a coarser discretization, compared to the second-order model. However, second-order derivatives of the phase field are required in the fourth-order model. Hence, at least a minimum of C1 continuous basis functions are essential, which is achieved using hierarchical cubic B-splines in IGA. PHT-splines enable the refinement to remain local at singularities and high gradients, consequently reducing the computational cost greatly. Unfortunately, when modeling complex geometries, multiple parameter spaces (patches) are joined together to describe the physical domain and there is typically a loss of continuity at the patch boundaries. This decrease of smoothness is dictated by the geometry description, where C0 parameterizations are normally used to deal with kinks and corners in the domain. Hence, the application of the fourth-order model is severely restricted. To overcome the high computational cost for the second-order model, we develop a dual-mesh adaptive h-refinement approach. This approach uses a coarser discretization for the elastic field and a finer discretization for the phase field. Independent refinement strategies have been used for each field. The next contribution is based on physics informed deep neural networks. The network is trained based on the minimization of the variational energy of the system described by general non-linear partial differential equations while respecting any given law of physics, hence the name physics informed neural network (PINN). The developed approach needs only a set of points to define the geometry, contrary to the conventional mesh-based discretization techniques. The concept of `transfer learning' is integrated with the developed PINN approach to improve the computational efficiency of the network at each displacement step. This approach allows a numerically stable crack growth even with larger displacement steps. An adaptive h-refinement scheme based on the generation of more quadrature points in the damage zone is developed in this framework. For all the developed methods, displacement-controlled loading is considered. The accuracy and the efficiency of both methods are studied numerically showing that the developed methods are powerful and computationally efficient tools for accurately predicting fractures.show moreshow less

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Metadaten
Document Type:Doctoral Thesis
Author: Somdatta GoswamiORCiD
DOI (Cite-Link):https://doi.org/10.25643/bauhaus-universitaet.4384Cite-Link
URN (Cite-Link):https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20210304-43841Cite-Link
Series (Serial Number):ISM-Bericht // Institut für Strukturmechanik, Bauhaus-Universität Weimar (2021,1)
Referee:Prof. Stephane BordasGND, Prof. Magd Abel Wahab
Advisor:Prof. Dr.-Ing. Timon RabczukORCiDGND
Language:English
Date of Publication (online):2021/03/02
Date of first Publication:2021/03/02
Date of final exam:2020/12/17
Release Date:2021/03/04
Publishing Institution:Bauhaus-Universität Weimar
Granting Institution:Bauhaus-Universität Weimar, Fakultät Bauingenieurwesen
Institutes and partner institutions:Fakultät Bauingenieurwesen / Institut für Strukturmechanik
Pagenumber:168
Tag:Isogeometric Analysis; Physics informed neural network; brittle fracture; deep neural network; phase field
GND Keyword:Phasenfeldmodell; Neuronales Netz; Sprödbruch
Dewey Decimal Classification:500 Naturwissenschaften und Mathematik
600 Technik, Medizin, angewandte Wissenschaften
BKL-Classification:30 Naturwissenschaften allgemein
31 Mathematik
52 Maschinenbau, Energietechnik, Fertigungstechnik
Licence (German):License Logo Creative Commons 4.0 - Namensnennung-Nicht kommerziell-Keine Bearbeitung (CC BY-NC-ND 4.0)