Meshless methods: theory and application in 3D fracture modelling with level sets

Accurate analysis of fracture is of vital importance yet methods for effetive 3D calculations are currently unsatisfactory. In this thesis, novel numerical techniques are developed which solve many of these problems. This thesis consists two major parts: firstly an investigation into the theory of meshless methods and secondly an innovative numerical framework for 3D fracture modelling using the element-free Galerkin method and the level set method. The former contributes to some fundamental issues related to accuracy and error control in meshless methods needing to be addressed for fracture modelling developed later namely, the modified weak form for imposition of essential boundary conditions, the use of orthogonal basis functions to obtain shape functions and error control in adaptive analysis. In the latter part, a simple and efficient numerical framework is developed to overcome the difficulties in current 3D fracture modelling. Modelling cracks in 3D remains a challenging topic in computational solid mechanics since the geometry of the crack surfaces can be difficult to describe unlike the case in 2D where cracks can be represented as combinations of lines or curves. Secondly, crack evolution requires numerical methods that can accommodate the moving geometry and a geometry description that maintains accuracy in successive computational steps. To overcome these problems, the level set method, a powerful numerical method for describing and tracking arbitrary motion of interfaces, is used to describe and capture the crack geometry and forms a local curvilinear coordinate system around the crack front. The geometry information is used in the stress analysis taken by the element-Free Galerkin method as well as in the computation of fracture parameters needed for crack propagation. Examples are tested and studied throughout the thesis addressing each of the above described issues

A CNN-based surrogate model of isogeometric analysis in nonlocal flexoelectric problems

We proposed a convolutional neural network (CNN)-based surrogate model to predict the nonlocal response for flexoelectric structures with complex topologies. The input, i.e. the binary images, for the CNN is obtained by converting geometries into pixels, while the output comes from simulations of an isogeometric (IGA) flexoelectric model, which in turn exploits the higher-order continuity of the underlying non-uniform rational B-splines (NURBS) basis functions to fast computing of flexoelectric parameters, e.g., electric gradient, mechanical displacement, strain, and strain gradient. To generate the dataset of porous flexoelectric cantilevers, we developed a NURBS trimming technique based on the IGA model. As for CNN construction, the key factors were optimized based on the IGA dataset, including activation functions, dropout layers, and optimizers. Then the cross-validation was conducted to test the CNN’s generalization ability. Last but not least, the potential of the CNN performance has been explored under different model output sizes and the corresponding possible optimal model layout is proposed. The results can be instructive for studies on deep learning of other nonlocal mech-physical simulations