Topology optimization for cracks

In a broad scope, the project addresses to the problem of structure optimization and identification of defects with emphasize on cracks. The problem consists of proper modeling, theoretical analysis, and construction of efficient computational tools. Our concrete aim is to get a consistent mathematical description of singular geometric structures like cracks with respect to their topological properties.This task is motivated by bifurcation phenomena appearing in a wide range of real world applications. The crucial point of all kind of structure design concerns topology changes of variable geometric objects. In the abstract sense, the generic topology change is produced by varying the degree of connectedness which is responsible for the number of connected components of a structure, or, respectively, voids inside a domain. A rough structure of the desired object is determined with a reasonable topology optimization procedure. This is a challenging mathematical problem. For refining the obtained structure, the design problem proceeds within the classic shape optimization context. The specialty of the topology optimization problem is focused on singular geometric objects. In the following we refer to topology changes represented by splitting or merging of cracks, and the kink phenomenon. Our aims are to obtain the following results: 1. To derive adequate kinematic description of singular geometric objects which represents the topology changes of bifurcation, branching, and alike. 2. To obtain topological characteristics for sensitivity analysis of the energy-type objective functionals dependent on singular domains. 3. To get the shape and topology optimization formulation for the problems describing bifurcation and identification of defects like cracks. In order to attain these results we propose to use the velocity method relaxed on nonsmooth velocities and the implicit surface description, as well as the Lagrange approach for a generalized sensitivity analysis. Our theoretical investigations will be supported by developing the corresponding numerical algorithms and codes for computations on irregular domains.

In a broad scope, the project addressed to the problem of structure design aiming at fracture resistance in the context of topology optimization. The specialty of the optimization problems was focused on singular geometric objects with emphasize on cracks. This specific task was motivated by cracking phenomena appearing in a wide range of real world applications. The conducted research involved proper modelling, theoretical analysis, and construction of computational tools for efficient solution of the underlying nonstandard problems. With the help of non-smooth optimization and topology sensitivity methods, the necessary mathematical description of singular geometric structures like cracks with respect to their topological properties was attained. In particular, the topology changes due to kink phenomenon, which of primary importance for fracture, were investigated. The theoretical results were supported by the corresponding numerical algorithms and codes suitable for computations on irregular domains. The results obtained under the project advance our understanding important for fracture mechanics, destructive testing, identification of defects, and the related engineering applications in geo-, bio-, and material sciences.