A Scaled Boundary Approach to Forward and Inverse Problems with Applications in Computational Fracture Mechanics, Damage Localization and Topology Optimization

Abstract

The demand for sustainable design in, e.g., the aerospace, automotive and construction industries has lead to the development of lighter, stronger and more resilient structures, spawning the need to guard against failure processes by leveraging robust, economical and high-fidelity numerical simulations. Since its inception, the finite element method (FEM) has been advanced to handle a multitude of structural analysis problems ranging from linear to nonlinear, static to dynamic, fracture and contact problems among others. Within the context of fracture mechanics, it has been demonstrated that modeling of damage-related phenomena such as crack initiation, crack propagation and delamination can successfully be accomplished by means of the FEM. Nonetheless, various undesirable characteristics persist, which render this method computationally prohibitive for more involved analyses. As a result, alternative methods have been pursued; the scaled boundary finite element method (SBFEM), is a little explored, yet highly capable alternative within the domain of linear elastic fracture mechanics (LEFM). Hence, the objective of this thesis is to accelerate computationally intensive numerical problems harnessing the merits and further extending the capabilities of the SBFEM to a wide class of forward and inverse problems, specifically with applications in computational fracture mechanics, damage localization and topology optimization. The increasing importance of sustainability implies the prudent use of existing resources, such that efficient computation schemes are sought. This thesis proposes several novel schemes to accomplish this goal. The Hamiltonian Schur decomposition is first adopted, to reverse the computational toll incurred during an SBFEM analysis, due to the linearization of the underlying quadratic eigen-problem. Further, an efficient recovery based error estimator is proposed, which additionally permits calculating the generalized stress intensity factors (gSIFs) at increased accuracy, using fewer degrees of freedom (DOF). The use of linear quadtree (QT) meshes, pioneered by previous authors, to overcome SBFEM's unique meshing requirements, can lead to reduced accuracy in calculated gSIFs for crack propagation problems. A method of internally elevating the approximation space of a crack tip element is proposed, which is shown to greatly improve the accuracy with which gSIFs are calculated on highly coarse QT meshes. These approaches are exploited to develop the multiscale scaled boundary finite element method (MSBFEM), which harnesses the SBFEM to incorporate fracture on the fine scale and the enhanced multiscale finite element method (EMsFEM) to construct a coarse scale representation, where the governing equations are solved at a reduced computational cost. The newly developed MSBFEM is then extended to a highly efficient crack propagation scheme, which resolves only regions directly surrounding the crack tip, and incorporates the remaining domain via coarse scale macro-elements. In doing so, the amount of DOFs present during analysis are drastically reduced, while the crack path is still accurately captured. These novel insights in accelerating the forward problem are then applied to inverse analyses. Due to its domain specific advantage, SBFEM is subsequently applied to damage localization schemes. Taking advantage of the parallel nature with which heuristic algorithms approach damage localization, combined with precomputation of the undamaged domain by SBFEM and updating the effects of varying crack candidates by reanalysis techniques, a highly efficient and effective scheme is devised to accelerate damage localization analyses to near real-time levels. Topology optimization (TO), which similarly to damage localization, is often marred by the repeated solution of an expensive forward problem, stands to benefit from efficient solvers. Automated adaptive analysis-ready meshes are achieved by harnessing image compression techniques. The proposed drop-in replacement for the forward solver, reduces the amount of DOF during present during analysis by over an order of magnitude. This approach is successfully extended to 3D problems.