Title: Efficient Computational Methods in Coupled Thermomechanical Problems
Description: Dynamic loading of polycrystalline metallic materials can result in brittle or ductile fracture depending on the loading rates, geometry, and material type. At high strain rates, mechanical energy due to plastic deformation may lead to significant temperature rise and shear localization due to thermal softening. These shear bands reduce the stress-bearing capacity of the material and act as a precursor to ductile fracture (e.g. cracks that develop rapidly on top of a shear band). Reliable models are needed to predict the response of metals subject to dynamic loads. Understanding the heat transfer physics in thermo-mechanical problems when cracks are developed is of great importance. In particular, capturing the interplay between heat conduction and crack propagation is still an open research field. To accurately capture the heat transfer physics across crack surfaces, damage models degrading thermal-conductivity are necessary. In this thesis, a novel set of isotropic thermal-conductivity degradation functions is derived based on a micro-mechanics void extension model of Laplace's equation.

The key idea is to employ an analytical homogenization process to find the effective thermal-conductivity of an equivalent sphere with an expanding spherical void. The closed-form solution is obtained by minimization of the flux differences at the outer surfaces of the two problems, which can be achieved using the analytical solution of Laplace's equations, so-called spherical-harmonics. Additionally, a new anisotropic approach is proposed in which thermal-conductivity, which depends on the phase-field gradient, is degraded solely across the crack. We show that this approach improves the near-field approximation of temperature and heat flux compared with isotropic degradation when taking the discontinuous crack solutions as reference. To demonstrate the viability of the proposed (isotropic and anisotropic) approaches, a unified model, which accounts for the simultaneous formation of shear bands and cracks, is used as a numerical tool. In this model, the phase-field method is used to model crack initiation and propagation and is coupled to a temperature-dependent visco-plastic model that captures shear bands.

Benchmark problems are presented to show the necessity of the anisotropic thermal-conductivity approach using physics-based degradation functions in dynamic fracture problems. On the other hand, the computational burden in dynamic fracture problems with localized solution features is highly demanding. Iterative methods used for their analysis often require special treatment to be more efficient. Specifically, the nonlinear thermomechanical problems we study in this thesis lead to strain localizations, such as shear bands and/or cracks, and iterative solvers may have difficult time converging. To address this issue, we develop a novel updating domain decomposition preconditioner for parallel solution of dynamic fracture problems. The domain decomposition method is based on the Additive Schwarz Method (ASM). The key idea is to decompose the computational domain into two subdomains, a localized subdomain that includes all localized features of the solution and a healthy subdomain for the remaining part of the domain.

In this way, one can apply different solvers in each subdomain, i.e. focus more effort in the localized subdomain. In this work, an LU solver is applied in both subdomains, however, while the localized subdomain is solved exactly at every nonlinear iteration, the healthy subdomain LU operator is reused and only selectively updated. Hence, significant CPU time savings associated with the setup of the preconditioner can be achieved. In particular, we propose a strategy for updating the preconditioner in the healthy subdomain. The strategy is based on an idealized performance-based optimization procedure that takes into account machine on-the-fly execution time. Three dynamic fracture problems corresponding to different failure modes are investigated. Excellent performance of the proposed updating preconditioner is reported in serial and parallel simulations.