大象传媒

Advisors: Eirik Keilegavlen, Inga Berre and聽Jan Martin Nordbotten

Short description of project:

Fracture-induced failure is an important consideration in engineering聽design, both from a standpoint of structural safety and as an active聽ingredient in realizing the necessary conditions for operational聽efficiency of engineered systems, for instance in the case of subsurface聽applications related to energy production and waste disposal. In recent聽years, phase-field models derived from the variational theory of聽fracture have been shown to possess a lot of potential for modeling聽fracture propagation, particularly in the context of multiphysics聽applications. In the phase-field approach, fractures are not modeled as聽discrete surfaces but rather as diffuse entities, resulting in the聽smoothing out of jump discontinuities associated with sharp cracks聽according to a length scale parameter. Phase-field models for fracture聽have found wide application in many areas of research, and are generally聽seen as useful for providing qualitative insight into complex processes聽related to fracture. On the other hand, the quantitative value of聽results obtained with fracture simulations using phase-field approaches聽has not yet been established due to outstanding questions related to聽interpretation of the phase-field length scale, along with apparent聽inaccuracies observed when using standard phase-field models to predict聽failure loads in benchmark problems with known analytical solutions.

The work done in this dissertation is aimed at improving the聽quantitative accuracy and efficiency of phase-field models for brittle聽fracture under quasi-static loading. On the issue of quantitative聽accuracy, we demonstrate via numerical experiments that a phase-field聽model utilizing the classical quadratic degradation function fails to聽reproduce thecorrect failure load for the propagation of cracks that are modeled as聽existing features in the geometry/discretization, which confirms the聽conclusions of a prior study. In particular we show that for the case of聽stable fracture, such inaccuracy cannot be remedied by increasing or聽decreasing the regularization of cracks. In addition for linear elastic聽materials, simulations using the standard second order phase-field model聽are plagued by premature growth of damage that lead to a departure from聽linearity in the load response curve prior to fracture. Under the聽hypothesis that said errors arise from a mismatch in the magnitude of聽bulk and surface energy increments, we introduce a novel parametric聽family of degradation functions whose shape can be adjusted based on聽actual material parameters as well as the choice of regularization聽parameter. The proposed family of degradation functions is able to聽effectively eliminate premature damage evolution and is also shown to聽reproduce the correct failure loads for differing values of the聽regularization parameter, thus yielding a phase-field model that is聽length scale insensitive.

The above concept of length-scale insensitivity further serves as a聽motivation for developing a new extension of phase-field brittle聽fracture in poroelastic materials. Said formulation is obtained by first聽deriving expressions for damage-dependent poroelastic parameters that聽are consistent with assumptions regarding regularization of elasticity聽coefficients pertaining to the porous medium skeleton. The diffuse聽representation of cracks in the coupled mechanics/fracture equations are聽combined with discrete representations of the same for purpose of flow聽calculations in fractures; the latter are then upscaled into the聽surrounding medium, to allow for use of discretizations in which the聽size of cell edges are orders of magnitude larger than the computed聽fracture apertures.

On the other hand, the computational expense associated with fracture聽phase-field models can be traced to two main challenges. The first has聽to do with the analytical solution of the standard second order model,聽which for fully developed cracks features sharp cusps in the phase-field聽profile. This is very challenging to reproduce with continuous聽discretization schemes, resulting in the need for aggressive mesh聽refinement in order to achieve acceptable accuracy. To alleviate this burden, we propose the combination of聽different numerical formulations for discretizing the nonlinear coupled聽system of partial differential equations. In particular, we investigate聽the prospect of a pairing linear finite elements with cell-centered聽finite volumes, in which the latter is used to discretize the governing聽equation for phase-field evolution in order to capitalize on the fact聽that two-point flux approximations implicitly allow for gradient聽discontinuities within control volumes. We show through numerical聽simulations of well known benchmark problems that accuracy of a pure聽linear finite element discretization can be matched by the proposed聽scheme with the use of relatively coarser meshes, yielding up to 80聽percent reduction in run time for 2D problems.

The second issue contributing to the computational cost of phase-field聽simulations of fracture stems from non-convexity of the regularlized聽energy functional from which the governing equations of the model are聽derived. This precludes the use of classical Newton-type algorithms;聽instead, alternate minimization schemes are often adapted which achieve聽only linear convergence. On the other hand, run times can be聽considerably shortened by maximizing use of available computing power.聽This is becoming more and more relevant given the current manufacturing聽trend in computer processors that is focused on improving performance by聽increasing the number of cores available with each processor. To聽capitalize on this, a new object-oriented framework has been developed聽as part of this thesis with the goal of enabling the efficient聽combination of different numerical methods in a single software聽framework. This allows for tight coupling and the use of monolithic聽solution schemes where applicable, and eliminates overhead associated聽with external coupling of separate codes. The source code is written in聽C++ and makes use of functionality available with C++11 standard.聽Parallel execution on shared memory architecture is achieved by means of聽OpenMP directives, and in addition the framework contains wrappers for聽integrating high performance libraries for dense linear algebra and聽solution of sparse systems. Its effectiveness and versatility is聽demonstrated through the simulation of varied problems in solid聽mechanics, poroelasticity and fracture. These highlight the framework鈥檚聽ability to handle non-standard degrees of freedom, multi-point聽constraints and partitioning of the total simulation domain into聽subdomains governed by different sets of equations.