A coupled phase-field model for ductile fracture in crystal plasticity

The objective of this work is to present a simplified, nonetheless representative first stage of a phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional well- stablished elasto-plastic crystal plasticity model and a well-known phase-field model (PFM) modified to predict ductile fracture. A two-dymensional initial boundary-value problem of ductile fracture is introduced considering a single crystal Nickel-base superalloy material. the model is implemented into the finite element context subjected to a one-dimensional tension test (displacement-controlled). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A polyconvex strain-energy split for a high-order phase-field approach to fracture

This contribution focuses on a novel phase-field model for a high-order phase-field approach to brittle fracture in the range finite deformation. In particular, two different challenges are tackled in this study: First, we want to establish a polyconvex free energy density to ensure the existence of a minimizer for the coupled problem, second, we have to deal with a fourth-order Cahn-Hilliard type equation for the approximation of the phase-field. Phase-field methods employ a variational framework for brittle fracture and have proven to predict complex fracture patterns in two and three dimensional examples. Basis of the model are the conjugate stresses of the three strain measures deformation gradient (line map), its cofactor (area map) and its determinant (volume map). The introduction of the tensor cross product simplifies the presentation of the first Piola-Kirchhoff stress tensor and its derivatives in elegant manner. The proposed Cahn-Hilliard type equation requires global -continuity. Therefore, we apply an isogeometric framework using NURBS basis functions. Moreover, a general hierarchical refinement scheme based on subdivision projection is used here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization. We finally demonstrate the accuracy and the robustness with a series of benchmark problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of hydraulic fracture of porous materials using the phase-field modeling approach

In this contribution, the numerical simulation of hydraulic fracture of fluid-saturated porous materials is carried out on a continuum-mechanical scale using the theory of porous media (TPM), extended by a phase-field modeling (PFM) approach. Following this, behaviors such as crack nucleation and propagation, solid matrix deformation and interstitial-fluid flow change from Darcy to Stokes-like flow in the cracked region can be realized. Moreover, permanent changes of the local physics due to occurrence of the crack, such as of the volume fractions and the permeability, are taken into consideration. The mathematical modeling of this problem yields a strongly coupled system of differential algebraic equations (DAE). Thus, special descretization schemes for a stable and efficient solution are needed. To reveal the ability of the proposed model to simulate the important features of hydraulic cracking, a two-dimensional example using the finite element method is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The phase-field model with an auto-calibrated degradation function based on general softening laws for cohesive fracture

Phase-field models have become popular to simulate cohesive failure problems because of their capability of predicting crack initiation and propagation without additional criteria. In this paper, a new phase-field damage model coupled with general softening laws for cohesive fracture is proposed based on the unified phase-field theory. The commonly used quadratic geometric function in the classical phase-field model is implemented in the proposed model. The modified degradation function related to the failure strength and length scale is used to obtain the length scale insensitive model. Based on the analytical solution of a 1-D case, general softening laws in cohesive zone models can be considered. Parameters in the degradation function can be calibrated according to different softening curves and material properties. Numerical examples show that the results obtained by the proposed model have a good agreement with experimental results and the length scale has a negligible influence on the load-displacement curves in most cases, which cannot be observed in classical phase-field model.     

Ductile failure with gradient plasticity coupled to the phase-field fracture at finite strains

Most metals fail in a ductile fashion, i.e, fracture is preceded by significant plastic deformation. The modeling of failure in ductile metals must account for complex phenomena at micro-scale, such as nucleation, growth and coalescence of micro-voids. In this work, we start with von-Mises plasticity model without considering void generation. The modeling of macroscopic cracks can be achieved in a convenient way by the continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities and can account for complex crack patterns. The key aspect of this work is the extension of the energetic and the stress-based phase field driving force function in brittle fracture to account for a coupled elasto-plastic response in line with our recent work [3]. We develop a new theoretical and computational framework for the phase field modeling of ductile fracture in elastic-plastic solids. To account for large strains, the constitutive model is constructed in the logarithmic strain space, which simplify the model equations and results in a formulation similar to small strains. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by representative simulations of ductile failure mechanisms in metals. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-local temperature-dependent phase-field models for non-isothermal phase transitions

We propose a model for non-isothermal phase transitions withnon-con-served order parameter driven by a spatially non-localfree energy with respect to both the temperature and the orderparameter. The resulting system of equations is shown to bethermodynamically consistent and to admit a strong solution.     

Microscopical investigation of wave propagation phenomena in residual saturated porous media

Wave propagation is used in many fields for measurement and characterization. Corresponding multiphase models usually use a continuous approach. Nevertheless, systems like wetted rocks may be saturated residually in certain situations. In such cases, one fluid is distributed as clusters, each different in size and shape. One single, continuous phase cannot account for a variety of fluid clusters, either disconnected from each other or connected only about thin liquid films. Therefore, we present a model that considers a heterogeneous distribution of disconnected fluid clusters in the form of harmonic oscillators. These oscillators are described and distinguished by their mass, damping and eigenfrequency. Hence, the model allows to characterize different clusters and includes an additional damping mechanism due to oscillations of the fluid clusters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximal attractors for the phase-field equations of penrose-fife type

In this paper, the dynamics for the phase-field equations of Penrose-Fife type arising from the study of phase transitions is investigated. One of important features of this problem is that the metric space H we work with is incomplete.     

On a nonlocal phase-field system

We study the existence, uniqueness and continuous dependence on initial data of the solution to a nonlocal phase-field system on a bounded domain. The system is a gradient flow for a free energy functional with nonlocal interaction. Also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space.     

Block thresholding for density estimation: local and global adaptivity

We consider wavelet block thresholding method for density estimation. A block-thresholded density estimator is proposed and is shown to achieve optimal global rate of convergence over Besov spaces and simultaneously attain the optimal adaptive pointwise convergence rate as well. These results are obtained in part through the determination of an optimal block length.     

A posteriori error estimation and adaptivity for degenerate parabolic problems

Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

     

Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S_{ɛ, σ}(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors for S_{ɛ, σ}(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from tends to 0 in an explicitly controlled way.     

Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.     

On numerical schemes for phase-field models for electrowetting with electrolyte solutions

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase-field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parallel anisotropic mesh adaptivity with dynamic load balancing for cardiac electrophysiology

Simulations in cardiac electrophysiology generally use very fine meshes and small time steps to resolve highly localized wavefronts. This expense motivates the use of mesh adaptivity, which has been demonstrated to reduce the overall computational load. However, even with mesh adaptivity performing such simulations on a single processor is infeasible. Therefore, the adaptivity algorithm must be parallelised. Rather than modifying the sequential adaptive algorithm, the parallel mesh adaptivity method introduced in this paper focuses on dynamic load balancing in response to the local refinement and coarsening of the mesh. In essence, the mesh partition boundary is perturbed away from mesh regions of high relative error, while also balancing the computational load across processes. The parallel scaling of the method when applied to physiologically realistic heart meshes is shown to be good as long as there are enough mesh nodes to distribute over the available parallel processes. It is shown that the new method is dominated by the cost of the sequential adaptive mesh procedure and that the parallel overhead of inter-process data migration represents only a small fraction of the overall cost.     

Analysis of weak solutions for the phase-field model for lithium-ion batteries

We study a phase-field model for lithium-ion batteries of olivine LiFePO₄. During electrochemical cycling the fundamental behavior of the crystal is the diffusion of Li which controls the movement of the phase boundary without changing the olivine topology. This model with diffusive phase interfaces consists of two nonlinear parabolic equations of second order. We first prove the existence of global solutions to an initial-boundary value problem of this model. Numerical experiments of the model are then performed to simulate the evolution of lithium concentration and of phase interfaces.     

Simulating cardiac electrophysiology using anisotropic mesh adaptivity

The simulation of cardiac electrophysiology requires small time steps and a fine mesh in order to resolve very sharp, but highly localized, wavefronts. The use of very high resolution meshes containing large numbers of nodes results in a high computational cost, both in terms of CPU hours and memory footprint. In this paper an anisotropic mesh adaptivity technique is implemented in the Chaste physiological simulation library in order to reduce the mesh resolution away from the depolarization front. Adapting the mesh results in a reduction in the number of degrees of freedom of the system to be solved by an order of magnitude during propagation and 2–3 orders of magnitude in the subsequent plateau phase. As a result, a computational speedup by a factor of between 5 and 12 has been obtained with no loss of accuracy, both in a slab-like geometry and for a realistic heart mesh with a spatial resolution of 0.125 mm.     

Global and exponential attractors for a Caginalp type phase-field problem

We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.