A note on stress-driven anisotropic diffusion and its role in active deformable media

We propose a new model to describe diffusion processes within active deformable media. Our general theoretical framework is based on physical and mathematical considerations, and it suggests to use diffusion tensors directly coupled to mechanical stress. A proof-of-concept experiment and the proposed generalised reaction-diffusion-mechanics model reveal that initially isotropic and homogeneous diffusion tensors turn into inhomogeneous and anisotropic quantities due to the intrinsic structure of the nonlinear coupling. We study the physical properties leading to these effects, and investigate mathematical conditions for its occurrence. Together, the experiment, the model, and the numerical results obtained using a mixed-primal finite element method, clearly support relevant consequences of stress-assisted diffusion into anisotropy patterns, drifting, and conduction velocity of the resulting excitation waves. Our findings also indicate the applicability of this novel approach in the description of mechano-electrical feedback in actively deforming bio-materials such as the heart

A three-dimensional multiscale model of intergranular hydrogen-assisted cracking

We present a three-dimensional model of intergranular hydrogen-embrittlement (HE) that accounts for: (i) the degradation of grain-boundary strength that arises from hydrogen coverage; (ii) grain-boundary diffusion of hydrogen; and (iii) a continuum model of plastic deformation that explicitly resolves the three-dimensional polycrystalline structure of the material. The polycrystalline structure of the specimen along the crack propagation path is resolved explicitly by the computational mesh. The texture of the polycrystal is assumed to be random and the grains are elastically anisotropic and deform plastically by crystallographic slip. We use the impurity-dependent cohesive model in order to account for the embrittling of grain boundaries due to hydrogen coverage. We have carried out three-dimensional finite-element calculations of crack-growth initiation and propagation in AISI 4340 steel double-cantilever specimens in contact with an aggressive environment and compared the predicted initiation times and crack-growth curves with the experimental data. The calculated crack-growth curves exhibit a number of qualitative features that are in keeping with observation, including: an incubation time followed by a well-defined crack-growth initiation transition for sufficiently large loading; the existence of a threshold intensity factor K_(Iscc) below which there is no crack propagation; a subsequent steeply rising part of the curve known as stage I; a plateau, or stage II, characterized by a load-insensitive crack-growth rate; and a limiting stress-intensity factor K_(Ic), or toughness, at which pure mechanical failure occurs. The calculated dependence of the crack-growth initiation time on applied stress-intensity factor exhibits power-law behavior and the corresponding characteristic exponents are in the ball-park of experimental observation. The stage-II calculated crack-growth rates are in good overall agreement with experimental measurements

Achieving the Way for Automated Segmentation of Nuclei in Cancer Tissue Images through Morphology-Based Approach: a Quantitative Evaluation

In this paper we address the problem of nuclear segmentation in cancer tissue images, that is critical for specific protein activity quantification and for cancer diagnosis and therapy. We present a fully automated morphology-based technique able to perform accurate nuclear segmentations in images with heterogeneous staining and multiple tissue layers and we compare it with an alternate semi-automated method based on a well established segmentation approach, namely active contours. We discuss active contours’ limitations in the segmentation of immunohistochemical images and we demonstrate and motivate through extensive experiments the better accuracy of our fully automated approach compared to various active contours implementations

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density

Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations

The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems

Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based "auxiliary regions". We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Further, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including, filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.Comment: 29 pages, 14 figures, 2 table

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods