Evolving discontinuities and cohesive fracture

Multi-scale methods provide a new paradigm in many branches of sciences, including applied mechanics. However, at lower scales continuum mechanics can become less applicable, and more phenomena enter which involve discon- tinuities. The two main approaches to the modelling of discontinuities are briefly reviewed, followed by an in-depth discussion of cohesive models for fracture. In this discussion emphasis is put on a novel approach to incorporate triaxi- ality into cohesive-zone models, which enables for instance the modelling of crazing in polymers, or of splitting cracks in shear-critical concrete beams. This is followed by a discussion on the representation of cohesive crack models in a continuum format, where phase-field models seem promising

A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media

In this paper we consider a multiscale phase-field model for capillarity-driven flows in porous media. The presented model constitutes a reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model, valid in situations where interest is restricted to dynamical and equilibrium behavior in an aggregated sense, rather than a precise description of microscale flow phenomena. The model is based on averaging of the equation of motion, thereby yielding a significant reduction in the complexity of the underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its macroscopic dynamical and equilibrium properties. Numerical results are presented for the representative 2-dimensional capillary-rise problem pertaining to two closely spaced vertical plates with both identical and disparate wetting properties. Comparison with analytical solutions for these test cases corroborates the accuracy of the presented multiscale model. In addition, we present results for a capillary-rise problem with a non-trivial geometry corresponding to a porous medium

Bayesian uncertainty quantification for transient heat conduction problems with temperature-dependent conductivity

We present an inverse analysis for the estimation and uncertainty quantification of a temperature dependent thermal conductivity. For that, we consider the one-dimensional problem of a slab made of steel with a temperature-dependent thermal conductivity, to which a constant heat flux is applied at both edges. After defining the mathematical formulation, we can numerically solve the direct problem and derive transient temperature values across the slab. A third-degree polynomial is used to model the temperature-dependence of the conductivity, so the estimation of this physical property is obtained through the estimation of the polynomial's coefficients. We use a Bayesian framework to model the uncertainties involved, consisting of a likelihood and a prior. The likelihood models the temperature measurements and their uncertainties, and is chosen to be a normal distribution with mean at the solution of the direct problem and a standard deviation characterizing the errors. The prior models our pre-existing knowledge about the coefficients that describe the thermal conductivity. To gain insights about the impact of using different priors, we investigate both uniform and normal distributions. The prior and likelihood are combined into a posterior, which describes the coefficients for a given set of temperature measurements. This posterior cannot be computed analytically, and therefore we solve the inverse problem with the Metropolis–Hastings algorithm, a Markov chain Monte Carlo method. Results showed that our approach could deal with all uncertainties involved, and not only provided an estimation of the thermal conductivity curve but also delivered uncertainty quantification using credible intervals. It can furthermore be modified to estimate thermal conductivity values instead of coefficients, allowing for a more physical formulation of the prior

Skeleton-stabilized ImmersoGeometric Analysis for incompressible viscous flow problems

A Skeleton-stabilized ImmersoGeometric Analysis technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed formulation fits within the framework of the finite cell method, where essential boundary conditions are imposed weakly using a Nitsche-type method. The key idea of the proposed formulation is to stabilize the jumps of high-order derivatives of variables over the skeleton of the background mesh. The formulation allows the use of identical finite-dimensional spaces for the approximation of the pressure and velocity fields in immersed domains. The stability issues observed for inf-sup stable discretizations of immersed incompressible flow problems are avoided with this formulation. For B-spline basis functions of degree $k$ with highest regularity, only the derivative of order $k$ has to be controlled, which requires specification of only a single stabilization parameter for the pressure field. The Stokes and Navier-Stokes equations are studied numerically in two and three dimensions using various immersed test cases. Oscillation-free solutions and high-order optimal convergence rates can be obtained. The formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut, and hence it does not require the existence of interior cells. In terms of the sparsity pattern, the algebraic system has a considerably smaller stencil than counterpart approaches based on Lagrange basis functions. This important property makes the proposed skeleton-stabilized technique computationally practical. To demonstrate the stability and robustness of the method, we perform a simulation of fluid flow through a porous medium, of which the geometry is directly extracted from 3D $\mu{CT}$ scan data