Cut Finite Elements for Convection in Fractured Domains

We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain which is a union of manifolds of different dimensions such that a $d$ dimensional component always resides on the boundary of a $d+1$ dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem can be formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is based on using a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples

Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. We explore the capabilities of a recovery technique based on an enhanced MLS fitting, which directly provides continuous interpolated fields, to obtain estimates of the error in energy norm as an alternative to the superconvergent patch recovery (SPR). Boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results show the high accuracy of the proposed error estimator

A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

AbstractThis paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented

Strain injection techniques in dynamic fracture modeling

A computationally affordable modeling of dynamic fracture phenomena is performed in this study by using strain injection techniques and Finite Elements with Embedded strong discontinuities (E-FEM). In the present research, classical strain localization and strong discontinuity approaches are considered by injecting discontinuous strain and displacement modes in the finite element formulation without an increase of the total number of degrees of freedom. Following the Continuum Strong Discontinuity Approach (CSDA), stress–strain constitutive laws can be employed in the context of fracture phenomena and, therefore, the methodology remains applicable to a wide number of continuum mechanics models. The position and orientation of the displacement discontinuity is obtained through the solution of a crack propagation problem, i.e. the crack path field, based on the distribution of localized strains. The combination of the above mentioned approaches is envisaged to avoid stress-locking and directional mesh bias phenomena. Dynamic simulations are performed increasing the loading rate up to the appearance of crack branching, and the variation in terms of failure modes is investigated as well as the influence of the strain injection together with the crack path field algorithm. Objectivity of the presented methodology with respect to the spatial and temporal discretization is analyzed in terms of the dissipated energy during the fracture process. The dissipation at the onset of branching is studied for different loading rate conditions and is linked to the experimental maximum velocity observed before branching takes place.Fil: Lloberas Valls, Oriol. Universidad Politecnica de Catalunya; España. Centre Internacional de Metodes Numerics en Enginyeria; EspañaFil: Huespe, Alfredo Edmundo. Centre Internacional de Metodes Numerics en Enginyeria; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones en Métodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Métodos Computacionales; ArgentinaFil: Oliver, J.. Centre Internacional de Metodes Numerics en Enginyeria; España. Universidad Politecnica de Catalunya; EspañaFil: Dias, I.F.. Laboratório Nacional de Engenharia Civil; Portuga

A phase-field model for fractures in incompressible solids

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented

Formulation and optimization of the energy-based blended quasicontinuum method

We formulate an energy-based atomistic-to-continuum coupling method based on blending the quasicontinuum method for the simulation of crystal defects. We utilize theoretical results from Van Koten and Luskin [32] and Ortner and Van Koten [24] to derive optimal choices of approximation parameters (blending function and finite element grid) for microcrack and di-vacancy test problems and confirm our analytical predictions in numerical tests

An embedded formulation with conforming

Use of strong discontinuities with satisfaction of compatibilit

Numerical Computations with H(div)-Finite Elements for the Brinkman Problem

The H(div)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in previous work of the authors. Furthermore, the results are extended to cover a non-constant permeability. A hybridization technique for the problem is presented, complete with a convergence analysis and numerical verification. Finally, the numerical convergence studies are complemented with numerical examples of applications to domain decomposition and adaptive mesh refinement.Comment: Minor clarifications, added references. Reordering of some figures. To appear in Computational Geosciences, final article available at http://www.springerlink.co