35,593 research outputs found
A phase-field model for cohesive fracture
In this paper, a phase-field model for cohesive fracture is developed. After casting the cohesive zone approach in an energetic framework, which is suitable for incorporation in phase-field approaches, the phase-field approach to brittle fracture is recapitulated. The approximation to the Dirac function is discussed with particular emphasis on the Dirichlet boundary conditions that arise in the phase-field approximation. The accuracy of the discretisation of the phase field, including the sensitivity to the parameter that balances the field and the boundary contributions, is assessed at the hand of a simple example. The relation to gradient-enhanced damage models is highlighted, and some comments on the similarities and the differences between phase-field approaches to fracture and gradient-damage models are made. A phase-field representation for cohesive fracture is elaborated, starting from the aforementioned energetic framework. The strong as well as the weak formats are presented, the latter being the starting point for the ensuing finite element discretisation, which involves three fields: the displacement field, an auxiliary field that represents the jump in the displacement across the crack, and the phase field. Compared to phase-field approaches for brittle fracture, the modelling of the jump of the displacement across the crack is a complication, and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack. The sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated, as well as how the results depend on the orders of the discretisation of the three fields. Finally, examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure, the latter example being somewhat limited because only straight crack propagation is considered
A Relaxation result for energies defined on pairs set-function and applications
We consider, in an open subset Ω
of RN, energies depending on the perimeter of a subset
E С Ω
(or some equivalent surface integral) and on a function u which is defined only on
E. We compute the lower semicontinuous envelope of such energies. This relaxation has
to take into account the fact that in the limit, the “holes”
Ω \ E may collapse into a
discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss
some situations where such energies appear, and give, as an application, a new proof of
convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah
functional
Efficient solution of 3D electromagnetic eddy-current problems within the finite volume framework of OpenFOAM
Eddy-current problems occur in a wide range of industrial and metallurgical
applications where conducting material is processed inductively. Motivated by
realising coupled multi-physics simulations, we present a new method for the
solution of such problems in the finite volume framework of foam-extend, an
extended version of the very popular OpenFOAM software. The numerical procedure
involves a semi-coupled multi-mesh approach to solve Maxwell's equations for
non-magnetic materials by means of the Coulomb gauged magnetic vector potential
and the electric scalar potential. The concept is further extended on the basis
of the impressed and reduced magnetic vector potential and its usage in
accordance with Biot-Savart's law to achieve a very efficient overall modelling
even for complex three-dimensional geometries. Moreover, we present a special
discretisation scheme to account for possible discontinuities in the electrical
conductivity. To complement our numerical method, an extensive validation is
completing the paper, which provides insight into the behaviour and the
potential of our approach.Comment: 47 pages, improved figures, updated references, fixed typos, reverse
phase shift, consistent use of inner produc
Phase-field approximation for a class of cohesive fracture energies with an activation threshold
We study the -limit of Ambrosio-Tortorelli-type functionals
, whose dependence on the symmetrised gradient is
different in and in , for a
-elliptic symmetric operator , in terms of the
prefactor depending on the phase-field variable . This is intermediate
between an approximation for the Griffith brittle fracture energy and the one
for a cohesive energy by Focardi and Iurlano. In particular we prove that
functions with bounded -variation are
On the plane wave Riemann Problem in Fluid Dynamics
The paper contains a stability analysis of the plane-wave Riemann problem for
the two-dimensional hyperbolic conservation laws for an ideal compressible gas.
It is proved that the contact discontinuity in the plane-wave Riemann problem
is unstable under perturbations. The implications for Godunovs method are
discussed and it is shown that numerical post shock noise can set of a contact
instability. A relation to carbuncle instabilities is established.Comment: 27 pages, 1 figur
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