688 research outputs found
Convergence of a cell-centered finite volume discretization for linear elasticity
We show convergence of a cell-centered finite volume discretization for
linear elasticity. The discretization, termed the MPSA method, was recently
proposed in the context of geological applications, where cell-centered
variables are often preferred. Our analysis utilizes a hybrid variational
formulation, which has previously been used to analyze finite volume
discretizations for the scalar diffusion equation. The current analysis
deviates significantly from previous in three respects. First, additional
stabilization leads to a more complex saddle-point problem. Secondly, a
discrete Korn's inequality has to be established for the global discretization.
Finally, robustness with respect to the Poisson ratio is analyzed. The
stability and convergence results presented herein provide the first rigorous
justification of the applicability of cell-centered finite volume methods to
problems in linear elasticity
Stable cell-centered finite volume discretization for Biot equations
In this paper we discuss a new discretization for the Biot equations. The
discretization treats the coupled system of deformation and flow directly, as
opposed to combining discretizations for the two separate sub-problems. The
coupled discretization has the following key properties, the combination of
which is novel: 1) The variables for the pressure and displacement are
co-located, and are as sparse as possible (e.g. one displacement vector and one
scalar pressure per cell center). 2) With locally computable restrictions on
grid types, the discretization is stable with respect to the limits of
incompressible fluid and small time-steps. 3) No artificial stabilization term
has been introduced. Furthermore, due to the finite volume structure embedded
in the discretization, explicit local expressions for both momentum-balancing
forces as well as mass-conservative fluid fluxes are available.
We prove stability of the proposed method with respect to all relevant
limits. Together with consistency, this proves convergence of the method.
Finally, we give numerical examples verifying both the analysis and convergence
of the method
Robust Discretization of Flow in Fractured Porous Media
Flow in fractured porous media represents a challenge for discretization
methods due to the disparate scales and complex geometry. Herein we propose a
new discretization, based on the mixed finite element method and mortar
methods. Our formulation is novel in that it employs the normal fluxes as the
mortar variable within the mixed finite element framework, resulting in a
formulation that couples the flow in the fractures with the surrounding domain
with a strong notion of mass conservation. The proposed discretization handles
complex, non-matching grids, and allows for fracture intersections and
termination in a natural way, as well as spatially varying apertures. The
discretization is applicable to both two and three spatial dimensions. A priori
analysis shows the method to be optimally convergent with respect to the chosen
mixed finite element spaces, which is sustained by numerical examples
Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
In this paper we develop adaptive iterative coupling schemes for the Biot
system modeling coupled poromechanics problems. We particularly consider the
space-time formulation of the fixed-stress iterative scheme, in which we first
solve the problem of flow over the whole space-time interval, then exploiting
the space-time information for solving the mechanics. Two common
discretizations of this algorithm are then introduced based on two coupled
mixed finite element methods in-space and the backward Euler scheme in-time.
Therefrom, adaptive fixed-stress algorithms are build on conforming
reconstructions of the pressure and displacement together with equilibrated
flux and stresses reconstructions. These ingredients are used to derive a
posteriori error estimates for the fixed-stress algorithms, distinguishing the
different error components, namely the spatial discretization, the temporal
discretization, and the fixed-stress iteration components. Precisely, at the
iteration of the adaptive algorithm, we prove that our estimate gives
a guaranteed and fully computable upper bound on the energy-type error
measuring the difference between the exact and approximate pressure and
displacement. These error components are efficiently used to design adaptive
asynchronous time-stepping and adaptive stopping criteria for the fixed-stress
algorithms. Numerical experiments illustrate the efficiency of our estimates
and the performance of the adaptive iterative coupling algorithms
Functional Analysis and Exterior Calculus on Mixed-Dimensional Geometries
We are interested in differential forms on mixed-dimensional geometries, in
the sense of a domain containing sets of -dimensional manifolds, structured
hierarchically so that each -dimensional manifold is contained in the
boundary of one or more dimensional manifolds.
On any given -dimensional manifold, we then consider differential
operators tangent to the manifold as well as discrete differential operators
(jumps) normal to the manifold. The combined action of these operators leads to
the notion of a semi-discrete differential operator coupling manifolds of
different dimensions. We refer to the resulting systems of equations as
mixed-dimensional, which have become a popular modeling technique for physical
applications including fractured and composite materials.
We establish analytical tools in the mixed-dimensional setting, including
suitable inner products, differential and codifferential operators, Poincar\'e
lemma, and Poincar\'e--Friedrichs inequality. The manuscript is concluded by
defining the mixed-dimensional minimization problem corresponding to the
Hodge-Laplacian, and we show that this minimization problem is well-posed
High-accuracy phase-field models for brittle fracture based on a new family of degradation functions
Phase-field approaches to fracture based on energy minimization principles
have been rapidly gaining popularity in recent years, and are particularly
well-suited for simulating crack initiation and growth in complex fracture
networks. In the phase-field framework, the surface energy associated with
crack formation is calculated by evaluating a functional defined in terms of a
scalar order parameter and its gradients, which in turn describe the fractures
in a diffuse sense following a prescribed regularization length scale. Imposing
stationarity of the total energy leads to a coupled system of partial
differential equations, one enforcing stress equilibrium and another governing
phase-field evolution. The two equations are coupled through an energy
degradation function that models the loss of stiffness in the bulk material as
it undergoes damage. In the present work, we introduce a new parametric family
of degradation functions aimed at increasing the accuracy of phase-field models
in predicting critical loads associated with crack nucleation as well as the
propagation of existing fractures. An additional goal is the preservation of
linear elastic response in the bulk material prior to fracture. Through the
analysis of several numerical examples, we demonstrate the superiority of the
proposed family of functions to the classical quadratic degradation function
that is used most often in the literature.Comment: 33 pages, 30 figure
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