1,293 research outputs found
Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations
Domain decomposition based time integrators allow the usage of parallel and
distributed hardware, making them well-suited for the temporal discretization
of parabolic systems, in general, and degenerate parabolic problems, in
particular. The latter is due to the degenerate equations' finite speed of
propagation. In this study, a rigours convergence analysis is given for such
integrators without assuming any restrictive regularity on the solutions or the
domains. The analysis is conducted by first deriving a new variational
framework for the domain decomposition, which is applicable to the two standard
degenerate examples. That is, the -Laplace and the porous medium type vector
fields. Secondly, the decomposed vector fields are restricted to the underlying
pivot space and the time integration of the parabolic problem can then be
interpreted as an operators splitting applied to a dissipative evolution
equation. The convergence results then follow by employing elements of the
approximation theory for nonlinear semigroups
Phase field modeling and computer implementation: A review
This paper presents an overview of the theories and computer implementation
aspects of phase field models (PFM) of fracture. The advantage of PFM over
discontinuous approaches to fracture is that PFM can elegantly simulate
complicated fracture processes including fracture initiation, propagation,
coalescence, and branching by using only a scalar field, the phase field. In
addition, fracture is a natural outcome of the simulation and obtained through
the solution of an additional differential equation related to the phase field.
No extra fracture criteria are needed and an explicit representation of a crack
surface as well as complex track crack procedures are avoided in PFM for
fracture, which in turn dramatically facilitates the implementation. The PFM is
thermodynamically consistent and can be easily extended to multi-physics
problem by 'changing' the energy functional accordingly. Besides an overview of
different PFMs, we also present comparative numerical benchmark examples to
show the capability of PFMs
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Cracking and damage from crystallization in pores: Coupled chemo-hydro-mechanics and phase-field modeling
Cracking and damage from crystallization of minerals in pores center on a wide range of problems, from weathering and deterioration of structures to storage of CO2 via in situ carbonation. Here we develop a theoretical and computational framework for modeling these crystallization-induced deformation and fracture in fluid-infiltrated porous materials. Conservation laws are formulated for coupled chemo-hydro-mechanical processes in a multiphase material composed of the solid matrix, liquid solution, gas, and crystals. We then derive an expression for the effective stress tensor that is energy-conjugate to the strain rate of a porous material containing crystals growing in pores. This form of effective stress incorporates the excess pore pressure exerted by crystal growth – the crystallization pressure – which has been recognized as the direct cause of deformation and fracture during crystallization in pores. Continuum thermodynamics is further exploited to formalize a constitutive framework for porous media subject to crystal growth. The chemo-hydro-mechanical model is then coupled with a phase-field approach to fracture which enables simulation of complex fractures without explicitly tracking their geometry. For robust and efficient solution of the initial–boundary value problem at hand, we utilize a combination of finite element and finite volume methods and devise a block-partitioned preconditioning strategy. Through numerical examples we demonstrate the capability of the proposed modeling frameworkfor simulating complex interactions among unsaturated flow, crystallization kinetics, and cracking in the solid matrix
High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems
We design and compute first-order implicit-in-time variational schemes with
high-order spatial discretization for initial value gradient flows in
generalized optimal transport metric spaces. We first review some examples of
gradient flows in generalized optimal transport spaces from the Onsager
principle. We then use a one-step time relaxation optimization problem for
time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes.
Their minimizing systems satisfy implicit-in-time schemes for initial value
gradient flows with first-order time accuracy. We adopt the first-order
optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite
element methods in spatial discretization to compute the one-step optimization
problem. This allows us to derive the implicit-in-time update of initial value
gradient flows iteratively. We remark that the iteration in ALG2 has a
simple-to-implement point-wise update based on optimal transport and Onsager's
activation functions. The proposed method is unconditionally stable for convex
cases. Numerical examples are presented to demonstrate the effectiveness of the
methods in two-dimensional PDEs, including Wasserstein gradient flows,
Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species
reversible reaction-diffusion systems
Convergence of iterative methods based on Neumann series for composite materials: theory and practice
Iterative Fast Fourier Transform methods are useful for calculating the
fields in composite materials and their macroscopic response. By iterating back
and forth until convergence, the differential constraints are satisfied in
Fourier space, and the constitutive law in real space. The methods correspond
to series expansions of appropriate operators and to series expansions for the
effective tensor as a function of the component moduli. It is shown that the
singularity structure of this function can shed much light on the convergence
properties of the iterative Fast Fourier Transform methods. We look at a model
example of a square array of conducting square inclusions for which there is an
exact formula for the effective conductivity (Obnosov). Theoretically some of
the methods converge when the inclusions have zero or even negative
conductivity. However, the numerics do not always confirm this extended range
of convergence and show that accuracy is lost after relatively few iterations.
There is little point in iterating beyond this. Accuracy improves when the grid
size is reduced, showing that the discrepancy is linked to the discretization.
Finally, it is shown that none of the three iterative schemes investigated
over-performs the others for all possible microstructures and all contrasts.Comment: 41 pages, 14 figures, 1 tabl
A review of nonlinear FFT-based computational homogenization methods
Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform
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