arXiv subjects

Jakub Wiktor Both

Publications and source records attributed to Jakub Wiktor Both.

26 records · Page 2Linked to original sources

arXiv2020

An accelerated staggered scheme for phase-field modeling of brittle fracture

There is currently an increasing interest in developing efficient solvers for phase-field modeling of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver, hence, the implementation can be done with minor modifications to already available software. Moreover, the cost of combining the two acceleration schemes is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

math.NA↗

arXiv2019

The fixed-stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.

math.NA↗

arXiv2019

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-linear models. In the second part, the gradient flow structures are utilized for constructing efficient discrete approximation schemes for thermo-poro-visco-elasticity -- in particular robust, physical splitting schemes. Applying alternating minimization to naturally arising minimization formulations of (semi-)discrete models is proposed. For such, the energy decrease per iteration is quantified by applying abstract convergence theory only utilizing convexity and Lipschitz continuity properties of the problem -- a fairly simple but flexible machinery. By this approach, e.g., the widely used undrained and fixed-stress splits for the linear Biot equations are derived and analyzed. By application of the framework to more advanced models, novel splitting schemes with guaranteed theoretical convergence rates are naturally derived. Moreover, based on the minimization character of the (semi-)discrete equations, relaxation of splitting schemes by line search is proposed; numerical results show a potentially great impact on the acceleration of splitting schemes for both linear and nonlinear problems.

math.NA↗

arXiv2019

On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces

In this paper, the convergence of alternating minimization is established for non-smooth convex optimization in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth and a non-smooth part is considered with the latter being block-separable, e.g., corresponding to convex constraints or regularization. For the smooth part, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. Linear convergence is established for the first two cases, generalizing and improving previous results for strongly convex problems; sublinear convergence is established for the third case, also improving previous results from the literature. All the convergence results have in common, that opposing to previous corresponding results for the general block coordinate descent, the performance of the alternating minimization is beneficially governed by properties of the single blocks, instead of global properties. Ultimately, not only the better conditioned block determines the performance, as has been similarly observed in the literature. But also the worse conditioned problem enhances the performance additionally, resulting in potentially significantly improved convergence rates. Furthermore, by solely using the convexity and smoothness properties of the problem, the results immediately apply in general Banach spaces.

math.OC↗

arXiv2019

Global existence of a weak solution to unsaturated poroelasticity

In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biot's quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards' equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.

math.AP↗

arXiv2018

On the optimization of the fixed-stress splitting for Biot's equations

In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization/tuning parameter. In this work, we propose a new approach to optimize this parameter. We show theoretically that it depends also on the fluid flow properties and not only on the mechanics properties and the coupling coefficient. The type of analysis presented in this paper is not restricted to a particular spatial discretization. We only require it to be inf-sup stable. The convergence proof applies also to low-compressible or incompressible fluids and low-permeable porous media. Illustrative numerical examples, including random initial data, random boundary conditions or random source terms and a well-known benchmark problem, i.e. Mandel's problem are performed. The results are in good agreement with the theoretical findings. Furthermore, we show numerically that there is a connection between the inf-sup stability of discretizations and the performance of the fixed-stress splitting scheme.

math.NA↗

arXiv2018

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, employing the equivalent pore pressure. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators, which is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration provides a robust linearization scheme, meeting the above criteria.

math.NA↗

arXiv2018

Numerical investigation on the fixed-stress splitting scheme for Biot's equations: Optimality of the tuning parameter

We study the numerical solution of the quasi-static linear Biot's equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.

math.NA↗