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

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).publishedVersio

Numerical modelling of convection-driven cooling, deformation and fracturing of thermo-poroelastic media

Convection-driven cooling in porous media influences thermo-poro-mechanical stresses, thereby causing deformation. These processes are strongly influenced by the presence of fractures, which dominate flow and heat transfer. At the same time, the fractures deform and propagate in response to changes in the stress state. Mathematically, the model governing the physics is tightly coupled and must account for the strong discontinuities introduced by the fractures. Over the last decade, and motivated by a number of porous media applications, research into such coupled models has advanced modelling of processes in porous media substantially. Building on this effort, this work presents a novel model that couples flow, heat transfer, deformation, and propagation of fractures with flow, heat transfer, and thermo-poroelasticity in the matrix. The model is based on explicit representation of fractures in the porous medium, and discretised using multi-point finite volume methods. Frictional contact and non-penetration conditions for the fractures are handled through active set methods, while a propagation criterion based on stress intensity factors governs fracture extension. Considering both forced and natural convection processes, the numerical results show the intricate nature of thermo-poromechanical fracture deformation and propagation

Numerical Modelling of Convection‑Driven Cooling, Deformation and Fracturing of Thermo‑Poroelastic Media

Convection-driven cooling in porous media influences thermo-poro-mechanical stresses, thereby causing deformation. These processes are strongly influenced by the presence of fractures, which dominate flow and heat transfer. At the same time, the fractures deform and propagate in response to changes in the stress state. Mathematically, the model governing the physics is tightly coupled and must account for the strong discontinuities introduced by the fractures. Over the last decade, and motivated by a number of porous media applications, research into such coupled models has advanced modelling of processes in porous media substantially. Building on this effort, this work presents a novel model that couples fracture flow and heat transfer and deformation and propagation of fractures with flow, heat transfer and thermo-poroelasticity in the matrix. The model is based on explicit representation of fractures in the porous medium and discretised using multi-point finite volume methods. Frictional contact and non-penetration conditions for the fractures are handled through active set methods, while a propagation criterion based on stress intensity factors governs fracture extension. Considering both forced and natural convection processes, numerical results show the intricate nature of thermo-poromechanical fracture deformation and propagation.publishedVersio

A fully coupled numerical model of thermo-hydro-mechanical processes and fracture contact mechanics in porous media

Various phenomena in the subsurface are characterised by the interplay between deforming structures such as fractures and coupled thermal, hydraulic and mechanical processes. Simulation of subsurface dynamics can provide valuable phenomenological understanding, but requires models which faithfully represent the dynamics involved; these models therefore are themselves highly complex. This paper presents a mixed-dimensional thermo-hydro-mechanical model designed to capture the process–structure interplay using a discrete–fracture–matrix framework. It incorporates tightly coupled thermo-hydro-mechanical processes based on balance laws for momentum, mass and energy in subdomains representing the matrix and the lower-dimensional fractures and fracture intersections. The deformation of explicitly represented fractures is modelled by contact mechanics relations and a Coulomb friction law, with a novel formulation consistently integrating fracture dilation in the governing equations. The model is discretised using multi-point finite volume methods for the balance equations and a semismooth Newton scheme for the contact conditions and is implemented in the open-source fracture simulation toolbox PorePy. Finally, simulation studies demonstrate the model’s convergence, investigate process–structure coupling effects, explore different fracture dilation models and show an application of the model to stimulation and long-term cooling of a three-dimensional geothermal reservoir.publishedVersio

A convergent entropy diminishing finite volume scheme for a cross-diffusion system

We study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy dissipation estimate. Numerical results illustrate the behavior of our scheme