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• Brun, Mats Kirkesæther; Ahmed, Elyes; Nordbotten, Jan Martin & Stenseth, Nils Christian (2022). Modeling the Process of Speciation Using a Multiscale Framework Including A Posteriori Error Estimates. SIAM Journal on Applied Mathematics. ISSN 0036-1399. 82(2), p. 450–475. doi: 10.1137/21M1405228. Full text in Research Archive Show summary

  This paper concerns the modeling and numerical simulation of the process of speciation. In particular, given conditions for which one or more speciation events within an ecosystem occur, our aim is to develop the necessary modeling and simulation tools. Care is also taken to establish a solid mathematical foundation on which our modeling framework is built. This is the subject of the first half of the paper. The second half is devoted to developing a multiscale framework for eco-evolutionary modeling, where the relevant scales are that of species and individual/population, respectively. The species level model we employ can be considered as an extension of the classical Lotka--Volterra model, where in addition to the species abundance, the model also governs the evolution of the species mean traits and species trait covariances and in this sense generalizes the purely ecological Lotka--Volterra model to an eco-evolutionary model. Although the model thus allows for evolving species, it does not (by construction) allow for the branching of species, i.e., speciation events. The reason for this is related to that of separate scales; the unit of species is too coarse to capture the fine-scale dynamics of a speciation event. Instead, the branching species should be regarded as a population of individuals moving along a selection of trait axes (i.e., trait-space). For this, we employ a trait-specific population density model governing the dynamics of the population density as a function of evolutionary traits. At this scale there is no a priori definition of species, but both species and speciation may be defined a posteriori as, e.g., local maxima and saddle points of the population density, respectively. Hence, a system of interacting species can be described at the species level, while for branching species a population level description is necessary. Our multiscale framework thus consists of coupling the species and population level models where speciation events are detected in advance and then resolved at the population scale until the branching is complete. Moreover, since the population level model is formulated as a PDE, we first establish the well-posedness in the time-discrete setting and then derive the a posteriori error estimates, which provides a fully computable upper bound on an energy-type error, including also for the case of general smooth distributions (which will be useful for the detection of speciation events). Several numerical tests validate our framework in practice.

• Bjørnestad, Maria; Kalisch, Henrik; Abid, Malek; Kharif, Christian & Brun, Mats (2021). Wave Breaking in Undular Bores with Shear Flows. Water Waves. ISSN 2523-367X. 3, p. 473–490. doi: 10.1007/s42286-020-00046-6. Full text in Research Archive Show summary

  It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

• Brun, Mats Kirkesæther; Wick, Thomas; Berre, Inga; Nordbotten, Jan Martin & Radu, Florin Adrian (2020). An iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 361:112752, p. 1–22. doi: 10.1016/j.cma.2019.112752. Full text in Research Archive Show summary

  This paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi-static brittle fracture propagation models, where the fracture evolution is tracked by a phase field variable. The model we consider is a two-field variational inequality system, with the phase field function and the elastic displacements of the solid material as independent variables. Using a penalization strategy, this variational inequality system is transformed into a variational equality system, which is the formulation we take as the starting point for our algorithmic developments. The proposed scheme involves a partitioning of this model into two subproblems; phase field and mechanics, with added stabilization terms to both subproblems for improved efficiency and robustness. We analyze the convergence of the proposed scheme using a fixed point argument, and find that under a natural condition, the elastic mechanical energy remains bounded, and, if the diffusive zone around crack surfaces is sufficiently thick, monotonic convergence is achieved. Finally, the proposed scheme is validated numerically with several bench-mark problems.

• Brun, Mats Kirkesæther; Ahmed, Elyes; Berre, Inga; Nordbotten, Jan Martin & Radu, Florin Adrian (2020). Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport. Computers and Mathematics with Applications. ISSN 0898-1221. 80(8), p. 1964–1984. doi: 10.1016/j.camwa.2020.08.022. Full text in Research Archive Show summary

  This paper concerns monolithic and splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE’s, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare six approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from the mechanics, suitable for varying coupling strength between the three physical processes. The approaches have in common a simultaneous application of the so-called -scheme, which works both to stabilize iterative splitting as well as to linearize nonlinear problems, and can be seen as a generalization of the Undrained and Fixed-Stress Split algorithms. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes.

• Brun, Mats Kirkesæther; Ahmed, Elyes; Nordbotten, Jan Martin & Radu, Florin Adrian (2019). Well-posedness of the fully coupled quasi-static thermo-poroelastic equations with nonlinear convective transport. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 471(1-2), p. 239–266. doi: 10.1016/j.jmaa.2018.10.074. Full text in Research Archive Show summary

  This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence proof for this algorithm is then inferred by a contraction of successive difference functions of the iterates using suitable norms.

• Brun, Mats Kirkesæther & Kalisch, Henrik (2018). Convective wave breaking in the KdV equation. Analysis and Mathematical Physics. ISSN 1664-2368. 8(1), p. 57–75. doi: 10.1007/s13324-017-0163-y. Full text in Research Archive
• Brun, Mats Kirkesæther; Berre, Inga; Nordbotten, Jan Martin & Radu, Florin Adrian (2018). Upscaling of the coupling of hydromechanical and thermal processes in a quasi-static poroelastic medium. Transport in Porous Media. ISSN 0169-3913. 124(1), p. 137–158. doi: 10.1007/s11242-018-1056-8. Full text in Research Archive Show summary

  We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity.We also establish the symmetry and positivity for the homogenized coefficients.

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• Goksøyr, Anders; Eide, Marta; Brun, Nadja Rebecca; Yadetie, Fekadu; Zhang, Xiaokang & Hylland, Ketil [Show all 14 contributors for this article] (2022). Complex mixture and omics responses in Atlantic cod: a time- and dose-response study. Full text in Research Archive
• Brun, Mats Kirkesæther; Ahmed, Elyes; Radu, Florin Adrian; Berre, Inga & Nordbotten, Jan Martin (2019). Monolithic and Splitting Based Solution Schemes for Nonlinear Quasi-static Thermo-poroelasticity. Full text in Research Archive
• Brun, Mats Kirkesæther; Radu, Florin Adrian; Berre, Inga; Nordbotten, Jan Martin & Ahmed, Elyes (2018). Upscaling of coupled geomechanics, flow, and heat in a poroelastic medium in the quasi-static situation. Full text in Research Archive
• Brun, Mats Kirkesæther; Radu, Florin Adrian; Berre, Inga & Nordbotten, Jan Martin (2019). Upscaling, analysis, and iterative numerical solution schemes for thermo-poroelasticity. Universitetet i Bergen. Full text in Research Archive Show summary

  The main objectives of this research project is to provide part of the mathematical models and simulation technology required to assess large-scale deployment of thermomechanical subsurface energy storage in the context of intermittent renewable energy. Within this overarching framework, two main topics are discussed in this dissertation: Thermo-poroelasticity, i.e., the coupling of geomechanics, flow, and heat within a porous material, and phase field brittle fracture propagation, i.e., the temporal and spatial tracking of brittle fracture evolution by creating a diffusive zone around fracture surfaces through an auxiliary variable known as a phase field. These subjects are highly relevant for e.g., applications involving injection of heated fluids at high pressures into the subsurface, where the increased pressure may cause a fracturing of the rock matrix, in addition to the induced temperature gradient affecting the poroelastic properties of the surrounding medium. The part of this dissertation focused on thermo-poroelasticity can again be separated into three parts; (1) modeling, (2) analysis and (3) numerical implementation. In part (1), formal upscaling techniques (i.e., homogenization) are employed in order to derive the constitutive thermo-poroelastic equations from the known equations governing the physical processes at the pore-scale. Homogenization is a well-known and trusted technique, which is applicable in situations where the physical processes in question can be viewed from several scales, and where there is some uniformity or periodicity on the smaller scale. Within the context of porous media, the two relevant scales are the pore-scale and the macro-scale. Viewed from the pore scale, a porous medium consists of solid grains (obeying the laws of solid mechanics), and a fluid which is saturating the space in between the grains (obeying the laws of fluid mechanics), where the two processes are coupled at the mutual interface (i.e., at the grain surfaces). At the macroscale, however, the porous medium is considered as a homogenous material obeying its own set of physical laws, but which is still somehow implicitly dependent on (or rather, the culmination of) the processes which are taking place at the pore-scale. Assuming some uniformity in the distribution of grains (formally, periodicity), the technique of homogenization allows for deriving the macro-scale model equations from the microscale model equations. Part (2) is concerned with analyzing the thermo-poroelastic system derived in part (1). The existence and uniqueness of a weak solution to this model problem is established in the fully mixed formulation, under some natural assumptions on the regularity of the source and initial data, as well as imposing some constraints on the effective coefficients. Since the upscaled thermo-poroelastic system is a nonlinear one, this analysis is done in two steps: First, a linearized system is analyzed using a standard Galerkin technique together with the weak compactness properties of the relevant function spaces. Finally, the previously analyzed linearized system is used to design an iterative scheme in order to approximate the original nonlinear system. By employing a contraction argument, the convergence of this iterative scheme is proved, which implies the existence and uniqueness of a weak solution to the full nonlinear problem. Part (3) covers the numerical analysis and implementation of the thermo-poroelastic system derived in part (1). Here, six different iterative algorithms are proposed, all based on the linearization technique employed in part (2) when analyzing the full nonlinear system, as well as the stabilized splitting scheme known informally as the ‘𝐿-scheme’. These six algorithms involve different combinations of coupling / decoupling of the three subproblems involved (flow, mechanics and heat), i.e., at each iteration either a linearized system is solved monolithically, two subproblems are solved together decoupled from the third, or all three subproblems are decoupled. As such, these six algorithms exhaust all possibilities of coupling / decoupling of the three subproblems. The convergence of all six algorithms are proved using a contraction argument, similar to the one employed in part (2), except that only the fully discrete formulation is considered. Several numerical tests validate the robustness and efficiency of the algorithms. Furthermore, the performance of all six algorithms are compared with respect to a wide range of different physical regimes (i.e., with respect to various coupling strengths between the three subproblems). Finally, the last part of this dissertation concerns brittle fracture propagation in a quasi-static elastic medium, where the fracture evolution is tracked by a phase field variable. In particular, the numerical approximation of such models. Phase field brittle fracture problems are notoriously difficult to solve, and currently no universally accepted method exists. Hence, this work involves designing a novel iterative algorithm for brittle fracture phase field models, analyzing its convergence, and testing it in detail with several numerical benchmark problems. The proposed algorithm is based on a linearization as well as stabilization of the model, where the two subproblems (phase field and mechanics) are solved separately at each iteration, while sharing updated solution information. Under the natural conditions that the mechanical elastic energy remains bounded, and that the diffusive zone around crack surfaces must be sufficiently thick, monotonic convergence of the proposed scheme is proved. These properties are also confirmed by the extensive numerical tests.

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