Mikael Mortensen

Miroslav Kuchta, Magne Nordaas, Joris C. G. Verschaeve et al.

We study preconditioners for a model problem describing the coupling of two elliptic subproblems posed over domains with different topological dimension by a parameter dependent constraint. A pair of parameter robust and efficient preconditioners is proposed and analyzed. Robustness and efficiency of the preconditioners is demonstrated by numerical experiments.

Miroslav Kuchta, Kent-Andre Mardal, Mikael Mortensen

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work we consider a simplified model problem of a 3d-1d coupling and the main objective is to construct algorithms that may utilize stan- dard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well-defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the sys- tem. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.

Colin Vignon, Jean Rabault, Joel Vasanth et al.

Rayleigh-Bénard convection (RBC) is a recurrent phenomenon in several industrial and geoscience flows and a well-studied system from a fundamental fluid-mechanics viewpoint. However, controlling RBC, for example by modulating the spatial distribution of the bottom-plate heating in the canonical RBC configuration, remains a challenging topic for classical control-theory methods. In the present work, we apply deep reinforcement learning (DRL) for controlling RBC. We show that effective RBC control can be obtained by leveraging invariant multi-agent reinforcement learning (MARL), which takes advantage of the locality and translational invariance inherent to RBC flows inside wide channels. The MARL framework applied to RBC allows for an increase in the number of control segments without encountering the curse of dimensionality that would result from a naive increase in the DRL action-size dimension. This is made possible by the MARL ability for re-using the knowledge generated in different parts of the RBC domain. We show in a case study that MARL DRL is able to discover an advanced control strategy that destabilizes the spontaneous RBC double-cell pattern, changes the topology of RBC by coalescing adjacent convection cells, and actively controls the resulting coalesced cell to bring it to a new stable configuration. This modified flow configuration results in reduced convective heat transfer, which is beneficial in several industrial processes. Therefore, our work both shows the potential of MARL DRL for controlling large RBC systems, as well as demonstrates the possibility for DRL to discover strategies that move the RBC configuration between different topological configurations, yielding desirable heat-transfer characteristics. These results are useful for both gaining further understanding of the intrinsic properties of RBC, as well as for developing industrial applications.

Miroslav Kuchta, Kent-Andre Mardal, Mikael Mortensen

The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a non-singular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, while a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. Based on this observation a modification to the conjugate gradient method is proposed that achieves optimal error convergence of the computed solution.

Joel Vasanth, Jean Rabault, Francisco Alcántara-Ávila et al.

Deep reinforcement learning (DRL) has found application in numerous use-cases pertaining to flow control. Multi-agent RL (MARL), a variant of DRL, has shown to be more effective than single-agent RL in controlling flows exhibiting locality and translational invariance. We present, for the first time, an implementation of MARL-based control of three-dimensional Rayleigh-Bénard convection (RBC). Control is executed by modifying the temperature distribution along the bottom wall divided into multiple control segments, each of which acts as an independent agent. Two regimes of RBC are considered at Rayleigh numbers $\mathrm{Ra}=500$ and $750$. Evaluation of the learned control policy reveals a reduction in convection intensity by $23.5\%$ and $8.7\%$ at $\mathrm{Ra}=500$ and $750$, respectively. The MARL controller converts irregularly shaped convective patterns to regular straight rolls with lower convection that resemble flow in a relatively more stable regime. We draw comparisons with proportional control at both $\mathrm{Ra}$ and show that MARL is able to outperform the proportional controller. The learned control strategy is complex, featuring different non-linear segment-wise actuator delays and actuation magnitudes. We also perform successful evaluations on a larger domain than used for training, demonstrating that the invariant property of MARL allows direct transfer of the learnt policy.

Kristian Holme, Jean Rabault, Ricardo Vinuesa et al.

Rotating detonation engines (RDEs) are a promising propulsion concept that may offer higher thermodynamic efficiency and specific impulse than conventional systems, but nonlinear phenomena, including transitions to oscillatory or chaotic propagation modes, can hinder practical operation. Deep Reinforcement Learning (DRL) has emerged as a promising method for controlling complex nonlinear dynamics such as those observed in RDEs. However, the multi-timescale nature of the RDE system makes direct application of DRL challenging. We address this challenge by reformulating the DRL problem in a moving reference frame that follows the detonation-wave pattern, making the wave structure appear quasi-steady to the agent. This reformulation enables scale separation between fast detonation propagation and slower operating-mode dynamics. We train DRL controllers to modulate spatially segmented injection pressure in a one-dimensional reduced-order RDE model and induce rapid transitions between different mode-locked states. Across a range of actuation periods, initial states, and target modes, controllers trained in the moving frame learn more reliably than those trained in a stationary frame and remain effective over a broader range of actuation periods. These results suggest that symmetry-aware moving reference frame formulations may be useful for related multiscale flow-control problems and that scale separation should be exploited whenever possible to enable DRL control of multi-timescale systems.

Hans Harder, Jean Rabault, Ricardo Vinuesa et al.

We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance.