On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behavior of the solutions of the autonomous third order nonlinear differential equation $f'''+(m+2)ff''-(2m+1)f'^2=0$ on $\mathbb{R}^+$ with the boundary conditions $f(0)=-\gamma$, $f'(\infty)=0$ and $f''(0)=-1$. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing-up coordinates to obtain a plane dynamical system.Comment: v2: new page-settin

Canards in stiction: on solutions of a friction oscillator by regularization

We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Therefore we introduce the concept of stiction solutions: these are the Carath\'eodory solutions that are physically relevant, i.e. the ones that follow the stiction law. However, we find that some of the stiction solutions are forward non-unique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further the understanding of the non-unique dynamics, we introduce a regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. We identify a repelling slow manifold that separates the forward slipping to forward sticking solutions, leading to a high sensitivity to the initial conditions. On this slow manifold we find canard trajectories, that have the physical interpretation of delaying the slip onset. We show with numerics that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure

An adaptive graph for volumetric mesh visualization

AbstractThis work presents an adaptive strategy in order to visualize volumetric data generated from numerical simulations of partial differential equations. The mesh is represented by a graph data structure. Moreover, the Autonomous Leaves Graph is extended to the three-dimensional case. This scheme intends to achieve better transversal cost than a treelike (e.g., bintree, quadtree and octree) space arrangement approach. Furthermore, this strategy intends to reduce the computational cost of constructing the discretization and the visualization of data. The total-ordering of the mesh volumes used in the discretization and the visualization processes is by the 3D Modified Hilbert space-filling Curve. To evaluate the performance, the strategy is applied on a Heat Conduction simulation problem using finite difference discretizations and the experimental results are discussed. Comparisons are made between numerical results obtained when using the Hilbert Curve and its modified version. In addition, experiments are shown when visualization is made from inside and outside the volume. The results expose the efficiency of using this strategy

Deep Reinforcement Learning for Swarm Systems

Recently, deep reinforcement learning (RL) methods have been applied successfully to multi-agent scenarios. Typically, these methods rely on a concatenation of agent states to represent the information content required for decentralized decision making. However, concatenation scales poorly to swarm systems with a large number of homogeneous agents as it does not exploit the fundamental properties inherent to these systems: (i) the agents in the swarm are interchangeable and (ii) the exact number of agents in the swarm is irrelevant. Therefore, we propose a new state representation for deep multi-agent RL based on mean embeddings of distributions. We treat the agents as samples of a distribution and use the empirical mean embedding as input for a decentralized policy. We define different feature spaces of the mean embedding using histograms, radial basis functions and a neural network learned end-to-end. We evaluate the representation on two well known problems from the swarm literature (rendezvous and pursuit evasion), in a globally and locally observable setup. For the local setup we furthermore introduce simple communication protocols. Of all approaches, the mean embedding representation using neural network features enables the richest information exchange between neighboring agents facilitating the development of more complex collective strategies.Comment: 31 pages, 12 figures, version 3 (published in JMLR Volume 20