Entropy Viscosity Method for Lagrangian Hydrodynamics and Central Schemes for Mean Field Games

In this dissertation we consider two major subjects. The primary topic is the Entropy Viscosity method for Lagrangian hydrodynamics, the goal of which is to solve numerically the Euler equations of compressible gas dynamics. The second topic is concerned with applications of second order central differencing schemes to the Mean Field Games equations. The Entropy Viscosity method discretizes all kinematic and thermodynamic variables by high-order finite elements and solves the resulting discrete problem on a computational mesh that moves with the material velocity. The method is based on two major concepts. The first one is producing high order convergence rates for smooth solutions even with active viscosity terms. This is achieved by using high order finite element spaces and, more importantly, entropy based viscosity coefficients that clearly distinguish between smooth and singular regions. The second concept is providing control over oscillations around contact discontinuities as well as oscillations in shock regions. Achieving this requires adding extra viscosity terms in a way that the resulting system is still in agreement with generalized entropy inequalities, the minimum principle on the specific entropy and the general requirements for artificial tensor viscosities like orthogonal transformation invariance, radial symmetry, Galilean invariance, etc. We define a fully-discrete finite element algorithm and present numerical results on model Lagrangian hydro problems. We also discuss possible extensions of the method, e.g. length scale independent viscosity coefficients, incorporating mass diffusion into the mesh motion, and handling of different materials. In addition we present approaches to the different stages of arbitrary Lagrangian-Eulerian (ALE) methods, which can be used to extend the Entropy Viscosity method. That is, we discuss mesh relaxation by harmonic smoothing schemes, advection based solution remap, and multi-material zones treatment. The Mean Field Games (MFG) equations describe situations in which a large number of individual players choose their optimal strategy by considering global (but limited) incentive information that is available to everyone. The resulting system consists of a forward Hamilton-Jacobi equation and a backward convection-diffusion equation. We propose fully discrete explicit second order staggered finite difference schemes for the two equations and combine these schemes into a fixed point iteration algorithm. We discuss the second order accuracy of both schemes, their interaction in time, memory issues resulting from the forward-backward coupling, stopping criteria for the fixed point iteration, and parallel performance of the method

Weak Boundary Conditions for Lagrangian Shock Hydrodynamics: A High-Order Finite Element Implementation on Curved Boundaries

We propose a new Nitsche-type approach for weak enforcement of normal velocity boundary conditions for a Lagrangian discretization of the compressible shock-hydrodynamics equations using high-order finite elements on curved boundaries. Specifically, the variational formulation is appropriately modified to enforce free-slip wall boundary conditions, without perturbing the structure of the function spaces used to represent the solution, with a considerable simplification with respect to traditional approaches. Total energy is conserved and the resulting mass matrices are constant in time. The robustness and accuracy of the proposed method are validated with an extensive set of tests involving nontrivial curved boundaries

Currency risk of public debt in Serbia: Current status and European lessons

Currency risk, and its effect on public debt, is becoming more important in economic analysis, particularly in highly dollarized countries like Serbia. The aim of this paper is to analyse the currency structure of the public debt in Serbia and the associated risks, and to present the recommendations for improving public debt management based on international experiences. This paper analyses the currency structure of public debt of Serbia over the period 2004-2016, the comparative indicators of dollarization in Serbia and selected countries of the region, as well as the currency risk movements. The currency structure of the public debt of Serbia characterizes the predominance of three currencies - dinar, dollar and euro. Serbia is the most euroised country in the region, along with a marked highest share of dollar debt in total public debt. As a result, due to currency mismatch, the public debt of Serbia is exposed to high currency risk. Based on the presented successful examples of public debt management in Hungary and Poland, as countries with a low level of dollarization, the recommendations for improving public debt management system in Serbia are presented in the final section of our paper

High-Order Mesh Morphing for Boundary and Interface Fitting to Implicit Geometries

We propose a method that morphs high-orger meshes such that their boundaries and interfaces coincide/align with implicitly defined geometries. Our focus is particularly on the case when the target surface is prescribed as the zero isocontour of a smooth discrete function. Common examples of this scenario include using level set functions to represent material interfaces in multimaterial configurations, and evolving geometries in shape and topology optimization. The proposed method formulates the mesh optimization problem as a variational minimization of the sum of a chosen mesh-quality metric using the Target-Matrix Optimization Paradigm (TMOP) and a penalty term that weakly forces the selected faces of the mesh to align with the target surface. The distinct features of the method are use of a source mesh to represent the level set function with sufficient accuracy, and adaptive strategies for setting the penalization weight and selecting the faces of the mesh to be fit to the target isocontour of the level set field. We demonstrate that the proposed method is robust for generating boundary- and interface-fitted meshes for curvilinear domains using different element types in 2D and 3D.Comment: 30 pages, 16 figure

hr-adaptivity for nonconforming high-order meshes with the target matrix optimization paradigm

We present an $hr$-adaptivity framework for optimization of high-order meshes. This work extends the $r$-adaptivity method for mesh optimization by Dobrev et al., where we utilized the Target-Matrix Optimization Paradigm (TMOP) to minimize a functional that depends on each element's current and target geometric parameters: element aspect-ratio, size, skew, and orientation. Since fixed mesh topology limits the ability to achieve the target size and aspect-ratio at each position, in this paper we augment the $r$-adaptivity framework with nonconforming adaptive mesh refinement to further reduce the error with respect to the target geometric parameters. The proposed formulation, referred to as $hr$-adaptivity, introduces TMOP-based quality estimators to satisfy the aspect-ratio-target via anisotropic refinements and size-target via isotropic refinements in each element of the mesh. The methodology presented is purely algebraic, extends to both simplices and hexahedra/quadrilaterals of any order, and supports nonconforming isotropic and anisotropic refinements in 2D and 3D. Using a problem with a known exact solution, we demonstrate the effectiveness of $hr$-adaptivity over both $r-$ and $h$-adaptivity in obtaining similar accuracy in the solution with significantly fewer degrees of freedom. We also present several examples that show that $hr$-adaptivity can help satisfy geometric targets even when $r$-adaptivity fails to do so, due to the topology of the initial mesh.Comment: 29 pages, 15 figure

Adaptive Tangential Relaxation and Surface Fitting for High-Order Mesh Optimization

empt

Reinforcement Learning for Adaptive Mesh Refinement

Large-scale finite element simulations of complex physical systems governed by partial differential equations crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL in that both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence scale to arbitrarily large and complex simulations. We demonstrate in comprehensive experiments on static function estimation and the advection of different fields that RL policies can be competitive with a widely-used error estimator and generalize to larger, more complex, and unseen test problems.Comment: 14 pages, 13 figure