On the Singular Neumann Problem in Linear Elasticity

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 Lam\'e constant are constructed for the pure displacement formulations, while a preconditioner that is robust in both Lam\'e 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

Termální konvekce s volným povrchem v rotujícím ledovém měsíci

Thermal convection with evolving surface in a rotating icy satellite Master's Thesis author: Miroslav Kuchta∗ supervisor: Doc. RNDr. Ondřej Čadek, CSc.† Keywords: Stokes-Fourier system, Free surface, Finite-differences Abstract This thesis is concerned with modeling the surface deformations and thermal convection in a rotating icy satellite. The system of gov- erning equations, that we derive from general balance laws, is solved numerically using the finite-difference method on a staggered grid. Free surface is understood as implicitly described interface between the satellite and an almost massless medium with viscosity orders of magnitude smaller than ice. We design a numerical method capable of tracking the deforming surface. The numerical method is applied to models with temperature-dependent viscosity. ∗ Mathematical Institute of Charles University, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. [email protected] † Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. [email protected] 1Termální konvekce s volným povrchem v rotujícím ledovém měsíci Diplomová práce author: Miroslav Kuchta∗ školitel: Doc. RNDr. Ondřej Čadek, CSc.† Klíčová slova: Stokes-Fourier systém, Vol'ný povrch, Konečné diferencie Abstrakt Táto práca sa zaoberá modelovaním povrchových deformácií a termálnej konvekcie v rotujúcom l'adovom mesiaci. Systém riadiacich rovníc, ktoré odvodíme z obecných zákonov zachovania, riešime nume- ricky pomocou metódy konečných diferencií na posunutých siet'ach. Vol'ný povrch chápeme ako implicitne popísané rozhranie medzi me- siacom a takmer nehmotným médiom s rádovo menšou viskozitou ako l'ad. Vytvoríme numerickú metódu schopnú sledovat' deformovaný po- vrch. Numerickú metódu aplikujeme na príklady s teplotne závislou viskozitou. ∗ Matematický ústav UK, Matemeticko-fyzikální fakulta, Univerzita Karlova v Praze, Česká republika. [email protected] † Katedra geofyziky, Matemeticko-fyzikální fakulta, Univerzita Karlova v Praze, Česká republika. [email protected] 1Mathematical Institute of Charles UniversityMatematický ústav UKFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes--Darcy problem. Three different formulations and their discretizations in terms of conforming and nonconforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.publishedVersio

Artificial Neural Networks trained through Deep Reinforcement Learning discover control strategies for active flow control

We present the first application of an Artificial Neural Network trained through a Deep Reinforcement Learning agent to perform active flow control. It is shown that, in a 2D simulation of the Karman vortex street at moderate Reynolds number (Re = 100), our Artificial Neural Network is able to learn an active control strategy from experimenting with the mass flow rates of two jets on the sides of a cylinder. By interacting with the unsteady wake, the Artificial Neural Network successfully stabilizes the vortex alley and reduces drag by about 8%. This is performed while using small mass flow rates for the actuation, on the order of 0.5% of the mass flow rate intersecting the cylinder cross section once a new pseudo-periodic shedding regime is found. This opens the way to a new class of methods for performing active flow control

HAZniCS -- Software Components for Multiphysics Problems

We introduce the software toolbox HAZniCS for solving interface-coupled multiphysics problems. HAZniCS is a suite of modules that combines the well-known FEniCS framework for finite element discretization with solver and graph library HAZmath. The focus of the paper is on the design and implementation of a pool of robust and efficient solver algorithms which tackle issues related to the complex interfacial coupling of the physical problems often encountered in applications in brain biomechanics. The robustness and efficiency of the numerical algorithms and methods is shown in several numerical examples, namely the Darcy-Stokes equations that model flow of cerebrospinal fluid in the human brain and the mixed-dimensional model of electrodiffusion in the brain tissue

Rational approximation preconditioners for multiphysics problems

We consider a class of mathematical models describing multiphysics phenomena interacting through interfaces. On such interfaces, the traces of the fields lie (approximately) in the range of a weighted sum of two fractional differential operators. We use a rational function approximation to precondition such operators. We first demonstrate the robustness of the approximation for ordinary functions given by weighted sums of fractional exponents. Additionally, we present more realistic examples utilizing the proposed preconditioning techniques in interface coupling between Darcy and Stokes equations

Encoder–decoder neural networks for predicting future FTIR spectra – application to enzymatic protein hydrolysis

In the process of converting food-processing by-products to value-addedingredients, fine grained control of the rawmaterials, enzymes and process conditionsensures the best possible yield and eco-nomic return. However, when raw mate-rial batches lack good characterization andcontain high batch variation, online or at-line monitoring of the enzymatic reac-tions would be beneficial. We investigate the potential of deep neural networks inpredicting the future state of enzymatic hydrolysis as described by Fourier-trans-form infrared spectra of the hydrolysates. Combined with predictions of averagemolecular weight, this provides a flexible and transparent tool for process moni-toring and control, enabling proactive adaption of process parameters.publishedVersio