Semi-implicit Euler method

Summary

In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method. Setting The semi-implicit Euler method can be applied to a pair of differential equations of the form :\begin{align} {dx \over dt} &= f(t,v) \ {dv \over dt} &= g(t,x), \end{align} where f and g are given functions. Here, x and v may be either scalars or vectors. The equations of motion in Hamiltonian mechanics take this form if the Hamiltonian is of the form : H = T(t,v) + V(t,x). , The differential equations are to be solved with the initial condition : x(t_0) = x_0, \qquad v(t_0) = v_0. The method The semi-implicit Euler met

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Nonlinear study on a rigid rotor supported by herringbone grooved gas bearings: Theory and validation

Philipp Kaspar Bättig, Wanhui Liu, Jürg Alexander Schiffmann, Patrick Hubert Wagner

The nonlinear behavior of a rigid rotor supported by herringbone grooved journal gas bearings (HGJBs) was investigated in this study. The two-dimensional narrow groove theory (2D-NGT) was adopted to model the HGJBs. A set of integrated rotor-bearing state equations were built by coupling the rotor motion equations and the bearing Reynolds equation. An implicit integrator with adaptive time step method was used to solve those state equations continuously. Two low-stability HGJBs were implemented to experimentally demonstrate and analyze the appearance of self-excitation motions. The theoretical model was successfully validated by the experimental data on predicting the onset speed of the sub-synchronous vibration of the HGJB-rotor system and the whirl frequency ratio. The predicted limit cycle amplitude increases as the speed increases until the rotor contacts with the bearing surface, which leads to a bearing failure. Forward conical mode dominates the self-excited motion during the whole speed range of self-excited motion. The prediction shows that the HGJB-rotor system can still operate in a stable, even though the rotor is installed vertically, i.e., without static load on the bearings. This is a distinct advantage in comparison to plain bearings. As the static load applies on the bearings increases, the onset speed of sub-synchronous vibration increases as well. For the investigated rotor-bearing system, an increase of the onset speed of sub-synchronous vibration from 36 krpm to 75 krpm is predicted as the static load increases from 0 to 4 times of the rotor weight. This indicates an increased HGJB stability with increased static load. The rotor orbits show complex shapes when the imbalance excitation is considered in the simulation. Both synchronous frequency and whirl frequency are shown in the spectral analysis. Moreover, the speed range of self-excited motion reduces from [38,48] krpm to [40,42] krpm as the imbalance increases from 0 to 40 mgmm.

2021

This thesis explores various approaches of studying the long-range colour order of antiferromagnetic SU(N) Heisenberg models with the linear flavour-wave theory (LFWT). The LFWT is an extension of the well-known SU(2) spin-wave theory to SU(N), and this semi-classical method has been used to study SU(N) models with colour-ordered ground states in the fundamental irreducible representation (irrep). The aim of this thesis is to study various SU(N) Heisenberg models in different irreps and in various colour configurations, in part by extending the LFWT method to different irreps of SU(N). This will be achieved by using three different methods all using different bosonic representations. First, the LFWT willbe performed using the multiboson method that introduces a boson for each state of a given irrep. Then, a different SU(3) bosonic representation first introduced by Mathur & Sen will be presented and used to perform the LFWT calculations. Finally, another way of applying the LFWT will be shown using the bosons first used by Read & Sachdev for SU(N) irreps with rectangular Young tableaux. The specific models that will be treated by these methods are the fully antisymmetric SU(N) models on the square, honeycomb, and triangular lattices with m>1 particles per site, as well as the bipartite SU(3) Heisenberg chain in the adjoint irrep to show how the LFWT can be used for mixed symmetries. The last chapter of the thesis will be dedicated to another problem related to the accidental line of zero modes in the harmonic spectrum of the antiferromagnetic SU(3) Heisenberg model on the square lattice with one particle per site in a three-sublattice order. The objective will be to try to lift the accidental zero modes in the spectrum.

EPFL2019

We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, at each iteration the fluid velocity is computed separately from the coupled pressure-structure velocity system, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid- structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system.

2008

Nonlinear study on a rigid rotor supported by herringbone grooved gas bearings: Theory and validation

Philipp Kaspar Bättig, Wanhui Liu, Jürg Alexander Schiffmann, Patrick Hubert Wagner

2021