고차 정확도 라그랑지안 표면 추적 문제에 대한 연구

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 2. 강명주.In this thesis, we proposed the method for tracking the interface with higher order accuracy, when given the normal velocity. Our proposed model combines the well-known Lagrangian surface tracking method with the high order interpolation method for discontinuity capturing. The proposed method not only tracks the surface with higher accuracy than the conventional method, but also depends little on the geometric parameters. Furthermore, the method accurately detects the local shapes of the surface, which is an essential part for a stable interface tracking method. The model developed on the two-dimensional interface can be extended naturally on the three-dimensional interface using the high order interpolation method on triangular meshes.1 Introduction 1 2 Previous works 4 2.1 Level set method 4 2.1.1 Basic equations 4 2.1.2 Numerical discretization 5 2.1.3 Reinitialization 6 2.2 Face offsetting method 8 2.2.1 Advection type 8 2.2.2 Wavefrontal type 11 2.2.3 Null-space smoothing 13 2.3 Weighted essentially non-oscillatory scheme 14 2.3.1 Polynomial reconstruction 14 2.3.2 ENO reconstruction 15 2.3.3 WENO reconstruction 16 2.4 WENO scheme on triangular meshes 20 2.4.1 Third order reconstruction 20 2.4.2 Fourth order reconstruction 22 2.4.3 Positivity of linear weights 23 2.4.4 Smoothness indicators and nonlinear weights 25 2.5 Total variation diminishing Runge-Kutta method 26 3 Proposed models 28 3.1 FOM-WENO scheme 28 3.1.1 Motivations for high order FOM scheme 28 3.1.2 High order reconstruction of normals 30 3.1.3 Modified normal vector and error analysis 32 3.1.4 Setting functions reflecting geometric shocks 35 3.1.5 Mesh smoothing method 37 3.1.6 FOM-WENO algorithm 37 3.2 FOM-WENO scheme in three dimension 39 3.3 Numerical experiments 42 3.3.1 Accuracy for normal vector approximation 43 3.3.2 Long time accuracy for corner propagation 51 3.3.3 Geometric stability of FOM-WENO scheme 60 3.3.4 Comparison of volume loss 62 3.3.5 Propagating under non-uniform normal velocity 66 3.3.6 Results in three dimension 68 4 Conclusion 73 Abstract (in Korean) 80Docto

Simulating liquids on dynamically warping grids

We introduce dynamically warping grids for adaptive liquid simulation. Our primary contributions are a strategy for dynamically deforming regular grids over the course of a simulation and a method for efficiently utilizing these deforming grids for liquid simulation. Prior work has shown that unstructured grids are very effective for adaptive fluid simulations. However, unstructured grids often lead to complicated implementations and a poor cache hit rate due to inconsistent memory access. Regular grids, on the other hand, provide a fast, fixed memory access pattern and straightforward implementation. Our method combines the advantages of both: we leverage the simplicity of regular grids while still achieving practical and controllable spatial adaptivity. We demonstrate that our method enables adaptive simulations that are fast, flexible, and robust to null-space issues. At the same time, our method is simple to implement and takes advantage of existing highly-tuned algorithms

Meshless methods: theory and application in 3D fracture modelling with level sets

Accurate analysis of fracture is of vital importance yet methods for effetive 3D calculations are currently unsatisfactory. In this thesis, novel numerical techniques are developed which solve many of these problems. This thesis consists two major parts: firstly an investigation into the theory of meshless methods and secondly an innovative numerical framework for 3D fracture modelling using the element-free Galerkin method and the level set method. The former contributes to some fundamental issues related to accuracy and error control in meshless methods needing to be addressed for fracture modelling developed later namely, the modified weak form for imposition of essential boundary conditions, the use of orthogonal basis functions to obtain shape functions and error control in adaptive analysis. In the latter part, a simple and efficient numerical framework is developed to overcome the difficulties in current 3D fracture modelling. Modelling cracks in 3D remains a challenging topic in computational solid mechanics since the geometry of the crack surfaces can be difficult to describe unlike the case in 2D where cracks can be represented as combinations of lines or curves. Secondly, crack evolution requires numerical methods that can accommodate the moving geometry and a geometry description that maintains accuracy in successive computational steps. To overcome these problems, the level set method, a powerful numerical method for describing and tracking arbitrary motion of interfaces, is used to describe and capture the crack geometry and forms a local curvilinear coordinate system around the crack front. The geometry information is used in the stress analysis taken by the element-Free Galerkin method as well as in the computation of fracture parameters needed for crack propagation. Examples are tested and studied throughout the thesis addressing each of the above described issues

Toward a Simple, Accurate Lagrangian Hydrocode.

Lagrangian hydrocodes play an important role in the computation of transient, compressible, multi-material flows. This research was aimed at developing a simply constructed cell-centered Lagrangian method for the Euler equations that respects multidimensional physics while achieving second-order accuracy. Algorithms that can account for the multidimensional physics associated with acoustic wave propagation and vorticity transport are needed in order to increase accuracy and prevent mesh imprinting. Many of the building blocks of traditional finite volume schemes, such as Riemann solvers and spatial gradient limiters, have their foundations in one-dimensional ideas and so were not used here. Instead, multidimensional point estimates of the fluxes were computed with a Lax-Wendroff type procedure and then nonlinearly modified using a temporal flux limiting mechanism. The linear acoustic equations were used as a simplified test environment for the Lagrangian Euler system. Here Lax-Wendroff methods that exactly preserve vorticity were investigated and found to resist mesh imprinting. However, the dispersion properties of the schemes were poor and so third-order accurate vorticity preserving methods were developed to remedy the problem. The third-order methods guided the construction of a temporal limiting mechanism, which was then used in a vorticity preserving flux-corrected transport scheme. While the acoustic work was interesting in its own right, it also proved to be a useful stepping stone to Lagrangian hydrodynamics. The acoustics algorithms were extended to produce the Simple Lagrangian Method (SLaM). Standard test problems have shown that a first-order accurate version of the method is able to resist mesh imprinting and spurious vorticity despite its minimalistic structure. SLaM is capable of second-order accuracy with a simple parameter change and some preliminary work was done to extend the temporal flux limiting ideas from acoustics to the Lagrangian case. The limited SLaM method converges at second-order for smooth data and is able to capture shocks without producing large unphysical oscillations.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113577/1/tblung_1.pd

Physically-based simulation of ice formation

The geometric and optical complexity of ice has been a constant source of wonder and inspiration for scientists and artists. It is a defining seasonal characteristic, so modeling it convincingly is a crucial component of any synthetic winter scene. Like wind and fire, it is also considered elemental, so it has found considerable use as a dramatic tool in visual effects. However, its complex appearance makes it difficult for an artist to model by hand, so physically-based simulation methods are necessary. In this dissertation, I present several methods for visually simulating ice formation. A general description of ice formation has been known for over a hundred years and is referred to as the Stefan Problem. There is no known general solution to the Stefan Problem, but several numerical methods have successfully simulated many of its features. I will focus on three such methods in this dissertation: phase field methods, diffusion limited aggregation, and level set methods. Many different variants of the Stefan problem exist, and each presents unique challenges. Phase field methods excel at simulating the Stefan problem with surface tension anisotropy. Surface tension gives snowflakes their characteristic six arms, so phase field methods provide a way of simulating medium scale detail such as frost and snowflakes. However, phase field methods track the ice as an implicit surface, so it tends to smear away small-scale detail. In order to restore this detail, I present a hybrid method that combines phase fields with diffusion limited aggregation (DLA). DLA is a fractal growth algorithm that simulates the quasi-steady state, zero surface tension Stefan problem, and does not suffer from smearing problems. I demonstrate that combining these two algorithms can produce visual features that neither method could capture alone. Finally, I present a method of simulating icicle formation. Icicle formation corresponds to the thin-film, quasi-steady state Stefan problem, and neither phase fields nor DLA are directly applicable. I instead use level set methods, an alternate implicit front tracking strategy. I derive the necessary velocity equations for level set simulation, and also propose an efficient method of simulating ripple formation across the surface of the icicles

Numerical and experimental investigation of droplet actuation by surface acoustic waves

Surface acoustic waves (SAWs) technology for manipulating small volumes of liquids has received much attention in recent years. SAW-based manipulation can be used for different bio-sampling functions, such as mixing, heating, pumping, jetting, separation, and atomization of droplets with volumes in the scale of microliters. Most studies in recent years have mainly focused on investigating SAW potential in different real-world microfluidics applications. However, the underlying physics of the droplet deformation by SAW still remains controversial. This thesis aims to investigate droplet deformations subjected to SAWs using both numerical and experimental methods. Different types of SAW devices with various resonant frequencies and different substrates are fabricated to carry out droplet actuation experiments. The experimental models are developed for three main reasons. First, to analyse the droplet deformation; second, to accurately define the contact angle boundary condition needed for simulations; and third, to validate the computational model. A Coupled Level Set Volume of Fluid (CLSVOF) mathematical model is developed to investigate the large deformation of sessile droplet induced by SAWs. A dynamic contact angle boundary condition is implemented to model the droplet three-phase contact line (TPCL) movement. The numerical and experimental results are quantitatively and qualitatively compared, and a remarkable agreement is achieved, which proves that the developed computational model can be used to simulate different droplet actuation scenarios. After validation of the computational model, it is used for analysing the physics of the droplet jetting and pumping. The effects of important factors such as droplet volume and SAW frequency and power on droplet pumping are investigated. Moreover, the model is used to analyse the energy budget of a droplet jetting. An investigation into the optimization of the interdigital transducers (IDT) location of the SAW devices for different microfluidic applications is also carried out using the computational model. The experimental and computational models are then employed to investigate a novel application of SAW devices to control the droplet impact. SAWs devices are used to manipulate and control the droplet dynamics. The experimental results revealed that characteristic impact parameters such as impact regime, contact time, maximum spreading and re-bouncing angle could be modified and controlled by SAWs. By changing the SAW direction and power, droplets impact behaviour can be altered. The maximum reduction of contact time up to ∼50% can be achieved, along with alterations of droplet spreading, re-bouncing angle, and movement along the inclined surfaces. On the other hand, numerical results revealed that the SAWs could be used to modify and control the internal velocity fields inside the droplet. By breaking the symmetry of the internal recirculation patterns inside the droplet during the impact on flat surfaces, the kinetic energy recovered from interfacial energy during the retraction process is increased, and the droplet can be entirely separated from the surface with a much shorter contact time. Also, numerical results revealed that applying SAWs modifies the energy budget inside the liquid medium on both flat and inclined surface, leading to different impact behaviours. This innovative paradigm opens up new opportunities to actively program and controls the droplet impact on smooth or planar and curved surfaces, as well as rough or textured surfaces

IST Austria Thesis

Computer graphics is an extremely exciting field for two reasons. On the one hand, there is a healthy injection of pragmatism coming from the visual effects industry that want robust algorithms that work so they can produce results at an increasingly frantic pace. On the other hand, they must always try to push the envelope and achieve the impossible to wow their audiences in the next blockbuster, which means that the industry has not succumb to conservatism, and there is plenty of room to try out new and crazy ideas if there is a chance that it will pan into something useful. Water simulation has been in visual effects for decades, however it still remains extremely challenging because of its high computational cost and difficult artdirectability. The work in this thesis tries to address some of these difficulties. Specifically, we make the following three novel contributions to the state-of-the-art in water simulation for visual effects. First, we develop the first algorithm that can convert any sequence of closed surfaces in time into a moving triangle mesh. State-of-the-art methods at the time could only handle surfaces with fixed connectivity, but we are the first to be able to handle surfaces that merge and split apart. This is important for water simulation practitioners, because it allows them to convert splashy water surfaces extracted from particles or simulated using grid-based level sets into triangle meshes that can be either textured and enhanced with extra surface dynamics as a post-process. We also apply our algorithm to other phenomena that merge and split apart, such as morphs and noisy reconstructions of human performances. Second, we formulate a surface-based energy that measures the deviation of a water surface froma physically valid state. Such discrepancies arise when there is a mismatch in the degrees of freedom between the water surface and the underlying physics solver. This commonly happens when practitioners use a moving triangle mesh with a grid-based physics solver, or when high-resolution grid-based surfaces are combined with low-resolution physics. Following the direction of steepest descent on our surface-based energy, we can either smooth these artifacts or turn them into high-resolution waves by interpreting the energy as a physical potential. Third, we extend state-of-the-art techniques in non-reflecting boundaries to handle spatially and time-varying background flows. This allows a novel new workflow where practitioners can re-simulate part of an existing simulation, such as removing a solid obstacle, adding a new splash or locally changing the resolution. Such changes can easily lead to new waves in the re-simulated region that would reflect off of the new simulation boundary, effectively ruining the illusion of a seamless simulation boundary between the existing and new simulations. Our non-reflecting boundaries makes sure that such waves are absorbed

Statistical and temporal analysis of shock-driven instability through simultaneous density and velocity measurements

The effects of initial conditions (single- and multi-mode) and nondimensional density ratio (Atwood number, A) on dynamics of mixing in Richtmyer--Meshkov Instability evolution are studied using high resolution results from ensembles of experiments as well as temporally resolved measurements, all utilizing simultaneous PLIF and PIV. Campaigns were undertaken at an incident shock Mach number of 1.55 on both single and multi-mode perturbed interfaces between two gas pairs before and after reshock. This was done to hold constant as many parameters as possible between the cases. The gas pairs used were Nitrogen/Carbon Dioxide and Nitrogen/Sulfur Hexafluoride, where the incident shock travels from light (Nitrogen) to heavy gas. These gas pairs yield Atwood numbers of 0.22 and 0.67, respectively. This constitutes the first work where turbulence statistics resulting from ensemble averaging are collected on both of these Atwood numbers, and also the first comparison of Atwood numbers with otherwise parametric consistency where turbulence statistics from ensemble averaging can be compared. Furthermore, the high speed measurements in this flow are the first temporally resolved simultaneous PLIF and PIV measurements collected for RMI flows. This allows calculation of time-resolved quantities and time-correlated analysis of features from robust measurements.Ph.D

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described