쌍곡 보존 법칙들을 풀기 위한 고차정확도 수치기법에 대한 연구

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 강명주.In this thesis, we develop efficient and high order accurate numerical schemes for solving hyperbolic conservation laws such as the Euler equation and the ideal MHD(Magnetohydrodynamics) equations. The first scheme we propose is the \textit{wavelet-based adaptive WENO method}. The Finite difference WENO scheme is one of the popular numerical schemes for application to hyperbolic conservation laws. The scheme has high order accuracy, robustness and stable property. On the other hand, the WENO scheme is computationally expensive since it performs characteristic decomposition and computes non-linear weights for WENO interpolations. In order to overcome the drawback, we propose the adaptation technique that applies WENO differentiation for only discontinuous regions and central differentiation without characteristic decomposition for the other regions. Therefore continuous and discontinuous regions should be appropriately classified so that the adaptation method successfully works. In the wavelet-based WENO method, singularities are detected by analyzing wavelet coefficients. Such coefficients are also used to reconstruct the compressed informations. Secondly, we propose \textit{central-upwind schemes with modified MLP(multi-dimensional limiting process)}. This scheme decreases computational cost by simplifying the scheme itself, while the first method achieve efficiency by skipping grid points. Generally the high-order central difference schemes for conservation laws have no Riemann solvers and characteristic decompositions but tend to smear linear discontinuities. To overcome the drawback of central-upwind schemes, we use the multi-dimensional limiting process which utilizes multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications.1 Introduction 1 2 Governing Equations 7 2.1 Hyperbolic Conservation Laws 7 2.2 Euler equation 9 2.2.1 Model equation 9 2.2.2 Eigen-structure 10 2.3 Ideal MHD equation 14 2.3.1 Model equation 14 2.3.2 Eigen-Structure 15 2.4 The r B = 0 Constraint in MHD Codes 20 2.4.1 Constraints Transport Method 20 2.4.2 Divergence cleaning technique 23 3 Wavelet-based Adaptation Strategy with Finite Dierence WENO scheme 28 3.1 Finite Dierence WENO scheme 28 3.1.1 Characteristic Decomposition 28 3.1.2 WENO-type Approximations 30 3.2 Wavelet Analysis 32 3.2.1 Multi-resolution Approximations 32 3.2.2 Orthogonal Wavelets 36 3.2.3 Constructing Wavelets 37 3.2.4 Biorthogonal Wavelets 38 3.2.5 Interpolating Scaling Function 40 3.3 Adaptive wavelet Collocation Method 45 3.3.1 Interpolating Wavelets 47 3.3.2 Lifting Scheme 52 3.3.3 Lifting Donoho wavelets family 56 3.3.4 The Lifted interpolating wavelet transform 58 3.3.5 Compression 64 3.4 Wavelet-based Adaptive WENO scheme 65 3.4.1 Adjacent Zone 65 3.4.2 Methodology for Spatial discretizations 66 3.4.3 Time Integration 67 3.4.4 Conservation error and boundary treatment 68 3.4.5 Overall Process 69 3.5 Numerical results 69 3.5.1 1-dimensional equations 70 3.5.2 2-dimensional Euler equations 71 3.5.3 2-dimensional MHD equations 83 4 Combination of Central-Upwind Method and Multi-dimensional Limiting Process 90 4.1 Review of Central-Upwind method 92 4.2 Review of Multi-dimensional Limiting Process 95 4.3 Central-Upwind method with Modied MLP limiter 98 4.4 Numerical results 104 4.4.1 Linear advection equation 105 4.4.2 Burger's equation 106 4.4.3 2D Euler system - Four shocks 106 4.4.4 2D Euler system - Rayleigh-Taylor instability 107 4.4.5 2D Euler system - Double Mach reection of a strong shock 109 5 Conclusions 111 Abstract (in Korean) 121Docto

Turbulence: Numerical Analysis, Modelling and Simulation

The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is available for calibration, the importance of a sound mathematical foundation that addresses the needs of practical computing increases. This Special Issue is directed at this crossroads of rigorous numerical analysis, the physics of turbulence and the practical needs of turbulent flow simulations. It seeks papers providing a broad understanding of the status of the problem considered and open problems that comprise further steps

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

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

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Novel Paradigms in Physics-Based Animation: Pointwise Divergence-Free Fluid Advection and Mixed-Dimensional Elastic Object Simulation

This thesis explores important but so far less studied aspects of physics-based animation: a simulation method for mixed-dimensional and/or non-manifold elastic objects, and a pointwise divergence-free velocity interpolation method applied to fluid simulation. Considering the popularity of single-type models e.g., hair, cloths, soft bodies, etc., in deformable body simulations, more complicated coupled models have gained less attention in graphics research, despite their relative ubiquity in daily life. This thesis presents a unified method to simulate such models: elastic bodies consisting of mixed-dimensional components represented with potentially non-manifold simplicial meshes. Building on well-known simplicial rod, shell, and solid models, this thesis categorizes and defines a comprehensive palette expressing all possible constraints and elastic energies for stiff and flexible connections between the 1D, 2D, and 3D components of a single conforming simplicial mesh. For fluid animation, this thesis proposes a novel methodology to enhance grid-based fluid animation with pointwise divergence-free velocity interpolation. Unlike previous methods which interpolate discrete velocity values directly for advection, this thesis proposes using intermediate steps involving vector potentials: first build a discrete vector potential field, interpolate these values to form a pointwise potential, and apply the continuous curl to recover a pointwise divergence-free flow field. Particles under these pointwise divergence-free flows exhibit significantly better particle distributions than divergent flows over time. To accelerate the use of vector potentials, this thesis proposes an efficient method that provides boundary-satisfying and smooth discrete potential fields on uniform and cut-cell grids. This thesis also introduces an improved ramping strategy for the “Curl-Noise” method of Bridson et al. (2007), which enforces exact no-normal-flow on the exterior domain boundaries and solid surfaces. The ramping method in the thesis effectively reduces the incidence of particles colliding with obstacles or creating erroneous gaps around the obstacles, while significantly alleviating the artifacts the original ramping strategy produces