Entropy Stable Summation-by-Parts Methods for Hyperbolic Conservation Laws on h/p Non-Conforming Meshes

In this work we present high-order primary conservative and entropy stable schemes for hyperbolic systems of conservation laws with geometric (h) and algebraic (p) non-conforming rectangular meshes. Throughout we rely on summation-by-parts (SBP) operators which discretely mimic the integration-by-parts rule to construct stable approximations. Thus, the discrete proofs of primary conservation and entropy stability can be done in a one-to-one fashion to the continuous analysis. Here, we consider different SBP operators based on finite difference as well as discontinuous Galerkin approaches. We derive non-conforming schemes by extending ideas of high-order primary conservative and entropy stable SBP methods on conforming meshes. Here, special attention is given to the coupling between non-conforming elements. The coupling is instructed to entropy stable projection operators. However, these projection operators suffer from a suboptimal degree. Therefore, we develop degree preserving SBP operators where the norm matrix has a higher degree compared to classical SBP operators. With these operators it is possible to construct entropy stable projection operators which have the same degree as the SBP differentiationmatrix. Typically, high-order primary conservative and entropy stable schemes are semi-discrete methods with a discretized spatial domain and assuming continuity in time. Therefore, temporal errors are introduced when integrating the conservation laws in time with standard methods, e.g. Runge-Kutta schemes, for which the entropy can have an unpredictable temporal behaviour. Thus, we extend high-order primary conservative and entropy stable semi-discrete methods to fully-discrete schemes on conforming and non-conforming meshes. This results in an implicit space-time method. We introduce a simple mesh generation strategy to obtain quadrilateral meshes surrounding two dimensional complex geometries. Finally, with the generated meshes we simulate a flow around a NACA0012 airfoil to demonstrate the benefits of considering non-conforming elements for a practical simulation

Simulation-based determination of systematic errors of flow meters due to uncertain inflow conditions

Computational fluid dynamics (CFD) provides well-established tools for the prediction of the velocity profiles in turbulent pipe flows. As far as industrial pipe and district heating systems are concerned, combinations of elbows are the most common pipe assemblies. Among the different pipe combinations, double elbows out-of-plane are of special interest, since they introduce strong disturbances into the flow profile and have a strong influence on many common types of flow meters. In front of a double elbow there is often another flow-disturbing installation. As a result the upstream conditions are unknown and an investigation of the resulting systematic bias on the measurement of the flow rate and the associated contribution to its measurement uncertainty is necessary. We demonstrate here that this can be achieved by a variation of the inlet profile in terms of swirls and asymmetry components. In particular, an ultrasonic and an electromagnetic flow meter are modeled in order to quantify the systematic errors stemming from uncertain inflow conditions. For this purpose, a generalized non-intrusive polynomial chaos method has been used in conjunction with a commercial CFD code. As the most influential parameters on the measured volume flow, the distance between the double elbow and the flow meter as well as the orientation of the flow meter are considered as random variables in the polynomial chaos approach. This approach allowed us to obtain accurate prediction of the systematic error for the ultrasonic and electromagnetic meter as functions of the distance to the double elbow. The resulting bias in the flow rate has been found to be in the range of 1.5–4.5% (0.1–0.5%) with a systematic uncertainty contribution of 2–2.4% (0.6–0.7%) for the ultrasonic (electromagnetic) flow meter if the distance to the double elbow is smaller than 40 pipe diameters. Moreover, it is demonstrated that placing the flow meters in a Venturi constriction leads to substantial decrease of the bias and the contribution to the measurement uncertainty stemming from the uncertain inflow condition

Modelling and Validation of Robot Manipulators

There are many methods to describe manipulator dynamics, the iterative Newton-Euler dynamic formulation and the Lagrange-Euler formulation are two of them. Between these two well known methods, the former has been regarded as computationally efficient, and the latter as understandable in representing manipulator dynamics. It is hard and dull to generate robot manipulator dynamic equations manually from either the iterative Newton-Euler dynamic formulation or the Lagrange-Euler formulation. Therefore, the two general programmes, which are based on these two formulations respectively and suited to rotary joint manipulators, have been written in REDUCE. After running the programmes, we find that the calculation time for generating the dynamic equations of a rotary joint manipulator by the programme based on the Lagrange-Euler formulation is much shorter than the one by the programme based on the other

Numerical Study Of The Atmospheric Radiative Transfer Process With Application To The Arctic Energy Balance

Thesis (Ph.D.) University of Alaska Fairbanks, 1986A high-order discrete-ordinate approximation is utilized to solve the radiative transfer equation for both solar and terrestrial spectra. The solutions have been compared with other methods and found to be reliable and efficient. These solutions have been used to construct a complete and comprehensive radiation model for the arctic atmosphere. The bulk radiative properties (e.g. fluxes and heating/cooling rates) as well as the angular distribution of intensity can be computed as functions of wavelength at various levels in vertically inhomogeneous atmospheres. The radiation model treats Rayleigh scattering, gaseous absorption/emission, scattering and absorption/emission by cloud droplets and haze particles. Snow conditions of the arctic region are simulated by snow grains and soot contamination in the surface layers. A unified treatment of shortwave and longwave radiative transfer is achieved. Use has been made of the five McClatchey atmospheres and of data from the Arctic Stratus Clouds Experiment collected in 1980. Results are compared among broad-band, narrow-band and line-by-line (restricted to gases) computations. We find that at the expense of accuracy by a few watts.m('-2) for flux or a few tenth (DEGREES)C/day for heating/cooling rate computations, the broad-band models are very fast and suitable for certain types of climate modelling. During the arctic summer, stratus clouds are a persistent feature and decrease largely the downward flux at the surface. Arctic haze is important if it is above the cloud layer or in air with low relative humidity and also decreases the downward flux at the surface. The greenhouse effect of doubling the CO(,2) amount can be offset by the haze condition or by the increase in cloudiness of about 4%. In late June, we find that a clear sky condition results in more available downward flux for snow melt than does a cloudy sky condition. This is because the increase of infrared radiation diffused back to surface by the cloud can not compensate the reduction of solar radiation. If the snow starts to melt, the decreasing snow albedo further accelerates the melting process

An adaptive space-time boundary element method for impulsive wave propagation in elastodynamics

Wave propagation in natural or man-made bodies is an important problem in civil engineering, electronic engineering and ocean engineering etc. Common examples of wave problems include earthquake wave modeling, ocean wave modeling, soil- structure interaction, geological prospecting, and acoustic or radio wave diffraction. The Boundary Element Method (BEM) is a widely-used numerical method to solve such problems in both science and engineering fields. However, conventional BEM modeling of wave problems encounters many difficulties. Firstly, the method is expensive since influence matrices are computed at each time step and BEM solutions at every former time step have to be stored. Secondly, if large time steps are used, inaccuracies arise in BEM solutions; but if small time steps are used, computational costs become impractical. Thirdly, the dimensionless space-time ratio must be limited to a narrow range to produce a stable solution. In this thesis, we attack these problems by introducing adaptive schemes and mesh refinement. Instead of using uniform meshes and uniform time steps, error indicators are employed to locate high-gradient areas; then mesh refinement in space-time is used to improve the resolution in those areas only. Another strategy is to introduce the space-time concept to track moving wave fronts. In wave problems, wave fronts move in space-time, and high gradients arise both in space and in time. It is thus inadequate to refine the mesh in space only because there are high gradients in time as well. Hence, besides a locally mesh refinement scheme employed in space, local time stepping is also used to improve the accuracy and efficiency of the algorithm. This adaptive scheme is implemented in the C language and used to solve scalar and electrodynamic 2D and 3D wave propagation problems in a open and closed field. Gradient-based and resolution-based error indicators are employed to locate these moving high-gradient areas. A space mesh refinement scheme and the local time stepping is used to refine the area to achieve higher accuracy. The adaptive BEM solver is 1.4 ~ 1.8 times faster than the conventional BEM solver. It is also more stable than the conventional BEM. We also parallelize the BEM solver to further improve its efficiency. Compared with the non-parallel code, using a 8-processor Linux cluster, a speed-up factor of four is achieved. This suggests that substantial further gains can be obtained if a larger parallel computer is available. July 11. 2007

High-Capacity Directional Graph Networks

Deep Neural Networks (DNN) have proven themselves to be a useful tool in many computer vision problems. One of the most popular forms of the DNN is the Convolutional Neural Network (CNN). The CNN effectively learns features on images by learning a weighted sum of local neighborhoods of pixels, creating filtered versions of the image. Point cloud analysis seems like it would benefit from this useful model. However, point clouds are much less structured than images. Many analogues to CNNs for point clouds have been proposed in the literature, but they are often much more constrained networks than the typical CNN. This is a matter of necessity: common point cloud benchmark datasets are fairly small and thus require strong regularization to mitigate overfitting. In this dissertation we propose two point cloud network models based on graph structures that achieve the high-capacity modeling capability of CNNs. In addition to showing their effectiveness on point cloud classification and segmentation in typical benchmark scenarios, we also propose two novel point cloud problems: ATLAS Detector segmentation and Computational Fluid Dynamics (CFD) surrogate modeling. We show that our networks are much more effective than others on these new problems because they benefit from deeper networks and extra capacity that other researchers have not pursued. These novel networks and datasets pave the way for future development of deeper, more sophisticated point cloud networks

A Divergence-Free and H(div)H(div)-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations

We proposed a divergence-free and $H(div)$-conforming embedded-hybridized discontinuous Galerkin (E-HDG) method for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG method is computationally far more advantageous over the hybridized discontinuous Galerkin (HDG) counterpart in general. The benefit is even significant in the three-dimensional/high-order/fine mesh scenario. On a simplicial mesh, our method with a specific choice of the approximation spaces is proved to be well-posed for the linear case. Additionally, the velocity and magnetic fields are divergence-free and $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-conformity can be directly applied to the HDG version of the proposed approach. The HDG or E-HDG method for the linearized MHD equations can be incorporated into the fixed point Picard iteration to solve the nonlinear MHD equations in an iterative manner. We examine the accuracy and convergence of our E-HDG method for both linear and nonlinear cases through various numerical experiments including two- and three-dimensional problems with smooth and singular solutions. For smooth problems, the results indicate that convergence rates in the $L^2$ norm for the velocity and magnetic fields are optimal in the regime of low Reynolds number and magnetic Reynolds number. Furthermore, the divergence error is machine zero for both smooth and singular problems. Finally, we numerically demonstrated that our proposed method is pressure-robust

Quantum plasmonic theory of hot carriers generated in metallic nanoparticles

A detailed understanding of the interaction between electrons and light at the nanoscale is used to gain insight into the wide range of plasmonic phenomena observed experimentally for the exploration of novel applications in nanotechnology. In particular, the decay of the plasmon excitation into energetic electrons and holes has been proposed as a promising energy conversion mechanism with applications in photovoltaics and photocatalysis. In the first part of this thesis a material-specific quantum model is introduced for both the plasmon and the electrons in a metallic nanoparticle. For the description of the optical properties of the nanoparticle we use linear response time-dependent density functional theory while the decay of the plasmon into energetic carriers is described using many-body perturbation theory. We find that hot-carrier generation rates differ significantly from semiclassical predictions, which treat the plasmon as a classical dipole field induced by the charge oscillation on the surface of the nanostructure. We also study the decay of non-plasmonic excitations, such as electron-hole pairs, and find that they can result in similar hot-carrier generation rates as plasmonic excitations. This quantum model can also capture both the external screening induced by the dielectric environment surrounding the nanoparticle and the internal screening induced by the polarizable d-band electrons. This is achieved by using an effective screened electron-electron interaction that modifies the calculation of the electron-plasmon coupling as well as the plasmon resonance. We present results for silver nanoparticles embedded in five different dielectrics (air, SiO2, SiN, TiO2 and GaP) and predict that large generation rates can be achieved by choosing a host material that shifts the localised plasmon energy such that it coincides with the maximum in joint density of states. Also, a large number of relatively low-energy carriers are obtained by embedding in strongly screening environments, such as GaP. In the second part of the thesis, a semiclassical approach is introduced to study the contribution of the d-bands to the generation of plasmon-induced hot carriers in noble metals. This description combines atomistic and continuum theories using the envelope function technique. Fermi’s golden rule is applied to calculate the plasmonic hot-carrier rates due to transitions either from a d-band state to an sp-band state (interband transition) or from an sp-band state to another sp-band state (intraband transtion). We apply this formalism to silver nanoparticles with radii up to 20 nm. We find that for small nanoparticles intraband transitions dominate while interband transitions give the largest contribution for larger nanoparticles.Open Acces