On singularity formation of a 3D model for incompressible Navier–Stokes equations

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.     

Quaternionic representation of the 3D elastic and thermoelastic boundary problems

This paper contains new representations of the 3D elasticity and thermoelasticity problems, expressed in terms of regular quaternion functions, which in 3D and 4D have properties analogous to those of complex analytical functions in 2D. The known and some new results, described in the paper, are used to formulate boundary 3D problems, related to the theory. The formulae obtained are very similar to those given by Muskhelishvili for 2D boundary problems. The obtained representations can be used for development of a 3D boundary element method in numerical analysis.     

Approximating three‐dimensional potential problems using the complex variable boundary element method (CVBEM)

In the current research, the primary focus is to extend the CVBEM to solving potential problems in three dimensions (3D). This is achieved by applying the CVBEM to three coupled projections of the 3D problem domain, in 2D planes, and then superimposing the resulting corresponding 2D CVBEM solutions. The new 3D CVBEM technique is also applied towards improving 3D problem approximations, which are based on the usual 3D boundary element method (BEM) techniques, by approximating the 3D BEM residual error. Finally, a technique to extend a 3D problem geometry into higher geometric dimensions is introduced, and a corresponding numeric error reduction technique is advanced for use in superimposing multiple dimension approximations to improve 3D approximations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 535–560, 2000.     

A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy

In this paper we develop a new efficient and stable Lagrangian numerical method for computing the evolution of 3D curves driven in the normal plane by a driving force and curvature. This new method contains asymptotically uniform tangential redistribution of grid points designed originally for 3D curve evolution in this paper which makes our computations stable and is crucial for the presented application. Together with the design of new tangentially stabilized algorithm for 3D evolving curves, we develop a new method for the fully automatic finding of the optimal trajectory of the camera in the virtual colonoscopy. The proposed method consists of three steps: 3D segmentation of the colon from CT images, finding an initial trajectory guess inside the segmented 3D subvolumes, and driving the initial 3D curve to its optimal position. To that goal, a suitable intrinsic advection-diffusion partial differential equation with a driving force is designed and solved numerically in fast and robust way in order to find a smooth uniformly discretized 3D curve representing the ideal path of the camera in the virtual colonoscopy.     

Simulation of complex regimes of Rayleigh-Benard convection

2D and 3D turbulent convectional flows of viscous and incompressible fluids heated from below in a rectangular parallelepiped are numerically simulated. The horizontal boundaries are stress-free in 3D simulation and stress-free or rigid in 2D simulation. It is shown that although there is a quantitative discrepancy between the results of the 3D simulation and the experimental data, the 3D simulation shows correct power laws for the temperature and vertical velocity pulsations versus supercriticality greater than 150. In the 2D simulation, a similar correspondence is observed at relatively low supercriticality (approximately up to 250). At high supercriticality, in 2D convection there exists a large-scale structure that determines the flow properties.     

On a new 3D model for incompressible euler and navier-stokes equations

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.     

Coupling 3D and 1D fluid-structure interaction models for blood flow simulations

Three-dimensional (3D) simulations of blood flow in medium to large vessels are now a common practice. These models consist of the 3D Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall structure. However, it is still computationally unaffordable to simulate very large sections, let alone the whole, of the human circulatory system with fully 3D fluid-structure interaction models. Thus truncated 3D regions have to be considered. Reduced models, one-dimensional (1D) or zero-dimensional (0D), can be used to approximate the remaining parts of the cardiovascular system at a low computational cost. These models have a lower level of accuracy, since they describe the evolution of averaged quantities, nevertheless they provide useful information which can be fed to the more complex model. More precisely, the 1D models describe the wave propagation nature of blood flow and coupled with the 3D models can act also as absorbing boundary conditions. We consider in this work the coupling of a 3D fluid-structure interaction model with a 1D hyperbolic model. We study the stability of the coupling and present some numerical results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21], [22], [23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.     

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.      

基于外场实验的风力机叶片三维效应研究

针对一台33 kW水平轴风电机组开展了外场实验,得到其叶片7个断面翼型的压力分布曲线;基于求解时均N-S方程对风轮进行三维数值模拟,以及将叶片各断面作为二维翼型进行数值计算,分别得到各断面翼型的压力分布曲线及升阻力系数．通过将外场实验、三维和二维数值计算所得压力分布曲线及升阻力系数进行对比分析,研究了三维效应对风力机气动性能的影响．研究表明,从叶尖到叶根各断面翼型的压差先增大后逐渐减小,叶片表面压力分布曲线比较明显地反映了从叶尖到叶根流动分离的变化;叶片表面压力分布的三维数值计算结果较二维计算结果更加接近于外场实验值;风力机叶片表面的三维流动对叶片的气动性能影响较大,在叶尖和叶根部分尤为突出.     

Shape-matching and classification using structural descriptors

Technological improvements related to object acquisition, visualisation and modelling, have caused a considerable growth of the number of 3D models in digital form. Digital 3D models are now available in large databases of shapes, ranging from unstructured repositories, like the web, to specialised catalogues used in engineering and simulation. In this panorama it is clear that methods for the retrieval and automatic classification of 3D content will play a crucial role in the development of efficient applications for the organisation and filtering of 3D data. In this paper, a methodology based on the coupling of geometric and structural information will be discussed for the global and partial matching of 3D shapes and for their automatic classification. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations

We prove new a priori estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.     

The 3‐D bifurcation and limit cycles in a food‐chain model

In this paper, by using a corollary to the center manifold theorem, we show that the 3‐D food‐chain model studied by many authors undergoes a 3‐D Hopf bifurcation, and then we obtain the existence of limit cycles for the 3‐D differential system. The methods used here can be extended to many other 3‐D differential equation models. Copyright © 2008 John Wiley & Sons, Ltd.     

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

Uniformly Distributed Circular Porous Pattern Generation on Surface for 3D Printing

We present an algorithm for uniformly distributed circular porous pattern generation on surface for three-dimensional (3D) printing using a phase-field model. The algorithm is based on the narrow band domain method for the nonlocal Cahn–Hilliard (CH) equation on surfaces. Surfaces are embedded in 3D grid and the narrow band domain is defined as the neighborhood of surface. It allows one can perform numerical computation using the standard discrete Laplacian in 3D instead of the discrete surface Laplacian. For complex surfaces, we reconstruct them from point cloud data and represent them as the zero-level set of their discrete signed distance functions. Using the proposed algorithm, we can generate uniformly distributed circular porous patterns on surfaces in 3D and print the resulting 3D models. Furthermore, we provide the test of accuracy and energy stability of the proposed method.     

Asymptotics for thin elastic fibers in unilateral contact

What is the contact condition in a 1D beam-model and is it possible to obtain the frictional moments and forces from the 3D traction? If it is possible does the cross-section of the beams influence these values? These questions motivate to study the dimension reduction of a 3D contact problem for beams. This paper is a continuation of [1]. In [1] the asymptotic dimension reduction of a Robin-type elasticity boundary value problem was presented. In this work the explicit relation between a 3D contact problem and a 3D Robin-type elasticity boundary value problem are established and the 1D equations derived in [1] are interpreted as 1D contact conditions, further some numerical examples are shown. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-Dimensional Variational Analysis of Composite Structures Using Bernstein Polynomial Approximations. Report 1

The paper overviews the previous 3D variational analysis approaches and the numerical results obtained with the use of piecewise polynomial splines. A recently developed 3D displacement-assumed variational analysis approach is also described, where the Bernstein approximation polynomials of arbitrary degree are used as the shape functions for a 3D hexahedral discrete element of a composite body.     

3D curvelet transforms and astronomical data restoration

This paper describes two new 3D curvelet decompositions, which are built in a way similar to the first generation of curvelets (Starck et al., 2002 [35]). The first one, called BeamCurvelet transform, is well designed for representing 1D filaments in a 3D space, while the second one, the RidCurvelet transform, is designed to analyze 2D surfaces. We show that these constructions can be useful for different applications such as filament detection, denoising or inpainting. Hence, they could lead to alternative approaches for analyzing 3D cosmological data sets, such as catalogs of galaxies, λCDM simulation or weak lensing data.