共查询到20条相似文献,搜索用时 435 毫秒
1.
On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations
Thomas Y. Hou Zhen Lei Guo Luo Shu Wang Chen Zou 《Archive for Rational Mechanics and Analysis》2014,212(2):683-706
In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects. 相似文献
2.
Thomas Y. Hou Congming Li Zuoqiang Shi Shu Wang Xinwei Yu 《Archive for Rational Mechanics and Analysis》2011,199(1):117-144
We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional
system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) 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. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model
by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation
of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a
class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of
the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model. 相似文献
3.
Dongho Chae 《Archive for Rational Mechanics and Analysis》2009,194(3):1011-1027
In this paper we exclude the scenario of the apparition of finite time singularity in the form of self-similar singularities
in the ideal magnetohydrodynamic equations, assuming suitable integrability conditions on the vorticity and the magnetic field.
We also consider the more refined possibility of asymptotically self-similar singularities, where the local classical solution
converges to the self-similar profile as we approach the possible time of singularity. The scenario of asymptotically self-similar
singularity is also excluded under suitable conditions on the profile. In the two-dimensional magnetohydrodynamics the magnetic
field evolution equations reduce to a divergence free transport equation for a scalar stream function. This helps us to improve
the above nonexistence theorems on the self-similar singularities, in the sense that we require merely weaker integrability
conditions on the profile in order to prove the results. 相似文献
4.
热机载荷共同作用下双材料、复合材料中的裂纹扩展往往发生在界面处,并且工程中实际遇到的裂纹大多数是三维裂纹,传统的求解热冲击及机械载荷共同作用下界面裂纹应力强度因子的数值方法如有限元、边界元法计算量大,计算效率低。由于通用权函数仅仅与裂纹体的几何形状有关,与载荷、时间无关,求解应力强度因子时避免了反复的应力分析,计算效率大大提高, 通用权函数法非常适合计算复杂冲击载荷下应力强度因子分布的过渡过程。根据Betti互易原理,本文推导出了三维界面裂纹问题通用权函数法的普遍表达式,并给出了热机载荷共同作用下三维界面I型、Ⅱ型和Ⅲ型裂纹问题通用权函数法的有限元格式. 通过实例计算比较,表明此方法得到的结果可以达到满意的工程应用精度。 相似文献
5.
We study metric properties of trajectory attractors for infinite-dimensional dissipative systems. Under natural conditions
we show that in the appropriate topology the functional dimension of this attractor is not greater than 1 and the metric order
is 0. We also prove that every finite (in time) “piece” of the trajectory attractor has finite fractal dimension. As examples
we consider a reaction-diffusion system, the 2D Navier-Stokes equation and also 3D Navier-Stokes equation under an additional
regularity assumption concerning the corresponding trajectory attractor which is valid in the case of thin domains
2000 Mathematics Subject Classification: 37C45; 37L30. 相似文献
6.
《International Journal of Solids and Structures》2014,51(15-16):2908-2919
In the present study, a conservative integral based on the Betti reciprocal principle is formulated to determine the intensity of singularity at a vertex of the interface in three-dimensional dissimilar material joints with one real singularity. Eigenanalysis formulated using a three-dimensional finite element method (FEM) is used to calculate the order of stress singularity, angular functions of displacements and stresses. Models with various element sizes and various integral areas are used to investigate the effect of the integration area on the accuracy of the results. The results are compared with those obtained from the boundary element method (BEM) using a curve-fitting technique to calculate the intensity of singularity. In addition, models of various lengths and various material combinations are used to investigate the stress singularity characteristics in three-dimensional dissimilar material joints. The results of the present study indicate that the conservative integral can be used to determine the intensity of singularity in three-dimensional bi-material joints. The accuracy of the results can be improved by mesh refinement. Finally, the relationships among the intensity of singularity, the order of stress singularity and the model geometry are discussed. 相似文献
7.
E. P. Chen 《Theoretical and Applied Fracture Mechanics》1985,3(3):257-262
In this investigation, the enriched element method developed by Benzley was extended to treat the stress analysis problem involving a bimaterial interface crack. Unlike crack problems in isotropic elasticity, where the stress singularity at the crack tip is of the inverse square root type, the interface crack contains an additional oscillatory singularity. Although the effect of this oscillatory characteristic is confined to a region very close to the crak tip, it nevertheless requires proper treatment in order to obtain accurate predictions on the stress intensity factors. Using appropriate crack tip stress and displacement expressions, the enriched element method can model the stress singularity for an interface crack exactly. The finite element implementation of this method has been made on the code APES. Stress intensity factor results predicted by the modified APES program compare favorably with those available in the literature. This indicates tha the enriched element technique provides an accurate and efficient numerical tool for the analysis of bimaterial interface crack problems. 相似文献
8.
Angel Castro Diego Córdoba Charles Fefferman Francisco Gancedo 《Archive for Rational Mechanics and Analysis》2016,222(1):213-243
In this paper we show the existence in finite time of splash singularities for the one-phase Muskat problem. 相似文献
9.
Céline Grandmont Matthieu Hillairet 《Archive for Rational Mechanics and Analysis》2016,220(3):1283-1333
We study an unsteady nonlinear fluid–structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action–reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain, in particular that contact between the viscoelastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, and of the existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure. 相似文献
10.
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang 《Archive for Rational Mechanics and Analysis》2010,198(1):39-123
The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions
because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily
for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the
hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the
commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect
in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For
completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity
variable, there exists a unique local solution with the same regularity, so that this solution acquires the C
∞ regularity for any positive time. 相似文献
11.
《International Journal of Non》2005,40(2-3):361-371
The objective of this paper is to describe a different approach to modeling the material symmetry associated with singularities that can occur in curvilinear anisotropic elastic symmetries. In this analysis, the intrinsic non-linearity of a cylindrically anisotropic problem is demonstrated. We prove that a simple homogenization process applied to a representative volume element containing the cylindrical anisotropic singularity removes the singularity. This geometric and interpretive approach is an aid to better modeling of material symmetry associated with these singularities. 相似文献
12.
《International Journal of Solids and Structures》2001,38(1):91-113
The characteristic equations for the order of stress singularity of anisotropic bimaterial wedges subjected to traction boundary conditions are investigated. For an angle-ply bimaterial wedge, both fully bonded and frictional interfaces are considered, whereas for a monoclinic bimaterial wedge, a frictional interface is considered. Here, the Stroh formalism and the separation of variables technique are used. In general, the order of stress singularity can be real or complex, but for the special geometry of a crack along the frictional interface of a monoclinic composite, it is always real. Explicit characteristic equations for the order of singularity are presented for an aligned orthotropic composite with a frictional interface. Numerical results are given for an angle-ply bimaterial wedge and a monoclinic bimaterial wedge consisting of a graphite/epoxy fiber-reinforced composite. 相似文献
13.
14.
Russel E. Caflisch 《Fluid Dynamics Research》1988,3(1-4):75-77
The perturbations in a nearly flat vortex sheet will initially grow due to Kelvin-Helmholtz instability. Asymptotic analysis and numerical computations of the subsequent nonlinear evolution show several interesting features. At some finite time the vortex sheet develops a singularity in its shape; i.e. the curvature becomes infinite at a point. This is immediately followed by roll-up of the sheet into an infinite spiral. This paper presents two mathematical results on nonlinear vortex sheet evolution and singularity formation: First, for sufficiently small analytic perturbations of the flat sheet, existence of smooth solutions of the Birkhoff-Rott equation is proved almost up to the expected time of singularity formation. Second, we present a construction of exact solutions that develop singularities (infinite curvature) in finite time starting from analytic initial data. These results are derived within the framework of analytic function theory. 相似文献
15.
《International Journal of Solids and Structures》2002,39(8):2235-2255
Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method. 相似文献
16.
In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems. 相似文献
17.
随着复合材料的应用和发展,不同材料组成的界面结构越来越受到人们的重视。界面层两侧材料的性能相异会引起材料界面端奇异性,同时界面和界面附近存在裂纹会引起裂尖处的应力奇异性。因此双材料界面附近的力学分析是比较复杂的。本文建立双材料直角界面模型,在材料界面附近预设初始裂纹,计算了有限材料尺寸对界面应力场及其附近裂纹应力强度因子的影响。运用弹性力学中的 Goursat 公式求得直角界面端在有限尺寸下的应力场以及其应力强度系数。通过叠加原理和格林函数法进一步得到在直角界面端附近的裂纹尖端应力强度因子。计算结果表明,在适当范围内改变材料内裂纹与界面之间的距离,界面附近裂纹尖端的应力强度因子随着裂纹与界面距离的增加而减少,并且逐渐趋于稳定。分析结果可以为预测双材料结构复合材料界面失效位置提供参考。 相似文献
18.
Summary The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical
continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of
NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of
bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the
singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found
near the dislocation core.
Received 27 May 1997; accepted for publication 1 July 1997 相似文献
19.
In the present paper, the homogenized mechanical response of an interface in a microsystem interconnection is established on the basis of micropolar theory. The interface is treated as a finite RVE (representative volume element), across which macroscopic discontinuities occur as expressed in terms of the regularized discontinuous displacement and rotation fields. For the microstructure within the interfacial RVE, the micro-macro kinematical coupling is introduced as a second-order Taylor series expansion, along with a fluctuation term representing the microscopic displacement variation. In the second-order term of the expansion a restriction for the curvature is made, which motivates the adopted micropolar kinematics. Explicit expressions for the homogenized traction vector and the couple stress traction, associated with displacement and rotational jumps across the interface surface, are derived. A planar elastic interface is subjected to three basic deformation modes, i.e. the standard modes I, II and a non-conventional rotation mode, which are considered in the numerical examples representing a typical interconnect. A comparison to the results from the Taylor assumption is made, which shows that the Taylor assumption method produces an overstiffening of the interface. 相似文献