Non-singular Part Research Articles

Investigation of Buckling Behavior of Cracked FG Cylindrical Panels Reinforced by Graphene Platelets

The buckling behavior of a functionally graded graphene-platelet-reinforced composite (FG-GPLRC) was traditionally investigated, mostly with respect to its undamaged structures. In this context, the current study investigated the buckling behavior of an FG-GPLRC cylindrical panel with an anti-symmetric central crack by introducing a 2-D extended natural element method (XNEM). The displacement was basically expressed with the first-order shear deformation theory (FSDT) and approximated using Laplace interpolation functions (for the non-singular displacement part) and crack-tip singular functions (for the singular displacement part) without grid refinement around the crack tips. The complex numerical manipulation on the curved shell surface was resolved by geometrically transforming the curved shell surface to a 2-D planar rectangular NEM grid. The painstaking numerical locking was suppressed by adopting the concept of a stabilized MITC3+ shell element. The validity of the developed numerical method was examined through a benchmark test, and the fundamental buckling loads of cracked FG-GPLRC cylindrical panels were investigated in depth by changing the major parameters. The numerical results also included a comparison with the FG-CNTRC. The numerical results indicated that the developed numerical method effectively predicts the buckling loads with reasonable accuracy, and that the fundamental buckling load of cracked FG-GPLRC cylindrical panels are remarkably influenced by the inclination angle and length of the crack as well as the other associated parameters.

Rf cavity point-charge wakefield computation through a hybrid analytical and numerical approach

We study short-range geometric wakefields of a normal conducting third harmonic cavity, developed by ALBA, which is currently under investigation for the BESSY II main storage ring upgrade. As is commonly done to enhance the shunt impedance, the single cell cavity shape includes smooth nose-cone transitions to the outside beam pipe, which significantly affects the wakefields. Utilizing the general approach developed by the earlier work [B. Podobedov and G. Stupakov, Phys. Rev. ST Accel. Beams 16, 024401 (2013)], a model of point-charge wakefield for cavities of this type is suggested. The wakefield model consists of two parts: the singular part, inherently difficult for numerical codes, is found analytically, and the nonsingular part, which could be retrieved from relatively nondemanding numerical wakefield computations. Based on this hybrid approach, the wakefields due to arbitrary short bunches can be readily obtained both efficiently and accurately. The model was cross-checked against direct numerical simulations and a good agreement was found within the entire range of bunch lengths relevant to potential applications, from sub-millimeter down to $1\text{\ensuremath{-}}\ensuremath{\mu}\mathrm{m}\text{ }\text{ }\mathrm{rms}$. We also observed that the wakefield model of a single cavity applies equally well to a small number of coupled cavities after some straightforward parameter scaling.

Bulk viscosity of hard sphere fluids by equilibrium and nonequilibrium molecular dynamics simulations.

The bulk viscosity, ηb, of the hard sphere (HS) fluid is computed by equilibrium and nonequilibrium molecular dynamics (NEMD) simulations, the latter using an adaptation of the time-stepping method for continuous potential systems invented by Hoover et al. [Phys. Rev. A 21, 1756 (1980)], which employs an imposed cyclic density variation on the system by affine scaling of the particle coordinates. The time-stepping method employed for HS is validated against exact event-driven hard sphere methodology for a series of equilibrium quantities over a wide density range, including the pressure, singular parts of the hard sphere viscosities, and the nonsingular parts of the shear viscosity time correlation functions. The time steps used are typically only a little smaller than those employed in continuous potential simulations. Exact pressure tensor fluctuation expressions are derived for the singular (or infinite limiting frequency) equilibrium parts of the viscosities, which were employed in the simulations. The values obtained agree well with the predictions of the Enskog theory for all densities considered. The bulk viscosity obtained by NEMD is shown to be noticeably frequency dependent for densities in excess of ∼0.8, decaying approximately exponentially to the Enskog and equilibrium simulation values at all densities considered for frequencies in excess of ∼5 in hard sphere units. Temperature profiles during the cycle and the effects of strain amplitude on the computed frequency dependent bulk viscosity are presented. The bulk viscosity increases with the maximum density amplitude.

Planar potential flow on Cartesian grids

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

Pansu–Wulff shapes in ℍ1

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

The semianalytical analysis of nearly singular integrals in 2D potential problem by isogeometric boundary element method

AbstractBenefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density.

Determination of stress intensity factors of V-notch structures by characteristic analysis coupled with isogeometric boundary element method

To describe the physical fields of the V-notch structure accurately, the structure is first divided into two parts, the notch tip sector containing singular stress field and the non-singular remained part. A twofold procedure is then carried out to obtain the stress intensity factors: (I) the asymptotic expansions are introduced to describe physical fields of the singular region and transform the elastic governing equation to the characteristic ordinary differential equations. The interpolating matrix method is then implemented to provide stress singular orders and related characteristic angular functions of this singular area. (II) The boundary integral equation for the left part is then discretized by the non-uniform rational B-splines (NURBS) elements and is solved under the isogeometric framework. The amplitudes of the asymptotic expansions in process (I) are calculated by coupling procedures (I) and (II) through utilizing the interfacial continuity conditions. The stress intensity factors are finally obtained and compared with the referenced results for both symmetric and inclined V-notch structures. Benefiting from the isogeometric analysis, satisfied results can be provided by discretizing the artificial arc boundary between the singular and non-singular parts with fewer NURBS elements. This method can also be integrated with the preprocessing of computer aided design (CAD) conveniently.

Localization of Bott‐Chern classes and Hermitian residues

We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies in a unified way and obtain a long exact sequence involving the three. We then study the localization problem of characteristic classes in the relative Bott-Chern cohomology. For this we define the cup product and integration in our framework and we discuss local and global duality homomorphisms. After reviewing some materials on connections, we give a vanishing theorem relevant to our localization. With these, we prove a residue theorem for a vector bundle admitting a Hermitian connection compatible with an action of the non-singular part of a singular distribution. As a typical case, we discuss the action of a distribution on the normal bundle of an invariant submanifold (so-called the Camacho-Sad action) and give a specific example.

Efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels

In this paper, we present a special Clenshaw–Curtis Filon (CCF) type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels. The non-oscillatory and nonsingular part of the integrand is replaced by a special Hermite interpolation polynomial. Error bounds with respect to the frequency k and the number of the Clenshaw–Curtis points N are considered. The overall computational complexity for the scheme is O(Nlog (N)). Numerical experiments support the theoretical analysis.

Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals

This paper introduces and analyzes quadrature rules and asymptotic expansions of a few highly oscillatory infinite integrals. We first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter ω, which clarify the large ω behavior of these integrals. Then, based on the resulting asymptotic expansions, two different interpolatory quadrature rules are given. One is the so-called Filon-type methods based on standard Hermite interpolation of the non-oscillatory and non-singular part of the integrands at equidistant nodes. The other is the Filon–Clenshaw–Curtis-type method (FCC) by using special Hermite interpolation at N+1 Clenshaw–Curtis points and the fast computation of modified moments. The interpolation coefficients needed in the FCC method, can be computed by a numerically stable algorithm in O(Nlog N) operations based on fast Fourier transform (FFT). The required modified moments, can be accurately and efficiently calculated by some recurrence relation formulae. Moreover, for these quadrature rules, their error analyses in inverse powers of the frequency ω, are provided. The presented methods share the advantageous property that the accuracy improves greatly, for fixed N, as ω increases. Numerical examples show the accuracy and efficiency of the proposed methods.

The semi-analytical evaluation for nearly singular integrals in isogeometric elasticity boundary element method

Benefiting from improvement of accuracy in modeling complex geometry and integrity of discretization and simulation, the isogeometric analysis in the boundary element method (IGABEM) has now been implemented by several groups. However, the difficulty of evaluating the nearly singular integral in IGABEM for elasticity has not yet been effectively solved, which will hinder the application of IGABEM in engineering structure analysis. Herein, the nearly singular integrals in IGABEM are separated to the non-singular part and singular part by the subtraction technique. The integral kernels in singular part are approximated by the Taylor series polynomial expressions, in which different orders of derivatives are interpolated by the non-uniform rational B-splines (NURBS). Furthermore, the analytical formulations for the singular part with the approximated kernels are derived by a series of integration by parts, while the non-singular part is calculated with Gaussian quadrature. In this way, a semi-analytical method is proposed for the nearly singular integrals in the IGABEM. Comparing with the conventional IGABEM, the present method can yield accurate displacement and stress for inner points much closer to the boundary. It can obtain effective results with fewer elements than the finite element method because of the precise simulation of geometry and boundary-only discretization.

On the convergence rate of DGMRES

The DGMRES algorithm was designed to compute the Drazin inverse solution of consistent or inconsistent linear systems Ax=b, where A is a square singular matrix with arbitrary index. In this paper we compare the convergence rate of DGMRES with that of GMRES applied to the nonsingular part of A.

On the Induction of the First-Order Phase Magnetic Transitions by Acoustic Vibrations in MnSi

The main result of the paper contains the conclusion that the magnetic phase transition in MnSi always remains first order at any temperature and magnetic field. In these aims, a model of coupling of an order parameter with other degrees of freedom is used. The coupling of magnetic order parameters with long-wave acoustic phonons, in the presence of the nonsingular parts of the bulk and shear moduli, a first-order transition occurs, participle near the transition the heat capacity and the compressibility remain finite, if the heat capacity becomes infinite in the system disregarding the acoustic phonons. The role of the Frenkel heterophase fluctuations is discussed. The impurity effect shows that, for some phases, the heat capacity of the system remains continuous and finite at the transition point. It is supposed that the transition is progressively smoothed by these fluctuations at the application of the magnetic field.

On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N \log N)$ operations, which requires the accurate computation of modified moments. We first give a method for the derivation of recurrence relation for the modified moments, which can be applied to the derivation of recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using the recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.

A combined dislocation fan-finite element (DF-FE) method for stress field simulation of dislocations emerging at the free surfaces of 3D elastically anisotropic crystals

A combined dislocation fan-finite element (DF-FE) method is presented for efficient and accurate simulation of dislocation nodal forces in 3D elastically anisotropic crystals with dislocations intersecting the free surfaces. The finite domain problem is decomposed into half-spaces with singular traction stresses, an infinite domain, and a finite domain with non-singular traction stresses. As such, the singular and non-singular parts of the traction stresses are addressed separately; the dislocation fan (DF) method is introduced to balance the singular traction stresses in the half-spaces while the finite element method (FEM) is employed to enforce the non-singular boundary conditions. The accuracy and efficiency of the DF method is demonstrated using a simple isotropic test case, by comparing it with the analytical solution as well as the FEM solution. The DF-FE method is subsequently used for calculating the dislocation nodal forces in a finite elastically anisotropic crystal, which produces dislocation nodal forces that converge rapidly with increasing mesh resolutions. In comparison, the FEM solution fails to converge, especially for nodes closer to the surfaces.

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

The collinearly-improved Balitsky–Kovchegov equation

The high-energy evolution in perturbative QCD suffers from a severe lack-of-convergence problem, due to higher order corrections enhanced by double and single transverse logarithms. We resum double logarithms to all orders within the non-linear Balitsky-Kovchegov equation, by taking into account successive soft gluon emissions strongly ordered in lifetime. We further resum single logarithms generated by the first non-singular part of the splitting functions and by the one-loop running of the coupling. The resummed BK equation admits stable solutions, which are used to successfully fit the HERA data at small x for physically acceptable initial conditions and reasonable values of the fit parameters.

Resumming large higher-order corrections in non-linear QCD evolution

Linear and non-linear QCD evolutions at high energy suffer from severe issues related to convergence, due to higher order corrections enhanced by large double and single transverse logarithms. We resum double logarithms to all orders by taking into account successive soft gluon emissions strongly ordered in lifetime. We further resum single logarithms generated by the first non-singular part of the splitting functions and by the one-loop running of the coupling. The resulting collinearly improved BK equation admits stable solutions, which are used to successfully fit the HERA data at small-x for physically acceptable initial conditions and reasonable values of the fit parameters.

The 3-Block KSOR Method for Full Rank Rectangular Systems

A new version of the KSOR method is considered to introduce the least squares solution of a full rank over determinant system of linear algebraic equations. The treatment depends on introducing an augmented non-singular square system through splitting the coefficient matrix <i>A</i> into two matrices <i>A₁, A₂</i> with non-singular part, <i>A₁</i>. Accordingly, a new version of the 3-block SOR method is introduced, the 3-block KSOR method. Selection of the relaxation parameter which guarantees the convergence in the sense of reducing the spectral radius of the iteration matrix is considered. Application of the theoretical results to a numerical example has confirmed the expected behaviour of the 3-block KSOR method.