Corner Boundary Research Articles

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Boundary conforming mesh to T-NURCC surface conversion

We present a method for converting an open triangle mesh, whose boundary is constrained to a set of smooth curves, to a T-NURCC surface approximating seamlessly both the mesh and the curve geometry. The curves represent boundaries of B-Rep faces adjacent to the mesh and include intersections, trim-curves, patch boundaries, etc. This conversion problem arises in generative design and topology optimization, where “organic” meshes are synthesized by a physics solver to complement an existing mechanical B-Rep model. It can also occur in reverse engineering, where a scanned mesh is converted to a collection of adjacent analytical and free-form surfaces.The core of our algorithm is a novel transfinite interpolation construction defining a C0 patch network interpolating the curves at the boundaries and coincident with the mesh in the interior. We convert this network to a T-NURCC surface by applying an approximation algorithm designed to produce a very accurate boundary fit and reproduce exactly all boundary corners. These properties allow a B-Rep modeling kernel to stitch the output T-NURCCs with adjacent B-Rep faces into a watertight, low tolerance B-Rep CAD model. We demonstrate our method’s results and applicability on several examples synthesized by topology optimization and generative design applications.

The mineralogy, mineral chemistry, and origin of the Wuyang banded iron formations, North China Craton

Several banded iron formations (BIFs) occur in Precambrian strata of the North China Craton (NCC). Two sub-types of BIF occur in the Wuyang area, quartz-magnetite (QMB) and pyroxene-magnetite BIFs (PMB). However, the genesis of pyroxene and the how the BIFs formed are not well understood. Magnetite grains in QMB are relatively large with a straight boundary and sharp corners, and display a pronounced alignment, whereas magnetite in PMB is disseminated and has a smaller grain size. X-ray diffraction (XRD) analyses indicate that the crystal parameters of magnetite in QMB are: a = b = c = 8.396 Å and z = 8, whereas those of the magnetite in PMB are: a = b = c = 8.394 Å and z = 8. These parameters are consistent with magnetite forming in sedimentary iron formation. Based on scanning electron microscopy and energy dispersive spectroscopy (EDS) results, magnetite of both BIFs possesses high stabilities of geochemical property. Compared with magnetite in PMB, magnetite in QMB has higher concentrations of SiO2, Na2O, Al2O3, CaO, K2O, MgO, Sc, Co, Y, Nb, Ta and U. The enrichment of these elements is consistence with their derivation from seawater due to the strong adsorption capability of precursor iron hydroxide. A spider diagram of trace elements reveals that magnetite in QMB is enriched in Ta, whereas magnetite of PMB is depleted in Ce. Based on Raman and electron probe micro-analysis of pyroxene in PMB, synchronous and compositional zoning are found in the degrees of crystallinity (FWHM values of Gaussian fitting for Raman shift next to 1014 cm−1) and the concentration of SiO2, Al2O3, MgO, MnO, CaO and FeO of pyroxene from core to rim. The two types of BIF formed in different sedimentary environments from different precursors. The precursor to the QMB BIF is almost pure chemical sediment, whereas a that for PMB contains a volcanic ash component. The QMB was likely deposited during a period of hydrothermal sedimentation in the Wuyang basin that was associated with Neoarchean magmatism activity, while the PMB was deposited during periods of volcanic paroxysm.

Algebraic entropy of (integrable) lattice equations and their reductions

We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler–Bobenko–Suris equations and Viallet’s and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonlinearities

A singularly perturbed parabolic equation $${{\varepsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\varepsilon )$$ is considered in a rectangle with the boundary conditions of the first kind. At the corner points of the rectangle, the monotonicity of the function $$F$$ with respect to the variable $$u$$ in the interval from the root of the degenerate equation to the boundary value is not required. The asymptotic approximation of the solution is constructed under the assumption that the principal term of the corner part exists. A complete asymptotic expansion of the solution as $$\varepsilon \to 0$$ is constructed, and its uniformity in a closed rectangle is proved.

Novel Boundary Edge Detection for Accurate 3D Surface Profilometry Using Digital Image Correlation

Digital image correlation (DIC) has emerged as a popular full-field surface profiling technique for analyzing both in-plane and out-of-plane dynamic structures. However, conventional DIC-based surface 3D profilometry often yields erroneous contours along surface edges. Boundary edge detection remains one of the key issues in DIC because a discontinuous surface edge cannot be detected due to optical diffraction and height ambiguity. To resolve the ambiguity of edge measurement in optical surface profilometry, this study develops a novel edge detection approach that incorporates a new algorithm using both the boundary subset and corner subset for accurate edge reconstruction. A pre-calibrated gauge block and a circle target were reconstructed to prove the feasibility of the proposed approach. Experiments on industrial objects with various surface reflective characteristics were also conducted. The results showed that the developed method achieved a 15-fold improvement in detection accuracy, with measurement error controlled within 1%.

Significant influence of sharp grain boundary corner on tensile elongation of copper bars with columnar grains and its mechanism

Abstract The reason why elongation of copper bars with columnar grains drops significantly after small cold-drawing was explored. The copper bars were prepared by warm-mould continuous casting. Tensile test was interrupted at various tensile strains in order to detect crack origin. Electron backscattered diffraction (EBSD) was used to analyze dislocation slip bands. It is found that the as-cast microstructure contains sharp grain boundary (GB) corners nearly parallel to the solidification direction (SD). Elongation of the copper bars drops significantly from 68.8% in as-cast state to 18.8% in as-drawn state. It is revealed that plastic deformation becomes severer in the vicinity of sharp GB corners. Locally accumulated internal stress even activates a slip system with very low Schmid factor of 0.17. The localized plastic deformation near sharp grain boundary corner promotes crack initiation and propagation, which eventually leads to the significant drop of elongation.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Monotonic Nonlinearity

For a singularly perturbed parabolic equation $${{\epsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\epsilon )$$ in a rectangle, a problem with boundary conditions of the first kind is considered. It is assumed that, at the corner points of the rectangle, the function $$F$$ is cubic in the variable $$u$$ . A complete asymptotic expansion of the solution at $$\varepsilon \to 0$$ is constructed, and its uniformity in a closed rectangle is substantiated.

Singularly Perturbed System of Parabolic Equations in the Critical Case

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

Revised Three-Dimensional Navier-Stokes Characteristic Boundary Conditions for Intense Reactive Turbulence

The three-dimensional Navier-Stokes characteristic boundary conditions (3D-NSCBC), although physically reasonable and popular in many applications, may encounter the instability problem in simulating complex flows, especially for large Reynolds number reactive turbulence where locally the strong reversed flow appears at the outflow boundary surfaces. In the present work, a revised 3D-NSCBC strategy is proposed based on the kinematic relation in different moving coordinate systems. Following this strategy, a systematic formulation is presented for the outflow surface with local reversed flow and can be easily extended to the coupled edge and corner boundaries. Direct numerical simulation (DNS) tests of flow with different turbulence intensities are carried out. Compared with the conventional 3D-NSCBC, the newly proposed method exhibits satisfactory performance to confine numerical instability in the strong reversed flow region. The results confirm the robustness and effectiveness of this newly proposed algorithm.

Conjugate natural convection of Al2O3–water nanofluid in a square cavity with a concentric solid insert using Buongiorno’s two-phase model

The problem of conjugate natural convection of Al2O3–water nanofluid in a square cavity with concentric solid insert and isothermal corner boundaries using non-homogenous Buongiorno’s two-phase model is studied numerically by the finite difference method. An isothermal heater is placed on the left bottom corner of the square cavity while the right top corner is maintained at a constant cold temperature. The remainder parts of the walls are kept adiabatic. Water-based nanofluids with Al2O3 nanoparticles are chosen for the investigation. The governing parameters of this study are the nanoparticle volume fraction (0 ≤ ϕ ≤ 0.04), the Rayleigh number (102 ≤ Ra ≤ 106), thermal conductivity of the solid block (kw=0.28, 0.76, 1.95, 7 and 16) (epoxy: 0.28, brickwork: 0.76, granite: 1.95, solid rock: 7, stainless steel: 16) and dimensionless solid block thickness (0.1 ≤ D ≤ 0.7). Comparisons with previously experimental and numerical published works verify good agreement with the proposed method. Numerical results are presented graphically in the form of streamlines, isotherms and nanoparticles volume fraction as well as the average Nusselt number and fluid flow rate. The results show that the thermal conductivity ratio and solid block size are very good control parameters for an optimization of heat transfer inside the partially heated and cooled cavity.

Singularly Perturbed Parabolic Problems with Multidimensional Boundary Layers

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter e multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x1, x2) = 0 and tends to zero as e → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.

Length effects on interactive buckling in thin-walled rectangular hollow section struts

A variational model formulated using analytical techniques describing the nonlinear coupling between local and global buckling modes within an elastic thin-walled rectangular hollow section strut is presented. A system of nonlinear differential and integral equations subject to boundary conditions is derived and solved using numerical continuation techniques. The nonlinear behaviour of four representative lengths is investigated, which are characterized by the post-buckling equilibrium paths. The numerical results from the variational model are validated using a nonlinear finite element model and largely show excellent comparisons, particularly for the practically important ultimate load and the initial post-buckling behaviour. Boundaries for the four distinct length-dependent zones are identified and the most unstable zone is demonstrated to have a considerably narrower length range than previously determined for practical corner boundary conditions within the cross-section.

Mechanism and Morphology of Formation of Micropores in the Structure of DC Cast AlMgSi Alloy

The phenomena defects in the triple junction grain boundaries affect the properties of the microstructure of aluminum alloys. For that purpose the microstructure of the alloy EN AW 6060 on the billet sample cast by DC (Direct Chill) technology was analysed in order to gain insight into the origin and form pores in the final stages of solidification. Mechanisms of solidification shrinkage, thermal contraction and low permeability of interdendritic channels networks were discussed in moment when melt becomes isolated in separate locations forming new structure. Under such conditions, the tensile stress caused by anisotropic thermal contraction of a coherent dendrites network, causing the formation of pores in the corners of the grain boundaries whose morphology is determined by SEM / EDS analysis.

Features of development process displacement of earth’s surface when dredging coal in Eastern Donbas

The results of studies of the process of the earth’s surface displacement due to the influence of the adjacent longwalls are presented. It is established that the actual distributions of soil subsidence in the fall and revolt of the reservoir with the same boundary settlement processes differ both from each other and by the distribution of subsidence, recommended by the rules of structures protection. The application of the new boundary criteria – the relative subsidence of 0.03 – allows one to go from two distributions to one distribution, which is also different from the sedimentation distribution of protection rules. The use of a new geometrical element – a virtual point of the mould – allows one to transform the actual distribution of subsidence in the model distribution of rules of constructions protection. When transforming the curves of subsidence, the boundary points vary and, consequently, the boundary corners do.

Asymptotic of weak solutions for linear elliptic nonlocal Robin problem without Dini-continuity condition in a plane angle domain

We investigate the behaviour of weak solutions to the nonlocal Robin problem for linear elliptic divergence second order equations in a neighbourhood of the boundary corner point. We find the exponent of the solution decreasing rate under the assumption that the leading coefficients of the equations do not satisfy the Dini-continuity condition.

Quantitative description of non-equilibrium turbulent phenomena in compressors

A description of non-equilibrium turbulence in compressors is introduced by using the equation of Lagrangian velocity gradient correlation. This description is consistent to other traditional definitions in homogeneous isotropic turbulence and can be easily extended to complex anisotropic flows in compressors. We then show by post-processing in a linear cascade, that the non-equilibrium phenomena are obvious in the regions of corner separation, wake and boundary layer. Theoretical predictions show good agreement in magnitude with numerical results. We also show that the non-equilibrium phenomena cannot be eliminated even when the flow is far downstream, which indicates the importance of non-equilibrium phenomena in compressors. The aim of the present contribution is to provide a method to quantitatively describe the non-equilibrium phenomena in compressors, and to inspire future improvement on turbulence models by considering these non-negligible non-equilibrium properties in the future.

A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection–diffusion problems

A higher-accuracy discontinuous Galerkin method with Lagrange multipliers (DGLM) is presented for the solution of the advection–diffusion equation with a spatially varying advection field in the high Péclet number regime, where the classical polynomial finite element method (FEM) produces spurious oscillations in the solution at practical mesh resolutions. The proposed DGLM method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes. It overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free-space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by Prandtl’s boundary layer theory. These enrichment functions are capable of resolving exponential, parabolic, and corner boundary layers at relatively coarse mesh resolutions. The proposed method enforces a weak continuity of the solution approximation across inter-element boundaries using polynomial Lagrange multipliers, which makes it a hybrid method. However, unlike other hybrid methods, it operates directly on the second-order form of the advection–diffusion equation and does not require any stabilization. Its intrinsic performance and its superiority over the higher-order polynomial FEM are demonstrated for several test problems at Péclet numbers ranging from one thousand to one billion.

Boundary layer mesh generation on arbitrary geometries

SummaryVolume boundary layer mesh generation on non‐smooth geometries with multiple normals is considered in this work. A new highlight is given to corner and ridge boundary layer mesh generation through the generalized Voronoi diagram in general, and the spherical Voronoi diagram in particular. This provides the keystone to allow for the first time boundary layer mesh generation at arbitrary corner configurations. The work proposed in is revisited and compared with the new approach. A detailed description of the new algorithm is provided. The spherical Voronoi diagram delivers geometrically the optimal normals as well as topologically the optimal connectivities, as far as normality is concerned. Numerical examples illustrate the accuracy and robustness of the method. This approach seems to handle arbitrary geometry boundary layer mesh generation, in theory as well as in practice. Copyright © 2017 John Wiley & Sons, Ltd.

Краевая задача для уравнения эллиптического типа в области с углом в математическом моделировании магнитных систем

Modern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demands solving a nonlinear boundary-value problem of magnetostatic. The region in which the boundary-value problem is solved, consists of two sub-domains: a domain of vacuum and a domain of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle. Thereby, the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirements, the corresponding solution of the boundary value problem will have a limited gradient. In this paper, an upper estimate of maximum possible growth of the magnetic field in the corner domain is given. In terms of this estimate, a method of condensing the differential mesh near the corner domain is proposed. This work represents an algorithm of constructing an adaptive mesh in the domain with a boundary corner point of ferromagnetic taking into account the character of behavior of the solution of the boundary value problem. An example of calculating a model problem in the domain containing a corner point is given.