Singular Corners Research Articles

A new three-dimensional rock strength criterion based on shape function in deviatoric plane

Rock strength criteria are the theoretical grounding of geotechnical design and stability estimation, the Mohr–Coulomb (MC) and Hoek–Brown (HB) criteria are the widely accepted criteria at present, due to their reasonability and unambiguous concept, however they overlook the effect of intermediate principal stress, and contain six singular corners in π plane. Aimed at overcoming those limitations, the MC and normal parabolic criterion (NPC) were improved to their 3D versions that lead to smooth and convex for a wide range of strength parameters. The extended 3D strength criteria coincide with corresponding original forms in the triaxial compression and triaxial extension states, which not only take intermediate principal stress into account, but also provide great convenient in numerical calculation. Multigroup of poly-axial strength datasets gathered from the references are used to check the prediction accuracy of the proposed 3D criteria by the least absolute deviation method. Research proved that the 3D NPC criterion has a relatively larger deviation on poly-axial strength data prediction, but the proposed 3D MC criterion can describe peak strength with low misfit for soft or hard rocks. Peak strength σ1 increases first and then decreases with the increase of σ2, whether increasing or decreasing σ2, both will result in rock failure. Moreover, the 3D MC can fit the poly-axial strength data well for lower or higher values of σ3, which strongly suggests the proposed 3D MC criterion is adequate. Applicability of the proposed strength criterion will be discussed in further research.

Resolving black-hole microstructure with new momentum carriers

All known horizonless black-hole microstate geometries correspond to brane sources that acquire a finite size, and hence break the spherical symmetry of the black hole. We construct, for the first time, solutions with zero horizon area that have the same charges as a three-charge F1-NS5-P Type-IIA black hole and preserve this spherical symmetry. The momentum of these solutions is carried by longitudinal D0-D4 density fluctuations inside the NS5-branes. We argue that these solutions should be interpreted as the long-throat limit of a family of smooth, horizonless microstate geometries, called superstrata, where such geometries degenerate. The existence of these geometries indicates that a finite-size horizon does not appear even in the singular corners of the moduli space of three-charge microstate geometries.

Lightning Stokes Solver

Gopal and Trefethen recently introduced “lightning solvers” for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner.

Singular enrichment for multi-material corners with application to assessing the risk of fracture in semiconductor devices

A computational method for multi-material corners that enables accurate estimations of generalized stress intensity factors with relatively coarse and uniform discretizations is described in this paper. Specifically, explicitly modeled geometries of material junctions, crack tips and debonded interfaces are isoparametrically and hierarchically enriched to construct approximations with the known local behavior. The analysis procedure is termed enriched isogeometric analysis since the enrichments are constructed directly (explicitly) on the parametric spline geometry of the vertex or crack face unlike in procedures where the influence of the enriching entity is implicitly inferred on the underlying mesh. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is first validated through analysis of a bi-material corner and growth of an inclined crack in a homogeneous solid. Next, a detailed analysis of a representative back end of line structure of a semiconductor chip with multiple singular corners is carried out to identify critical corners susceptible to crack initiation. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly refined coarse discretizations.

Computation of Tight Enclosures for Laplacian Eigenvalues

Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

Topology of three-dimensional steady cellular flow in a two-sided anti-parallel lid-driven cavity

The structure of the incompressible steady three-dimensional flow in a two-sided anti-symmetrically lid-driven cavity is investigated for an aspect ratio $\unicode[STIX]{x1D6E4}=1.7$ and spanwise-periodic boundary conditions. Flow fields are computed by solving the Navier–Stokes equations with a fully spectral method on $128^{3}$ grid points utilizing second-order asymptotic solutions near the singular corners. The supercritical flow arises in the form of steady rectangular convection cells within which the flow is point symmetric with respect to the cell centre. Global streamline chaos occupying the whole domain is found immediately above the threshold to three-dimensional flow. Beyond a certain Reynolds number the chaotic sea recedes from the interior, giving way to regular islands. The regular Kolmogorov–Arnold–Moser tori grow with increasing Reynolds number before they shrink again to eventually vanish completely. The global chaos at onset is traced back to the existence of one hyperbolic and two elliptic periodic lines in the basic flow. The singular points of the three-dimensional flow which emerge from the periodic lines quickly change such that, for a wide range of supercritical Reynolds number, each periodic convection cell houses a double spiralling-in saddle focus in its centre, a spiralling-out saddle focus on each of the two cell boundaries and two types of saddle limit cycle on the walls. A representative analysis for $\mathit{Re}=500$ shows chaotic streamlines to be due to chaotic tangling of the two-dimensional stable manifold of the central spiralling-in saddle focus and the two-dimensional unstable manifold of the central wall limit cycle. Embedded Kolmogorov–Arnold–Moser tori and the associated closed streamlines are computed for several supercritical Reynolds numbers owing to their importance for particle transport.

A fourth order block-hexagonal grid approximation for the solution of Laplace’s equation with singularities

The hexagonal grid version of the block-grid method, which is a difference-analytical method, has been applied for the solution of Laplace’s equation with Dirichlet boundary conditions, in a special type of polygon with corner singularities. It has been justified that in this polygon, when the boundary functions away from the singular corners are from the Hölder classes \(C^{4,\lambda}\), \(0<\lambda<1\), the uniform error is of order \(O(h^{4})\), h is the step size, when the hexagonal grid is applied in the ‘nonsingular’ part of the domain. Moreover, in each of the finite neighborhoods of the singular corners (‘singular’ parts), the approximate solution is defined as a quadrature approximation of the integral representation of the harmonic function, and the errors of any order derivatives are estimated. Numerical results are presented in order to demonstrate the theoretical results obtained.

The discrete compactness property for anisotropic edge elements on polyhedral domains

We prove the discrete compactness property of the edge elements of any order on a class of anisotropically refined meshes on polyhedral domains. The meshes, made up of tetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl. Sci. 21 (1998) 519–549]. They are appropriately graded near singular corners and edges of the polyhedron.

Failure criteria for rocks based on smooth approximations to Mohr–Coulomb and Hoek–Brown failure functions

Existing experimental evidence from true triaxial test (also referred to as poly-axial test) indicates that the influence of the intermediate principal stress (σ2) on rock strength is substantial. In order to consider this effect in rock engineering design, an adequate 3D failure criterion, which takes into account the effect of σ2, is required. Such a criterion should not only be a convex but also a smooth function of stress state; the latter to avoid singularities and to ensure the numerical stability. The most popular failure criteria used in rock mechanics, i.e., Mohr–Coulomb (M–C) and Hoek–Brown (H–B), both ignore the effect of σ2 and also have singular corners. Hence, in this paper, the M–C and H–B criteria are modified to their 3D versions based on geometric approximations of deviatoric sections that lead to smooth and convex surfaces for a wide range of strength parameters. The derived criteria inherit all the key properties of the original ones and are formulated in terms of stress invariants introduced by Nayak and Zienkiewicz. The latter provides an easy way to identify the strength parameters from non-linear multiple regression analysis using the poly-axial test data. In order to validate the proposed criteria, eight sets of poly-axial tests selected from literature are fitted. The results clearly show that the new criteria are quite accurate for all rock types considered. Finally, an extension of the proposed criteria to account for anisotropic rock conditions is discussed and an illustrative example is provided incorporating the functional forms of H–B and modified H–B criteria.

An efficient Wave Based Method for 2D acoustic problems containing corner singularities

The Wave Based Method is an efficient alternative prediction technique based on a Trefftz approach which, as compared to element-based methods, can be applied in a wider frequency range at a reasonable cost. This paper discusses the use of this method for the particular case of 2D acoustic problems where singularities are present in the corners of the acoustic domain. The conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners. This enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred. This is illustrated for various problem settings, including different combinations of both homogeneous and non-homogeneous boundary conditions and bounded as well as unbounded problems are considered.

Shock wave-induced vortex loops emanating from nozzles with singular corners

The focus of the current study is to examine experimentally the diffracted shock wave pattern and the consequent vortex loop formation, propagation, and decay from nozzles having singular corners. Non-intrusive qualitative and quantitative techniques: schlieren, shadowgraphy, and particle image velocimetry (PIV) are employed to analyze the induced flow-fields. Eye-shaped nozzles were used with the corner joints representing singularities. The length of the minor axes are a = 6 and 15 mm, with the major axis b = 30 mm for both cases. The experiments are performed for flow Reynolds numbers in the range 0.8 × 105 and 4.6 × 105. Air is used in both driver and driven sections of the shock tube.

A Refined Mixed Finite Element Method for Stationary Navier—Stokes Equations with Mixed Boundary Conditions Using Lagrange Multipliers

Abstract This paper is concerned with a dual mixed formulation of the Navier — Stokes system in a polygonal domain of the plane with mixed boundary conditions and its numerical approximation. The Neumann boundary condition is imposed using a Lagrange multiplier corresponding to the velocity field. Moreover, the strain tensor and the antisymmetric gradient tensor (vorticity), quantities of practical interest, are introduced as new unknowns. The problem is then approximated by a mixed finite element method. Quasi-optimal error estimates are finally obtained using refined meshes near singular corners.

Two-dimensional curl-conforming singular elements for FEM solutions of dielectric waveguiding structures

This paper proposes a curl-conforming singular element for modeling electromagnetic fields around singular points. Similar to the Nedelec types of regular vector elements, the space of the proposed singular elements consists of gradient and rotational subspaces. The proposed singular elements have arbitrary singularity orders that are precomputed analytically according to local geometry and material properties. The singularity orders of the gradient bases depend on the electric-field behavior; the rotational bases on magnetic-field behavior. Assigning integer singularity orders transforms singular elements into regular elements. Since the gradient subspace is properly modeled, the proposed singular elements are free from contamination by spurious modes. By including the singular elements in the solution space, deterioration of convergence rates often encountered with waveguides containing singular corners is avoided. Validation of the proposed singular elements is provided both theoretically in terms of the de Rham diagram and numerically by solving canonical singular dielectric waveguiding structures.

A FIRST STEP IN APPLYING THE SINC COLLOCATION METHOD TO THE NONLINEAR NAVIER-STOKES EQUATIONS

Different numerical approaches have been proposed in the past to solve the Navier-Stokes equations. Conventional methods have often relied on finite-difference, finite-element, and boundary-element techniques. Multigrid methods have been recently introduced because they help to obtain a faster convergence rate of the error residual. A difficulty plaguing numerical methods today is the inability to treat singularities at or near boundaries. Such difficulties become even more pronounced when coupled with the need to handle semi-infinite and infinite domains. Sinc-based numerical algorithms have the advantage of handling singularities, boundary layers, and semi-infinite domains very effectively. In addition, they typically require fewer nodal points and are proven to provide an exponential convergence rate in solving linear differential equations. This study involves a first step in applying the Sinc-based algorithm to solve a nonlinear set of partial differential equations. The example we consider arises in the context of a driven-cavity flow in two space dimensions. As such, the steady and incompressible Navier-Stokes equations are solved by means of two-dimensional Sinc collocation in conjunction with the primitive variable method and a pressure correction algorithm based on artificial compressibility. Simulations are also carried out using forward differences, central differences, and a commercial code. Results are compared with one another and with the Sinc-collocation approximation. It is found that the error in the Sinc-collocation approximation outperforms other solutions, especially near the singular corners of the cavity.

Joule heat in a two-dimensional electron gas exposed to a normal non-homogeneous magnetic field of a ‘chess’ configuration

We have calculated analytically the spatial distribution of the electric field and current density, and obtained the Joule heat generation rate in a two-dimensional electron gas (2DEG) subjected to a non-homogeneous magnetic field of a ‘chess’ configuration. The generation of the Joule heat from the 2DEG is mainly concentrated near the singular corners of each ‘chess’ field and tends to zero in other corners.

Accurate solutions for steady plane flow in the driven cavity. I. Stokes flow

The problem considered here is the steady, incompressible plane Stokes flow in a rectangular cavity generated by uniform translation of the upper wall. An exact analytical solution of the governing biharmonic equation is derived which not only contains the leading term of the required singularities at the upper corners, but also approximately satisfies the boundary conditions at all four walls. A standard numerical algorithm is employed to correct the small deviations in the boundary conditions satisfied by the analytical solution. This technique enables accurate computation of the solution uniformly throughout the domain; in particular, near the singular corners and in those regions where the flow is extremely weak, for example, in the secondary vortex regions of the deep cavity. The method is illustrated for the square cavity and also for a deep cavity with a depth-to-width ratio of five, and the results are compared with previous analytical and numerical solutions.