Corner Singularities Research Articles

The influence of substrate stiffness on interfacial stresses for adhesive microfibrils

Bioinspired microfibrillar surfaces adhere by exploiting the presence of intermolecular forces at the contact interface. Experimental work has shown that compliant mushroom shaped fibrils can facilitate optimal adhesion across a range of substrates. This work evaluates numerically how fibril contact tip shape and the degree of elastic mismatch at the fibril-substrate interface influence adhesion. In a finite element simulation of fibrils subject to a remote stress, the interfacial stress distributions that define the detachment behaviour are computed for a range of fibril cap diameters, thicknesses, and substrate stiffnesses. Depending on the substrate stiffness, there is a singular stress field present at the edge of contact for smaller mushroom cap and straight punch fibrils while the stress at the corner for larger mushroom fibrils tends to zero. This variation in fibril-substrate interfacial stress has implications for the most probable defect type to initiate detachment and the stability of this crack growth. For fibrils within this corner singularity regime, the adhesion strength of a patch of fibrils is determined using interfacial fracture mechanics. The simulations are compared with experimental data of pull-off strengths and the biomedical applications of adhesive microfibrillar patches are considered.

Steady Two-Dimensional Conduction: Simple and Double Layer Potentials, Corner Singularities, and Induced Heat Flux

Abstract Two-dimensional steady-state heat conduction is possible outside closed boundaries on which two isothermal segments at different temperatures are separated by two adiabatic segments. Remarkably, previous research showed that the conduction shape factor for the region exterior to the boundary is equal to that for the interior region, despite the asymmetry and singularity of the boundary heat flux distributions. In this study, classical potential theory is used for the temperature and heat flux distributions as a combination of simple-layer and double-layer potentials, including the relationships between the values inside and outside the boundary curve. Isothermal boundaries exhibit an induced heat flux that varies from point-to-point on the boundary. The induced flux integrates to zero over each isothermal edge. Singularities of the heat flux are identified and resolved. Computations that validate the theory are provided for mixed boundary conditions on a disk and a square. Numerical fits to both the simple-layer and double-layer densities are given for the disk and the square. The analysis explains why the interior and exterior conduction shape factors are equal despite wildly differing heat flux distributions, and the results are compared to a previous study of this configuration. This paper also develops fundamental concepts of potential theory and can serve as a tutorial on the subject.

Local well-posedness of unsteady potential flows near a space corner of right angle

In this paper we are concerned with the local well-posedness of the unsteady potential flows near a space corner of right angle, which could be formulated as an initial-boundary value problem of a hyperbolic equation of second order in a cornered-space domain. The corner singularity is the key difficulty in establishing the local well-posedness of the problem. Moreover, the boundary conditions on both edges of the corner angle are of Neumann-type and fail to satisfy the linear stability condition, which makes it more difficult to establish a priori estimates on the boundary terms in the analysis. In this paper, extension methods will be updated to deal with the corner singularity, and, based on a key observation that the boundary operators are co-normal, new techniques will be developed to control the boundary terms.

Theoretical Analysis of Sheet Beam Electron Gun for Terahertz Vacuum Electron Devices

In the terahertz band, the vacuum electron devices with planar and miniaturized structures require the ultrahigh-precision description of high-density electron beams for formation and transportation with the theoretical analysis and simulation. Thus, the absence of the available theory for the sheet beam electron is put forward to investigate the 3-D spatial potential using a differential geometry and tensor analysis. The equation of electron beam potential can be derived by establishing a moving coordinate frame, carrying out the numerical calculations, and doing series expansion. To solve the complex sheet beam description of the existence of four corner singularities, the Riemann method and conformal mapping are adopted to overcome this difficulty. Then, a 0.22-THz vacuum electron tube using a sheet beam electron gun and corresponding optical system are designed. The beam voltage and current are 16.5 kV and 0.5 A, respectively, with a beam channel size of 1.4 mm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times0.14$ </tex-math></inline-formula> mm and a uniform magnetic field of 5980G. With the 3-D simulation, the high-density sheet electron beam can transport 18 mm successfully. The spatial potential of numerical calculation with the obtained sheet beam theoretical function and the simulation model are compared to verify the theoretical description. Both the results are with good consistency, which preliminarily proves the effectiveness of the theoretical function. This work will give the deep physical mechanics vision for the development of high-power and high-efficiency sheet beam terahertz tube in the future.

An error analysis of the finite element method of the Stokes equations with parameter based on the singular function expansion

AbstractIn this paper, we introduce a mixed finite element method of the Stokes system with parameter on a non‐convex polygon. This model problem is considered in the time‐discretization of the unstationary Stokes equations, and it is mentioned in Kweon's study that the velocity field and the pressure function are composed of singular parts and smoother remainders due to the corner singularities of the solution near each re‐entrant corner. We propose a stable numerical scheme to approximate the coefficients in singular part and the regular part of solution, based on the extraction formulae of the coefficients in singular part. Furthermore, to show the reliability of the proposed method, we derive the absolute error estimates of the approximations of coefficients, and the and error estimates of the finite element solution corresponding to the regular part.

Optimal multilevel adaptive FEM for the Argyris element

The main drawback for the application of the conforming Argyris FEM is the labourious implementation on the one hand and the low convergence rates on the other. If no appropriate adaptive meshes are utilised, only the convergence rate caused by corner singularities (Blum and Rannacher, 1980), far below the approximation order for smooth functions, can be achieved. The fine approximation with the Argyris FEM produces high-dimensional linear systems and for a long time an optimal preconditioned scheme was not available for unstructured grids. This paper presents numerical benchmarks to confirm that the adaptive multilevel solver for the hierarchical Argyris FEM from Carstensen and Hu (2021) is in fact highly efficient and of linear time complexity. Moreover, the very first display of optimal convergence rates in practically relevant benchmarks with corner singularities and general boundary conditions leads to the rehabilitation of the Argyris finite element from the computational perspective.

On the Existence and Uniqueness of an Rν-Generalized Solution to the Stokes Problem with Corner Singularity

We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to construct numerical methods for finding an approximate solution without loss of accuracy. In the paper, the existence and uniqueness of an Rν-generalized solution in weighted sets is proved.

New Numerical Approach for the Steady-State Navier–Stokes Equations with Corner Singularity

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

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.

Reconstructing the neutrally-buoyant particle flow near a singular corner

Reconstructing the neutrally-buoyant particle flow near a singular corner

On the Properties of Operators of the Stokes Problem with Corner Singularity in Nonsymmetric Variational Formulation

The weighted finite element method makes it possible to find an approximate solution of a boundary value problem with corner singularity without loss of accuracy. The construction of this numerical method is based on the introduction of the concept of an Rν-generalized solution for a boundary value problem with a singularity. In this paper, special weighted sets based on the corresponding operators from the definition of the Rν-generalized solution of the Stokes problem in a nonsymmetric variational formulation are introduced. The properties and relationships of these weighted sets are established.

Rates of Convergence for the Continuum Limit of Nondominated Sorting

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order, and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton-Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.

Computational semi-analytic code for stress singularity analysis

Problems of stress singularities in single or multi-material corners have been addressed by many authors over the years. Most of the authors presented closed-form corner-eigenequations for special cases, and often there is no easy way to check if the solution is correct. In this work, we present a general computational tool that can solve many different cases of stress singularity problems for multi-material corners under generalized plane strain. The semi-analytic code is based on the matrix formalism presented in Mantič et al. (1997, 2014); Barroso et al. (2003); Herrera-Garrido et al. (2022) and is developed in MATLAB. The following boundary conditions are implemented: stress-free, fixed, some restricted or allowed direction of displacements (defined either in the reference frame aligned with the cylindrical coordinate system or in an inclined reference frame), or frictional sliding. The following interface condition between two consecutive materials are implemented: perfectly bonded, and frictionless or frictional sliding. The code can analyze both open and closed (periodic) corners, composed of one or multiple materials with isotropic, transversely isotropic or orthotropic (with any orientation) constitutive laws. The code has proven to be a reliable, very accurate, robust and easy-to-use tool, which has been verified by comparing the results computed with those obtained by other authors. A summary of the corner singularity problems solved is presented. The results of the corner singularity analysis obtained by the code can be further used for prediction of crack onset at the corner tip by the Coupled Criterion of Finite Fracture Mechanics and FEM, see García and Leguillon (2012) and references therein.

On singularities of solution of the elasticity system in a bounded domain with angular corner points

This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξμ. We derive the transcendental equations for all 10 possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, that is, tangential and normal direction, respectively, which depends on ξμ. So a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ωcritical are listed such that for interior angles ω &lt; ωcritical the H2‐regularity of solution can be guaranteed. Moreover, the characterization of stress singularities in terms of the inner angle of a corner point is studied, and the regularity results are given.

Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner‐edge singularities

This article concerns with the thermoelastic corner‐edge‐type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted p‐Sobolev spaces on manifolds with corner‐edge singularities. Then, we prove the corner‐edge‐type Sobolev inequality, Poincar inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner‐edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo–Galerkin approximations to obtain the global weak solutions for the thermoelastic corner‐edge‐type system 1.1.

A diffusion Monte Carlo method for charge density on a conducting surface at non-constant potentials

Abstract We develop a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials. In the previous researches, last-passage Monte Carlo algorithms on conducting surfaces with a constant potential have been developed for charge density at a specific point or on a finite region and a hybrid BIE-WOS algorithm for charge density on a conducting surface at non-constant potentials. In the hybrid BIE-WOS algorithm, they used a deterministic method for the contribution from the lower non-constant potential surface. In this paper, we modify the hybrid BIE-WOS algorithm to a last-passage Monte Carlo algorithm on a conducting surface at non-constant potentials, where we can avoid the singularities on the non-constant potential surface very naturally. We demonstrate the last-passage Monte Carlo algorithm for charge densities on a circular disk and the four rectangle plates with a simple voltage distribution, and update the corner singularities on the unit square plate and cube.

Ritz Neural Network (RitzNN) Method for $H$ (curl) Problems

In this paper, we propose to use deep neural network method for numerically solving $H$ (curl) problems through the Ritz loss functional formulation, where discretization of a PDE is set up as an optimization problem through the natural minimization principles. A two-dimensional test problem shows that the resulting RitzNN method is capable of approximating $H$ (curl) problem with corner singularity accurately.

Error estimate of a finite element method for an optimal control problem with corner singularity using the stress intensity factor

We consider an optimal control problem for the Poisson equation on a non-convex polygonal domain with the corner singularity. Previously, we proposed a novel algorithm for the accurate numerical solution for the Poisson equation on a polygonal domain with the domain singularity. Then, we investigated the error estimate and its efficient procedure for the numerical algorithm. In this article, we propose an efficient algorithm and perform an error estimate for a distributed optimal control problem of the Poisson equation. The solutions of the optimality system with such singularity have singular decompositions: regular part plus singular part for each state variable and adjoint variable. The coefficient of the singular function is usually called stress intensity factor and can be computed by the extraction formula. We introduced a modified optimality system which has “zero” stress intensity factors using this stress intensity factor, from whose solutions we can compute very accurate solution of the original optimality system simply by adding a singular part. We give a precise error analysis and provide numerical results which justify the results therein.

The Detachment of an Inclined Micro-Pillar Adhered to a Dissimilar Substrate

Abstract We investigate the mechanics of the detachment of an inclined micro-pillar adhered to a dissimilar substrate when subjected to a combination of an axial load and end moment. When the micro-pillar has adhered to the substrate, singular stress fields exist at the bi-material corners. The order of singularity is estimated using asymptotic analysis. The first two terms in the asymptotic expansion lead to singular stress fields. The magnitude of the singularity is evaluated in terms of the elastic mismatch between the pillar and substrate and the micro-pillar inclination. The asymptotic stress due to the moment loading is more sensitive to the micro-pillar inclination when compared to that due to the axial loading. They are insensitive to the micro-pillar inclination when the micro-pillar is rigid when compared to the substrate. A short interfacial crack is further assumed to exist at the bi-material corner. This crack is embedded in the corner singularity region and is loaded by the singular fields due to axial and bending loads. A boundary layer analysis is performed on the singular zone to estimate the stress intensity factor when a short crack embedded in it is subjected to the singular fields. The stress intensity factors are also calculated for a long interfacial crack at the bi-material corner, which extends beyond the singular zone. By using the aforementioned results, we investigate the detachment of the inclined micro-pillar under the combination of an axial load and end moment.