Singular Boundary Research Articles

An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application

An accurate numerical method of solving singular boundary value problems for the stationary flow of granular materials and its application

Regularized singular boundary method for calculating wave forces on three-dimensional large offshore structure

How to quickly calculate the singularity of the special Green's function at origin has been a difficult problem that has plagued ocean dynamics for many years. This letter proposes a regularized singular boundary method (RSBM) based on the special Green's function and origin intensity factor (OIF) technique. The source points of the RSBM only need to be arranged on the boundary of the obstacle. By linearly combining the OIF of the modified Helmholtz equation and the OIF of the Helmholtz equation, an explicit OIF suitable for three-dimensional ocean dynamics is derived. The derived OIF does not contain singular integrals and is easy to use. The RSBM replaces the Green's function at origin with the OIF. Therefore, it is no longer necessary to reuse the integral form of Green's function as basis function at origin. The computational efficiency and accuracy have been significantly improved. Numerical experiments show that the RSBM can accurately calculate the wave forces on large three-dimensional offshore structure.

On the numerical integration of singular initial and boundary value problems for generalised Lane–Emden and Thomas–Fermi equations

We propose a geometric approach for the numerical integration of singular initial and boundary value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

Confidence Contours: Uncertainty-Aware Annotation for Medical Semantic Segmentation

Medical image segmentation modeling is a high-stakes task where understanding of uncertainty is crucial for addressing visual ambiguity. Prior work has developed segmentation models utilizing probabilistic or generative mechanisms to infer uncertainty from labels where annotators draw a singular boundary. However, as these annotations cannot represent an individual annotator's uncertainty, models trained on them produce uncertainty maps that are difficult to interpret. We propose a novel segmentation representation, Confidence Contours, which uses high- and low-confidence ``contours’’ to capture uncertainty directly, and develop a novel annotation system for collecting contours. We conduct an evaluation on the Lung Image Dataset Consortium (LIDC) and a synthetic dataset. From an annotation study with 30 participants, results show that Confidence Contours provide high representative capacity without considerably higher annotator effort. We also find that general-purpose segmentation models can learn Confidence Contours at the same performance level as standard singular annotations. Finally, from interviews with 5 medical experts, we find that Confidence Contour maps are more interpretable than Bayesian maps due to representation of structural uncertainty.

Performance analysis and material distribution optimization for sound barriers using a semianalytical meshless method

AbstractWith the increase in car ownership, traffic noise pollution has increased considerably and is one of the most severe types of noise pollution that affects living standards. Noise reduction by sound barriers is a common protective measure used in this country and abroad. The acoustic performance of a sound barrier is highly dependent on its shape and material. In this paper, a semianalytical meshless Burton–Miller‐type singular boundary method is proposed to analyze the acoustic performance of various shapes of sound barriers, and the distribution of sound‐absorbing materials on the surface of sound barriers is optimized by combining a solid isotropic material with a penalization method. The acoustic effect of the sound‐absorbing material is simplified as the acoustical impedance boundary condition. The objective of optimization is to minimize the sound pressure in a given reference plane. The volume of the sound‐absorbing material is used as a constraint. The density of the nodes covered with the sound‐absorbing material is used as the design variable. The method of moving asymptotes was used to update the design variables. This model completely avoids the mesh discretization process in the finite element method and requires only boundary nodes. In addition, the approach also does not require the singular integral calculation in the boundary element method. The method is illustrated and validated using numerical examples to demonstrate its accuracy and efficiency.

The first passage problem of a stochastic wheelset system

In this paper, the first passage problem of a wheelset system under Gaussian white noise excitation is studied. Mainly from two aspects of analysis, one is the so-called first passage damage. The influence of noise intensity on mean first passage time and the sensitivity of yaw damper to wheelset system at different operating speeds are analyzed. The other type of damage is caused by the accumulation of fatigue damage represented by a physical quantity. In addition, according to the stochastic averaging method of quasi-non-integrable Hamiltonian system is proposed, the Hamiltonian function is approximated as a one dimensional Markov diffusion process dominated by Itoˆ equation. The global stability of wheelset system is analyzed by using singular boundary theory. The instability condition of the wheelset system is analyzed theoretically from the perspective of energy, and the response of the wheelset system under different running speeds is analyzed numerically, and the critical speed is determined. Numerical simulation verifies the correctness of theoretical analysis.

Acoustic simulation using singular boundary method based on loop subdivision surfaces: A seamless integration of CAD and CAE

This paper firstly combines the Burton-Miller-type singular boundary method (BMSBM) with the Loop subdivision surfaces (LSS) for the acoustic simulation of 3D complicated structures. As a modern modeling technique, the LSS provides a flexible and simple means to represent model surfaces by controlling meshes and subdivision rules. Thus, it can achieve arbitrary topology and multi-resolution structures. As a semi-analytical and boundary-type meshless technique, the BMSBM using the fundamental solutions is simple, accurate, and straightforward for solving acoustic radiation and scattering problems. The new approach constructs a geometry model using a subdivision scheme, and therefore can create a multi-resolution hierarchical model meeting various accuracy requirements without a repetitive meshing procedure. The proposed method effectively breaks through the technical bottleneck that the BMSBM makes it difficult to collate boundary nodes of complex structures and achieves a seamless integration between CAD and CAE. The numerical results of three examples demonstrated the effectiveness and accuracy of the proposed method for acoustic simulation. The present study provides an innovative idea for the subsequent optimization of materials and shapes of acoustic structures, which has the potential to enhance precision and efficiency significantly.

An efficient hybrid collocation scheme for vibro-acoustic analysis of the underwater functionally graded structures in the shallow ocean

In this paper, a novel hybrid collocation scheme based on the generalized finite difference method (GFDM) and the Burton–Miller singular boundary method (BMSBM) is developed to study the vibration and acoustic radiation characteristics of an underwater functionally graded (FG) structure immersed in the shallow ocean. By utilizing the individual strengths of both methods, the GFDM is adopted for the vibration analysis of the FG structures, while the BMSBM is applied for the acoustic wave radiation analysis. Numerical examples verify the accuracy and efficiency of the proposed hybrid collocation scheme for the natural frequency and acoustic radiation behavior of the underwater structures consisting of the functionally graded materials (FGMs). The influences of the material parameters, structural geometries and shallow ocean environment on the natural frequencies of the structural vibration and the underwater acoustic wave radiation field are investigated in detail. Numerical study shows that the variation trend of the natural frequencies of the structural vibration is consistent with that of the resonance peak frequencies of the underwater acoustic radiation field. The present results may provide valuable references for the design and application of underwater marine structures consisting of the FGMs.

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. Mathematics subject classification03B80, 65L10, 65L12

Numerical solution of third order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation

In this paper, we use the nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation methods for the numerical solution of singular third-order boundary value problems. The novelty of the approach is based on using the new non-classical weight function for sinc method instead of the classic ones. The convergence and error estimation of our methods is discussed. Several examples are solved and numerical results are compared with the existed methods, the obtained results demonstrate the validity of the obtained theoretical results and the efficiency of the our methods.

A Unified Trapezoidal Quadrature Method for Singular and Hypersingular Boundary Integral Operators on Curved Surfaces

A Unified Trapezoidal Quadrature Method for Singular and Hypersingular Boundary Integral Operators on Curved Surfaces

A pile–soil interaction model for ground-borne vibration problems based on the singular boundary method

An efficient three-dimensional approach for solving pile–soil interaction problems is proposed. In the approach, the soil is modelled as an elastic half-space, and its response in the presence of the pile’s corresponding cavity is computed by employing the singular boundary method. The pile is modelled analytically using the classic rod and Euler–Bernoulli beam theories. For the coupling with the soil, the pile is divided in a set of rigid segments that interact along their circumference with the soil. The methodology allows the rotational motions and reaction torques at these segments to be accounted for and their contribution in the accuracy of the scheme is assessed. A criterion to define the minimum number of collocation points that offers an acceptable trade-off between accuracy and numerical performance is also proposed. The method is validated against well-established methodologies and using the reciprocity principle that relates the wave radiation from the pile to the ground field with the incident wave problem due to a load on the ground surface. Results are shown for different soil stiffnesses and different pile length to diameter ratios. The employment of the singular boundary method is shown to provide strong computational advantages to detailed modelling approaches such as the three-dimensional finite element-boundary element method, as well as overcoming the fundamental limitations of plane-strain and axisymmetric methods.

A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation 1rβrβ1rβ(rβΘ′)′′′=12rβK11μ′Θ′2+2μΘ′Θ″+K12μ′φ′2+2μφ′φ″+λ1G1(r),1rβrβ1rβ(rβφ′)′′′=12rβK21μ′Θ′2+2μΘ′Θ″+K22μ′φ′2+2μφ′φ″+λ2G2(r), where λ1≥0 and λ2≥0 are two parameters, μ=pr2β−2,p∈R+, G1,G2∈L1[0,1] such that M1*≥G1(r)≥M1>0,M2*≥G2(r)≥M2>0 and K12>0, K11≥0, and K21>0, K22≥0 are constants that are connected by the relation (K12+K22)≥(K11+K21) and β>1. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation t=r1+β1+β to deduce the second-order singular boundary value problem. Also, for β=p=G1(r)=G2(r)=1, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonexistence. We also conclude bounds are symmetric with respect to two different transformations.

A Pair of Optimized Nyström Methods with Symmetric Hybrid Points for the Numerical Solution of Second-Order Singular Boundary Value Problems

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized technique is obtained after imposing the vanishing of some of the local truncation errors, which results in a set of symmetric hybrid points. By solving an algebraic system of equations, our proposed approach generates simultaneous approximations at all grid points, resulting in a highly effective technique that outperforms several existing numerical methods in the literature. To assess the efficiency and accuracy of our approach, we perform some numerical tests on diverse real-world problems, including singular boundary value problems (SBVPs) from chemical kinetics.

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions.

A novel semi-analytical meshless method for the thickness optimization of porous material distributed on sound barriers

This paper presents a novel semi-analytical meshless method to optimize the thickness of porous material distributed on sound barriers, by employing the Burton–Miller-type singular boundary method in conjunction with the method of moving asymptotes. Firstly, the Delany–Bazley–Miki model is utilized to characterize the acoustic property of sound barrier with a porous sound-absorbing material. Then, the sensitivity formula with respect to the thickness of the porous layer is derived based on the analytical computation and the adjoint variable formula, in which the design variable is a thickness parameter distributed between [0,1]. Finally, the optimal thickness distribution is achieved by solving the optimization model using the method of moving asymptotes. Compared to traditional algorithms, the proposed new method is simple, accurate, easy-to-program, and free of mesh and integration. Numerical experiments demonstrate the feasibility and effectiveness of the proposed algorithm.

Some novel analyses of the Caputo-type singular three-point fractional boundary value problems

Some novel analyses of the Caputo-type singular three-point fractional boundary value problems

Nonlinear interactions of two-kink-breather solution in Yu-Toda-Sasa-Fukuyama equation by modulated phase shift

The waveforms and nonlinear interactions of a two-kink-breather solution of the (2 + 1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation are studied by modulated phase shift. First, we obtain the parameter relations that respective affect the amplitudes of the kink and the breather solutions in kink-breather solution. Then, it is proved that the solutions in the regions near the singular boundaries of the phase shift can be divided into three kinds of solutions with repulsive or attractive interactions, in addition to the two-kink-breather solution. Interestingly, a breather soliton acts as a messenger to transfer energy during the repulsive interaction between the two kink-breather solutions with small amplitudes.

A Singularly Perturbed Dirichlet Problem for a Ring With Singular Boundaries

Uniform asymptotic expansions of small-parameter solutions of bisingular Dirichlet problems for a ring with any degree of accuracy are constructed by a generalized method of boundary functions and a small-parameter method. The investigated problems have two features: equations with a small parameter in the first derivatives and external solutions simultaneously have increasing features on the boundaries of the domain, i. e. limit equations have singularities at the same time on both boundaries of the ring. Formal asymptotic expansions are based on the maxima principle and the method of differential inequalities. The resulting asymptotic series are represented by the Puiseux series. The main term of the asymptotic expansions of the solutions has a negative fractional power with respect to the small parameter.