Natural Boundary Element Method Research Articles

Coupling of direct discontinuous Galerkin method and natural boundary element method for exterior interface problems with curved elements

Coupling of direct discontinuous Galerkin method and natural boundary element method for exterior interface problems with curved elements

The Natural Boundary Element Method of the Uniform Transmission Line Equation in 2D Unbounded Region

Herein, we are mainly concerned with the natural boundary element (NBE) method of the uniform transmission line (UTL) equation defined in the two-dimensional (2D) boundless region, which has a real physical background. We first create the time semi-discretized scheme of the UTL equation, as well as analyze the convergence and stability for the series of time semi-discretized solutions. Then, we create a fully discretized NBE format by means of a natural boundary reduction and analyze the stability and errors between the fully discretized NBE solutions and the analytical solution. Lastly, we employ two numerical examples to verify the effectiveness of the NBE method.

A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

In this article, we devote ourselves to the research of order reduction of natural boundary element (NBE) based on a proper orthogonal decomposition (POD) for the parabolic equation in the two-dimensional (2D) unbounded domain. For this purpose, we first build a NBE format for the parabolic equation in the 2D unbounded domain and discuss the existence, stability, and convergence of the NBE solutions. And then, we build a reduced-order NBE extrapolated (RONBEE) format based on POD, analyze the errors between the classical NBE and RONBEE solutions, and supply the implementation procedure for the RONBEE format. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the RONBEE format are further verified.

A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.

The Schwartz Alternating Algorithm for Solving 3D Exterior Helmholtz Problems

Based on the natural boundary reduction, an overlapping domain decomposition method is discussed for solving exterior Helmholtz problem over a three-dimensional (3D) domain. By introducing two different artificial boundaries, the original unbounded domain is divided into a bounded subdomain and a typical unbounded region, and a Schwartz alternating method is presented. The finite element method and natural boundary element method are alternately applied to solve the problems in the bounded subdomain and the typical unbounded subdomain. Moreover, the convergence of the Schwartz alternating algorithm is studied. Finally, some numerical experiments are presented to show the performance of this method.

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and analyze the errors between the exact solution and the fully discretized NBE solutions. Finally, we use some numerical experiments to verify that the NBE method is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

Temperature field distribution of coal seam in heat injection

In this article, we present a natural boundary element method (NBEM) to solve the steady heat flow problem with heat sources in a coal seam. The boundary integral equation is derived to obtain the temperature filed distribution of the coal seam under the different injecting conditions.

Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries.

The Schwarz alternating algorithm, which is based on natural boundary element method, is constructed for solving the exterior anisotropic problem in the three-dimension domain. The anisotropic problem is transformed into harmonic problem by using the coordinate transformation. Correspondingly, the algorithm is also changed. Continually, we analysis the convergence and the error estimate of the algorithm. Meanwhile, we give the contraction factor for the convergence. Finally, some numerical examples are computed to show the efficiency of this algorithm.

Thermal stress singularity analysis for V-notches by natural boundary element method

The accuracy of thermal stresses at internal points by the conventional boundary element method becomes deteriorate when the points are approaching to the boundary due to the inaccuracy of the calculation of nearly singular integrals by the Gaussian integration. Herein, a thermal stress natural boundary integral equation is proposed, in which the nearly hyper-strongly singular integral is reduced to a nearly strongly singular one and then dealt by the regularization method. Thus, it can be applied to model the high stress gradient in the vicinity very close to the V-notch vertex. The stress method is subsequently introduced to calculate the stress singularity orders and thermal stress intensity factors once the thermal stresses along the bisector and very close to the notch tip are yielded. The mathematical and physical singularity difficulties, i.e., the evaluation of nearly singular integrals and singular stress fields, are both overcome in this paper. After a benchmark model being given out to verify the efficiency of the present method, the thermal stress singularities for a symmetrical V-notch and an inclined one are respectively analyzed. The benchmark example manifests that the thermal stress nature boundary integral equation can be successfully used to calculate the thermal stresses much closer to the boundary by the comparison with the conventional thermal stress boundary integral equation. The accuracy of the stress singularity orders and thermal stress intensity factors by the present method is confirmed and the computational effort is dramatically decreased by comparing with the finite element method.

The direct coupling of local discontinuous Galerkin and natural boundary element method for nonlinear interface problem in [formula omitted

In this article, we use the direct coupling of local discontinuous Galerkin (LDG) and natural boundary element method (NBEM) to solve a class of three-dimensional interface problem, which involves a nonlinear problem in a bounded domain and a Poisson equation in an unbounded domain. A spherical surface as an artificial boundary is introduced. The coupled discrete primal formulation on a bounded domain is obtained. The well-posedness of the primal formulation is verified. The optimal error order with respect to energy norm is given. Numerical examples are presented to demonstrate the optimal convergent rates.

Hybrid Natural Element Method-Boundary Element Method for Unbounded Problems

This paper presents a hybrid approach coupling natural element method (NEM) and boundary element method (BEM) for the treatment of unbounded problems. The goal is to combine the accuracy of the NEM and ability of the BEM in modeling linear and deformable domains without a mesh. The performance of the proposed scheme is compared with a finite-element method-BEM coupling in terms of accuracy. Results show that for a given number of degrees of freedom, the proposed approach (NEM-BEM) is able to provide more accurate solutions.

A NBEM for the Time-Dependent Anisotropic Problem in an Exterior Elliptic Domain

In this paper, the natural boundary element method~(NBEM) for an anisotropic hyperbolic problem in an exterior elliptic domain is investigated. By the theory of the natural boundary reduction~(NBR), the natural integral equation~(NIE) and the Poisson integral formula of the problem considered are obtained, and the numerical method of the NIE is given. Finally, some numerical examples are presented to demonstrate the performance of the method in the paper.

The Coupling of NBEM and FEM for Quasilinear Problems in a Bounded or Unbounded Domain with a Concave Angle

Based on the Kirchhoff transformation and the natural boundary element method, we investigate a coupled natural boundary element method and finite element method for quasi-linear problems in a bounded or unbounded domain with a concave angle. By the principle of the natural boundary reduction, we obtain natural integral equation on circular arc artificial boundaries, and get the coupled variational problem and its numerical method. Moreover, the convergence of approximate solutions and error estimates are obtained. Finally, some numerical examples are presented to show the feasibility of our method. Our work can be viewed as an extension of the existing work of H.D. Han et al..

A Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary

In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.

A Coupling of Local Discontinuous Galerkin and Natural Boundary Element Method for Exterior Problems

In this paper, we apply the coupling of local discontinuous Galerkin (LDG) and natural boundary element method(NBEM) to solve a two-dimensional exterior problem. As a consequence, the main features of LDG and NBEM are maintained and hence the coupled approach benefits from the advantages of both methods. Referring to Gatica et al. (Math. Comput. 79(271):1369---1394, 2010), we employ LDG subspaces whose functions are continuous on the coupling boundary. In this way, the primitive variables become the only boundary unknown, and hence the total number of unknown functions is reduced. We prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm. Some numerical examples conforming the theoretical results are provided.

On the Coupled of NBEM and FEM for an Anisotropic Quasilinear Problem in Elongated Domains

In this paper, based on the Kirchhoff transformation, the coupling of natural boundary element method and finite element method are discussed for solving exterior anisotropic quasilinear problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation on elliptic artificial boundaries, the coupled variational problem and its numerical method. Moreover, the convergence and error estimate of the approximate solutions are obtained. Finally, some numerical examples are presented to illuminate the feasibility of the method.

Natural Boundary Element Method for Stress Field in Rock Surrounding a Roadway with Weak Local Support

Weak local support is a very common phenomenon in roadway sup- port engineering. It is a problem that needs to be studied thoroughly at the theoret- ical level. So far, the literature on stress field theory of rock surrounding a roadway is largely restricted to analytical solutions of stress for roadways with a uniform support or no support at all. The corresponding stress solution under conditions of local or weak local support has not been provided. Based on a mechanical model of weak local support at the boundary of a circular roadway and the boundary el- ement method on boundary value problems of bi-harmonic equations of a circular exterior domain, the boundary integration formula of an Airy stress function is de- duced for roadways with weak local support. Using the boundary surface force of the roadway to calculate the boundary stress function and its normal derivative and substituting them into an integration formula, we derived a concrete expression of the stress function under various levels of support resistance. Furthermore, we have provided the rules of the distribution of stress and strain in the rock surrounding a roadway with different areas of weak support and support resistance. From the integration formula, the solutions for analytical stress can be directly obtained for the rock surrounding a roadway with uniform support or without any support at all. The solutions are exactly the same as those given in the literature.

Coupling of Element Free Galerkin and Natural Boundary Element Method for Problems in Unbounded Domains

Coupling of Element Free Galerkin and Natural Boundary Element Method for Problems in Unbounded Domains

Theoretical study on multiple holes grouting with natural boundary element method

Aiming at the cylinder grouting manner, according to the character of complex function, and based on the natural boundary element theory and Fourier series method, interaction problems of multiple holes grouting with arbitrary forms of hole-sitting are investigated for osmotic grouting. Considering the wall of the vertical shaft as impermeable stratum, the seepage pressure solution of exterior vertical shaft is deduced. The serum pressure distribution and relationships between serum pressure and consolidating distance are obtained. The results show that the grouting pressure of the shaft wall will increase while the radius of hole-sitting is small, and accordingly, the shaft wall will suffer from bad influence; the stratum can not be consolidated well if the radius of hole-sitting is large; when the consolidating width keeps the same, there is an optimal consolidating distance and the relative consolidating distance will be 2∼4.

Compatible wavelet-Galerkin method to solve stokes interior problem with circular boundary

This article addresses Neumann boundary value interior problem of Stokes equations with circular boundary. By using natural boundary element method, the Stokes interior problem is reduced into an equivalent natural integral equation with a hyper-singular kernel, which is viewed as Hadamard finite part. Based on trigonometric wavelet functions, the compatible wavelet space is constructed so that it can serve as Galerkin trial function space. In proposed compatible wavelet-Galerkin method, the simple and accurate computational formulae of the entries in stiffness matrix are obtained by singularity removal technique. It is also proved that the stiffness matrix is almost a block diagonal matrix, and its diagonal sub-blocks all are both symmetric and circulant submatrices. These good properties indicate that a 2 J+3 × 2 J+3 stiffness matrix can be determined only by its 2 J + 3J + 1 entries. It greatly decreases the computational complexity. Some error estimates for the compatible wavelet-Galerkin projection solutions are established. Finally, several numerical examples are given to demonstrate the validity of the proposed approach.