Degenerate Scale Research Articles

Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method

In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.

Are the direct and indirect BEM/BIEMs equivalent?

For a long time, direct and indirect BEM/BIEMs were always unified and seemed equivalent in real implementation for ensuring the solution. Regarding the solution space and the numerical aspects in linear algebra, it is interesting to find that they are not fully equivalent. To demonstrate this finding, antiplane shear problems containing the elliptical rigid inclusion and hole as well as rigid-line inclusion and line crack were given to demonstrate the non-equivalence between the direct and indirect BEM/BIEMs. It is interesting to find that nonunique boundary densities may yield the unique field solution in both analytical derivation and numerical implementation. The mechanism was analytically studied and numerically performed by using the degenerate kernel and the singular value decomposition technique, respectively. Besides, the Fredholm alternative theorem was employed to examine the solvability condition in both the indirect boundary integral equation method and indirect boundary element method. For continuous system and discrete system, we can explain why the boundary density does not match the exact solution but the field solution is always acceptable when setting ξ0 = 0 in the beginning of the derivation of BIE. Two approaches, direct and indirect BEM/BIEMs, show the non-equivalence of solution in case of degenerate boundary and degenerate scale.

Steady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scale

AbstractRegarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.

On the solution arising in two-cylinders electrostatics.

The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski[2] and Lekner[4] for identity.

Degenerate scales for thin elastic plates with Dirichlet boundary conditions

A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We extend several results about degenerate scales previously found for the Laplace and Lamé equation to the biharmonic equation in 2 dimensions. The degenerate scales are obtained from a \(4\times 4\) discriminant matrix whose shape is provided for different kinds of domain symmetry. We show that degenerate scales can be obtained from a minimization problem. Then, we compare the degenerate scales of two boundaries, one included within the other. For smooth star-shaped curves, we show that there are only two degenerate scales, give sufficient conditions for not being at a degenerate scale and produce bounds to the degenerate scales. For symmetric smooth simply connected curves there are also only two degenerate scales. For these symmetric cases, the use of complex variables allows us to go further and to link the problem of biharmonic equation to the one of plane elasticity and to give information, including bounds and exact values of degenerate scales, for many cases, while until now only very few ones were known. Our results have some consequences for the biharmonic outer radius defined by Pólya and Szegö. The numerical computation of degenerate scales by using boundary elements confirms the theoretical results.

Numerical solution of the degenerate scale problem with arbitrary hole configuration in 2D Laplace equation

This paper provides a numerical solution for the degenerate scale of arbitrary hole configuration in 2D Laplace equation. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant boundary integral equation (BIE) in the ordinary scale. The formulated BIE in the ordinary scale represents a weaker singular equation. After solving the weaker singular equation, the degenerate scale can be evaluated immediately.

Numerical solution for the degenerate scale for two unequal circles in thermal conduction problem

The degenerate scale problem for two unequal circles in thermal conduction is studied in this paper. In addition, this paper provides a numerical solution for the addressed problem. Coordinate transform technique is suggested in the derivation. By using this technique, the problem formulated in the degenerate scale can be converted into similar one formulated in the normal scale. The integral operator formulated in the normal scale is invertible. By using this property, the degenerate scale can be evaluated for the case of two unequal circles. Many computed results for the addressed degenerate scale problem are provided.

Study on the double-degeneracy mechanism of BEM/BIEM for a plane elasticity problem with line segments

There are four degenerate problems by using the BEM/BIEM. Only the degenerate scale and degenerate boundary may appear at the same time. This is called double degeneracy. A double-degeneracy mechanism of BEM/BIEM for the plane elasticity problem with line segments is studied, and the degenerate scale is analytically examined and numerically performed. Different from the past result with complex variables, we propose a new idea to deal with the problem, and obtain the analytical solution of the degenerate scale. This analytical derivation can clearly show why the BEM/BIEM suffer the degenerate scale in the plane elasticity problem with the line segment. Only rigid inclusion faces this problem instead of the crack due to the use of single-layer kernel. Double degeneracy of degenerate boundary and degenerate scale in the BEM are numerically examined. The double-degeneracy mechanism is clearly displayed through numerical results by showing the number of zero singular values in the influence matrix. Following the result of single line segment, we can extend to multiple line segments. Finally, the analytical and numerical results show consistency.

An indirect BIE free of degenerate scales

<p style='text-indent:20px;'>Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.</p>

Численные эксперименты двойственным методом нулевого поля в задаче Дирихле для уравнения Лапласа в эллиптических областях с эллиптическими отверстиями

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). In the first part of the two-part article, our efforts were focused on studying the theoretical aspects of this problem, including the analysis of errors and the study of stability. We provided the theoretical analysis for Laplace equation in elliptic domains with elliptic holes. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the null field methods (NFM) are used for the exterior and the interior boundaries, simultaneously. This approach is called the dual null field method (DNFM). This paper is the second part of the study. Numerical results for degenerate models of an elliptic domain with one elliptic hole at 𝑎 + 𝑏 = 2 are carried out to verify the theoretical analysis obtained. The collocation Trefftz method (CTM) is also designed for comparisons. Both the DNFM and the CTM can provide excellent numerical performances. The convergence rates are the same but the stability of CTM is excellent and can achieve the constant condition numbers, Cond = 𝑂(1).

Degenerate scale problem for some configurations with rigid line tips in antiplane elasticity or Laplace equation

This paper provides several solutions of the degenerate scale for some configurations with rigid line tips in antiplane elasticity. For solving the problem, it is necessary to investigate the conformal mapping function in advance. It is found that when a particular dimension reaches its critical value, a closed form solution for complex potential on the mapping plane can be found. This solution takes zero value for displacement along the boundary. However, it is not vanishing in the exterior region to the contour.

Solution for the degenerate scale for a rigid curve in antiplane elasticity by using a weakly singular integral equation

This paper provides a numerical solution for the degenerate scale for a rigid curve in antiplane elasticity. The degenerate scale problem for the rigid curve is formulated on the usage of the logarithmic potential. After assuming the displacement to be a vanishing value along the rigid curve, the boundary integral equation (BIE) is formulated. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant BIE in the ordinary scale. Finally, a numerical solution is achieved. Several numerical examples are provided. In addition, the degenerate scale problem for the multiple rigid curves is also solved.

On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain

ABSTRACT In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) and discrete system (boundary element method, BEM) for a circular case. The infinite-dimensional degree of freedom for the continuous system can be reduced to finite-dimensional space using the generalized Fourier coordinates. The property of the second-order tensor for the influence matrix under different observers is also examined. On the other hand, the discrete system in the BEM can be analytically studied, thanks to the spectral property of circulant matrix. We adopt the circulant matrix of odd dimension, (2N + 1) by (2N + 1), instead of the previous even one of 2N by 2N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.

On the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernels

The stress concentration factor (SCF) along the boundary of a hole and a rigid inclusion in an infinite isotropic solid under the anti-plane shear is revisited by using degenerate kernels in the boundary integral equation (BIE) although this result was obtained by invoking the extended circle theorem of Milne-Thomson as well as the complex variable approach. The degenerate kernel of series form for the closed-form fundamental solution is used for the circle and the ellipse in terms of polar and elliptic coordinates, respectively. The slender ratio of the ellipse and the orientation are two parameters for our study. The strain energy density along the boundary is increased or decreased due to the different types of loading and various aspect ratios of the ellipse. An analytical solution for the SCF is then derived for any orientation of the ellipse relative to the applied load. The reciprocal relation for the SCF between a hole and a rigid inclusion with respect to different loading is also addressed. Besides, this analytical derivation can clearly show the appearing mechanism why the BEM/BIEM suffers the degenerate scale in the rigid inclusion.

Анализ методов двойного нулевого поля в задаче Дирихле для уравнения Лапласа в эллиптических областях с эллиптическими отверстиями: проблема алгоритмической сингулярности

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). Our efforts are paid to explore theoretical analysis, including error and stability analysis, to fill up the gap between theory and computation. Our group provides the analysis for Laplace’s equation in circular domains with circular holes and in this paper for elliptic domains with elliptic holes. The explicit algebraic equations of the first kind and second kinds of the null field method (NFM) and the interior field method (IFM) have been studied extensively. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual null field method (DNFM) in this paper. Optimal convergence rates and good stability for the DNFM can be achieved from our analysis. This paper is the first part of the study and mostly concerns theoretical aspects; the second part is expected to be devoted to numerical experiments.

Study on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problems

The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind of problem by using the degenerate kernels in terms of the elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can’t be clearly examined. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.

Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM

Degenerate scale of line segments is derived in this paper. It is found that two formulas in the Rumely's book are inconsistent after comparing with our results. We confirm our formulas not only by testing the value of degenerate scale in the BEM/BIEM but also by finding the unit logarithmic capacity. Double degeneracy due to the degenerate boundary (line segment) and degenerate scale in the BEM/BIEM is also examined. By increasing the number of boundary elements, the value of degenerate scale converges to the corresponding one derived by using the complex variables instead of Rumely's formulas. Three cases in the Rumely's book, include one segment, two equal segments and three segments (L, 2 L, L) with the spacing 3 L, are re-confirmed. However, we found that there are two inconsistent cases, two unequal segments (L, 5 L) with the spacing 3 L and three segments (L, 2 L, L) with the spacing L, respectively, after analytically deriving by using the complex variables and numerically examining by detecting the degenerate scale in the BEM/BIEM.

An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered.The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.

A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems

ABSTRACTThe influence matrix may be of deficient rank in the specified scale when we have solved the 2D elasticity problem by using the boundary element method (BEM). This problem stems from in the 2D Kelvin solution. On the other hand, the single-layer integral operator can not represent the constant term for the degenerate scale in the boundary integral equation method (BIEM). To overcome this problem, we have proposed the enriched fundamental solution containing an adaptive characteristic length to ensure that the argument in the logarithmic function is dimensionless. The adaptive characteristic length, depending on the domain, differs from the constant base by adding a rigid body mode. In the analytical study, the degenerate kernel for the fundamental solution in polar coordinates is revisited. An adaptive characteristic length analytically provides the deficient constant term of the ordinary 2D Kelvin solution. In numerical implementation, adaptive characteristic lengths of the circular boundary, the regular triangular boundary and the elliptical boundary demonstrate the feasibility of the method. By employing the enriched fundamental solution in the BEM/BIEM, the results show the degenerate scale free.

The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

The degenerate scale issue for D boundary integral equations and boundary element methods has been already investigated for Laplace equation, antiplane and plane elasticity, bending plate for Dirichlet boundary condition. Recently, the problem of Robin and mixed boundary conditions and of piecewise heterogeneous domains have been considered for the case of Laplace equation. We investigate similar questions for plane elasticity for more general boundary conditions. For interior problems, it is shown that the degenerate scales do not depend on the boundary condition. For exterior problems, the two degenerate scales (homogeneous medium) or two of them (heterogeneous medium) are tightly linked with the behavior at infinity of the solutions. The dependence of this behavior on the boundary conditions is investigated. We give sufficient conditions for the uniqueness of the solution. Numerical applications are provided and validate the set of theoretical results.