Sinh Transformation Research Articles

Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

AbstractConsider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.

A transformation method evaluate for near singular boundary integrals in the structural analysis of thin structure

The computational accuracy of near singular integrals is the key to determine the overall accuracy and stability of BEM in solving elastic variables, and even the success or failure of BEM analysis. In this paper, an alterative sinh transformation method is developed for solving the near singular boundary integrals in the structural analysis of thin structure. Firstly, the original sinh method is modified into the form include minimum distance and approximate expansion of local coordinates, and Newton-Raphson and cubic equation solving methods are used to obtain the position of the near singular projection point. To facilitate the implementation of the near singular integral, the integral element in the local coordinate space is mapped to the integral parameter space, and then the element integration is performed directly by sinh transformation in this space. The results of several numerical examples show that the proposed method can be used to regularize the nearly singular boundary integral effectively, and to accurately solve the elastic physical variables in the analysis of thin structures.

Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning

In the boundary element method (BEM), the sinh transformation method is an effective method for evaluating nearly singular integrals, but a relationship between the integration accuracy and the number of Gaussian points is needed to achieve adaptive computation. Based on deep learning, we propose a novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), which can determine the number of Gaussian points according to the required accuracy. First, a large number of integration data samples of the sinh transformation method are generated in different cases, and the neural network is trained to establish the relationship between the number of Gaussian points and the integration accuracy. Then, based on the improved loss function and evaluation index, a better network model is obtained to ensure that the actual integration accuracy is slightly higher than the requirement of using the minimum Gaussian points. In this way, when the trained neural network is used in the sinh transformation method, the higher accuracy requirement can be met at a lower cost. Numerical examples demonstrate that, compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency when evaluating the nearly singular integrals for very thin coatings and other structures.

A semi-analytical treatment for nearly singular integrals arising in the isogeometric boundary element method-based solutions of 3D potential problems

The nearly singular integral, arising in simulating thin coatings or close-boundary physical quantities, are not adequately dealt with in the isogeometric boundary element method (IGABEM), especially in 3D problems. In this paper, we propose a semi-analytical approach for the nearly singular integrals of 3D potential problems. We first expand all the kernel items by Taylor series up to second order accuracy. In order to employ the semi-analytical formulae when integrating in parametric space, coordinate (ξ,η) is further transformed to polar coordinate (ρ,θ). We then use the subtraction technique to separate the integrals to near-singular parts and regular parts. For the near-singular parts, a semi-analytical treatment is performed where the integrations with respect to ρ are expressed by analytical formulae recursively, while the ones related to θ are computed by Gaussian quadrature. The remaining regular integrals are treated numerically by the sinh transformation method. By adding them together, we could efficiently handle the nearly singular integrals and therefore obtain accurate close-boundary potentials and flux densities in 3D potential problems. The accuracy of the presented method for nearly singular integrals to a curved element with different orders of singularities, namely the nearly weakly, strongly and highly singular integrals, are first tested. We then further consider potential problems of three typical 3D structures. All the presented results are compared with the recently proposed improved sinh transformation method and analytical solutions. The above numerical examples fully show the efficiency and competitiveness of the presented semi-analytical schemes.

BEM analysis of multilayer thin structures using a composite transformation method for boundary integrals

In this paper, a composite transformation method is developed to evaluate the singular and nearly singular boundary integrals in the boundary element analysis for multilayer thin structures. The composite transformation method is implemented as follows: Firstly, a sinh transformation method is developed to eliminate the influence of nearly singular integrals in the multilayer thin-structural problem; and then a complex transformation method is improved to deal with the weakly singular integrals. The complex transformation formulations of weakly singular integrals are derived from the composition of the (α, β) coordinate transformation and the sinh transformation in β direction. With this method for weakly singular integrals, the weak singularity in the radial direction and the potentially near singularity in the circumferential direction can be removed. Finally, the proposed transformation formulations of singular and nearly singular integrals are used to analyze multi-domain elasticity problem for multilayer thin structures. The numerical results for several examples are presented to demonstrate the computational accuracy of the proposed method.

Boundary element analysis of static plane problems in size-dependent consistent couple stress elasticity

Based on a consistent couple stress elasticity theory, a two-dimensional (2D) boundary element method (BEM) is developed to solve the boundary value problems in the couple stress elastic materials. The displacement boundary integral equations (BIEs) are applied to compute the mechanical quantities on the boundary. The integral representations for the generalized strains and stresses at internal points are also presented together with the explicit expressions for the high-order fundamental solutions. The collocation method is adopted for the spatial discretization of the BIEs. Some special numerical and semi-analytical techniques are provided for the evaluation of weakly and strongly singular integrals. The sinh transformation is used to deal with the nearly singular integrals for near-boundary internal points or boundary nodes in thin structures. An error-controlled integration scheme is adopted for computing the non-singular integrals to improve the accuracy. Three representative 2D boundary value problems in the consistent couple stress elasticity theory are studied by the developed BEM. The correctness and accuracy of the present BEM results are verified by comparison with the corresponding analytical results. Then, multiple flat elliptical holes in a plate are numerically studied to show the interaction effects on the stress concentration.

Confidence intervals for the common odds ratio based on the inverse sinh transformation

ABSTRACT This paper proposes two new approximate confidence limit methods for the common odds ratio from multiple 2 × 2 tables. The two new procedures, based on the asymptotic distribution of Woolf estimator and Mantel-Haenszel estimator, associate with inverse sinh transformation. We employ three pseudo-frequency methods to calculate confidence intervals in order to avoid the interval failure caused by the presence of zero cells in multiple 2 × 2 tables. We develop the modified inverse sinh intervals for the common odds ratio which add one pseudo-frequency (c 1) to all the cells before computing the point estimate of common odds ratio and another pseudo-frequency (c 2) to all the cells before computing the standard error estimate. The simulation is to evaluate the 22 confidence intervals, including Woolf, Mantel-Haenszel, their inverse sinh intervals, and their pseudo-frequency modified inverse sinh intervals, in terms of their coverage probabilities and average log lengths. Simulation results demonstrate that the adjusted inverse sinh intervals by two different pseudo-frequencies perform quite well when c 2 is slightly greater than c 1 since the coverage probabilities of them are closer to confidence level of 95%. Larger values of c 2 lead to narrow intervals and low coverage probabilities. We also find that inverse sinh intervals are shorter than untransformed intervals based on Woolf estimator and Mantel-Haenszel estimator, respectively. These procedures were illustrated with two clinical trials.

A boundary weak singularity elimination method for multilayer structures

In the numerical discretization for the models of multilayer thin structures, the elements with narrow strip shape may be appeared. When using the previous methods to integrate these elements, the potentially near singularity will arise in the circumferential direction. In this paper, a weak singularity elimination method for multilayer structures of 3D boundary element method is presented to accurately calculate weakly singular integrals. Its numerical implementation is divided into the following two parts: (1) (α, β) coordinate transformation is applied to deal with the weakly singular integral arising in boundary integral equation and separate out the integral in the circumferential direction (β direction); (2) Based on the integral form of β, a corresponding sinh transformation method is developed to eliminate the near singularity in β direction. Finally, the proposed transformation formulation is extended to multi-domain potential problem and applied to the boundary analysis for multilayer structures. Several numerical examples are given to verify the computational accuracy of the proposed method. And the numerical results are promising and encouraging.

The Novel Rational Spectral Collocation Method for a Singularly Perturbed Reaction-Diffusion Problem with Nonsmooth Data

A singularly perturbed reaction-diffusion problem with a discontinuous source term is considered. A novel rational spectral collocation combined with a singularity-separated method for this problem is presented. The solution is expressed as u = w + v , where w is the solution of corresponding auxiliary boundary value problem and v is a singular correction with direct expressions. The rational spectral collocation method combined with a sinh transformation is applied to solve the weakened singularly boundary value problem. According to the asymptotic analysis, the sinh transformation parameters can be determined by the width and position of the boundary layers. The parameters in the singular correction can be determined by the boundary conditions of the original problem. Numerical experiment supports theoretical results and shows that compared with previous research results, the novel method has the advantages of a high computational accuracy in singularly perturbed reaction-diffusion problems with nonsmooth data.

Near singularity cancellation in weakly singular integrals of three-dimensional boundary element method

This paper focuses on near singularity in weakly singular integrals of three-dimensional boundary element method. In traditional methods, only one directional transformation is introduced to remove the weak singularity. However, near singularity is still existed. Thus in our paper, serval methods for removal of weak singularity are introduced firstly. Then, angular and sigmoidal transformations are introduced to remove the near singularity in the circumferential direction combined with polar transformation. Generalized Duffy transformation is introduced to remove weak singularity in u direction, which also separates the near singularity in v direction. Based on this method, sinh transformation is constructed with the integrand of v. Finally, several numerical examples considering large top opening angle of the singular integration element and large ratio of the two edge length of the angle (the max edge length/ the min edge length) are proposed. Results demonstrate that generalized Duffy transformation coupled with sinh transformation in v direction is a best choose for weakly singular integrals.

Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh+ scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh+ scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh+ scheme in the analysis of thermoelastic problems.

Bi-directional sinh transformations based on the generalized Duffy space for nearly singular integrals

This paper focuses on the sinh transformation in the generalized Duffy space for accurate computation of the nearly singular integral. The integral patch is normalized through the Duffy transformation. Near-singularity has been found in both directions of the transformed parametric space. Based on the complex function theory, these near-singularities of both variables in the integral are canceled by the bi-directional sinh transformation method. Comparing with the sinh transformation in a single direction, the bi-directional transformation achieves a much more accurate result. To further improve the computational accuracy, an iterative sinh transformation is applied. In the application of the bi-directional iterative sinh transformation, it has been found that the results are much less sensitive to the location of the nearest point. Thus, high accuracy and stability can be kept in most cases.

The semianalytical analysis of nearly singular integrals in 2D potential problem by isogeometric boundary element method

AbstractBenefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density.

Factors Influencing the Performance of Regression-Based Statistical Postprocessing Models for Short-Term Precipitation Forecasts

Abstract Statistical postprocessing models can be used to correct bias and dispersion errors in raw precipitation forecasts from numerical weather prediction models. In this study, we conducted experiments to investigate four factors that influence the performance of regression-based postprocessing models with normalization transformations for short-term precipitation forecasts. The factors are 1) normalization transformations, 2) incorporation of ensemble spread as a predictor in the model, 3) objective function for parameter inference, and 4) two postprocessing schemes, including distributional regression and joint probability models. The experiments on the first three factors are based on variants of a censored regression model with conditional heteroscedasticity (CRCH). For the fourth factor, we compared CRCH as an example of the distributional regression with a joint probability model. The results show that the CRCH with normal quantile transformation (NQT) or power transformation performs better than the CRCH with log–sinh transformation for most of the subbasins in Huai River basin with a subhumid climate. The incorporation of ensemble spread as a predictor in CRCH models can improve forecast skill in our research region at short lead times. The influence of different objective functions (minimum continuous ranked probability score or maximum likelihood) on postprocessed results is limited to a few relatively dry subbasins in the research region. Both the distributional regression and the joint probability models have their advantages, and they are both able to achieve reliable and skillful forecasts.

Rational Spectral Collocation Combined with the Singularity Separated Method for a System of Singularly Perturbed Boundary Value Problems

A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems. The solution is expressed as u=w+v, where w is the solution of the corresponding auxiliary boundary value problem and v is a singular correction with explicit expressions. The rational spectral collocation method in barycentric form with the sinh transformation is applied to solve the auxiliary third boundary problem. The parameters of the singular correction can be determined by the boundary conditions of the original problem. Numerical experiments are carried out to support theoretical results and provide a favorable comparison with research results of other work.

The Distance-Sinh combined transformation for near-singularity cancelation based on the generalized Duffy normalization

The singularity cancelation is usually of great importance in the calculation of nearly singular integral that is involved in the boundary element analysis of thin structures and in the extended finite element method for crack problems. The well-known distance transformation can be performed for various kinds of singular kernels to compute the near singular integral. However, in the case of that the source point is located nearby the boundary of the integral element, the result is usually very sensitive to the shape of the sub-triangles. A Distance-Sinh combined transformation method based on the generalized Duffy normalization is proposed in this paper to improve the accuracy of the nearly singular integral calculation. Furthermore, a two-level transformation method is proposed to cancel the singularity of the integrand. In first level of the method, the generalized Duffy transformation is implemented. In the transformed integral space (u, v), the singularity of the integrand along u-direction is eliminated by the distance transformation. The singularity of the integrand along the other direction is canceled by the Sinh transformation. Numerical examples are presented to verify the proposed method.

Evaluating 2D domain integrals by sinh transformation for transient heat conduction problem

The time-dependent boundary element method (BEM) is widely used to solve transient heat conduction problem. The integrands of the domain integrals in the time-dependent BEM are close to singular when small time step is used. A straightforward computation of these integrals using Gaussian quadrature will produce large errors. To evaluate domain integrals accurately and efficiently, the sinh transformation combined with element subdivision method is proposed in this paper. With the sinh transformation, the integrands of the domain integrals become smoother. Meanwhile, the Gaussian points are shifted towards the source point due to the element subdivision. Thus, the 2D domain integrals in the time-dependent BEM can be computed accurately. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures

We present an isogeometric boundary element method (IGABEM) capable of delivering accurate and efficient solutions in the heat transfer analysis of 2D coated structures such as those commonly found in turbomachinery. Although we consider very thin coatings (of thickness down to 10−7 m), they are modelled explicitly as BEM zones, and this is made possible by the development of a new integration scheme (sinh+) aimed particularly at the challenging nearly singular integrals that arise. Sinh+ is a hybrid of adaptive and sinh transformation approaches, and we make further extensions to the latter to improve its robustness. The scheme is tuned to deliver results of engineering accuracy in an optimal time. The scheme is adaptable, by changing a tolerance, to enable engineers to achieve a different balance between accuracy and computational efficiency as may be required for different applications. A set of numerical examples demonstrates the ability of the scheme to produce accurate temperature distributions efficiently in the presence of very thin coatings.

Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer

We are concerned with using high accuracy numerical methods for solving singularly perturbed problems whose solutions exhibit an interior layer. A rational spectral collocation method in barycentric form with sinh transformation is firstly applied to find the approximate solution to the given problem. The sinh transformation consists of two parameters: the location and the width of the interior layer. With the tool of the asymptotic analysis, the interior layer can be easily located. Then, in determining the width of the interior layer, we design a nonlinear unconstrained optimization problem which is solved by using the basic differential evolution (DE) algorithm. The feasibility and effectiveness of the proposed method is verified by numerical results. The proposed method can also be applied to other singularly perturbed problems. • We consider a high accuracy numerical method to solve singularly perturbed problems with an interior layer. • The presented method is based on the rational spectral collocation method in barycentric form and the DE algorithm. • We construct a nonlinear unconstrained optimization problem which is solved by using the DE algorithm. • It is noted that the method presented in this paper can be extended to other type of singularly perturbed problems.

Three-dimensional stress analysis of thin structures using a boundary element method with sinh transformation for nearly singular integrals

In this work a three dimensional (3D) boundary element method was established with an efficient nonlinear coordinate transformation scheme, namely sinh transformation, to evaluate nearly singular integrals in boundary integral formulations. Second-order quadrilateral surface elements were developed based on this method to accurately describe the geometry of thin structures. The elastic behaviors of selected thin structures were then computed by using the 3D boundary element model to demonstrate the accuracy and efficiency of this approach. A number of testing examples, i.e., the 3D Kirsch problem, the thin spherical shell problem, the ellipsoidal vessel problem with non-uniform thickness and the hollow circular cylinder problem, were numerically studied to test the established method. Remarkable accuracy and efficiency were found in the developed approach through the comparison to the numerical results and analytical solutions reported in the literature.