Boundary element analysis of thin structures using a dual transformation method for weakly singular boundary integrals

https://doi.org/10.1016/j.camwa.2022.03.014Get rights and content

Abstract

This paper focuses on the weakly singular integrals in thin structures of 3D boundary element method. In the pioneering woks for the weakly singular integrals, the singularities are eliminated through introducing transformations whose Jacobian at the singular point is zero. When using these methods for weakly singular boundary integrals on elements with a large-angle and large side length ratio, near singularity will appear. In this paper, a dual transformation method is proposed to accurately calculate weakly singular integrals in which the near singularity is also considered. The numerical implementation is divided into several steps. Firstly, the (α, β) coordinate transformation method is used to eliminate the weak singularity by introducing the α factors in the transformation Jacobian. Then extract the integral form with near singularities about β. Finally, a corresponding distance transformation is constructed to eliminate its near singularity. The results show that the proposed method can accurately integrate for arbitrary element shape and source point position. The dual transformation method may be a good choice to evaluate weakly singular integral for thin structural problem.

Introduction

Thin structures [1], [2], [3], [4], [5], [6] which have the advantages of light weight and high energy absorption efficiency are widely used in engineering, such as the coating of automobile, ship's shell, sensor, airplanes, high-speed railways, mechanical parts and so on. As the complex working environment of thin structures, the mechanical properties of thin structures easily become deformed under the action of various alternating loads, which will eventually lead to structural failure. Therefore, in order to improve the engineering application value of thin-walled structures, it is necessary to choose a numerical method to analyze their mechanical properties, which is of great engineering significance for their structural design and optimization.

At present, the numerical methods are widely used to analyze thin-walled structures, such as boundary element method (BEM) [7], [8], [9], [10], [11], [12], [13], meshless method [14], [15] and finite element method (FEM) [16], [17], [18], [19]. In these numerical methods, FEM is a most widely used method, such as the previous work of J. Jaskowiec for thin structures [3], [4], [18], [19], while the trial functions of FEM require C0 continuity which will reduce the computational accuracy of stress and flow. Moreover, the aspect ratio issues associated with the FEM which are easy to cause grid distortion limit its application to thin-structural problems [6], [8]. The meshless method does not need to generate mesh in numerical calculation, but constructs interpolation functions according to some arbitrarily distributed coordinate points to discrete governing equations, which can easily simulate the flow field of various complex shapes. BEM only needs boundary discretization and has the advantages of dimensionality reduction and highly computational accuracy, which is suitable for the analysis of thin structural problem. When using these numerical methods to analyze thin structures, as the singularity of the fundamental solutions and the singularity of the nodal shape functions of the extended element are existed in the integral equation, the numerical integration needs to be special attention. Especially in BEM, the computational accuracy of singular and nearly singular integrals directly affects the final solutions of BEM. Therefore, a special method is indispensable to eliminate the singularity and near singularity in the fundamental solutions.

Accurate calculation of near singular integrals is the key factor for BEM analysis of thin structural problem. There are many nearly singular techniques have been developed to eliminate the near singularity in the fundamental solutions and improve the computational accuracy, such as element subdivision method [5], [20], analytic and semi-analytic method [21], [22], sinh transformation [23], [24], [25], [26], [27], optimal coordinate transformation [28], [29], exponential transformation [30], [31], sigmoidal transformation [32], [33], distance transformation and other nonlinear coordinate transformation [34], [35], [36], [37], [38], [39], etc. With this method, the nearly singular integral can be accurately and effectively calculated. In terms of weakly singular integrals, the analytical integral method [40] is an effective method, which needs complex derivation of integral formula and has some difficulties to deal with the curve boundary element. Polar coordinate transformation [41], [42], [43], [44], [45] coupled with element subdivision [46], [47] is a widely used method at present. The Jacobian of the coordinate transformation can cancel the singularity in the denominator of integrand. However, the element subdivision method has poor computational efficiency. When the method is applied to subdivide the elements with poor shape, the shape of the sub-elements cannot be guaranteed. When the sub-elements include large angles and large side length ratios, although the singularity in the radial direction can be eliminated by the coordinate transformation, the near singularity in the circumferential direction still exists, which greatly reduce the integral accuracy. To solve this problem, the angular and sigmoidal transformations [48], [49] can be used to deal with the singularity in the circumferential direction, but the integral accuracy cannot be improved too much and the effect is not obvious.

In this paper, we apply a (α, β) coordinate transformation method which can separate the variables in two integral directions while eliminating the weak singularity in α direction. Then extract the integral form with near singularity in β direction, based on the theory of complex variable function, the distance transformation is constructed in β direction to eliminate its near singularity. Finally, the method is applied to the boundary element analysis of thin structural problem. With the proposed method, the weakly singular integrals over large-angle and narrow strip boundary elements in the discrete model of thin structures can be accurately evaluated.

This paper is organized as follows. Section 2 describes the boundary integral equation and its numerical discretization of 3D linear elasticity problems. In section 3, the implementation procedure of dual transformation method for weakly singular integrals is presented in detail. The processing scheme of nearly singular integrals is introduced in Section 4. Several numerical examples are given in Section 5. The conclusions are given in Section 6.

Section snippets

The singular boundary integral equation of 3D elasticity problems

The boundary integral equation (BIE) of three-dimensional elasticity problems for thin structures can be expressed as [50]cij(P)uj(P)=Γuij(P,Q)tj(Q)dΓ(Q)Γtij(P,Q)uj(Q)dΓ(Q) where P and Q are the source and field point, respectively. cij(P) is a coefficient matrix depending on the smoothness of the boundary Γ at the source node P. uj and tj represent the displacement and traction fields, respectively. uij(P,Q) and tij(P,Q) are the displacement and traction Kelvin fundamental solutions,

The dual transformation method

It can be seen from the Eq. (2), the weakly singularity is existed in the integral of the displacement fundamental solution, and there is strong singularity in the integral of the traction fundamental solution. As the proposed method focuses on weakly singular boundary integral, the integral of the displacement fundamental solution is taken into account. It can be simplified into the following form on one discrete element:I=Γef(P,Q)r(P,Q)dΓe(Q) in which, f(P,Q) represents a smooth function

The processing scheme of nearly singular integrals

From Eq. (2) and Eq. (3), we can see the nearly integral will arise when the source point P is close to the integral element Γe. To obtain more accurate results, the treatment of nearly singular integrals is unavoidable, especially for thin structural problem. There will be a large number of nearly singular integrals in the numerical implementation of BEM. To verify the validity of the proposed dual transformation method for weakly singular integrals, we don't apply other nonlinear

Numerical examples

To verify the computational accuracy of the dual transformation method, several test examples including the large angle and narrow strip elements are considered in this section. A relative error e is defined as followse=|InumIexactIexact| in which, Inum and Iexact stand for the numerical and exact solutions, respectively. Iexact is obtained by the method of combining element subdivision and polar coordinate transformation with large numbers of Gaussian points.

In the following Tables and

Conclusions

We have applied the dual transformation method to evaluate the weakly singular boundary integrals in this paper. The dual transformation method has considered the shape characteristics of the element and is used to deal with the weak singular integrals over large-angle and narrow strip boundary elements in the discrete model of thin structures. For the proposed method, both the weak singularity in α direction and the near singularity in the β direction can be eliminated. Moreover, the weakly

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (11602229, 52175256, 52075500 and 11701526), partly supported by the key scientific and technological project of Henan Province (212102210070), and partly supported by Key Scientific Research Projects of Institutions of Higher Learning in Henan Province (21A460029 and 21A460030).

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