2D Potential Problems Research Articles

Sensitivity to Gauss quadrature of isogeometric boundary element method for 2D potential problems

Sensitivity to Gauss quadrature of isogeometric boundary element method for 2D potential problems

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.

Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.

The consistent boundary element method for potential and elasticity: Part III — Topologically challenging numerical assessments for 2D problems

The collocation boundary element method is consistently outlined in a companion (Part I) paper for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. Simple code schemes are proposed in a second companion (Part II) paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high – actually only machine-precision dependent – computational accuracy of all results of interest independently of a problem’s geometry and topology. We present here numerical results and convergence assessments – including numerical illustration of convergence Theorem 1 of the companion paper I – for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source–field distances — maybe with approximations due to a coarse mesh discretization but never introducing unduly singularities. We dare say the present results cannot be replicated by any code implementation other than the ones of our own, which resort to no ad hoc means beyond the problem’s correct mathematics. In fact, we present the objective and controllable tools of separately assessing a mesh discretization for a given practical problem in terms of precision, accuracy and liability to round-off-errors. We also evaluate the distance threshold for a source point to be considered sufficiently close to demand the proposed exact mathematical treatment, as, for large distances, only Gauss–Legendre quadrature is required, be it in terms of quadrature adaptivity of even a fast multipole scheme.

The consistent boundary element method for potential and elasticity: Part I — Formulation and convergence theorem

The collocation boundary element method is outlined for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity, as already proposed in the literature. However, this is done in a consistent way that sheds light on some conceptual and implementation-oriented aspects that should be – but are missing – in the textbooks on the subject. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. A rather recreative example of a simple truss element is brought in an Appendix. It enables the demystification of apparently convoluted concepts that might be thought only pertaining to highly elaborated mechanics and mathematics for singularities related to infinity and to ungraspable linear algebra. Simple code schemes based on the outlined concepts are proposed in a companion paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high computational accuracy of all results of interest independently of a problem’s geometry or topology. Numerical results are shown in another companion paper and convergence features are assessed for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source-field distances. Summing up, whether a problem we are attempting to simulate consists of smooth fields or not, we are able to reproduce it — maybe with numerical approximations due to a coarse mesh discretization but never introducing unduly singularities.

A Novel Strategy for Eliminating the Boundary Layer Effect in the Regularized Integral Identity in PIES for 2D Potential Problem

The paper presents a new strategy for improving the accuracy of solutions near the boundary in the integral identity associated with the parametric integral equation system (PIES) for two-dimensional (2D) potential problems. A significant reduction in accuracy in the zone close to the boundary, also known as the boundary layer effect, is directly associated with the nearly singular properties of kernels present in the integral identity. The paper shows that these singularities can be efficiently eliminated by regularizing the integral identity with the help of the so-called regularizing function with appropriate coefficients. The analyzed examples demonstrate a significant improvement in accuracy, where all integrals of the regularized integral identity are accurately calculated using low-order standard Gauss–Legendre quadrature. The proposed regularization algorithm is independent of the actual boundary shape, its representation and assumed boundary conditions.

An Improved Element-Free Galerkin Method Based on the Dimension Splitting Moving Least-Squares Method for 2D Potential Problems in Irregular Domains

By introducing the dimension splitting (DS) method into the moving least-squares (MLS) approximation, a dimension splitting moving least-squares (DS-MLS) method is proposed in this paper. In the DS-MLS method, the operator splitting and independent variable splitting of the DS method are used to reduce the dimension, thereby reducing the computational complexity of the matrix. The shape function of the DS-MLS method has the advantages of simple derivation and high computational efficiency. Then, by coupling DS-MLS method and Galerkin weak form, and performing the coordinate transformation, an improved element-free Galerkin method (IEFGM) based on the DS-MLS method is proposed for two-dimensional (2D) potential problems on irregular domains. The effectiveness of the method in this paper is verified by some numerical examples. The numerical results show that, compared with the element-free Galerkin (EFG) method, the IEFGM based on the DS-MLS method in this paper consumes less CPU time and has higher computational accuracy for some 2D potential problems on irregular domains.

A multi-domain BEM based on the dual interpolation boundary face method for 3D potential problems

A multi-domain BEM based on the dual interpolation boundary face method for 3D potential problems

Boundary element method with particle swarm optimization for solving potential problems

In this paper, the boundary element method (BEM) based on the particle swarm optimization (PSO) algorithm is proposed for solving potential problems. At present, the traditional Gaussian elimination method still needs to improve for computing results accuracy. The PSO algorithms with substantial accuracy used to improve the accuracy of potential problems. So it is introduced the PSO to solve the system of matrix equations derived from the boundary integral equation (BIE). For the presented method, unknown solutions to the system of matrix equations are considered particles, and the optimal solution is sought through group collaboration. Some parameters in this model play a vital role, so confirming them and initializing the positions and velocities of the particles are needed. Then, the positions of the particles are constantly updated with the iteration formula until obtained the optimum solutions. Furthermore, the optimum solution are arrived at in which the cycle is jumped when meeting the conditions (i. e. given the maximum number of iterations or the fitness value). There are examples with constant element and linear element and quadratic element types of boundary, in turn. The three numerical models of 2D potential problems are analyzed and discussed. They certified that the PSO algorithm has feasibility and good convergence. Most importantly, it is proved that the PSO applied in BEM has higher accuracy compared with the Gaussian elimination method for solving potential problems.

Isoparametric singularity extraction technique for 3D potential problems in BEM

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantage.A particular series expansion of a singular kernel about a source point is investigated. The series in the parametric coordinates consists of functions of a type Rpxqyr, where R is a square root of a quadratic bivariate homogeneous polynomial with coefficients corresponding to the first fundamental form of a smooth surface, and p,q,r are integers, satisfying p≤−1 and q,r≥0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of Rpxqyr, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain high order of convergence for the governing singular integrals in 3D collocation BEM with respect to the number of quadrature nodes or with respect to size of the discretization mesh cells.

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.

A regularization of the parametric integral equation system applied to 2D boundary problems for Laplace’s equation with stability evaluation

The paper proposes the regularization of the parametric integral equation system (PIES) for 2D potential problems governed by Laplace’s equation. The regularization eliminates all singularities (weak and strong) in PIES associated with its kernels as a result of their modification and is based on introducing a regularizing function with unknown regularization coefficients. Hence, all integrals appearing in the regularized PIES formulation are non-singular and can be evaluated numerically with a standard low order Gauss–Legendre quadrature rule. Using this formulation, the stability of solutions for different distances between collocation points and quadrature nodes is investigated. The influence of the number of integration and collocation points on the accuracy and stability of solutions for the proposed regularization in comparison with dedicated procedures for evaluation of strongly and weakly singular integrals is also studied. Furthermore, we analyze the impact of the regularization in PIES on the results compared with other methods for evaluating singular integrals in non-regularized PIES.

A fast direct singular boundary method for three-dimensional potential problems

The main purpose of the study is to develop a fast singular boundary method for solving the three-dimensional (3D) potential problems. It is a difficult issue to simulate the large-scale problems by the traditional SBM, because of the fully populated coefficient matrix. A fast direct algorithm based on recursive skeletonization factorization (RSF) is combined in the SBM to reduce the CPU time and memory requirements in this study. For the dense coefficient matrix in the SBM, a hierarchically generalized LU decomposition is constructed to invert it fast and accurately. The numerical results demonstrate the great performance of the RSF-SBM for solving large-scale 3D potential problems.

Solving of multi-connected curvilinear boundary value problems by the fast PIES

This paper presents the fast parametric integral equations system (PIES) in solving two-dimensional (2D) multi-connected curvilinear potential boundary value problems. The fast PIES is a combination of the fast multipole technique with a modified binary tree and original PIES. It was successfully employed for exploring the efficiency of the method applied in modelling and solving single-connected 2D potential problems. The fast PIES allows reducing the time of numerical computations (CPU time), as well as RAM utilization. However, application to the multi-connected problems requires appropriate changes in the fast multipole computations due to the small distance between some segments in internal parts of the boundary. The method is demonstrated with the examples of curvilinear potential problems dealing with perforated plates (plates with many circular holes).

The Improved Interpolating Dimension Splitting Element-Free Galerkin Method for 3D Potential Problems

This paper proposes the improved interpolating dimension splitting element-free Galerkin (IIDSEFG) method based on the nonsingular weight function for three-dimensional (3D) potential problems. The core of the IIDSEFG method is to transform the 3D problem domain into a series of two-dimensional (2D) problem subdomains along the splitting direction. For the 2D problems on these 2D subdomains, the shape function is constructed by the improved interpolating moving least-squares (IIMLS) method based on the nonsingular weight function, and the finite difference method (FDM) is used to couple the discretized equations in the direction of splitting. Finally, the calculation formula of the IIDSEFG method for a 3D potential problem is derived. Compared with the improved element-free Galerkin (IEFG) method, the advantages of the IIDSEFG method are that the shape function has few undetermined coefficients and the essential boundary conditions can be executed directly. The results of the selected numerical examples are compared by the IIDSEFG method, IEFG method and analytical solution. These numerical examples illustrate that the IIDSEFG method is effective to solve 3D potential problems. The computational accuracy and efficiency of the IIDSEFG method are better than the IEFG method.

A simple formula for obtaining OIFs on Neumann boundary in 2D potential problems and its applications

Prior to this study, an empirical formula has been proposed to evaluate origin intensity factors on the Dirichlet boundary. In this work, a simple empirical formula for obtaining origin intensity factors on the Neumann boundary is provided. As a result, origin intensity factors both on Dirichlet boundary and Neumann boundary could be directly achieved. It makes the numerical implementation of the singular boundary method easier. Numerical results are carried out to verify the accuracy and stability of the present scheme.

The interpolating dimension splitting element-free Galerkin method for 3D potential problems

The interpolating dimension splitting element-free Galerkin method for 3D potential problems

A new global and direct integral formulation for 2D potential problems

A new global and direct integral formulation (GDIF) is presented for 2D potential problems. The 'global' and 'direct' mean that Gaussian quadrature can be applied directly to the entire body surface if its geometry description is mathematically available. This concept is simple and time-honored. The method has been long pursued by several researchers thanks to its accuracy and efficiency. However, the GDIF is based on the boundary integral equations (BIEs). The most crucial but difficult part in this method is to eliminate the singularities in BIEs, especially the source singularity. In this study, new non-singular boundary integral equations (NSBIEs) with indirect unknowns are developed in association with the average source technique without using the equi-potential method for source singularity. The integrands of all integrals in the NSBIEs are finite at any point on the body surface, which allows them to be considered as a normal function for computation. Based on this, with collocation points chosen in the NSBIEs being exactly the same as Gaussian points, an arbitrary order Gaussian quadrature can be directly applied to evaluate the integrals over the global elements. Three benchmark examples are tested to verify the efficiency and convergence of the proposed scheme.

On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

<abstract> This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters. </abstract>