Parametric Integral Equation System Research Articles

Near corner boundary regularization of the parametric integral equation system (PIES)

This paper presents the regularization of the parametric integral equation system (PIES) and the integral identity for 2D problems governed by the Laplace equation in polygonal and smooth domains. Due to the singularity of PIES and the existence of near-singularities in the integral identity, significant errors occur in the results near the boundary, even in the case of smooth boundaries. Additionally, polygonal domains have corner points where the boundary is not smooth, resulting in even greater errors when finding solutions within the domain near these points. These errors increase significantly as the distance to the corner points decreases. The purpose of the presented regularization is to eliminate singularities in PIES and nearly singularities in the integral identity that occur near the boundary, thus improving the accuracy of solutions. The main idea of the regularization is to introduce a regularizing function with two unknown regularization coefficients that need to be determined. As a result, the modified PIES and the integral identity become nonsingular. The paper includes several numerical examples with boundaries described by Bézier and NURBS curves that demonstrate the reliability of the regularization approach and highlight significant improvements in accuracy near the boundary and corner points.

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.

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.

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).

PIES for 2D elastoplastic problems with singular plastic strain fields

The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).

The strategy of modeling and solving the problems described by Laplace’s equation with uncertainly defined boundary shape and boundary conditions

This paper presents a new method for simultaneous modeling the uncertainty of measurement data (necessary to define the boundary shape and boundary conditions) in boundary problems. The interval parametric integral equation system (interval PIES) was developed for solving boundary problems with input data defined in this way. The motivation for conducting this research was that this topic (simultaneous consideration of uncertainties of all input data) has appeared sporadically in the literature (mainly with uncertainly defined boundary conditions or other parameters). In this paper, the uncertainty was defined using interval numbers and modeled using interval arithmetic. The direct application of both classical and directed interval arithmetic caused the overestimation and obtained solutions were useless in practice. Therefore, modification of the directed interval arithmetic was developed. The reliability of the interval PIES solutions obtained using such arithmetic was verified on 2D problems described by Laplace’s equation. The solutions were compared with the interval analytical solutions (differently obtained), as well as with the solutions of exactly defined (without the uncertainty) numerical methods. All performed tests indicated the high potential of the method. Obtained interval solutions occurred to be less overestimated and not as time-consuming as presented alternative methods.

Modeling the boundary shape of the problems described by Navier–Lamé equations using NURBS curves in parametric integral equations system method

This paper is an extended version of the ICCS conference paper (Kapturczak et al., 2020, [1]) and presents a way to improve the boundary shape modeling process in solving boundary value problems in elasticity. The inclusion of NURBS curves into the mathematical formalism of the parametric integral equations system method (PIES) is proposed. The advantages of such an application are widely discussed. Recently, the Bezier curves, mainly the cubic curves (of third-degree), were used. The segments of the boundary shape were modeled by such curves (with ensuring continuity at the connection points). Using NURBS curves, the boundary shape can be modeled with only one curve. So, continuity is automatically ensured. Additionally, the second degree NURBS curve is enough to obtain the shape with high accuracy (better than cubic Bezier curves). The NURBS curve is defined by points, their weights, and the knots vector. Such parameters significantly improve the shape modification process, which can directly improve e.g. the shape identification process. The examples of shape modification using such parameters are presented. The boundary shapes of the examples (even defined by both linear and curvilinear segments) can be defined using only one closed NURBS curve. The impact of modeling accuracy on the final PIES solutions is examined on examples described by the Navier–Lamé equations. To improve calculations, the PIES method using NURBS curves was implemented as a computer program. Then, it was decided to verify the accuracy of the obtained solutions. For comparison, the solutions were also obtained using analytical solutions, boundary element method, and PIES method (with the Bezier curves). An improvement in the boundary shape modeling was noticed. It significantly affects the accuracy of solutions. As a result, the consumption of computer resources was reduced, while the process of boundary shape modeling and the accuracy of the obtained results were improved.

The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries

The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC.

Parametric integral equation system (PIES) for solving problems with inclusions and non-homogeneous domains using Bézier surfaces

The paper presents the approach for solving 2D elastic boundary value problems defined in domains with inclusions with different material properties using the parametric integral equation system (PIES). The main feature of the proposed strategy is using Bézier surfaces for global modeling of inclusions. Polygonal inclusions are defined by bilinear surfaces, while others by bicubic surfaces. It is beneficial over other numerical methods (such as FEM and BEM) due to the lack of discretization. Integration over inclusions defined by surfaces is also performed globally without division into subareas. The considered problem is solved iteratively in order to simulate different material properties by applying initial stresses within the inclusion. This way of solving avoids increasing the number of unknowns and can also be used for elasto-plastic problems without significant changes. Some numerical tests are presented, in which the results obtained are compared with those calculated by other numerical methods. This paper is an extended version of author's conference paper [1]. It has been enriched with, among others, the description of modeling more complex inclusions, as well as additional results obtained by PIES compared to other numerical methods.

Interval arithmetic in modeling and solving Laplace’s equation problems with uncertainly defined boundary shape

The paper presents the problem of modeling the uncertainty of the boundary shape in boundary problems (described by Laplace’s equation) and proposes a method for solving so-defined problems. The uncertainty of the boundary shape is modeled using interval numbers. The interval coordinates of control points, defined using classical and directed interval arithmetic, are considered in this paper. However, because of some disadvantages of well-known interval arithmetic, the authors propose a modification of directed interval arithmetic. This arithmetic is also used in the proposed modification of the traditional parametric integral equation system (PIES) method (previously used for exactly defined problems) to solve so-defined problems. Control points of the appropriate curves, necessary to define the boundary shape, are directly included in the mathematical formalism of the mentioned method. The developed interval method is tested on problems of various shapes, modeled with linear and curvilinear segments. The correctness of the obtained solutions is verified using proposed alternative methods. The obtained solutions indicate a very high potential of the proposed method in solving problems with an uncertainly defined boundary shape.

Automatic generating and spread of a plastic region in PIES

The paper presents the strategy for automatic generating a plastic region in the parametric integral equation system (PIES) and its spread during the incremental process of solving elastoplastic problems. Instead of some preliminary research on the not known a priori plastic zone, it is found automatically on the basis of stresses on the boundary and then it propagates considering the current stress state. The presented strategy requires the PIES method as an efficient approach for solving boundary value problems and several algorithms of computational geometry. The effectiveness of PIES comes from the lack of discretization of the plastic zone, which is defined globally by surface patches known from computer graphics. Several examples provided confirm the validity of the proposed approach.

Optimization and Identification of the Shape in Elastoplastic Boundary Problems Using Parametric Integral Equation System (PIES)

Optimization and Identification of the Shape in Elastoplastic Boundary Problems Using Parametric Integral Equation System (PIES)

The fast parametric integral equations system in an acceleration of solving polygonal potential boundary value problems

The paper presents a new way of accelerating of polygonal potential boundary value problems solving by modified parametric integral equations system (PIES). For this purpose, known from the literature, the fast multipole method (FMM) was included into conventional PIES. Therefore the fast PIES method was obtained. FMM combined with conventional PIES also allows a significant reduction of utilization of random access memory (RAM) of the computer. It gives us a possibility of solving the large-scale engineering problems on a standard personal computer (PC). Finally, the paper presents the development of an algorithm for efficient solving of complex boundary value problems on a standard PC instead of supercomputers, which are usually used to accelerate calculations. The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions in a reasonable time on the PC, which was the main goal of our research.

A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

The paper introduces the analytical modification of the classic boundary integral equation (BIE) for Stokes equation in 3D. The performed modification allows us to obtain separation of the approximation boundary shape from the approximation of the boundary functions. As a result, the equations, called the parametric integral equation system (PIES) with formal separation between the boundary geometry and the boundary functions, are obtained. It enables us to independently choose the most convenient methods of boundary geometry modeling depending on its complexity without any intrusion into the approximation of boundary functions and vice versa. Furthermore, we investigated the possibility of the modeling of the domains bounded by rectangular and triangular parametric Bézier patches. The boundary functions are approximated by generalized Chebyshev series. In addition, numerical techniques for solving the obtained PIES have been developed. The effectiveness of the presented strategy for boundary representation by surface patches in connection with PIES has been studied in numerical examples.

Modeling the shape of boundary using NURBS curves directly in modified boundary integral equations for Laplace\u2019s equation

The main purpose of this study is the modification of the parametric integral equations systems (PIES) method, to include the NURBS curves into its mathematical formulas. We want to create an opportunity for defining the boundary shape using NURBS curves without the classical discretization process. We present an inclusion of the NURBS curves into PIES, and its application in process of modeling and solving the boundary value problems, on Laplace’s equation example. The correctness and accuracy of such modeling is confirmed by the examples. Obtained solutions are compared with these obtained using linear segments and B-spline curves. Additionally, we verify the correctness of modeling and modifying the shape using analytical solutions. The NURBS curves significantly expand the possibilities and increase the accuracy of the boundary shape modeling which is an important aspect of the numerical solving of boundary value problems.

A separation of the boundary geometry from the boundary functions in PIES for 3D problems modeled by the Navier–Lamé equation

In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 51–79, 2018

Modification of the Classical Boundary Integral Equation for Two-Dimensional Transient Heat Conduction With Internal Heat Source, With the Use of NURBS for Boundary Modeling

The most popular methods used for solving transient heat conduction problems, like finite element method (FEM) and boundary element method (BEM), require discretization of the domain or the boundary. The discretization problem escalates for unsteady issues, because an iterative process is required to solve them. An alternative to avoid the mentioned problem is parametric integral equations systems (PIESs), which do not require classical discretization of the boundary and the domain, while being numerically solved. PIES have been previously used with success to solve steady-state problems. Moreover, they have been recently tested also with success for transient heat conduction problems, without internal heat sources. The purpose of this paper is to generalize PIES based on analytical modification of classical boundary integral equation (BIE) for transient heat conduction with internal heat source and nonuniform rational basis spline (NURBS) for boundary modeling. The obtained generalization of PIES is tested on examples, mostly with defined exact solution.

2D elastoplastic boundary problems solved by PIES without strongly singular surface integrals

Abstract The paper presents the parametric integral equation system (PIES) without strongly singular surface integrals in elastoplastic boundary value problems. Plastic strains in PIES are approximated by interpolating polynomials and their derivatives instead of using the integral identity. Moreover, in the proposed method a boundary and a domain are not discretized by elements and cells, but are defined globally by the smallest number of curves and surfaces. Several examples are solved. The results are compared with exact values, numerical solutions obtained by other methods and also with PIES solutions obtained by the version with the singular integral identity. The results presented confirm the reliability and accuracy of the proposed approach.

Modification of the boundary integral equation for the Navier–Lamé equations in 2D elastoplastic problems and its numerical solving

SummaryThe paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2D elastoplastic problems. The modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements. This was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation. For approximation of plastic strains the Lagrange polynomials with various number and arrangement of interpolation nodes were used. Reliability of the modification was verified on examples with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.