Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation

A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomial

A comparison of techniques for overcoming non-uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering

The Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non-uniqueness of solution obtained from the conventional boundary integral equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non-uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so-called Combined Helmholtz Integral Equation Formulation (CHIEF) method and that of Burton and Miller. The latter method introduces a hypersingular integral, which may be treated in various ways. In this paper, we present the collocation partition of unity boundary element method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton–Miller formulation regularised using the approach of Li and Huang

BINN: A deep learning approach for computational mechanics problems based on boundary integral equations

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are approximated using neural networks and solved through a training process. The loss function is chosen as the residuals of the boundary integral equations. Regularization techniques are adopted to efficiently evaluate the weakly singular and Cauchy principle integrals in boundary integral equations. Potential problems and elastostatic problems are mainly concerned in this article as a demonstration. The proposed method has several outstanding advantages: First, the dimensions of the original problem are reduced by one, thus the freedoms are greatly reduced. Second, the proposed method does not require any extra treatment to introduce the boundary conditions, since they are naturally considered through the boundary integral equations. Therefore, the method is suitable for complex geometries. Third, BINN is suitable for problems on the infinite or semi-infinite domains. Moreover, BINN can easily handle heterogeneous problems with a single neural network without domain decomposition

The boundary element method for elasticity problems with concentrated loads based on displacement singular elements

The boundary element method (BEM) has been implemented in elasticity problems very successfully because of its high accuracy. However, there are very few investigations about BEM with concentrated loads. The displacement at the concentrated load point is infinite and traditional elements will lead inaccurate results near the concentrated load point. This paper proposes two types of displacement singular elements to approximate the displacement near concentrated load point, and high accuracy can be obtained without refinement. The second type of displacement singular element is a general element type, which can work well for both problems with or without boundary concentrated loads. Numerical examples have been studied and compared with results obtained by traditional BEM and finite element method (FEM) to show the necessary of the proposed methods

RBF interpolation of boundary values in the BEM for heat transfer problems

[Abstract]: This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10-5) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10-2) with the same boundary points employed. The IRBFN-BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h1.38) and O(h4.78) for the quadratic BEM and the IRBFN-based BEM, respectively, where h is the nodal spacing

PARTITION OF UNITY BOUNDARY ELEMENT AND FINITE ELEMENT METHOD: OVERCOMING NONUNIQUENESS AND COUPLING FOR ACOUSTIC SCATTERING IN HETEROGENEOUS MEDIA

The understanding of complex wave phenomenon, such as multiple scattering in heterogeneous media, is often hindered by lack of equations modelling the exact physics. Use of approximate numerical methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), is therefore needed to understand these complex wave problems. FEM is known for its ability to accurately model the physics of the problem but requires truncating the computational domain. On the other hand, BEM can accurately model waves in unbounded region but is suitable for homogeneous media only. Coupling FEM and BEM therefore is a natural way to solve problems involving a relatively small heterogeneity (to be modelled with FEM) surrounded by an unbounded homogeneous medium (to be modelled with BEM). The use of a classical FEM-BEM coupling can become computationally demanding due to high mesh density requirement at high frequencies. Secondly, BEM is an integral equation based technique and suffers from the problem of non-uniqueness. To overcome the requirement of high mesh density for high frequencies, a technique known as the ‘Partition of Unity’ (PU) method has been developed by previous researchers. The work presented in this thesis extends the concept of PU to BEM (PUBEM) while effectively treating the problem of non-uniqueness. Two of the well-known methods, namely CHIEF and Burton-Miller approaches, to overcome the non-uniqueness problem, are compared for PUBEM. It is shown that the CHIEF method is relatively easy to implement and results in at least one order of magnitude of improvement in the accuracy. A modified ‘PU’ concept is presented to solve the heterogeneous problems with the PU based FEM (PUFEM). It is shown that use of PUFEM results in close to two orders of magnitude improvement over FEM despite using a much coarser mesh. The two methods, namely PUBEM and PUFEM, are then coupled to solve the heterogeneous wave problems in unbounded media. Compared to PUFEM, the coupled PUFEM-PUBEM apporach is shown to result between 30-40% savings in the total degress of freedom required to achieve similar accuracy

Full-wave modeling of ultrasonic scattering for non-destructive evaluation

The physical modeling and simulation of nondestructive evaluation (NDE) measurements has a major role in the advancement of NDE and structural health monitoring (SHM). In ultrasonic NDE (UNDE) simulations, evaluating the scattering of ultrasound by defects is a computationally-intensive process. Many UNDE system models treat the scattering process using exact analytical methods or high-frequency approximations such as the Kirchhoff approximation (KA) to make the simulation effort tractable. These methods naturally have a limited scope. This thesis aims to supplement the existing scattering models with fast and memory-efficient full-wave models that are based on the boundary element method (BEM). For computational efficiency, such full-wave models should be applied only to those problems wherein the existing approximation methods are not suitable. Therefore, the adequacy of different scattering models for representing various test scenarios has to be studied. Although analyzing scattering models by themselves is helpful, their true adequacy is revealed only when they are combined with models of other elements of the NDE system, and the resulting predictions are evaluated against measurements. Very few comprehensive studies of this nature exist, particularly for full-wave scattering models. To fill this gap, two different scattering models-- the KA and a boundary-element method-- are integrated into a UNDE system model in this work, and their predictions for standard measurement outputs are compared with experimental data for various benchmark problems. This quantitative comparison serves as a guideline for selecting between the KA and full-wave scattering models for performing UNDE simulations. In accordance with theoretical expectations, the KA is shown to be inappropriate for modeling penetrable (inclusion-type) defects and non-specular scattering, such as diffraction from thin cracks above certain angles of incidence. A key challenge to the use of full-wave scattering methods in UNDE system models is the high computational cost incurred during simulations. Whereas the development of fast finite element methods (FEM) has inspired various applications of the FEM for ultrasound modeling in 3D heterogeneous and anisotropic media, very few applications of the BEM exist despite the progress in accelerated BEMs for elastodynamics. The BEM is highly efficient for modeling scattering from arbitrary shaped 3D defects in homogeneous isotropic media due to a reduction in the dimensionality of the scattering problem, and this potential has not been exploited for UNDE. Therefore, building on recent developments, this work proposes a fast and memory-efficient implementation of the BEM for elastic-wave scattering in UNDE applications. This method features three crucial elements that provide robustness and fast convergence. They include the use of (1) high-order discretization methods for fast convergence, (2) the combined-field integral equation (CFIE) formulation for overcoming the fictitious eigenfrequency problem, and (3) the multi-level fast-multipole algorithm (MLFMA) for reducing the computational time and memory resource complexity. Although numerical implementations based on a subset of these three elements are reported in the literature, the implementation presented in this thesis is the first to combine all three. Some numerical examples are presented to demonstrate the importance of these elements in making the BEM viable for practical applications in UNDE. This thesis contains the first implementation of the diagonal-form MLFMA for solving the CFIE formulation for elastic wave scattering without using any global regularization techniques that reduce hypersingular integrals into less singular ones