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Boundary equations and boundary element methods
Published in Levon G. Petrosian, Analysis of Structures on Elastic Foundation, 2022
The combination of both the main modern numerical methods of FEM and BEM in the program complexes seems to be a promising development. A combination of these methods will cover almost the entire range of boundary and initial-boundary value problems encountered in applications today. The first steps in this direction have been taken. There are a number of works and programs implementing the idea of connecting the BEM with other numerical methods, in particular with the FEM. The bibliography on this subject is listed in [48,53,54] and others.
EEG forward problem II
Published in Munsif Ali Jatoi, Nidal Kamel, Brain Source Localization Using EEG Signal Analysis, 2017
This chapter serves as a conclusive chapter for the discussion of the forward problem, which was divided into two sections to improve readability and for additional clarity. Hence, the discussion is provided for all techniques, which are normally employed for head modeling. It was seen that although numerical models are more complex, they have more resolution and good performance for source localization problems as compared with analytical methods. Hence, mostly the numerical techniques are applied to solve this problem. These numerical methods are carried out by a certain procedure that involves mesh generation, domain discretization, equation solution, and system assemblage. Hence, commonly FEM, BEM, and FDM are used to model the head for a high-resolution solution of brain source localization. Among them, BEM is simpler as compared with FEM as it is noniterative in nature and has less computational complexity as it uses the surface as the domain rather than volume as in the case of FEM and FDM, respectively. Hence, for most applications where low computational complexity is needed with good resolution, BEM is applied with certain software packages.
Introduction to Boundary Elements
Published in Darrell W. Pepper, Juan C. Heinrich, The Finite Element Method, 2017
Darrell W. Pepper, Juan C. Heinrich
The boundary element method (BEM) is a unique numerical technique that gives accurate solutions to a class of differential equations that are commonly used to model problems arising in physics and engineering. As in the finite element scheme, the boundary element method requires a problem defined in geometrical space (or domain), but it differs from the finite element method in that here only the boundary is subdivided into a finite number of elements.
Investigation on behaviors of acoustoelastic cavities using a novel reduced finite element–dual reciprocity boundary element formulation
Published in Engineering Applications of Computational Fluid Mechanics, 2021
Wei Liu, Saeed Bornassi, Yedan Shen, Mohammad Ghalandari, Hassan Haddadpour, Rohollah Dehghani Firouzabadi, Shahab S. Band, Kwok-wing Chau
It can be seen that the boundary element formulation of the Poisson equation leads to the conventional form of the boundary integral equation (Equation 13), which consists of two integral parts: one part is the boundary integral and the other is the domain integral part. Such domain integral terms have been a major difficulty in the development of the BEM, which weakens the main advantage of this method. As mentioned in Section 1, dealing with boundary integrals instead of domain integrals and meshing the boundary instead of the internal domain are the main benefits of the BEM compared to other numerical methods such as the FEM. The need to handle domain integrals has motivated many researchers to find solution methods which are general, simple and easy to use. Many solution methods, such as the internal cell method (ICM), particular solution method (PSM), MRM and, more recently, the DRM have been proposed as the result of considerable research efforts. The following subsection explains the outstanding performance of the DRM in addressing the solution of domain integrals as well as boundary integrals.
Improved Cuckoo Search Algorithm for Solving Inverse Geometry Heat Conduction Problems
Published in Heat Transfer Engineering, 2019
Hao-Long Chen, Bo Yu, Huan-Lin Zhou, Zeng Meng
There are different kinds of Inverse heat conduction problems (IHCP), such as the identification of heat transfer coefficients, the reconstruction of boundary conditions, and the estimation of boundary geometry shapes [1]–[3]. Tutcuoglu et al. [4] employed the finite difference scheme and gradient-based inverse scheme to estimate nonlinear thermal parameter. Chang and Chang [5] used the Taylor series approximation to estimate the thermal conductivity in one-dimensional domain. McMasters [6] utilized the regularization method to estimate the heat flux as a function of time. Nanthakumar et al. [7] adopted the extended Finite Element Method (FEM) for optimization of nanostructures. Amiri and Mansouri [8] estimated the boundary shape in enclosures with non-gray media by the Conjugate Gradient Method (CGM). Generally, the biggest challenge for a numerical method to deal with the inverse geometry problem is that the domain is needed to be redivided in the inversion progress. Compared with other numerical methods, the Boundary Element Method (BEM) can not only ensure the precision of the problems, but also avoid remeshing domain [9]–[11].
The Method of Fundamental Solutions for Solving the Inverse Problem of Magma Source Characterization
Published in Geomatics, Natural Hazards and Risk, 2019
Maryam Yazdanparast, Behzad Voosoghi, Farshid Mossaiby
MFS is a meshless boundary method applicable to certain boundary value problems and initial/boundary value problems (Golberg, 1995; Karageorghis et al. 2011b, 2013; Bin-Mohsin and Lesnic, 2017). Since its introduction as a numerical method, the method has become increasingly popular. This popularity is primarily due to the ease with which the MFS can be implemented in problems in various geometries, particularly three-dimensional problems. The advantages that the MFS has over the more classical domain or boundary discretization methods are discussed here. First, the MFS is a meshless method meaning that instead of mesh a mere collection of points is required for discretization of a domain. Second, it does not involve integration which could potentially be troublesome and complicated, as is the case with, for example, BEM. Finally, it is a boundary method which means that it shares all the advantages that the BEM has over domain discretization methods such as the finite-element method (FEM) or finite difference method (FDM). In addition to the benefits mentioned herein, unlike methods such as FEM, 3 D MFS does not need the very time consuming 3 D mesh generation. In mesh-based numerical methods, human interference can never be completely eliminated from the process of mesh generation (Mossaiby et al. 2016). As a result, the MFS is ideally suited for the solution of problems in which the boundary is of major importance or requires special attention (Karageorghis et al. 2011b).