A Variational Boundary Element Formulation for Shear-Deformable Plate Bending Problems

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Taha H. A. Naga ◽  
Youssef F. Rashed
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Energy Functional ◽  
Boundary Integral ◽  
Element Formulation ◽  
Plate Bending ◽  
Bending Problems ◽  
The One ◽  
Shear Deformable ◽  
Element Method

This paper presents the derivation of a new boundary element formulation for plate bending problems. The Reissner's plate bending theory is employed. Unlike the conventional direct or indirect formulations, the proposed integral equation is based on minimizing the relevant energy functional. In doing so, variational methods are used. A collocation based series, similar to the one used in the indirect discrete boundary element method (BEM), is used to remove domain integrals. Hence, a fully boundary integral equation is formulated. The main advantage of the proposed formulation is production of a symmetric stiffness matrix similar to that obtained in the finite element method. Numerical examples are presented to demonstrate the accuracy and the validity of the proposed formulation.

The Dual-Reciprocity Boundary Element Method Solution for Gas Recovery from Unconventional Reservoirs with Discrete Fracture Networks

SPE Journal ◽  
2020 ◽  
Vol 25 (06) ◽  
pp. 2898-2914
Author(s):  
Miao Zhang ◽  
Luis F. Ayala
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Nonlinear Partial Differential Equations ◽  
Boundary Integral ◽  
Element Formulation ◽  
Gas Reservoirs ◽  
Gas Recovery ◽  
Infinite Conductivity ◽  
Fractured Horizontal Wells ◽  
Element Method

Summary In this paper, we present a novel application of the dual-reciprocity boundary-element formulation (DRBEM) to model compressible (gas) fluid flow in tight and shale-gas reservoirs containing arbitrary distributed finite- or infinite-conductivity discrete fractures. Compared with the standard boundary-element method (BEM), the DRBEM transforms the nonlinear domain integrals at the righthand side (RHS) of BEM formulations for nonlinear partial differential equations into equivalent boundary integrals. This transformation allows retention of the domain-integral-free, boundary-integral-only character of standard BEM approaches. The proposed approach is based on coupling DRBEM with the finite-volume method (FVM) in which a multidimensional system is solved by integrating over a line with random fractures. The resulting system of equations is solved simultaneously for fracture and matrix boundary conditions by combining DRBEM and FVM without invoking any approximation for pressure-dependent nonlinear terms such as gas viscosity and compressibility. Numerical examples and field cases are presented to test the validity and showcase the capabilities of the proposed approach. The proposed model provides a general framework that can be applied to a variety of well and fracture geometries and operating schedules, and it is used to analyze production behavior for these complex systems. To the best of the authors’ knowledge, this is the first successful application of the dual-reciprocity principle to the BEM analysis of massively fractured horizontal wells (MFHWs) performance in natural-gas formations in which nonlinear, pressure-dependent gas properties are captured without approximation.

scholarly journals Numerical Solution of the Kirchhoff Plate Bending Problem with BEM

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
V. V. Zozulya
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Boundary Integral ◽  
Kirchhoff Plate ◽  
Reciprocal Theorem ◽  
Plate Bending ◽  
Linear Boundary ◽  
Analytical Formulas ◽  
Bending Problems ◽  
Plate Bending Problem

Direct approach based on Betty's reciprocal theorem is employed to obtain a general formulation of Kirchhoff plate bending problems in terms of the boundary integral equation (BIE) method. For spatial discretization a collocation method with linear boundary elements (BEs) is adopted. Analytical formulas for regular and divergent integrals calculation are presented. Numerical calculations that illustrate effectiveness of the proposed approach have been done.

A Novel Displacement Gradient Boundary Element Method for Elastic Stress Analysis With High Accuracy

1988 ◽  
Vol 55 (4) ◽  
pp. 786-794 ◽  
Author(s):  
H. Okada ◽  
H. Rajiyah ◽  
S. N. Atluri
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Numerical Differentiation ◽  
Boundary Integral ◽  
Displacement Gradient ◽  
Direct Integral ◽  
Displacement Boundary ◽  
Element Method

The boundary element method (BEM) in current usage, is based on the displacement boundary integral equation. The current practice of computing stresses in the BEM involves the use of a two-tier approach: (i) numerical differentiation of the displacement field at the boundary, and (ii) analytical differentiation of the displacement integral equation at the source point in the interior. A new direct integral equation for the displacement gradient is proposed here, to obviate this two-tier approach. The new direct boundary integral equation for displacement gradients has a lower order singularity than in the standard formulation, and is quite tractable from a numerical view point. Numerical results are presented to illustrate the advantages of the present approach.

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