New approaches for error estimation and adaptivity for 2D potential boundary element methods

In this work, a non-symmetric variational approach is derived to enforce C 1;a continuity at interelement nodes for the self-regular traction-BIE. This variational approach uses only Lagrangian C 0 elements. Two separate algorithms are derived. The first one enforces C 1;a continuity at smooth inter-element nodes, and the second enforces continuity of displacement derivatives in global coordinates at corner nodes, where C 1;a continuity cannot be enforced. The variational formulation for the traction-BIE is implemented in this work for two elastostatics problems with various discretizations and polynomial interpolants. Local and global measures of the discretization error are obtained by means of an error estimator recently derived by the authors. Comparisons are also made with the displacement-BIE, which does not require C 1;a continuity for the displacement. The lack of smoothness of the displacement derivatives at the inter-element nodes is shown to be an important source of both local and global error for the traction-BIE formulation, especially for quadratic elements. The accuracy of the boundary solution obtained from the traction-BIE improves significantly when C 1;a continuity is enforced where possible, i.e., at the smooth interelement nodes only. IntroductionRegularization of the traction-BIE eliminates the need for special numerical integration procedures for singular and hyper-singular integrals, but stricter continuity requirements must be satisfied by the density for the boundary solution to exist. The self-regular formulation, developed by Cruse and co-workers [1, 2, 3], based on conforming C 0 elements, uses a relaxed continuity approach and does not satisfy these continuity requirements, even though reasonably accurate numerical results are obtained. In this work, a variational formulation is presented for the selfregular traction-BIE, enforcing the necessary inter-element continuity conditions as subsidiary constraining equations.Somigliana's Stress Identity can be regularized by adding and subtracting appropriate terms [1]. This process is equivalent to imposing on the problem a simple linear solution for the displacement, or, equivalently, imposing a constant stress state [4]. The regularized SSI can be written with regularization based on a collocation boundary point [5]. When the limit is taken as the collocation point tends to a smooth point of the boundary and the resulting stress-BIE is multiplied with the surface normal at P, a self-regular traction-BIE is obtained [2].For problems in which the stress tensor is unique, the traction BIE gives a unique value for tractions at all smooth boundary points. A condition for the existence of a limit to the boundary of the traction BIE requires that the primary density (the displacement) must be Hölder continuous of the type C 1;a at the collocation points [6,7].The discretization of the self-regular traction-BIE into C 0 elements violates the C 1;a continuity requirement for the displacement at the interface between elements. Three approaches can...

show abstract

“…The external domain error estimator was derived in Jorge et al [19] for potential theory and extended to elastostatics in reference [20].…”

Section: Non-symmetric System Of Equations From Minimization Of Energmentioning

confidence: 99%

“…4. The numerical results for the local element errors were obtained using error estimators recently derived by the authors for potential theory [19] and extended to elastostatics problems [20].…”

Section: Introductionmentioning

confidence: 99%

A new variational self-regular traction-BEM formulation for inter-element continuity of displacement derivatives

Jorge

Cruse

Fisher

et al. 2003

Computational Mechanics

View full text Add to dashboard Cite

show abstract

“…Also in this category is the work of Abe 13, Paulino 14, Liang et al 15 and Chen et al 16. Still in this category is the research of Jorge et al 17 and Martinez‐Castro and Gallego 18, in which tangential derivatives are used to estimate the error. Some of these estimators require additional collocation points to compute residuals, which can be a drawback of this type of estimators.…”

Section: Introductionmentioning

confidence: 99%

Dual boundary‐element method: Simple error estimator and adaptivity

Portela

2011

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThis paper is concerned with the effective numerical implementation of the adaptive dual boundary-element method (DBEM), for two-dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single-region analysis. Taking advantage on the use of non-conforming parametric boundary-elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM.

show abstract

Tangential residual as error estimator in the boundary element method

Martínez-Castro

Gallego

2005

Computers & Structures

View full text Add to dashboard Cite

New approaches for error estimation and adaptivity for 2D potential boundary element methods

Cited by 6 publications

References 34 publications

A new variational self-regular traction-BEM formulation for inter-element continuity of displacement derivatives

A new variational self-regular traction-BEM formulation for inter-element continuity of displacement derivatives

Dual boundary‐element method: Simple error estimator and adaptivity

Tangential residual as error estimator in the boundary element method

Contact Info

Product

Resources

About