We present a new method for the computation of the coefficients of singularities
along the edges of a polyhedron for second order elliptic boundary value
problems. The class of problems considered includes:

• The heat transfer problem, in which case the edge coefficients
can be qualified as Edge Flux Intensity Functions (EFIF),

• The linear three-dimensional elasticity, in which case the
edge coefficients, the "Edge Stress Intensity Functions" (ESIF), represent
the stress concentration along edges or crack fronts.

Our method uses an incomplete construction of 3D dual singular functions,
which we call **Quasidual Functions**. They are based on explicitly
known dual singular functions of 2D problems **Psi_0(x,y)**
tensorized by test functions **b(z)**
along the edge, and combined with complementary terms **Psi_1(x,y)
b'(z) + ...** improving their orthogonality properties with
respect to the edge singularities.

Our method is aimed at the numerical computation of the stress intensity functions. It is suitable for a post-processing procedure in the finite element approximation of the solution of the boundary value problem. In a series of further works, we implement this method in order to compute EFIFs and ESIFs:

• In the paper presented , we implement the method for the Laplace operator and more general scalar second order operators.

Accepted June 2003.

*SIAM Journal on Mathematical Analysis* Volume **35**, Number
5 pp. 1177-1202 (2004)

Link
to Journal

Pdf file (288 k) |

The asymptotics of solutions to scalar second order elliptic boundary
value problems in three-dimensional polyhedral domains in the vicinity
of an edge involves:

(1) A family of eigen-functions with their shadows,

(2) The associated edge flux intensity functions (EFIFs), which
are functions along the edges.

Utilizing the explicit structure of the solution in the vicinity of
the edge we present a new method for the extraction of the EFIFs called
**Quasidual
Function Method,** whose mathematical foundation can be found in the
paper . This method can be interpreted as an extension of the dual
function contour integral method in 2-D domains, and involves the computation
of a surface integral **J[R] **
along a cylindrical surface of radius **R**
away from the edge.

The surface integral **J[R] **
utilizes special constructed extraction polynomials **b**
together with the dual eigen-functions for extracting EFIFs. This accurate
and efficient method provides a polynomial approximation of the EFIF along
the edge whose order is adaptively increased so to approximate the exact
EFIF. It is implemented as a post-solution operation in conjunction with
the **p**-version finite element
method.

Numerical realization of some of the anticipated properties of the **J[R]**
integral are provided, and it is used for extracting EFIFs associated with
different scalar elliptic equations in 3-D domains, including domains having
edge and vertex singularities. The numerical examples demonstrate the efficiency,
robustness and high accuracy of the proposed quasi-dual function method,
hence its potential extension to elasticity problems.

January 2004.

* International Journal of Fracture* **129** 97-130, 2004.

Pdf file (988 k) |