![](media/image1.jpeg){width="1.4791666666666667in"
height="0.4791666666666667in"} **Advanced Scientific Computing
Research**

**FY 2003 Accomplishment**

## **Galerkin Hypersingular Integration**

## L. J. Gray ^\*^, Oak Ridge National Laboratory

#### T. Kaplan, Oak Ridge National Laboratory

Summary

***Algorithms for the direct evaluation of hypersingular boundary
integrals have been developed. Although individual element integrals are
divergent, it has been proven that the complete boundary integration is
finite. The key advantages of the direct approach are that it is general
\-\-- applicable to an arbitrary Green\'s function \-- and that it forms
the basis for efficient gradient evaluation methods.***

Despite the fact that hypersingular equations are a well established and
essential component of boundary integral analysis, basic issues \-- the
definition and evaluation of hypersingular integrals \--  remain
difficult.  For a Galerkin formulation, singular integration is most
commonly accomplished by employing Stokes Theorem to reformulate the
equation, reducing the order of the singularities.  While successful,
the main drawback of this approach is that for complicated Green\'s
functions (e.g., functionally graded materials), even if the integration
by parts is possible, the numerical implementation can be quite onerous.

We have developed non-Stokes direct algorithms based upon defining the
(hyper)singular integrals as boundary limits \[1\].  This approach
provides a sound mathematical definition of the integrals, and moreover
establishes that the integral is finite, even though separate
integrations over individual elements are shown to be divergent.
Moreover, as the algorithms are based upon analytic evaluation at the
singular point, the accuracy is excellent.

There are two important aspects of the new algorithms, the first being
that they are applicable to a general Green\'s function.  As an example,
implementation for a thermal analysis in a functionally graded material
has been carried out \[2\]. In addition, preliminary work \[3\] suggests
that the methods will prove equally successful for the especially
difficult Green\'s function for anisotropic elasticity.  In this case,
the fundamental solution is not known in closed form, and thus new
techniques are needed to invoke the analytic integration and limit
process.

The second significant advantage is that the limit definition leads to a
highly efficient method for computing all function derivatives on the
boundary \[4\].  This is a key step in many applications, most
particularly for \'moving boundary\' simulations.

As an example, a Galerkin boundary integral analysis has been employed
to solve for the steady state temperature distribution in a graded
compressor blade. The thermal conductivity varies exponentially in the z
direction. These results (see Fig. 1) compared well to a corresponding
finite element analysis.

![](media/image2.wmf){width="2.9972222222222222in"
height="2.6645833333333333in"}

*Fig. 1. Contour plot of the temperature distribution in a graded
compressor blade.*

\[1\] L. J. Gray, J. M. Glaeser, and T. Kaplan, \"Direct Evaluation of
Hypersingular Galerkin Surface Integrals,\" *SIAM J. Scientific
Computing* (in press)

\[2\] A. Sutradhar, G. H. Paulino, and L. J. Gray, "Symmetric Galerkin
boundary element method for heat conduction in functionally graded
materials," *Int. J. Num. Meth. Eng*. (submitted)

\[3\] L. J. Gray, A. Griffith, L. Johnson and P. A. Wawrzynek,
"Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity:
Displacement Equation," *EJEE* (in press)

\[4\] L .J. Gray, A.-V. Phan, and T. Kaplan, \"Boundary Integral
Evaluation of Surface Derivatives,\" *SIAM J. Scientific Computing*
(submitted)

**For further information on this subject contact:**

Dr.Chuck Romine, Program Manager

Mathematical, Information, and Computational

Sciences Division

Office of Advanced Scientific Computing Research

Phone: 301-903-5152

romine@er.doe.gov