On the minimization of round-off errors induced by the
generalized finite element method
A. Sillem, A. Simone, L.J. Sluys
Delft University of Technology
Introduction
Notorious mechanical problems exist in which the solution of the corresponding boundary value problem contains dis-continuities, singularities and/or large gradients. An example hereof is fiber slip in composite materials of the type shown in Figure 1.
Figure 1. Fiber composites.
Applying the FEM to solve such problems necessitates
a computationally expensive mesh. The generalized
FEM (GFEM) solves this inefficiency problem but is per-turbed by round-off errors as a result of near linear depen-dent basis functions. The Stable GFEM (SGFEM) [2] can decrease the extent of round-off errors in particular cases, but fails to do so in the general sense. We propose a method which does not need a fine mesh, minimizes round-off errors and is applicable in a more general sense than the SGFEM.
Method
In [1], the authors interpolate the displacement field of a PDE with u(x ) =X j φj(x )aj + X k ,l [φk(x )χkl(x )bkl] ,
where φj are the partition of unity functions, χkl the
enrich-ment functions and aj and bklthe nodal degrees of freedom.
In this way, known information about the solution of the PDE can be included in the interpolation by means of the enrich-ment functions χ. Unfortunately, this approach induces ex-cessive round-off errors.
We propose a new method in which we modify the basis functions such that the unconstrained stiffness matrix is iden-tical to the identity matrix, thus leading to an improvement of the condition number of the constrained system matrix. As a result, round-off errors are minimized.
Results
To test the proposed method, we consider the setup depicted in Figure 2.
u = us u = ue
Figure 2. Bar under tension and constrained by springs.
The conservation of linear momentum for the bar is de-scribed by the one-dimensional Helmholtz equation
u00− α2u = 0,
where α is a problem dependent parameter. Solving the cor-responding discrete problem, with {χkl} :=x2, x4, x6, x8 as
global enrichment functions, we can compare the condition numbers and relative errors in the strain energy as depicted in Figures 3 and 4, respectively.
100 101 102 103 100 105 1010 1015 1020
number of degrees of freedom
condition n umber FEM GFEM SGFEM new method
Figure 3. Comparison of condition numbers for different methods (lower is better).
100 101 102 103 10−15 10−10 10−5 100 105
number of degrees of freedom
rel. error str ain energy FEM GFEM SGFEM new method
Figure 4. Comparison of convergence rates for different methods (lower is better).
The proposed method improves the condition number and improves the convergence rate in comparison with the FEM, GFEM and SGFEM.
Conclusion
The proposed method improves the condition number and in effect maintains the improvement in the convergence rate, as it is not perturbed by round-off errors. Similar improve-ments are realized when using multiple local and/or global (non)polynomial enrichment functions.
References
[1] I. Babuˇska, J.M. Melenk, The Partition of Unity Method, Int. J. Numer. Meth. Engng, 1997, 40, 727–758. [2] I. Babuˇska, U. Banerjee, Stable Generalized Finite
El-ement Method, Comput. Methods Appl. Mech. Engrg., 2012, 201-204, 91–111.
