A new algorithm for domain
decomposition of ﬁnite element
A. Kaveh, K. Laknegadi and M. Zahedi
Department of Civil Engineering, Iran University of Science and Technology,
Purpose – Domain decomposition of ﬁnite element models (FEM) for parallel computing are often
performed using graph theory and algebraic graph theory. This paper aims to present a new method
for such decomposition, where a combination of algebraic graph theory and differential equations is
Design/methodology/approach – In the present method, a combination of graph theory and
differential equations is employed. The proposed method transforms the eigenvalue problem involved
in decomposing FEM by the algebraic graph method, into a speciﬁc initial value problem of an
ordinary differential equation.
Findings – The transformation of this paper enables many advanced numerical methods for
ordinary differential equations to be used in the computation of the eigenproblems.
Originality/value – Combining two different tools, namely algebraic graph theory and differential
equations, results in an efﬁcient and accurate method for decomposing the FEM which is a
combinatorial optimization problem. Examples are included to illustrate the efﬁciency of the present
Keywords Finite element analysis, Differential equations, Graph theory, Parallel programming
Paper type Research paper
Most of the methods for ﬁnite element (FE) decomposition employ a graph model to
convert the decomposition problem into the partitioning of the corresponding model.
Ten different types of graphs are presented in Kaveh (2004, 2006) for transforming the
connectivity properties of the FE meshes into the topological properties of their graphs.
Algorithms for partitioning can be classiﬁed as:
topological graph theory methods;
algebraic graph theory methods; and
For the graph theory method, Farhat (1988) proposed an automatic FE domain
decomposer, which is based on a Greedy type algorithm and seeks to decompose a
ﬁnite element model (FEM) into balanced domains, sharing a minimum number of
common nodal points. In order to avoid domain splitting, Al-Nasra and Nguyen (1991)
incorporated geometrical information of the FEM into an automatic decomposition
algorithm similar to the one proposed by Farhat (1988). The Sparpak uses nested
dissection due to George and Liu (1978), which uses a level tree for dissecting a model.
Kaveh and Roosta (1994, 1995) employed different expansion processes for
decomposing space structures and FE meshes.
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Received 16 August 2007
Revised 30 January 2008
Accepted 30 January 2008
International Journal for
Computer-Aided Engineering and
Vol. 25 No. 5, 2008
q Emerald Group Publishing Limited