Comparative study of finite element nodal ordering methods

£)TgineeringStructures, Vol. 20,

ELSEVIER

PIh S0141-0296(97)00047-3

Nos I-2, pp. 86 96, 1998 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/98 $19.00 + 0.00

Comparative study of finite element nodal ordering methods A. Kaveh* and G. R. Roosta lran University of Science and Technology, Narmak, Tehran-16, lran, and Building and Housing Research Centre, Tehran- 14, lran (Received October 1995; revised version accepted January 1997)

Graph theoretical algorithms are presented for finite element nodal ordering to optimize the bandwidth of their stiffness matrices. Nine different graphs are defined for representing the connectivity of FE models and the problem of finite element nodal ordering is transformed into that of a graph nodal ordering. All the presented algorithms are analyzed and the efficiency of these methods are compared through examples of two- and three-dimensional finite element models. © 1997 Elsevier Science Ltd.

Keywords: stiffness matrices, sparsity, graphs, finite element nodal ordering, bandwidth optimization 1.

Fenves and Law s. A 'corner node method' is developed by Cassell et al. 6 and Kaveh and Ramachandran 7. The application of a 'element clique graph' is due to Sloan 8 and Livesley and Sabin 9. A comparative study of the application of these graphs has been made by Kaveh and Behfar ~°. In this paper, four additional graphs are introduced together with the five graphs presented in Kaveh and Behfar ~°. The connectivity properties of FE models are transformed to the topological properties of these nine graphs. A nodal ordering algorithm is then employed for numbering the nodes of these graphs, thus obtaining the element numbering of the FE models, followed by their nodal ordering. This process is summarized in the following page. The complexity analyses of the presented methods are provided in Section 4, for a logical comparison of their efficiency. The reader may skip this section and continue with the next section without interruption in the study of the paper. The interested reader, however, can refer to Baase ~~, Gary and Johnson j2 and Smith ~3 for concepts and definitions relevant to the detail study of Section 4. The presented methods are implemented and tested for efficiency by two- and three- dimensional FE models. For the models considered, the bandwidth and the corresponding computer time are presented for comparison.

Introduction

The finite element method is a powerful numerical procedure which has been applied to the solution of various problems in science and engineering. Application of these methods leads to large systems of linear equations of the form

Ax=b with A being symmetric, positive definite and highly sparse. Guassian elimination is one of the most popular methods for solution of these equations. For an efficient solution appropriate structures should be provided for A. Such a structure reduces the storage, computational time and increases the accuracy of the solution. Band form is one of such patterns with long history, which is obtained by resequencing the variables involved. Graph theoretical methods for nodal ordering to form different patterns for sparse matrices are well-advanced in the last two decades. A comprehensive review of the developed methods can be found in excellent books by Pissanetsky ~ and Duff et al. 2 In structural engineering, Cuthill and McKee 3 presented the first graph theoretical approach to reduce the bandwidth of stiffness matrices for skeletal structures, using a level tree. Kaveh 4 employed a shortest route tree and presented a four-step approach for nodal ordering of graph models. For finite element nodal ordering, different methods are developed. The application of a 'natural associate graph', in a two-step approach, have been suggested by Kaveh 4 and

2.

A graph theoretical nodal ordering algorithm

2.1. Definitions A simple graph S is defined as a set N(S) of nodes and a set M(S) of members together with a relation of incidence which associate two distinct nodes with each member, known as its ends'. Two nodes of a graph are called adjacent if these nodes are the end nodes of a member. A member

*This paper is dedicated to Professor F. Ziegler, Dean of The institut far Allgemeine Mechanik, TU-Wien, on the occasion of his 60th birthday.

86

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta

Clique

Ske,e,o. s. Graph Graph

Graph

[

Finite Element Model

]

[

Graph Model

]

IT..Graph la,od

V/q~el

Triangulated

Graph

Graph

I

Graph Nodal Orderin~

Associate Graph Graph

87

I Representative Graph

I

I Finite Element Nodal OrderinB I

Scheme 1 Topological transformation for FE nodal ordering is called incident with a node if it is an end node of the member. The degree (valency) of a node is the number of members incident with the node. The adjacency list of a graph S is a matrix containing N(S) rows and 6 columns, where 8 is the m a x i m u m degree of the nodes of S. The ith row contains the labels of the nodes adjacent to the node i. A subgraph Si of a graph S, is a graph for which N(Si) C N ( S ) and M(Si) C M(S), and each m e m b e r of Si has the same ends as in S. A path is a finite sequence Pk = {no, m~, n~ ..... mp, np} whose terms are alternately distinct nodes ni and distinct members m~ for 1 --< i <- p and ng_~ and n / a r e two end nodes of mg. The distance between two nodes is taken as the number of members in the shortest path between these nodes. A tree T of S is a subgraph which contains no cycle; a cycle being a closed path. A spanning tree is a tree containing all the node of S. A shortest route tree (SRT) rooted from a specified node (starting node) no, is a spanning tree for which the distance between every node nj of T and no is minimum. A shortest route subtree (SRsubtree) is an SRT, but does not contain all the nodes of the graph. A contour of an SRT contains all nodes of equi-distance from no. A transversal of an SRT is a connected path P, containing one distinct node from each contour of the selected SRT. The number of contours of an SRT is known as its length and the highest number of nodes in a contour specifies the width of the SRT. The stiffness matrix K of a skeletal structure is pattern equivalent to the node adjacency matrix D of its graph model. D is a (0,1)-matrix with d(i,j)= 1 if node i is adjacent to node j, and d(i,j) = 0 otherwise. The diagonal entries of D are also taken as unity. The bandwidth of K can be optimized by reducing the bandwidth of D. This in turn can be achieved by appropriate ordering of the nodes of the structural graph model. Let /xk = ]i-j] be the difference between the lowest and highest nodal numbers of the kth element of a FEM. Then the bandwidth of the adjaceny matrix D is given as b(D)=2xMax{/x~:

k=l,...,A}+l

and the bandwidth of the corresponding stiffness matrix is b(K) =/xb(D) where /~ and A are the nodal degree-of-freedom and the number of elements of the FEM, respectively. The graph theoretical concepts and definitions employed in this paper can be found in text books on graph theory or in Kaveh ~4.

Step 1: Find an appropriate starting node, i.e. a node corresponding the narrowest SRT. Method for selecting such a node can be found 16-19. Step 2: Decompose the node-set of S into ordered subsets (contours). Step 3: Select a transversal P (representative nodes for contours) of the selected SRT. Step 4: Order the nodes of each ordered subset sequentially, to obtain the nodal numbering of S. Step 1 can be carried out by generating several SRTs and detecting the one whose width is minimum. Step 2 should be executed by generating an SRT from the selected node in Step 1, i.e. the good starting node no. Using the adjacency list of a graph S, this step can be easily carried out as follows, however, any other list may also be used: (1) Select all nodes of the noth row of the adjacency list of S. (2) Take all the unselected nodes of the rows corresponding to the selected nodes in the previous step. (3) Repeat Step 2 until all the nodes are selected. In order to select a good transversal, the following process is employed: (1) Take a node nk with minimum degree from the last contour k. (2) Find a node nk_~ with minimum degree adjacent to nk from the ( k - 1)th contour. (3) Repeat (2) until a transversal P of the generated SRT (containing k + 1 nodes) is selected. Step 4 should be executed as follows: (1) Label no (the selected good starting node or root). (2) Label n~ (the selected node in P from the first contour). (3) Generate an SRsubtree rooted from n~ and label the nodes of the first contour according to their occurrence in the SRsubtree. (4) Repeat Steps 2 and 3 for other contours, in turn. For nodal ordering of graphs any other algorithm available can also be used (see e.g. n2 and Gibbs et al.2°).

2.3. 2.2.

A nodal ordering algorithm

An efficient algorithm for nodal ordering can be constructed by multiple use of an SRT algorithm ~s,~6. This algorithm consists of the following steps:

Example

The graph model S of a simple structure is considered, as depicted in Figure 1. Using the above algorithm, first no is found as a good starting node of S. An SRT is then generated from no and a transversal P is selected as shown in

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta

88 15

29

q

32

34

35

5

. 33

10~ %

method M ( S ) is high, therefore keeping the connectivity of S can take a large computer space. In an SRT all the nodes of an element will be contained in at most two adjacent contours, hence the bandwidth become dependent on the width of the SRT.

3.2. Skeleton graph method (SKGM) Definition: The l-skeleton graph, referred to as skeleton graph S of a FEM, is a graph whose nodes are the n,=l

2

4

7

ll

16

21

Figure I A nodal numbering of a simple graph S bigger dots. The nodal numbering corresponding to a banded node adjacency (stiffness) matrix is obtained.

3.

Finite element nodal ordering algorithms

Nine algorithms are presented in this section. The first six methods, order the nodes of a FEM directly and the last three approaches use a two-step process, in which first the elements are ordered, and then ordering within elements are performed, sequentially.

3.1. Element clique graph method (ECGM) Definition: The element clique graph S of a FEM, is a graph whose nodes are the same as those of the FEM and two nodes ni and nj of S are connected with a member if ni and nj belong to the same element in the FEM. The element clique graph of the F E M shown in Figure 2a is illustrated in Figure 2b. Algorithm: Step 1: Construct the element clique graph S of the con-

same as those of the FEM, and its members are the edges of the FEM. Figure 3 illustrates the skeleton graph of the FEM shown in Figure 2a.

Algorithm: Step l: Construct the skeleton graph S of the considered FEM. For each element i connect two end nodes of each edge which are not previously connected by a member.

Step 2: Order the nodes of S using any available nodal ordering algorithm (e.g. the method of Section 2.2), thus obtaining a nodal ordering of S. In order to generate the skeleton graph of a FEM it is necessary to list the nodes of each element in a suitable order. In this method the number of members of S is less than that of the ECGM, and in FEMs with triangular elements, these number are the same. Therefore, this method takes less computer space for keeping the connectivity of S. Generating an SRT in a skeleton graph may lead to allocation of the nodes of an element in three or more adjacent contours. Therefore, the width of the SRT being used together with the number of contours containing the nodes of an element of the FEM specifies the bandwidth.

sidered FEM.

Step 2: Use a nodal numbering algorithm available (e.g. the algorithm presented in Section 2.2) for numbering the nodes of S, thus obtaining the nodal ordering of the FEM. Step 1 of this algorithm can simply be executed using the following process: For each element i, connect by a member all pair of nodes of i which are not previously connected. For the formation of the element clique graph of a FEM no specific order is needed for the nodes of elements. In this

(o)

Figure3 The skeleton graph of the FEM shown in Figure2a

(b)

Figure2 A FEM and its element clique graph, a A FEM; b the element clique graph of the FEM

Study o f finite element nodal ordering methods: A. Kaveh and G. R. Roosta

3.3.

89

XI

Element star graph method (ESGM)

Definition: The element star graph S of a F E M has two sets of nodes; namely the main set containing the nodes of the F E M and the virtual set consisting of the virtual nodes in a one-to-one correspondence with the elements of the FEM. The m e m b e r set of S is constructed by connecting the virtual node of each element i to all nodes of the element i. The element star graph of the F E M shown in Figure 2a is illustrated in Figure 4. The virtual nodes are shown by bigger dots. Algorithm: Step 1: Construct the element star graph S of the considered FEM. For each element i generate a virtual node labeling with i + ~, and connect all the nodes of i to the node of i + c~, where a is the number of nodes of the FEM.

Step 2: Order the main nodes of S using a nodal numbering algorithm available, e.g. the method presented in Section 2.2. This step should be carried out similar to the previous methods, but virtual nodes need not be labeled in the process of numbering of the nodes. The virtual nodes can easily be identified by their labels being over c~. In order to generate the element star graph of a FEM, it is not necessary to list the nodes of each element in a special order. In this method M(S) is higher than that of S K G M and can also be higher than that of E C G M (e.g. for a F E M with triangular elements). N(S) of the star graph is equal to A + a, where A denotes number of elements of the FEM. Therefore, this method for most cases requires more computer space than ECGM, and always more than that of SKGM. Generation of an SRT in an element star graph makes the nodes of an element to be contained in at most three adjacent contours.

3.4.

Element wheel graph method (EWGM)

Definition: The element wheel graph S of a F E M is the union of the element star graph and the skeleton graph of the FEM. The element wheel graph of the F E M shown in Figure 2a is illustrated in Figure 5. The virtual nodes are shown by bigger dots. Algorithm: Step 1: Construct the element wheel graph S of the considered FEM. This can be done by generating the union of the element of the star graph and the skeleton graph.

Figure 5 The element wheel graph of the FEM of Figure2a

Step 2: Order the main nodes of S using a nodal numbering algorithm available, e.g. the method presented in Section 2.2. This step should be carried out similar to that of Step 2 in ESGM. In order to generate the element wheel graph of a FEM, it is necessary to list the nodes of each element in a special order. In this method M ( S ) is higher than that of E S G M and, therefore, it needs more computer space than ESGM. The nodes of an element of F E M are at most contained in three contours of the generated SRT of the element wheel graph.

3.5. Partially triangulated graph method ( PTGM) Definition: The partially triangulated graph S of a F E M is a graph whose nodes are the same as those of the FEM and a selected node of each element i is connected to all the nodes of i. The selected nodes of the elements are found by generating an SRsubtree from a good starting node in the skeleton graph of the F E M and taking the first node of an element included in the SRT at the process of generation. As an example, for the FEM shown in Figure 2a, an SRsubtree is routed from no and shown in bold lines in Figure 6a, and the selected nodes of the elements are shown by bigger dots. The partialy triangulated graph of the F E M is shown in Figure 6b. Algorithm: Step 1: Construct the partially triangulated graph S of the considered FEM.

Step 2: Order the nodes of S using a nodal numbering algorithm available, e.g. the algorithm presented in Section 2.2. In order to execute Step 1, the following process should be carried out: (1) generate the skeleton graph S" of the FEM; (2) find a good starting node no; (3) generate an SRT from no and calculate the distance between each node of S' and no; (4) for each element i select a node which is the nearest node to no; (5) form the partialy triangulated graph S by connecting selected node of each element i to all the nodes of i.

Figure 4 The element star graph of the FEM of Figure2a

In order to generate the partially triangulated graph of a

90

S t u d y o f finite e l e m e n t n o d a l o r d e r i n g m e t h o d s : A. Kaveh and G, R. Roosta

n,

(a)

(b)

Figure 6 The skeleton, an SRsubtree and the partially triangulated graph of the FEM of Figure 2a. a the skeleton graph and an SR subtree of the FEM; b the partially triangulated graph of the FEM

FEM, it is necessary to list the nodes of each element in a suitable order. In this method M(S) may or may not be higher than that of SKGM. Generating an SRT in a partially triangulated graph, the nodes of an element are contained in one, two or three adjacent contours.

3.6. Triangulated graph method (TRGM) Definition: The triangulated graph S of a FEM is the union of the partially triangulated graph and the skeleton graph of the FEM. The triangulated graph of the FEM shown in Figure 2a is illustrated in Figure 7. The selected nodes of the elements are the same as those of Figure 6b. Algorithm: Step 1: Construct the triangulated graph S of the considered FEM. This step can be carried out by generating the partially triangulated graph and the skeleton graph.

Step 2: Order the nodes of S using a nodal numbering algorithm. In order to generate the triangulated graph of a FEM, it is necessary to list the nodes of each element in a suitable order. In this method M(S) is higher than that of the PTGM. For an SRT in a triangulated graph, the nodes of an element of FEM are contained in at most three adjacent contours.

3. Z

Natural associate graph method (NAGM)

Definition: The natural associate graph S of a F E M has its nodes in a one-to-one correspondence with the elements of the FEM, and two nodes of S are connected

Figure 7 The triangulated graph of the FEM of Figure 2a

by a member if the corresponding elements have a common boundary. The natural associate graph of the FEM shown in Figure 2a is illlustrated in Figure 8.

Algorithm: Step 1: Construct the natural associate graph S of the given FEM.

Step 2: Order the nodes of S with a nodal ordering algorithm, to obtain an ordering for the elements of the FEM. Step 3: Order the nodes of the FEM, element by element, in the same sequence as decided in Step 2. Within each element priority is given to mid nodes, passive nodes and active nodes, respectively. A node is called passive if it has no incident new element, otherwise it is known as an active node. Efficient methods for dealing with mid-side nodes are introduced in Fenves and Law 5 and Livesley and Sabin 9. An efficient method is presented and analyzed to generate the natural associate graph of a FEM in Section 4.8. This method can be used to execute Step 1 of the algorithm. Step 3 of this method can be carried out using the following process: (1) Generate a matrix NE with c~ rows and e columns, in which its ith row contains the labels of the elements containing node i, where ~ is the maximum number of elements incident with a specified node. (2) For each element j (j = 1, ..., A) execute the following steps, in turn: (a) i f j has a mid-node, label it first. (b) detect the active and passive nodes of j using the matrix NE. It should be noted that N E only makes the process fast, however, one can discard this and search for finding whether a node of j is incident with a new element or not.

Figure 8 The natural associate graph of the FEM of Figure2a

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta (c) form a multiple root SRsubtree from the active nodes o f j . (d) label the passive nodes o f j when they are selected in the multiple root SRsubtree. (e) label the active nodes of j which are adjacent to the labeled nodes. (f) repeat Step (e) until all the active nodes of j are labeled. In order to generate the natural associate graph of a FEM, it is necessary to list the nodes of each element in a suitable order. In this method M(S) has the least value among the presented methods, thus it takes less computer space for keeping the data of the connectivity of S. However, this method has disadvantage from programming point of view. In order to check two elements for a common boundary, the nodes of each boundary of the elements or the dimensions of the elements should be given as data. If the dimensions of the elements are provided, it should be noted that in three-dimensional models the elements may have mid side nodes, so having three or more common nodes does not guarantee the existence of a common boundary. Thus, the mid side nodes in an irregular configuration of elements, or number of mid side nodes of an element in a regular configuration of elements, or other data required for elements should be provided.

3.8.

Incidence graph method (INGM)

Definition: The incidence graph S of a FEM has its nodes in a one-to-one correspondence with the elements of the FEM and two nodes are connected with a member if the corresponding elements have a common node. Figure 9 shows the incidence graph S of the finite element model of Figure 2a. Algorithm: This algorithm is the same as N A G M with the difference of using the incidence graph of the given FEM in place of its natural associate graph. An efficient method is presented and analyzed to generate the incidence graph of a FEM in Section 4.9. For execution of Step 1 of the algorithm, this method can be employed. In order to generate the incidence graph of a FEM, it is necessary to list the nodes of each element in a suitable order. In this method M(S) is higher and can be higher than those of ECGM and NAGM. Thus, it may take larger computer space to keep the data for the connectivity of the graph.

91

3.9. Representative graph method (REGM) Definition: Consider the skeleton graph and select an appropriate starting node, using any algorithm available (e.g. an algorithm t6 ~7). The nearest comer node of each element of the FEM is taken as the representative node of that element. The SRsubtree of the skeleton graph of the FEM containing all representative nodes of the elements is called a representative graph S of the FEM. The representative graph of the FEM shown in Figure 2a is illustrated in Figure 10. Algorithm: Step 1: Construct the representative graph of the given FEM and number its nodes, resulting in an ordering of the elements of the considered FEM.

Step 2: Use Step 3 of N A G M to number the nodes of the FEM. In Section 4.6, a method is presented and analyzed for generation of the representative graph of a FEM. For this formation it is necessary to list the nodes of each element in a suitable order. In this method M(S) is equal to c~-l, which is the least value among all the algorithms. It should be noted that the representative graph is an SRsubtree. This method is the most efficient approach from computational time and space points of view for most of the practical models.

4.

Analysis of the algorithms

One use of order arises when describing the number of numerical operations and/or the amount of storage needed to perform a certain calculation. This number typically depends on one or more integers defining the size of the problem (such as the row and column dimensions of a matrix). When the operation count or storage is expressed as a polynomial in the dimensions, order means the highest order term, sometimes with the coefficient included. For example, it might be said that solving an n-dimensional triangular system requires order n 2 or of order n 2 operations (which this is written as O(n2)). Because the higher-order terms dominate the value of the polynomial as the dimension increases a summary of work or storage in terms of order usually provides a reasonable estimate except for problems of small dimension 2~. In the following, the presented methods are analyzed for the worst-case.

no

Figure 9 The incidence graph of the FEM of Figure 2a

Figure 10 The representative graph of the FEM of Figure 2a

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta

92

4.1.

Analysis of the nodal ordering algorithm

All methods presented in this paper apply the nodal ordering algorithm, but the numbers of nodes q~ which should be ordered with this algorithm are different for the same FEM. As an example, for the clique graph q~= ~ and for the element star graph q~= ~ + A. In this part the nodal ordering algorithm is analyzed and in the next sections these differences are taken into account. Since degrees of nodes of the graph model of a FEM are usually not very different, the adjacency list is a good matrix for keeping the data (connectivity) of the graph model of a FEM. It is supposed that the adjacency list is used in all methods presented in this paper. It should be mentioned that the following method is adopted to keep the connectivity of the nodes of an element. For all elements of same type, the nodes should be listed in a specific order. Then the member list of the skeleton graph of an element is written in which in place of the nodal labels their positions in the list of nodes are considered.

Step 1: This step can be carried out using an arbitrary algorithm 14, however, the method used in the examples of this paper is as follows: (a) Form an SRT rooted from an arbitrary node no, select a node n~ from the last contour with the minimum degree and define the width Wo of the SRT. Let no be the good starting node. (b) Form an SRT rooted from nl, select a node n 2 in the last contour with the minimum degree and define the width w~ of the SRT. If w~ > Wo, then let n~ be the good starting node. (c) Form an SRT rooted from n2 and define the width w2 of the SRT. If w2 is higher than the width of the SRT whose root is the previously selected good starting node, then let n2 be the new good starting node. Each step takes O(q~6) operations, hence time complexity of Step 1 is O(q~6).

Step 2: Step 3:

Time complexity of this step is O(q~6). Time complexity of this step is also O(q~6).

Step 4: Time complexity of this step is O(~o2~). It is clear that this step is the critical step, hence only this step is considered in the analysis of the methods presented in this paper.

4.2.

Analysis of ECGM

Step 1: This step has time complexity O(A602), where 0 is as the same of the maximum number of nodes of an element. Step 2: This step has time complexity 0(a26). It should be mentioned that this time complexity is derived from time complexity of the critical step of the nodal ordering algorithm analyzed in the previous section, by changing q~ into O/.

4.3. Analysis of SKGM Step 1: Time complexity of this step is O(A6r), where ~- is the same as the maximum number of the edges of an element. Since the bound of 7 is 0/2(0 - 1 ), therefore, time complexity of this step is O(A602). Step 2:

This step has time complexity O(c~2~).

4.4. Analysis of ESGM Step 1: This step has time complexity O(A0). Step 2: Time complexity of this step is O(/326), where /3 = A + c~. 4.5. Analysis of EWGM Step 1: This step has time complexity O(A602). Step 2:

Time complexity of this step is 0(/320).

4.6. Analysis of PTGM Step 1: This step contains the following process: (a) generate the skeleton graph; (b) find a good starting node; (c) generate the partially triangulated graph. Hence, time complexity of this step is O(h~02 + c~6 + AS0) = 0(A802).

Step 2:

Time complexity of this step is O(c~26).

4.7. Analysis of TRGM Time complexity of this method is the same as the time complexity of PTGM.

4.8. Analysis of NAGM Step 1: The following method is used to generate the associate graph of a FEM which is efficient and has time complexity O(a~202), where ~ is the maximum number of elements containing a specified node. (a) Generate a matrix with a rows and E columns in which its ith row contains the labels of the elements which contain node i. (b) Check each two elements of each row for having more common nodes. (c) When two elements of equal or different dimensions have non-mid side interface nodes with dimension equal to or more than the smallest dimension of them, then the corresponding nodes of the graph are connected to each other by a member.

Step 2:

This step has time complexity

0()t26).

Step 3: This step has time complexity O(Ao-0), where ~r is the maximum value of {02,E}.

93

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta 4.9.

Analysis of INGM

iiii iiii

Step 1: The following method is used to generate the

Ilpmnlll

iiml

Illl mini

nil

Illl

mini

|il

mill

Nil

IIII

IIII

III ill Ill III

llll mill IIII IIII mill llll mill

illl

nil

mml

!!!! mlmllmmmm

lllllllll llmllllll

mill

Illl Illl llll Ilml

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mini

Nil

iiii !!!!

iiii

imlil

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(a) Generate a matrix similar to Step (a) of the previous section. (b) Each two nodes of the incidence graph corresponding to each two elements of every row of this matrix should be connected by a member.

4.10.

Illillll

!!!!

incidence graph of a FEM which is efficient and has time complexity O(c~6e2).

Steps 2 and 3 of this method have the same time complexities as Steps 2 and 3 of the NAGM.

iiJ

III ll| llhlllll Inllllll Illlllll llllllll lllllll|

mini

Figure 12 A planar FEM

Analysis of REGM

Step 1." This step contains the following process: (a) generate the skeleton graph; (b) find a good starting node; (c) define elements containing a specified node (this step makes the formation of the representative graph more efficient); (d) generate an SRT to label the elements in turn. Hence, time complexity of this step is O(A602 + c~8 + AE0+ o~Se)= 0(3"), where 3' is the maximum value of {AS0 ~ + aBE}.

Step 2: 4.11.

This step has time complexity O(A~r0).

Results of the analyses

It can be concluded from the above analyses that REGM takes the least operations, for the worst nearly the same time as PTGM and TRGM. However, for FEMs with higher-order elements, PTGM and TRGM take far less time than ECGM and SKGM. c~ is a constant for a FEM, but 8 differs from one graph model to another. However for the practical models studied, the following results are obtained: (a) the difference between the time required for EGGM and SKGM is small; (b) in general TRGM takes slightly more time than PTGM; (c) for a FEM with low order elements ECGM and SKGM in general takes less time than or nearly the same time as PTGM and TRGM, but for FEMs with high-order elements PTGM and TRGM take far less time than ECGM and SKGM. Time complexity of EWGM for worst case is the highest. For the practical models studied, the following results are obtained:

Figure 13 The FEM of a buttress dam

(a) in two-dimensional models with elements having less than 10 nodes, EWGM generally takes the highest computational time, but in models with higher-order elements SKGM takes the highest computer time; (b) in three-dimensional models, NAGM requires the highest computational time. If REGM is not considered, the following results are obtained from the practical models being studied for detecting the fastest approach: (a) in two-dimensional models with low-order elements ECGM and SKGM may be the fastest methods, but in FEMs with high-order elements, INGM is generally the fastest approach; (b) in three-dimensional models, ESGM is generally the most economical algorithm.

5.

Figure 11 A planar FEM

Computational results

The algorithms of Section 3 are implemented on a personal computer and many examples are examined, some of which are included in this section. The bandwidth of D and the relative computational time for nine algorithms are provided.

94

S t u d y o f finite e l e m e n t n o d a l o r d e r i n g m e t h o d s : A. Kaveh a n d G. R. Roosta

Table 1 Results of e x a m p l e 1 Method

ECGM

SKGM

ESGM

EWGM

PTGM

TRGM

NAGM

INGM

REGM

b(D) Time

313 29.77

497 27.02

313 21.92

457 36.09

513 20.65

515 22.03

447 18.29

451 15.71

491 9.72

Table2 Results of e x a m p l e 2

Element EGCM SKGM ESGM EWGM PTGM TRGM NAGM INGM REG M

4 nodes

b(D) Time b(D) Ti me b(D) Time b(D) Time b(D) Time b(D) Time b(D) Time b(D) Time b(D) Ti me

4 nodes with a mid node

111 4.12 95 4.29 111 4.72 97 7.31 159 4.29 167 4.17 95 5.22 113 4.39 95 2.70

8 nodes

217 7.25 179 7.14 217 7.03 185 10.60 309 6.59 327 6.98 177 6.87 225 5.77 177 4.18

8 nodes with a mid node

333 15.32 269 15.93 333 10.60 313 17.13 477 10.60 479 11.15 271 10.28 341 7.91 271 6.48

12 nodes

437 19.55 347 20.32 437 12.97 417 20.82 633 12.97 619 13.90 353 12.31 455 9,23 353 8.07

12 nodes with a mid node

553 30.49 439 33.23 553 15.98 541 26.04 807 16.70 791 17.74 447 16.03 569 11.10 447 10.05

657 37.90 519 39.71 657 19.00 639 29.94 963 19.45 945 21.09 529 18.84 687 12.96 529 12.24

Table 3 Results of e x a m p l e 3 Method

ECGM

SKGM

ESGM

EWGM

PTGM

TRGM

NAGM

INGM

REGM

b(D) Time

843 18.62

1173 7.08

843 5.93

787 8.12

1103 7.47

1103 7.85

1185 38.67

845 7.75

1195 5.93

Table 4 Results of e x a m p l e 4 Method

ECGM

SKGM

ESGM

EWGM

PTGM

TRGM

NAGM

INGM

REGM

b(D) Time

125 1.70

221 1.32

125 1.70

221 2.42

229 1.76

213 1.76

175 6.43

125 4.45

187 1.54

EXAMPLE 1 ECGM

SKGM

ESGM

EWGM

PTGM

TRGM

A planar F E M with three types of elements consisting of 4-, 8- and 12-node elements is considered as shown in Figure 11. This model contains 4959 nodes and 2250 elements. The combination of elements of this model may be not practical, however, it is purposely chosen to illustrate the generality of the methods in dealing with presence of different elements in a model. The results are presented in Table 1. EXAMPLE 2

NAGM Scheme 2

INGM

REGM

A planar FE model with two holes is considered as shown in Figure 12. Six FEMs with 1000 elements are studied with elements having 4 nodes, 4 nodes with a mid node, 8 nodes, 8 nodes with a mid node, 12 nodes, and 12 nodes with a mid node. These models contain 1134, 2134, 3269, 4269, 5404 and 6404 nodes, respectively. The results are depicted in Table 2.

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta

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Figure 14 Patterns of I) using different graphs

Figure 14 (continued)

EXAMPLE 3 A three-dimensional FEM consisting of 480 (5 x 8 z 12) 20-node cubic elements (each edge of elements has a mid side node) is considered having the total number of 2559 nodes. The results are depicted in Table 3.

matic members and each element contains six nodes. The results are depicted in Table 4, the patterns of the node adjacency matrices are illustrated in Figure l4 and Scheme 2.

EXAMPLE 4 The FE model of a buttress dam is considered the section of which is shown in Figure 13, consisting of 480 nodes and 603 elements. This model contains three layers of pris-

6. Concluding remarks The algorithms presented in Section 3 transform the connectivity properties of finite element models into the topological properties of different graphs. Then a nodal ordering

Study of finite element nodal ordering methods: A. Kaveh and G. R. Roosta

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compared. Such analyses is the most logical way of comparing the algorithms, since most of the combinatorial optimization algorithms are configuration dependent. The computational time and space for the algorithms are compared in Sections 3 and 4. Section 5 contains examples which verify the remarks. Each algorithm has preferences and disadvantages which become manifest while the algorithm is employed for models with different element types and connectivity properties. It should be noted that the relative performance of the algorithms depends also on the starting node selection algorithm and the nodal ordering algorithm being employed. Combination of different nodal ordering algorithms with nine graph presented in this paper, changes the efficiency of the methods discussed and leads to more powerful approaches.

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References Pissanetsky, S. Sparse Matrix Teehnology', Academic Press, New York, 1994 2 Duff, 1. S., Erisman, A. M. and Reid, J. K. Direct Methods.[or Sparse Matriees', Clarendon Press, Oxford, 1986 3 Cuthill, E. and McKee, J. 'Reducing the bandwidth of sparse sym metric matrices', Proc. 24th Nat. Confi ACM, 1969, pp. 157 172 4 Kaveh, A. 'Application of topology and matroid theory to the analysis of structures', Ph. D. thesis, IC, London University, 1974 5 Fenves, S. J. and Law, K. H. 'A two-step approach for finite element ordering,' Int. J. Numer. Meths Engng, 1983, 19, 891-911 6 Cassell, A. C. de C. Henderson, J. C. and Kaveh, A. 'Cycle bases for the flexibility analysis of structures', Int..L Numer. Meths Engng, 1974, 8, 521-528 7 Kaveh, A. and Ramachandran, K. 'Graph theoretical approach for bandwidth and frontwidth reductions', Proc. 3rd Int. Col~fi Space Struetures, Noosbin, H. (ed.), Surrey University, 1984, pp. 244-249 8 Sloan, S. W. "An algorithm for profile and wavefront reduction of sparse matrices*, Int. J. Numer. Meths Engng, 1986, 23, 239-251 9 Livesley, R. K. and Sabin, M. A. 'Algorithms for numbering the nodes of finite element meshes', Comput. Systems Engng, 1991, 2, 103-114 10 Kaveh, A. and Behfar, S. M. R. 'Finite element nodal ordering algorithms', Cmnmun. Appl. Numer. Meths 1995, 11, 995 1003 1 I Baase, S. Ck~mputer Algorithms: Introduction to Design and Analysis', 2nd edn, Addison-Wesley, Reading, MA, 1988 12 Garey, M. R. and Johnson, D. S. Computers and Intractability: Guide to the Theory qf NP-Completeness', W. H. Freeman, San Francisco, 1979 13 Smith, J. D. Design and Analysis of Algorithms, PWS-KENT, USA, 1989 14 Kaveh, A. Structural Mechanics: Graph and Matrix Methods, Wiley, Exeter, 2rid Edition, 1995 15 Kaveh, A. 'Multiple use of a shortest route tree for ordering', Commun. Appl. Numer. Meths, 1986, 2, 213 215 16 Kaveh, A. "Ordering for bandwidth reduction', Comput. Struct., 1986, 24, 413-420 17 Gibbs, N. E. Poole, W. G. and Stockmeyer, P. K. "An algorithm lbr reducing the profile and bandwidth of a sparse matrix', SlAM J. Numer. Anal., 1976, 13, 236-250 18 Cheng, K. Y. 'Note on minimizing the bandwidth of sparse symmetric matrices', Comput. Z, 1973, 11, 27-30 19 Grime, R. G. Pierce, D. J. and Simon, H. D. 'A new algorithm for finding a pseudo-peripheral node in a graph', SIAM J. Anal. Appl., 1990, I I , 323-334 20 Gibbs, N. E. Poole, W. G. and Stockmeyer. P. K. 'A comparison of several bandwidth and profile reduction algorithms', A('M Trans. Math. Software. 1976, 2, 322-330 21 Gill, P. E. Murray, W. and Wright, M. H. Numerical Linear Algebra attd Optimization', Vol. 1, Addison-Wesley, CA, 1991 I

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Figure 14 (continued)

algorithm undertakes numbering the nodes of the graphs, leading to nodal numbering of the FEMs. All the methods presented are low-order polynomial time algorithms. The analyses are considered in Section 4 for worst cases and