REFINING QUADRILATERAL AND BRICK ELEMENT MESHES

ROBERT SCHNEIDERS' AND JURGEN DEBYEt

A bstract. We consider the problem of refining unstructured quadrilateral and brick element meshes. We present an algorithm which is a generalization of an algorithm developed by Cheng et. a1. for structured quadrilateral element meshes. The problem is solved for the two-dimensional case. Concerning three dimensions we present a solution for some special cases and a general solution that introduces tetrahedral and pyramidal transition elements.

Key words. Mesh refinement, hexahedral elements, unstructured meshes

1. Introduction. Quadrilateral and brick elements are widely used for finite element analysis since their performance is very good. They are almost exclusively used for the simulation of metal forming processes that are commonly modeled by the theory of plasticity according to Levy and von Mises [4]. The reason is that a solution of the Levy-v.Mises-equations must fulfil an incompressibility condition which is achieved by choosing a linear displacement - constant pressure element pair.

Unfortunately, the generation of quadrilateral and especially brick el­ement meshes is much more difficult than the generation of triangular or tetrahedral element meshes. While the problem of generating quadrilateral element meshes has been solved [9], only few algorithms for the generation of brick element meshes are known (Blacker [1] and Armstrong [5]). At IBF and LAM I a new algorithm has been developed for this purpose [6], [7]. The generation of solution adaptive meshes is not yet feasible for the three-dimensional case.

Usually, the size of the elements should be chosen according to the numerical error. This criterion is not sufficient in metal forming simula­tion, because we often have situations where we need a finer mesh at the boundary of the geometry. The problem arises from the fact that the plas­tic deformation is simulated by the movement of the mesh nodes. Fig. 1.1 shows a situation where the material flow is not represented in an appro­priate manner: The mesh overlaps a part of the workpiece, so that an additional error is introduced in the simulation. The error can be reduced by inserting additional boundary nodes so that the contact zone is repre­sented more exactly (Fig. 1.Ib).

Fig. 1.2a shows another example where in reality the cavity would be filled with material. This does not occur in the finite element simulation .

• Lehrstuhl fur angewandte Mathematik insb. Informatik (LAMI), RWTH Aachen, Ahornstr. 55, 52056 Aachen, F.R. Germany (robert@feanor.informatik.rwth-aachen.de).

t Institut fur Bildsame Formgebung (IBF), RWTH Aachen, Intzestr. 10, 52056 Aachen, F.R. Germany (debye@ibf.rwth-aachen.de).

53

I. Babuska et al. (eds.), Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations© Springer-Verlag New York 1995

54 ROBERT SCHNEIDERS AND JURGEN DEBYE

FIG. 1.1. Workpiece-die-overlap

allSS;"~1 i 1::<

-Again we can cut the simulation error by adding boundary nodes so that the area that is not filled is evidently reduced (Fig. 1.2b).

FIG. 1.2. Cavity

- -An element can be refined into 4 = 22 or 9 = 32 smaller elements

(Fig. 1.3; we will refer to these choices as 2- and 3-refinement). In principle the first choice should be preferred since the number of new elements needed to fulfil a required mesh density is smaller. This also goes for the three­dimensional case (we can split a brick into 8 = 23 our 27 = 33 bricks). From an algorithmic point of view, however, the second choice is easier, especially in the case of three dimensions.

FIG. 1.3. 2- and 3-refinement

D The rest of the paper is organized as follows: In chapter 2 we de­

velop a general algorithm for 2- and 3-refinement. For the 2-refinement scheme we use triangular elements while for the 3-refinement we gener­ate all-quadrilateral-element meshes. In chapter 3 we give algorithms for 2- and 3-refinement; these are extensions of the two-dimensional schemes. The 2-refinement-scheme introduces tetrahedral and pyramidal elements. The algorithm for 3-refinement generates all-hexahedral-element meshes but is applicable to restricted cases only. In chapter 4 we give examples of applications in the field of metal forming simulation.

2. 2D-RefineIllent. First, we consider the problem of refining quadri­lateral element meshes. A finite element mesh can be regarded as a set M

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 55

of elements over a set V of nodes. A set E of edges can be constructed from the mesh. The mesh must be conforming, which means that two elements

• share two points and one edge, • share one point or • have no points or edges in common.

A mesh is structured if each internal node belongs to exactly four ele­ments. In the following we consider conforming, unstructured meshes.

In [2] Cheng et. al. present an algorithm for the 2-refinement of struc­tured quadrilateral element meshes. We use their basic ideas to develop al­gorithms for the 2- and 3-refinement of unstructured quadrilateral element meshes. Each element f E M is assigned a subdivision level S(f) E N that indicates the degree of refinement (f must be split up into 4s(I) resp. 9S (1)

new elements). Similarly we assign a subdivision level S(e) to each edge e E E (e must be split up into 2S(e) resp. 3S (e) edges).

FIG. 2.1. Subdivision level assignment and transition elements

Refining elements or edges requires a conforming refinement of the neighboring elements if the subdivision levels are not equal (cf. Fig. 2.1). This is done by creating appropriate transition elements. Fig. 2.2 shows a 2-refinement which has been performed two times. The choice of these transition elements yields bad results since elements with very acute angles are constructed. The reason for this is that in the second refinement step elements with sharp angles are split up which leads to the creation of elements with even more acute angles.

FIG. 2.2. Bad transition elements

~ /l r---/ "';---1-"" / o 0

2 2

Cheng et. al. [2] showed a way to overcome this difficulty for structured meshes. In our paper we propose a different approach to the 3-refinement of

56 ROBERT SCHNEIDERS AND JURGEN DEBYE

unstructured meshes. With the transition technique indicated in Fig. 2.3, elements without sharp angles are created even for every refinement level ( the elements to be split up are always squares).

FIG. 2.3. Good transition elements

2 2

The refinement algorithm uses a recursive template technique. In the first step the subdivision levels are transferred to the nodes. Each node is assigned the maximum subdivision level of its n adjacent edges and elements (Fig. 2.4):

(2.1 ) S(v) = max{max See;), maxS(/;)} ~=l)n z=l,n

FIG. 2.4. Nodal subdivision levels

1

1

Subsequently the elements can be split by choosing an appropriate template. Fig. 2.5 shows the cases which have to be considered. The dots indicate the nodes with positive subdivision levels ("marked" nodes).

The following qualities of the templates guarantee that the mesh re­finement process results in a conforming mesh:

• An edge is split up into three edges if its two nodes both have positive subdivision levels,

• an edge is split up into two edges if exactly one of its two nodes has a positive subdivision level and

• an edge is not split up if the subdivision levels of its nodes are zero. In each refinement step new subdivision levels are computed for the

nodes. Fig. 2.6a shows the refinement of an element f using template 2a (S( vt) > 0, S( V2) > 0, S( V3) = S( V4) = 0). In order to update the

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 57

FIG. 2.5. Templates

0, D "fZJ '"'tiH '"'I±B "Ern ., Em

subdivision levels, auxiliary subdivision levels are computed for the element and the edges:

S(f)

S(ed

!llin S( v;) .=1,4

min{S(vj), S(Vi+t)}

After this we can compute the subdivision levels for the new nodes:

i=1,.,4 => S(vj)=max{S(v;)-l,O} (2.2) i=5,.,8 => S(vi)=max{S(ej)-l,O} if Vi is adjacent to ej

i = 9,.,12 => S(v;) = max{S(f) -1, O} if Vi is an interior node

Fig. 2.6bshows an example with S(V1) = 2,S(V2) = 1,S(v3) = S(V4) = o.

FIG. 2.6. Updating the subdivision levels

b)

Vi, Vi, D-f±±] va v, Vio Vi

Vi Vs V. Vi 2 1 1 0 0 0

It is an easy task to state the (pseudo-code) procedure that performs the subdivision of an element f:

procedure subdivide(f); var f: element; begin

Case 1 node of f marked then

58 ROBERT SCHNEIDERS AND JURGEN DEBYE

template 1; Case 2 node of f marked then

if adjacent then template 2a

else template 2b;

Case 3 node of f marked then template 3;

Case 4 node of f marked then template 4;

else return;

end; end;

The following algorithm refines the mesh according to the prespecified subdivision levels. The procedure subdivide is called for each element f that has nodes with a non-zero subdivision level. The algorithm terminates if all nodal subdivision levels are zero.

Algorithm 3.1

begin {* Subdivision level assignment *} for all elements f

for all nodes v of f

end;

if LEVEL(f) > LEVEL(v) then LEVEL(v) = LEVEL(f) endif;

end;

for all edges e: F for all nodes v of e:

end;

if LEVEL(e) > LEVEL(v) then LEVEL(v) = LEVEL(e) endif;

end;

{* Refinement *}

while (nodes v with S(v»O exist) do for all elements f

end; end;

end; call subdivide(f);

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 59

It is obvious that the algorithm needs the time O( n) - n being the number of elements in the newly generated mesh. Fig. 2.4 and Fig. 2.7 show the refinement process for the mesh of Fig. 2.1.

FIG. 2.7. Recursive refinement

Algorithm 3.1 can easily be modified to perform 2-refinement by using another set of templates. Fig. 2.8 shows the templates for 2-refinement. Note that

• an edge is split up into two edges if at least one of its two nodes has a positive subdivision level,

• an edge is not split up if the subdivision levels of its nodes are zero. These qualities guarantee the conformity of the refined mesh.

FIG. 2.8. Templates for 2-refinement

"0 "ill '"'Ed 'b'lli "EB 'IE

We use algorithm 3.1 combined with the set of templates shown in Fig. 2.8 in order to fulfil arbitrary subdivision level assignments (the reason for the choice of the templates 1 and 2b will be explained when we describe the three-dimensional scheme). The new nodal sub di vision levels can be computed according to (2.2). Fig. 2.9 shows the resulting mesh for the example of Fig. 2.1.

An unpleasant effect of mesh refinement is that the stiffness matrices associated with the refined meshes have a relatively large bandwidth b. This effect follows from the unequality [3]

(2.3) n-l

b>--- {j

60 ROBERT SCHNEIDERS AND JURGEN DEBYE

FIG. 2.9. 2-refinement

n being the number of mesh nodes and 6 being the diameterl of the graph when the mesh is considered as a graph.

A 2-refinement can be achieved without the use of triangular elements. Fig. 2.10 examplifies the idea. In this case the transition between elements with different refinement levels cannot be handled separately for each ele­ment. Instead, it is done for element pairs using template 2b (Fig. 2.11). These pairs are determined in a preprocessing step, and the refinement levels of points adjacent to two pairs are temporarly set invalid.

FIG. 2.10. 2-refinement with quadrilateral elements

~ / ~ / 2 2 2 2 2 1 1 1 1 1 1 1 1 1 -

2 2 2 2

~ / ~ / /1"'.-1-/ i"'.- /1'.- /1"'.--

The mesh is refined by using only two templates (Fig. 2.11, elements with more than one marked node are split up into four quadrilaterals). This approach cannot be extend for the three-dimensional case.

3. 3D-Refinement. Algorithm 3.1 can easily be extended to an al­gorithm for the refinement of brick element meshes. In this section we give sets of templates for 2- and 3-refinement. It is not possible to construct

1 Let dij be the length of the shortest path between two nodes i and j. then fj = max1:Si,j:Sn dij.

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 61

FIG. 2.11. Templates for 2-refinement with quadrilaterals

all transition elements for the 3-refinement case. For 2-refinement the set of templates is complete, but it is necessary to introduce tetrahedral or pyramidal elements.

Let us give some definitions first. A brick element mesh consists of a set V of nodes and a set M of elements. A set F of faces and a set E of edges can be constructed from the mesh. The mesh must be conforming, which means that two elements

• share one face, four edges and four points, • share two points and one edge, • share one point or • have no points or edges in common.

Subdivision levels S can be assigned to edges, faces and elements. In the first step nodal subdivision levels are computed. Each node is assigned the maximum subdivision levels of its adjacent edges, faces and elements:

(3.1) S(v) = max{max See;), max S(I;), max S(h;)} 2.=l,nl Z=1,n2 z=1,n3

First we consider the 3-refinement. Fig. 3.1 shows the subdivision level assignments for which an equivalent refinement exists (admissible assign­ments).

FIG. 3.1. Set of admissible assignments

CdJUJOJJCDJLOJ I , I , I I , , , , , I I , , , , , 1 , , , , t ... ---- --- ... ---- --- ... ---- --- ... ---- --- .... --- ..

Fig. 3.2 shows the refinement strategy for these cases. The element faces are refined according to the templates of Fig. 2.5. This guarantees that the refinement of two adjacent bricks results in a conforming mesh.

The set of templates given in Fig. 3.1 is not complete; it follows that not every subdivision level assignment can be processed by the algorithm. One possibility is to increase nodal subdivision levels until each brick can be split according to one of the templates. Fig. 3.3 shows the resulting mesh for a special subdivision level assignment. Evidently, it is not ac­ceptable, because there is no template equivalent to template 3 in Fig. 2.5. The refinement algorithm produces meshes with too many nodes if the subdivision level assignment is "convex".

62 ROBERT SCHNEIDERS AND JURGEN DEBYE

FIG. 3.2. 9-refinement of brick elements

FIG. 3.3. Bad resulting mesh for "convex" subdivision level assignments

Generally the results are acceptable, and for many practical cases the algorithm works very well. An example is given in Fig. 3.4 where the "convex" corner has been removed. Other examples are given in section 4.

The 2-refinement-scheme can be extended to the three-dimensional case. A brick can be split up according to every subdivision level as­signment. However, it necessary to generate pyramidal and tetrahedral elements. We do not give the full set of templates here; Fig. 3.5 shows the templates for some cases, including the "convex" case (template 1).

The templates are constructed in a way that the faces of the bricks are split up according to Fig. 2.8; This guarantees the conformity of the refined mesh. Note that template 1 motivates the choice of template 1 in Fig. 2.8.

4. Examples. An important aspect in metal forming simulation is the modelling of boundary conditions (friction). In practice the surface of

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 63

FIG. 3.4. Better refined mesh

/

FIG. 3.5. 3-refinement with tetrahedral and pyramidal elements

CDJ~ [TIJ ~-~

the workpiece has a very irregular surface structure which is not represented by a coarse mesh. In order to investigate the stresses and strains near the surface, a two-dimensional simulation of a simple metal forming process was performed. The mesh refinement scheme was used to model the structure at the boundary. The mesh was constructed from an initial coarse mesh whose boundary edges at the top where refined with subdivision level 3. The nodes where subsequently projected onto the exact contour (Fig. 4.1).

In order to perform an equivalent three-dimensional simulation, the 3-refinement scheme of chapter 3 was used to construct a mesh which models the structure at the surface of a block. Using subdivision level 2, only admissible templates appeared. The result is shown in Fig. 4.2.

5. Conclusions. The problem of refining quadrilateral element meshes has been solved by using algorithm 3.1 with appropriate templates. There remain some problems for the three-dimensional case. For some appli­cations mixed-element-meshes can be employed so that we can use the 2-refinement scheme. However, the simulation of metal forming processes

64 ROBERT SCHNEIDERS AND JURGEN DEBYE

FIG. 4.1. 2D-modeling of fine boundary structures

FIG. 4.2. Modeling the surface of a block

can only be performed by using brick elements, so that the 3-refinement scheme must be applied.

Efforts are being made to extend the brick element mesh generation algorithm proposed in [6] and [7] for adaptive mesh generation by using an octree-approach. If we want to convert an oct tree-structure into a con­forming brick element mesh, we must solve a mesh refinement problem. However, there still has to be found a way to deal with subdivision level assignments that are not compatible to the templates in Fig. 3.l.

Acknowledgments. The authors thank the Deutsche Forschungsge­meinschaft and the Graduiertenkolleg "Informatik und Technik" for their support of the project. They also thank Birgit Bomanns who implemented the algorithms and Rolf Bunten who performed the simulations.

REFINING QUADRILATERAL AND BRICK ELEMENT MESHES 65

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