Adaptive mesh refinement for localised phenomena

Pergamorb Compuars & Svu~rures Vol. 63. No. 3, pp. 4lS495. 1997

8 1997 Elsevier Science Ltd Printed in Great Britam All rights reserved

PII: SOO45-7949(%)00372-O 00457949/97 $17.00 + 0.00

ADAPTIVE MESH REFINEMENT FOR LOCALISED PHENOMENA

A. Selman,? E. Hintont and N. BiCaniC$ TUniversitt de Technologie de CompiZlgne, GSM LG2mS MNM-BP20529, 60205 CompiBgne CZdex,

France

SUniversity of Wales Swansea, Department of Civil Engineering, Swansea SA2 8PP, U.K.

&Jniversity of Glasgow, Department of Civil Engineering, Glasgow G12 SQQ, U.K.

(Received 28 September 1995)

Abstract-The present work is concerned with the development, testing and use of finite element based methods in conjunction with adaptive mesh refinement procedures for the solution of three types of static and dynamic stress analysis r:oblems. First, an adaptive mesh refinement procedure is introduced and used in static plate bending finite element analysis based on Mindlin-Reissner assumptions to study the edge effects which occur in plates with certain types of boundary conditions. Several issues of finite element mesh dependence and adaptivity in strain localisation problems are then discussed and illustrated. Finally, adaptive finite element methods for the solution of two-dimensional transient dynamic stress analysis problems ar,: developed and tested on some benchmark examples. In all cases cited above there is a need to refine the finite element mesh locally in certain zones either within or at the boundary of the domain under consideration. In the transient dynamic analysis there is a further problem; these zones will move with time. (13 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The finite element (FE) method is now firmly established as an engineering tool of wide applica- bility. When carefully used by experienced and diligent engineers, FE based models can provide much valuable in.sight into structural behaviour. However, designers who do not possess extensive expertise in numerical analysis, may unquestionably believe that all FE results are highly accurate or even exact. This has led to recent emphasis on adaptive analysis or adaptive mesh refinement (AMR) procedures to improve the reliability of the FE method and to ensure that the results produced are of appropriate accuracy [l-5].

In general AMR procedures, three ingredients are required:

(a) a means for local refinement indication; (b) a technique for converting this refinement

information into a desired mesh density; and (c) an automatic mesh generation facility which

produces mesh.es of the desired mesh density.

The importance of AMR procedures in industrial applications has led to increased research on fully automatic mesh generators which require only the specification of the boundary and mesh size distribution over the domain under consideration. The success of AMR procedures depends to a large extent on the efficient coupling between the adaptive FE analysis and automatic mesh generation.

It is the aim of this work to develop, test and use

FE based methods in conjunction with AMR procedures for the solution of static and dynamic stress analysis problems. In particular, the following applications are considered:

l The AMR procedure is used to study edge effects in the static bending FE analysis in Mindlin-Reiss- ner (MR) plates.

l Several issues of FE mesh dependence and adaptivity for localised failure mode predictions in strain softening analysis are discussed and illustrated.

l The adaptive FE method for the solution of two-dimensional transient dynamic problems is considered.

In all cases cited above, there is a need to refine the FE mesh locally in certain zones either within or at the boundary of the domain under consideration.

Plate bending analysis based on MR formulations is complicated by the presence of boundary layers on plate edges subjected to certain types of boundary conditions. Furthermore, the solution depends on the plate thickness in a complicated way. The examples involving boundary layers presented in this work are very demanding and amply demonstrate the potential of AMR procedures.

In strain softening problems, it will be shown that mesh improvement by element densification alone, invariably leads to diffuse failure modes. It typically leads to an excessive number of elements, indicating that mesh improvement by realignment might be more efficient.

475

476 A. Selman et al.

MIn transient dynamic problems, there is a further problem; the zones of high gradients will move with time.

The applications considered in this work, consti- tute a set of stringent benchmark examples for the AMR procedures developed and their computer implementation.

We note two essential items which are required in

AMR algorithms:

l A mesh generator which is capable of producing meshes with a prescribed mesh density and arbitrary geometries.

l A means of evaluating an appropriate mesh density to be fed to the mesh generator to produce a mesh which should be better adapted to the solution.

The first item cited above is central to all adaptive procedures and will be presented in the next section.

However, the means for evaluating the new mesh density varies for different applications and this will be described at the appropriate part of this work.

2. AUTOMATIC MESH GENERATION

By far the most tedious and time consuming task in FE analysis lies in the preparation of input data. Extensive research has been carried out in the last two decades to automatically generate FE meshes in general domains without the need for extensive user interaction [6]. This has alleviated much of the effort involved in data preparation for FE analysis and

reduced the likelihood of input data errors. In recent years, a wide range of algorithms have

been developed for the generation of unstructured grids around complex geometrical shapes; of which a survey has been presented by Shephard [6].

The advancing front method (AFM) of Peraire et al. [7] appears to be one of the best approaches for mesh generation for problems involving adaptive analysis since it incorporates a remeshing facility to allow the possibility of (directional) refinement or de-refinement and allows significant variation of mesh spacing throughout the computational domain. Another important feature of Peraire’s implemen-

tation is that elements and nodes are generated simultaneously as compared to other algorithms which require the generation of all interior nodes in the domain before their connectivities can be defined.

The basic steps required for such mesh generation are:

(a) Specification of a background mesh. In the AFM it is necessary to specify a background mesh, the main purpose of which is to control the spatial distribution of element sizes i.e. mesh density throughout the domain. The background mesh consists of triangles with corner nodes at which the values of various mesh parameters (such as mesh size, stretching and orientation of the mesh) are specified. Naturally, the background mesh must completely cover the whole domain to be discretised. Generally,

Fig. 1. Background mesh.

one or two elements will be sufficient if only a “uniform” mesh density is required. An example of a background mesh is displayed in Fig, 1.

(b) Specification of mesh parameters. The geometrical characteristics of the mesh are locally defined by means of the mesh parameters. They represent the size and shape of an element and are taken to be a function of position. In the AFM, the element characteristics are represented by a set of orthogonal directions aI ,a2 with corresponding spacings 8, ,&, as shown in Fig. 2. The spacing bi is the maximum distance between projections of the nodes of the element onto the directions ~1, (i = 1,2). Alternatively, one can introduce the stretching parameter s (s = S,jS, 2 1) as the ratio between the spacing 6, and the reference (minimum) spacing & (or simply 6). In this case the mesh parameters are the value of S and the stretching parameter s.

3. AMR WITH MR ELEMENTS

3.1. Preamble

Over the last decade or so there has been a growing

tendency towards the use of MR FE plate formulations in place of Kirchhoff idealisations. Several benefits have accrued from the use of MR representations such as the automatic inclusion of transverse shear deformation effects. However, plate

Fig. 2. Mesh parameter definition for a typical triangular element: 6 is the

and aI and a2 the spacing; s the stretching parameter local orientation of the mesh.

Adaptive mesh refinement for localised phenomena 417

bending analysis based on MR formulations is complicated by the presence of boundary layers on plate edges subject to certain types of boundary conditions (BC). Kant and Hinton [8] have demonstrated this effect in the analysis of square plates with various BC using the segmentation method and other authors [9-161 have also discussed this problem.

In this section AMR processes for plate bending analysis using the 6-noded triangular MR element introduced by Zienkiewicz et al. [17] are developed. The implementation of the AMR scheme is slightly different to that of Zienkiewicz and Zhu [5] and follows the modifications suggested by Atamaz-Sibai and Hinton [18, 191. This scheme is used to study boundary layer effects in a series of rhombic and circular plates.

3.2. Background theory

In 1850 Kirchhoff [20] presented the well known governing biharmonic equation of classical thin plate theory (CPT). Associated with this equation are two BC along the edge of the plate. Instead of finding a displacement field w(x, y) to satisfy the biharmonic equation we can attempt to find the displacement field w(x, y) which minimises the total potential energy (TPE) of the plate

I-I = f t;D,,~bdQ - s 5

qw dQ (1) ‘L n

where a is the plane area of the plate, q is the distributed lateral loading, Db is the matrix of flexural rigidities and the vector of curvatures is

& = [a%/dX’, a2w/ay2, 2a2w/aXay]T. (2)

In 1945 Reissner [21] and later in 195 1 Mindlin [22] presented a higher-order theory which allows for transverse shear strain energy effects. The governing equation may be written as a sixth-order differential equation with three BC along the plate edge or in the following energy minimisation form: find the displacement field w(x, y) and the independent normal rotation fields 0,(x, y) and &(x, y), which minimise the TPE of the plate

where D, is the matrix of shear rigidities, the vector of curvatures is

(4)

t Except through the scaling factor t’ which is absorbed into the loading q.

Table 1. Boundary conditions in MR plates and the conditions assumed in displacement-based FE represen-

tations

Essential BC Natural BC

8. = 0 Hard clamped<,, 0, = 0

;Y = 0

Soft clamped-C 8. = 0

&=O w=o

Hard simple support-& M” = 0

8, = 0 w=o

h4,=0 Soft simple support-S, I%=0

w=o

Free-F lu” = 0 IV.,=0 S” = 0

and the vector of shear strains is

cs = [awjax - e,, aw/ay - e,y. (5) The constitutive relations may be written as follows: for the bending moments M = [M,, M,, M,,lT

M = D&b; (6)

and for the shear forces S = [S,, $1’

S = DICE = D,(Vw - 19) (7)

where V = [a/ax, a/+]’ and 8 = [e,, e,]‘. The BC for the MR mode1 are obtained in the

following way: if we insist that 0, and 0, (the normal and tangential components of 0) and w all vanish on the boundary of 52, we obtain a model of a clamped plate, which is called hard clamped. If we impose 8, = 0 and w = 0 on aR, but do not restrict 0,, we obtain another model of clamping, which is termed soft clamped. In general, we may impose any combination of the three essential BC 8, = 0, 8, = 0 and w = 0, thereby obtaining eight distinct boundary value problems. Table 1 summarizes the five BC with the greatest physical significance considered in this work and indicates how these conditions are enforced at boundary nodes in displacement-based FE. It also indicates the notation used in Section 3.6. Before discussing AMR procedures we first consider the existence of boundary layers in MR plates.

3.3. Boundary layer in MR plates

A fundamental difference between Kirchhoff and MR plate models is that the solution of the former is independent of the plate thickness t,t while the solution of the MR model depends on the thickness in a complicated way producing a narrow boundary layer.


In a detailed mathematical study of this problem, Arnold and Falk [l l] by expanding asymptotically (as t + 0), in powers of the plate thickness t, the transverse displacement w in a regular expansion and the rotation vector 8, in a similar expansion augmented by a boundary layer expansion, have identified situations in which a boundary layer, near smooth edges, may develop.

Haggblad and Bathe [ 151 used a different approach, they concentrated on the local behaviour of the MR solution near edges and corners in thin plate situations, and presented additional results. In their approach the deformation of the plate is represented by the transverse displacement of the mid-surface and the local twist of fibres, originally normal to the mid-surface.

In most cases, stress resultants are of prime interest in the FE analysis of plates. Areas of peak bending moments and shear forces are of particular importance. It is therefore of interest to note that the boundary layer has a more pronounced effect on quantities derived from 8. For example, the bending moments depend on the first derivatives of 0 and have a boundary layer of one order higher than that of 0. The shear force vector in the boundary region has a very marked edge effect for plates with soft simple supports or free edges. However, for hard clamped and hard simply-supported plates, the edge effect associated with the shear force vector? is of order one and for soft clamped plates, it is an order t effect. A detailed mathematical justification and a number of applications can be found in the papers by Arnold and Falk [l l] and Haggblad and Bathe [15].

3.4. Error estimation and adaptivity

In this section, after a description of the AMR algorithm, the two main ingredients of the algorithm, namely the error estimation and refinement procedure, are presented.

3.4.1. AMR algorithm. The AMR procedures are based on the following algorithm:

(1) Produce a starting mesh and carry out an initial FE analysis.

(2) Based on the results of the FE analysis produce an error estimate.

(3) If the error is acceptable stop; otherwise continue. (4) Using an automatic mesh generator re-mesh

based on information from the error estimator. (5) Carry out a further FE analysis based on the new

mesh and go to step 2.

t Scaled to be O(1) in the interior expansion. $ These “best guesses” may be obtained by least squares smoothing, nodal averaging of the FE results, some projection method [23] or alternatively by the recently presented superconvergent patch recovery (SPR) technique [24,25]. 9 The most efficient mesh is taken to be the one in which the error is equally distributed over the elements.

3.4.2. Error estimation. The first item required in AMR procedures is a reliable error estimator. In the present context we use an error energy norm expressed in terms of the bending moments and shear forces and of the following form

]le11*= [M--611TD;‘[M-@dR s n

+ s

[S - SITD;‘[S - S]dn (8) R

where M = [M,, My, A&IT and fi = [ax, Qy, I$?~,,]~ are the FE and exact bending moments, respectively, and S = [$,, $1’ and S = [&, $lT are the FE and exact shear forces, respectively.

As we do not have access to the exact solutions for the bending moments 6l and shear forces s, we substitute our “best guesses”$ M* and S*, so that

&I zM* and S xS*. (9)

Obviously, as will be seen in the next section, the determination of M* and S* depends on the type of element used.

The strain energy of the exact solution is estimated as

IIw*l12 = s [M*]rD;‘M* da + [S*]rD;‘S* dR. n

(10)

Both 11 e 11 and II w* I( can be evaluated at the element level so that

llell*= i lIelIt and IIw*ll*= i IIw*llf (11) i=l i=,

where n is the number of elements in the domain R. During the AMR process, when the relative global

error reaches a permissible value of ij (specified by the user) the solution is accepted. This condition may be expressed as

II e II G 4 II w* II (12)

3.4.3. ReJnement procedure. As the error is evaluated at the element level, we define a parameter & for each element i as$

6 = c II e II i/P (13)

where c is a relaxation factor and p = q 11 w* II/n”*.

Adaptive mesh refinement for local&d phenomena 479

Depending on the value of ci we can define the element state as

0 optimal element size if & = 1; l refinement is necessary if ci > 1; and l de-refinement is possible if 5, < 1.

By assuming th.at the rate of convergence of the adopted element is 0(/r’) we can design the new element size /; in terms of the existing size hi using the expression

The FE shear forces are also linear over the element. Since it is possible to get oscillations in the distribution of the shear forces along the element edges, the FE shear forces are evaluated at the element centroid and a constant distribution over the element is assumed. Averaged nodal values are subsequently obtained at the vertices and a linear interpolation is then assumed. Consequently, the terms in the second integral of (8) are quadratic polynomials and a three-point integration rule is adopted.

t;, = hi/t;” (14) 3.6. Examples

where I is the degree of the approximating polynomial or some constant which depends on the presence and nature of any singularities in the problem. Equation (14) is used to evaluate the design mesh density for the automatic mesh generator.

To demonstrate the AMR procedure with MR elements and to illustrate the occurrence of boundary layers, we consider the analysis of uniformly loaded, rhombic and circular plates.

3.5. Element selecrion

3.5.1. Characteristics of an ideal element. As the mesh generator referred to in Section 2 is based on triangles, it is necessary to select a triangular plate bending element. However, before describing the element used in the present studies the characteristics of an ideal MR element are considered:

3.6.1. Rhombic plates with soft simple supports. The analysis of a series of uniformly loaded rhombic plates with soft simple supports on which the lateral displacement w = 0, is considered. All plates in the initial study have a thickness-to-side length (t/L) ratio value of 0.05 and a Poisson’s ratio of 0.3.

In all cases, from considerations of symmetry only a symmetric quadrant is analysed as shown in Fig. 3. A convergence tolerance rj of 4% is chosen and the initial mesh has 9 elements.

l the element should not lock in thin plate situations; l the element should converge; l the formulation should not be based on numerically

adjusted factors; l the element should be capable of providing accurate

displacements, bending moments and shear forces and be relatively insensitive to element distortions and

l the element should be easy to implement and use.

3.5.2. Element used. Recently, a 6-noded triangular MR element, presented by Zienkiewicz et al. [ 171, was claimed to be highly competitive with available MR elements. The element has a total of 12 degrees of freedom, lateral displacement w and rotations 0, and 0, at the corner nodes and hierarchical tangential rotations A0, at the midsides. This new element has been independently implemented and tested [26] and found to give excellent all round performance. For this reason it has b,een chosen in the present studies.

For a couple of rhombic plates in which the angle c( = 45” (square plate) and a = 22.5”, Figs 4 and 5 illustrate the mesh progression and the associated distributions of twisting moments M,, and shear forces S, obtained using the AMR procedure.? The main feature is the gradual formation of the boundary layers in the mesh, twisting moment and shear force distributions on the outer edges of the plate. As the angle CI decreases from 45” down to 22.5” the obtuse corner acts as a pole of mesh attraction to capture peak values in that region. This conforms with mathematical results and confirms the potential of AMR procedures in dealing with such problems.

For thickness-to-side length ratio (t/L.) values of 0.1, 0.05 and 0.01, and a skew angle of 45”, Fig. 6 shows the dependence of the solution based on the MR plate model on the plate thickness. The width of the layer is shown to be of the order of the plate thickness.

3.5.3. Evaluation of the best guess solutions. As the bending moments resulting from the FE analysis are linear, the improved moments which are interpolated with quadratic shape functions from the averaged nodal values are used. Thus, the terms in the first integral of (8) are quartic polynomials and a seven-point integration rule is used.

7 Note that n and t coordinates are the normal and Fig. 3. Symmetric quadrant of a rhombic plate showing the tangential axes to the supported edge. initial mesh.

480 A. S&nan er ol.

For a chosen skew angle of 45” {square plate), a rotations 8, and 8, and displacements w, shear forces complete set of results is presented in Fig. 7. These S, and 3, and bending moments AZ,,, M,,,, M, and M2.

results consist of the final mesh and corresponding This amply demonstrates which variables exhibit

Mesh Ma,

Fig, 4. Symmetnc quadrant of a uniformly loaded rhombic (a = 45”) plate with soft simple-supports, sequence of adaptive analyses showing meshes, twisting moments M,, and shear farc.as SC, final mesh: 935

elements and 30% DOF.


Mesh

Fig. 5. Symmetric quadrant of a uniformly loaded rhombic (a = 22.5”) plate with soft simple-supports, sequence of adaptive analyses showing meshes, twisting moments M., and shear forces S,, final mesh: 65 1

elements and 2178 DOF.

boundary layers and gives an indication about the associated strength of the boundary layers.

3.6.2. Circular plate with soft simple supports and hard clamped supports. These examples are used to illustrate the special case of circular plates with axisymmetric loading and BC. In this case the edge effect disappears entirely for all types of boundary considered [I 11. Figure 8 shows the symmetric quadrant analysed while Fig. 9 shows results for a soft simply supported plate of radius R (for which t/R = 0.05 and v =: 0.3).t

Though this problem is often used as a benchmark test, the absence of boundary layers should be high1ighted.S Finally, we note that the final cost (CPU time) for these examples is less than 1.5 times the cost of the computation for the final mesh, which is quite reasonable.

t The hard clamped :plate shows similar contour plots. t As noted in Ref. [lti], care must be taken when modeling curved boundaries using plate bending elements with straight sides. The boundary layer may be unintentionally suppressed when analysing plates with hard simple-support and/or soft clamped support conditions.

4. AMR IN STRAIN SOFTENING ANALYSIS

4.1. Preamble Localisation of deformation refers to the emer-

gence of narrow regions in a structure where all further deformation tends to concentrate (localise), in spite of the fact that external actions continue to follow a monotonic loading programme. The remaining parts of the structure usually unload and behave in an almost rigid manner.

Geological materials are rich in examples of localised behaviour. The most prominent example of shear band formation in geotechnical engineering is, perhaps, the progressive failure of slopes. This problem has received considerable attention in the past. Another form of (extreme) localisation is the development of cracks in concrete, rock and other brittle materials.

The feasibility of capturing localised plastic deformation using the FE method has been demonstrated by a number of investigators for both finite and small deformations in various fields of application. There is, however, a tendency for these analyses to be highly sensitive to details of mesh design.

In order to remedy this situation, it appears to be extremely important to adopt some kind of mesh

482

Mesh

A. Selman et al.

Fig. 6. Symmetric quadrant of a uniformly loaded rhombic (a = 45”) plate with soft simple-supports and thickness-to-side length ratio values of 0.1, 0.05 and 0.01, final meshes and corresponding twisting

moments M., and shear forces S.

adaption strategy in order to enhance the possibility for a localised solution.

Different automatic mesh generators have herewith emerged, which are not only concerned with the change of domain geometry as a result of local&i deformation, the improved mesh for the whole domain (nodal spacing, element alignment, stretching) is constructed on the basis of the solution obtained [7]. Such developments are related to mesh adaptivity concepts used with error estimates analysis, in particular with developments in computational fluid dynamics, where the modelling of distinct shocks represents a direct counterpart to localised failure in the form of a tensile crack or a shear slip.

For localised failure predictions in strain softening analysis, an arbitrary FE mesh may not even be capable of kinematically reproducing a real failure mode. Moreover, an unbiased discretisation, i.e. an arbitrary mesh, may to some extent favour certain, possibly wrong, modes of failure. As will be seen, some form of mesh improvement is needed, once the numerical solution identifies that the actual mode of failure is not properly accommodated by the initial mesh.

4.2. Element selection

When the domain considered has a complex shape, the grid has to accommodate rapid solution changes and high solution gradients and when the topology is


Fig. 7. Symmetric quadrant of a uniformly loaded rhombic (tl = 45”) plate with soft simple-supports, final mesh, corresponding rotations 0. and B, and displacements w, shear forces S, and S, and bending moments

M., M,,, MI and MS.

severely restricted by the existence of internal In strain softening analysis, the choice of linear

boundaries, the use of triangular elements is FE interpolation functions adopted in constant

recommended [27,28]. strain triangles is a suitable means for the representation of strain discontinuities across inter-

TProvided that element atignment is used, as will be seen in element boundaries.7 Furthermore, an advantage of

Section 4.6. using constant strain triangles is that they are


Y

Fig. 8. Symmetric quadrant of a circular plate showing the initial mesh.

relatively simple to handle from a computational viewpoint.

4.3. Discretisation bias in failure modes

In the absence of some other criterion, a choice of a FE mesh is primarily governed by the problem geometry. In addition, the experience of the analyst dictates that mesh densification should occur in areas where large gradients in the solution are to be expected and care is often taken to minimise excessive element distortions, thus ensuring mesh quality.

A chosen FE mesh fixes the approximation space and the set of the available trial functions. For failure analysis, the same approximation space has to accommodate the solution both in the early stages, when the damage is very diffuse, as well as in the final stages, where the ultimate failure mechanism is formed.

Any differences in the solution, that might be recorded for different meshes in the initial nonlinear stages, can easily be attributed to different discretisa- tions. However, if the actual failure mode is highly localised, different meshes may lead to very different failure mechanisms. When the prediction of the failure mode is significantly affected by the mesh choice, it becomes questionable whether one can categorise these differences as discretisation errors, rather it is more appropriate to refer to them as a discretisation bias.

tThe elements bands in the first and second mesh shown on Fig. 11 are inclined to the exact angle (analytical solution [27]) of the slip lines for plane strain and plane stress analysis, respectively. 1: To advance the numerical solution in a truly adaptive fashion, it would be necessary to map all solution and state variables from the old to the new mesh. Such remeshing is done at every load increment and the difficulty arises as the mapping of the converged stress state on the old mesh (an inherent ingredient of a correct elasto-plastic solution procedure) may result in a nonconverged state of stress in the new mesh. Therefore, instead of a true continuous adaptivity process, a continuous remesh and reanalyse is adopted.

4.4. illustrative example-model problems

A rectangular sheet with a thickness of unity is subjected to a uniform prescribed extension with a displacement rate of i = 4.5 as shown in Fig. 10 [27]. The material and geometrical properties are also shown in Fig. 10. Both plane strain and plane stress conditions are considered for von Mises yield criterion with an isotropic softening rule, which is defined by the uniaxial test curve in Fig. 10. Ortiz et al. [29] analysed a similar model with enriched four-noded elements. On extension, the entire panel deforms uniformly. Due to the uniform state, the problem is of a bifurcation type and a family of possible failure mechanisms exist. However, if an imperfection is introduced at some point (in the form of a lower equivalent yield stress, for instance), the problem changes from a bifurcation one to a limit load problem and the deformation would be expected to localise in a definite localisation band.

The loading is applied proportionally in such a way that the sheet reaches the state just before the onset of yielding in the first displacement increment (Ad1 = 1.73 for plane strain and Ad, = 1.68 for plane stress). At the second increment, almost the entire sheet may enter the plastic range (Adz = 0.17 for both plane strain and plane stress).

The panel is discretised with two differently biased meshes as shown in Fig. 11. Each mesh favours a different localised failure mechanism and the solutions eventually follow that bias-although there is very little difference in the pre-peak response and the peak load, the final localised failure mode is different.? For a mesh with no obvious bias a diffused response is obtained. The same observations can be made for a panel compression model problem.

4.5. Mesh improvement by element denstjkation

In general it is not possible to predict where the localisation will develop. To avoid bias introduced by the initial mesh choice, some form of remeshing strategy is adopted. Such remeshing is usually done only once (typically near failure), followed by a complete reanalysis, using an improved mesh.f

Mesh redesign may be based on:

l Local material instabilities. l Plastic work density. l Contribution of predominant eigenvector near the

peak.

4.5.1. Remeshing using eigenvectors. Remeshing based on the contribution of the predominant eigenvector near the peak is used in the present work and is now presented. Eigenvalue analysis was used by de Borst [30] and Crisfield [31] to locate the lowest energy bifurcation branch and to obtain a localised solution. Here a limit problem is studied, as an imperfection is assumed to exist and the aim is to obtain fine localisation bands using conventional elements by redesigning meshes. The underlying


I I Fig. 9. Symmetric quadrant of a uniformly loaded circular plate with soft simple-supports, final mesh, twisting moments M,?, shear forces S,(S) and principal bending

moments MI (MS).

notion of this strategy is related to the fact that, in a fully developed localisation mode near the peak, the displacement pattern (slip or separation) will be represented, almost entirely, by the contribution of only one mode. The strategy is, therefore, to help the mesh locate the final localisation mode by cleaning up the deformation mode. This clean

UP should in turn lead to meshes where the localisation mode corresponds to somewhat lower eigenvalues, thereby making it easier to be excited.

t Using solution monitoring devices such as: l the current stiffness. parameter; or l the determinant of the tangent stiffness matrix.

‘V = 0.3

4

45.2. AMR algorithm. The procedure is based on the

(1)

(2)

(3)

following eigenvectors based algorithm [32]:

For a given mesh the problem is solved until the first signs* of the overall post-peak response are indicated. Determine the eigenvectors cpi of the tangent stiffness matrix KT at the peak. For displacement controlled analysis, the incremental displacement Ad can be written as

Ad = i uicpi i= I

(15)

where CC, are the participation factors. Premulti- plying the above equation by (p: and invoking the orthogonality condition for eigenvectors, the participation factors can be obtained as

The predominant eigenvector corresponds to the largest Itl, I. Let the predominant eigenvector and the corresponding participation factor be de- noted by cpO and CC~. Identify the modal contribution of the predominant eigenvector cpo to the actual displacement increment Ad to give

uo = g cpo. (17)

Treating u. as the displacement field obtain the corresponding stress field

00 = DBuo. (18)

(4) From the stress go obtain a scalar field represented by the effective stress level creff.

Uniaxial test curve

I E.&E = -0.067

1’1’1’1’1’1’1’1~1’ I 2 3 4 5 6 7 8 9

Fig. IO. Analysed panel and uniaxial test curve [27].

A. Selman et al.

Mesh

Displacement patterns

Plane stress Plane strain

Fig. II. Localised failure mode for a pronounced mesh bias.

(5) Redesign the mesh so that element sizes h are inversely proportional to the effective stress o,~, i.e.

h=z aetl

(19)

where c is some proportionality constant.? (6) Repeat the analysis (i.e. go to (1)) unless no mesh

redesign is needed.

tDue to computational limitations a minimum element size is imposed. A maximum element size is also set for adequate mesh generation. $ By raising the scalar field to a higher power than that present in the shape functions.

This algorithm leads to smaller elements in the failure region and larger elements outside of it. Such a procedure, with only one mesh control parameter (scalar field), leads only to densification of elements in the failure region, without any element alignment.

In such strategies the localised failure may not be possible at all, what is supposed to be a localised failure becomes a diffuse one. Figure 12 illustrates the improved meshes used for the prediction of the uniaxial extension failure of a panel, when the remeshing is driven by the predominant eigenvector. The procedure is relatively slow in converging to indicate a clear failure mode. However, mesh improvement marks the local&d failure mode more clearly when an overrelaxationS procedure is adopted (Fig. 12(d)). It should be noted that the failure modes


for all above meshes remain diffuse, despite mesh

(4 (4

Fig. 12. Mes.h densification based on the effective incremental stress corresponding to the predominant eigenvector: (a) initial mesh; (b),(c) successive mesh refinements; and (d) overrelaxed mesh.

refinement in the correct direction. In general, meshes designed with no preconceived

notion of failure modes locate their own, kinemati- tally most suitable, which may be different to the real failure mode [33]. Very often a final failure mode remains locked in t:he region favoured by the original mesh.

4.6. Mesh improvement by element alignment and densiJication

It has been shown that mesh improvement by element densificatiomn alone invariably leads to diffuse failure modes. It typically leads to an excessive number of elements, indicating that mesh improvement by realignment might be more efficient.

Mesh alignment strategies frequently utilise the construction of a characteristic surface (curve in 2D), along which failure is expected to occur [27]. A characteristic curve is typically formed by following the bifurcation directions through various points in the mesh. A similar characteristic curve, or rather a characteristic band can be constructed by tracing topologically the ridge of a suitable scalar field, for example the effective plastic strain [33].

Such a characteristic curve, which resembles discrete crack propagation, is then utilised as an internal boundary for automatic mesh generation, clearly leading to new element sides being aligned along the characteristic curve and any remeshing now recognizes zones outside the localisation band.

The mesh on Fig. 13(c) has been obtained by tracing the ridge of the scalar control field, in order

t This time a different loading programme is adopted: l the displacement is .mcremented by Ad, and l if convergence is not reached within a specified number of

iterations, then Ad is halved and the procedure started again until a converged solution is obtained.

to construct a localisation corridor. Such a corridor can in general have any shape. For the problem at hand, the extension panel, the effective incremental stress corresponding to the predominant eigenvector is used as a control scalar field, and a straight corridor is identified, leading to a fully localised solution, where the elements outside the corridor unload in the softening stage.

An improvement of the process just described may be obtained by the introduction of additional mesh control parameters into the adaptivity process [34]. For example, the orientation of mesh stretching and the aspect ratio of stretched elements for a new mesh is typically obtained by numerically evaluating the direction and the ratio of the maximum to minimum curvature of the control scalar field. Alternatively, stretching orientation may be defined from the bifurcation analysis [27]. Such additional mesh control parameters represent a combination of mesh densification and mesh alignment.

4.7. Mesh kinematics, loading strategy and Newton- Raphson (NR) scheme eficts

As already noted in Section 4.4, on extension (or compression) a panel would deform uniformly setting a uniform state of stress; the problem is of a bifurcation type and a family of possible failure mechanisms exist. However, as an imperfection is introduced, the problem changes from a bifurcation to a limit load problem.

In this section, it will be shown that a combination of mesh kinematics and the use of a NR scheme may send the solution to a possibly wrong mode of failure i.e. to a failure mode associated with a higher energy than the correct solution would require.

The panel in Fig. 10 is discretised with two differently biased meshes and extended into a plane strain analysis.?

A. Selman et al.

(a) (b)

Fig. 13. Mesh improvement by element alignment strategy, ridge tracing: (a) final (adapted) mesh; (b) corresponding effective plastic strain contours map; and (c) consequent mesh alignment.

Obviously, the introduction of an imperfection narrows down the number of possible localisation directions; a sliding failure mode along the analyti- cally correct direction (mesh bias 1), or along a wrong direction (mesh bias 2), or even a necking failure mode are all made possible through mesh bias and NR scheme effects.

Clearly, mesh 1 with a band of elements correctly aligned allows the lowest energy failure mode to be easily achieved and makes it difficult for any other failure mode such as necking to be excited. This has been shown to be true even for large load increments with a NR scheme which may be converging to a local energy minimum that is not necessarily an absolute energy minimum for the applied load increment. This is demonstrated in comparing Figs 11 and 14 and the reaction-displacement (P - 6)

FailummodeforMesb 1 Failuremode.forMesh2

Fig. 14. Mesh kinematics, loading strategy and NR scheme effects.

diagram in Fig. 15, which indicates the correctness of the failure mode for Mesh 1.

When utilising mesh 2, the story is quite different, the mesh bias may or may not allow a localised solution to take place. The kinematics of mesh 2 associated with the localised sliding failure mode is very close in terms of required energy to the necking failure mode. It is therefore possible, due to the load incrementation strategy and NR scheme, for the solution to converge to one or the other failure mode. The P-6 diagram (Fig. 16), which coincides with the P-6 curve for mesh 1 in the pre-peak response is now different in the post-peak response and indicates that the kinematics of mesh 2 prevented the lowest energy failure mode from taking place.

The attention of the analyst, who can, in general, predict neither the location of possible localisation

Rectangular sheet extension Plane strain: Mesh 1

0 2 4 6 6 10 12

Displacement

Fig. 15. P-6 diagram (mesh bias I).

Adaptive mesh refinement for local&d phenomena 489

Rectangular rheet extension Plane strain: Mesh 2

I I I I I I 0 2 4 6 a IO 12

Displacement

Fig. 16. 1’-6 diagram (mesh bias 2).

bands nor the form of the failure mode, is drawn to this effect as mesh bias may send the solution to a wrong mode of failure, which in turn, due to the loading incrementation strategy, the type of loading which may exclude certain modes of failure and the NR scheme, may yet converge to different modes of failure, making it even more difficult for the analyst to identify the ,correct solution in localisation problems.

Although it may be argued that the peak load is almost unaffected by the mesh bias, the correct prediction of the nature of the failure mode (ductile or brittle) is of paramount importance in assessing the load transfer characteristics of redundant structures.

5. AMR IN TRANSIENT DYNAMIC ANALYSIS

5.1. Preamble

In recent years there has been a growing interest in the use of adaptive procedures in the FE analysis of steady state problems [3-51. The incorporation of error estimators and mesh refinement techniques in FE codes has enabled user-specified accuracy to be achieved with a near optimal rate of convergence. In optimal meshes, the error in energy norm is equally distributed within each element; however, from a practical standpoint, it is not necessary to optimise the meshes exactly, instead we seek AMR strategies that are computationally efficient. Among the h-adaptive procedures, the most commonly used are (a) mesh enrichment and (b) mesh regeneration. Mesh regeneration is becoming more popular because of its inherent simplicity and also because it is easy to incorporate into existing FE codes. Most adaptive studies on steady state problems have dealt with static stress analysis [14, 191.

In transient problems, particularly those involving hypersonic fluid flow and multiphase heat flow, there have been major advances in the application of adaptive remeshing procedures [35-381. However,

CAS 63/3-E

there has been relatively little work done in applying such techniques in structural dynamics [3!J-43].

Typically, for transient problems involving shock waves, there are regions of the domain in which the solution is smooth and small regions where the solution changes rapidly. A fine mesh is needed in those areas of high gradient in order to successfully model the behaviour of the structure. To keep fronts sharp as they move around, it is extremely wasteful and impractical to operate on a grid that is uniform and fine. Consequently, in adaptive analysis it is necessary to use an error indicator (estimator) to identify the regions of high gradient which must be refined, leading to nonuniform meshes which change with time in the interest of computational accuracy and efficiency.

Adaptive refinement for transient and steady state problems operates on the same principles. However, for transient problems additional constraints have to be placed on the algorithms employed [38]:

(4

(b)

(cl

(4

As the grid adaption is performed frequently, the procedure needs to be very efficient in terms of its computer implementation if it is to be widely adopted in commercial FE programs. For steady state problems the grid adaption is done only a few times during a whole computation, so speed is not a relevant issue. As the grid adaption is an integrated part of the code, the algorithm should not involve a major storage overhead. Many transient calculations are currently performed with explicit time stepping schemes. This implies that the allowable time step will be governed by the smallest element in the mesh. Therefore the minimum element size achieved by the adaptive refinement algorithm should not be too much smaller than the desired minimum element size. This requirement does not appear for steady state calculations. A preset tolerance is specified once and for all at the beginning of a computation. No user intervention is possible as is the case in steady state simulations.

5.2. Error indication and adaptivity

In this section, a description of the AMR algorithm and error indication used, are presented.

5.2.1. AMR algorithm. The complete description of the algorithm for the advancement of the solution in time with adaptivity may be written as follows [35- 37,

(1)

(2)

401:

Generate an initial grid to represent the computational domain and to allow an adequate initial solution. Advance the solution for a prescribed number of time steps.


(3) Use the error indicator to compute a new mesh distribution.

(4) Obtain the nodal values of the solution on the new grid by direct interpolation from the previous grid.

(5) If the desired time interval has elapsed stop; otherwise go to step 2.

We note several essential items which are required in such an algorithm:

l A mesh generator which is capable of producing meshes with a prescribed mesh density and arbitrary geometries.

l An error indicator (estimator) which can be used to evaluate an appropriate mesh density.

l A method for mapping nodal values (displacements) data from one mesh to the next.

Having already presented the AFM based mesh generator in Section 2, the error indication and mapping method will now be discussed in detail.

5.2.2. Error indication (estimation). An essential item required in an AMR procedure for transient problems is a reliable error indicator (estimator). In the present context we use an error energy norm expressed in terms of stress?

I(e II2 = (u - ~?)~D-l(o - 6)dfi (20)

where D is the elastic modulus matrix, Q is the vector of FE stress and 6 is the vector of exact stress values. If we do not know the exact solution we may estimate the energy norm as

\(e II2 z 116 II* = s

(a - o*)‘V’(cr - a*)dR (21)

in which o* is an improved estimate of the stresses obtained using some stress recovery technique.

The strain energy of the exact solution is estimated as

II w II2 = s u*~D-%J* dR. (22)

Again, both (I e 11 and (( w II can be evaluated at the element level (see eqn (11)). The refinement procedure follows that already presented in Section 3.4.3.

t For a FE approximation the spatial error in kinetic energy I[ ek 11 is two orders smaller than the error in strain energy 11 e, 11, which means that the error in total energy lie, 11 G II er 1) and the same error estimator (indicator) can be used for both static and dynamic problems [42,44,45].

However, it should be noted that the error indicator is used here solely to identify areas where the mesh should be refined or coarsened.

5.3. Element selection

As already noted, triangular elements are best suited for adaption of FE meshes especially for the type of applications considered in this work.

5.3.1. Element used. As computational transient dynamics involve, generally, heavy computations, linear elements which are much preferred for their simplicity have been used in the adaptive transient dynamic structural analysis. However, for compara- tive performance, quadratic isoparametric triangular elements were also used in conjunction with the SPR technique [24, 251.

5.3.2. Evaluation of the best guess solutions. In the present work, a nodal averaging procedure is used to get a better stress estimate for the linear triangular elements. For the quadratic triangular elements, the best guess solutions were obtained using the recently presented SPR technique.

5.4. Mapping

When a new mesh is generated during the time-stepping procedure it is necessary to transfer data (such as displacements) from the nodes of the old grid to the nodes of the new grid. This is achieved by a mapping procedure as follows: when a new point p is generated by the triangulation process, the displacement values at that point are interpolated from the values in the previous (background) grid. This is done by identifying the element e, of the previous grid which contains the point under consideration. As this process should be efficient and economical, it is important to develop a searching algorithm, which given a point on the new mesh finds, in an efficient way, an element on the old mesh where that point is located. An overall searching algorithm is time consuming and should be avoided. In this work, we combined the advancing front technique with a searching technique using area-coordinates [7].

In this “dog sniffing” technique, for each element eb of the background grid, knowledge of the three surrounding elements which have sides in common with element eb is required. Given the coordinates of p and a starting element e$(i,j, k) of the background grid, the area coordinates (f+,(p), L,(p), L,(p)) of p are determined. If each area coordinate lies between zero and one then the element e, contains point p. If not, the node for which the area coordinate is minimum (see Fig. 17) is found and this indicates the next element to be checked. In this manner, the necessity of searching over all the background elements is avoided. The speed of this procedure will obviously depend on the position of point p in relation to the starting element e,.

When the advancing front generator creates a new

Adaptive mesh refinement for loealised phenomena 491

\----,, \

---.w__

\ \

\ \

\ \

\ l P

\

i

Fig. 17. Searching algorithm to locate point p on the background grid [7]. The area coordinates IL.,@), Lj(p) and L(J) of element e, are evaluated at p. Here L&r) < 4(p) < Lk*(p) and the next element to be

checked is e,.

point and an interpolation of the mesh parameters is required, the element used to start the searching process, as a result of a recent improvement to the searching algorithm in [7], is taken to be the element containing one of the points in the side of the front which is used as the base for the new triangle. Once identified, the number of the element containing the new point is stored in a vector. It is this vector that is used to supply the starting elements e, at the mapping stage. Consequently, a search will have to be performed over csnly a very few elements in the

0

II

2 I-42 pw ._ XI 1, = 100

t=1

x2

background grid making the interpolation of mesh parameters and solution values very efficient.

5.5. Numerical examples

To demonstrate the performance of the error indicator (estimator) and the spatial mesh adaption procedure, two examples involving shock and wave propagation will now be presented.

5.5.1. Rectangular bar subjected to a heavyside load [40]. Consider the uniform bar and edge loading

Heavyside load Half-sine pulse load

D I 2 Time

v-1 ’ I - I . , *

::o 40 60 80

Time

Fig. 18. Two dimensional uniform bar with a heavyside and half-sine edge loadings.


shown in Fig. 18. The bar is undamped and initially at rest. A plane stress condition is assumed. The material properties are: E = 30.0 x lo6 lb in-*, v = 0.3 and p = 7.4 x 10e41b s2inm4. It should be noted that only a relatively short time step enables accurate investigation in the case of shock waves on a structure.?

In this case, if a diagonal mass matrix is employed, then the central difference method is certainly a most attractive scheme to use, as the system of equations can be solved without factorising a matrix. Conse- quently, it is not necessary to assemble the stiffness or mass matrices, thereby reducing the computational cost.

In the adaptive analysis, due to computational limitations and the use of an explicit time integration scheme, a minimum element size of 1, /200 is enforced and the time step used with the central difference scheme is set accordingly [46] to 1.0 x 10e6 s. To allow an adequate mesh generation a maximum element size of /2 is also imposed, fI and f, are given in Fig. 18. Here the mesh is updated every 5 time steps.

Figure 19 illustrates the development of the mesh and the corresponding stress (or*) contours as the computation proceeds and shows the ability of the scheme to adapt the mesh to follow the wave front as it moves away from the excited edge, across the structure.

Note that when a uniform mesh (of element size II/ZOO) is used it is found that (a) the shock is poorly represented, (b) the solution overshoots by over 25% and (c) the solution oscillates at the back of the shock. The use of finer meshes does not improve the situation in (b) and (c).

5.5.2. Rectangular bar subjected to a half-sine pulse load [41,42]. A bar with the same geometry and geometrical constraints as in the previous example is subjected to a half-sine pulse load as shown in Fig. 18. The material properties are: E = 1 .O x 104, v = 0.3 and p = 10. Here, due to a less stringent loading, the mesh is updated every 10 time steps with a time step value of 0.01 s.

The predicted meshes, shown in Fig. 20, indicate clearly that the error indicator and the spatial adaption procedure are capable of monitoring the movement of steep stress regions. The (cr.:) stress maps plotted on the same figure testify to the reliability of the procedure.

Again, when using a uniform mesh, it is observed that, although the wave is better represented this time as the loading is less stringent, the oscillations at the back of the wave persist. It should be noted that the effectiveness of the adaptive procedure described in this work would be more attractive for larger or 3-D structures where the cost of a computation on a fine mesh is prohibitive if not impossible.

t The use of large time steps will smear the solution across the wave front.

I I 1

I 18 ‘1

lmll

Fig. 19. Mesh development and corresponding (a:) stress contours at some specific time stations for a bar under a

heavyside load.

5.5.3. Use of quadratic triangular elements. Quadratic trangular elements were also used with the SPR technique in both examples presented above. It is found that persistent oscillations in the solutions unfortunately overshadowed the otherwise excellent performance of the SPR technique already noted in static problems analyses leading to non-derefinement at the back of the shock or wave. Consequently, an increasing number of elements is demanded making the procedure less attractive than for linear elements.

6. CONCLUSIONS

Although it is true that significant advances in the theory of the FE method have been made, the discretisation of the FE model for a particular

problem is still largely based on the intuition and experience of the user (engineer). In designing a mesh, especially for large systems, a great deal of experience with similar problems is required. This requirement is crucial in linear problems (involving areas of high gradients), nonlinear and time dependent problems. It is, indeed, with such problems that the present work is concerned with the objective to develop AMR schemes for those problems to bring more confidence in the use of the FE method in those areas.

6.1. Boundary layers in MR plates

First, the potential of AMR procedures was demonstrated in the analysis of boundary layer effects in MR plates:

l It has been shown how solutions based on MR

I)I

L lllmll I

Fig. 20. Mesh development and corresponding (u.?) stress contours at some specific time stations for a bar under a

half-sine wave pulse load.

l The savings in computational cost are substantial especially when compared with equivalent costs for analyses based on (uniform) fine meshes. Extensions to three-dimensions are likely to produce even greater savings.


plate model depend on the plate thickness-to-span ratio.

l The behaviour predicted by Arnold and Falk has been confirmed: the transverse displacement exhibits no edge effect for any of the problems considered, while the rotation vector exhibits a weak boundary layer depending on the particular BC.

l No boundary layer may be detected for the soft clamped, the hard clamped and hard simply-supported edges, whereas a strong boundary layer de- velops at the free and soft simply-supported edges.

l The stress resultants (particularly the shear forces and twisting bending moments) exhibit stronger edge effects than the rotation vector.

The attention of structural analysts is drawn to the edge effects in MR plates as they can lead to analytical problems. The examples involving boundary layers and presented in Section 3 are very demanding and amply demonstrate the potential of AMR procedures.

6.2. Localised failure in strain softening analysis

Several issues of FE mesh dependence and mesh adaptivity for localised failure mode predictions in strain softening analysis were then discussed and illustrated:

l It has been shown that mesh choice for localised failure analysis should be treated with care. This topic will undoubtedly remain a subject of considerable research for some time to come.

l Adaptive strategies have to reconcile two seemingly opposing requirements. On the one hand they need to avoid the diffusion of localised failure modes by some form of element alignment and on the other hand they must avoid mesh bias towards some mode of failure associated with that very same element alignment.

6.3. Adaptive transient dynamic analysis

Finally, an adaptive FE method for the solution of two dimensional transient dynamic problems was described:

l It has been demonstrated that the use of AMR is a promising approach for the solution of transient dynamic problems. It not only reduces the computational cost by reducing the number of degrees of freedom, but as a bonus, it filters out spurious oscillations in the solution.

l The use of error indication procedures coupled with an automatic mesh generator capable of creating meshes with elements varying considerably in size throughout the domain proved to be extremely useful.


Although major advances have been made in the static and dynamic problems. Ph.D. thesis, Department

use of AMR procedures in linear steady state of Civil Engineering, University College of Swansea,

problems and the necessary confidence in the use of 1992.

adaptive procedures in those areas is now to some 17. Zienkiewicz, 0. C., Taylor, R. L., Papadopoulos, P. and

Ofiate, E., Plate bending elements with discrete extent secured, it is fair to say that this is not, as yet, constraints: new triangular elements. Computers and

the case for nonlinear and time dependent problems. Structures, 1990, 35, 505-522.

Much work is in progress to gain generality and 18. Atamaz-Sibai, W., Adaptive mesh refinement with the

universality in the use of AMR procedures in Morley thin plate element: static and free vibration

engineering problems. analysis. M.Sc. thesis, Department of Civil Engineering, University College of Swansea, 1989.

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Adaptive mesh refinement for localised phenomena