SCOTT KEATING* A N D CARLOS A. FELIPPA**

Department of Aerospace Engineering Sciences and Cender for Space Structures and Controls

Universiiy of Colorado, Boulder, CO 80309, USA

CARMELO MILITELLO***

Department of Theoretical and Experimental Physics Universidad de La Laguna, Ten,erife, Spain

Abstract

We investigate the formulation and application of element-level error indicators based on parametrized variational principles. The qualifier ‘element-level’ means that no information from adjacent elements is used for error estimation. This property is ideally suited to drive adaptive mesh refinement on parallel computers where access to neighboring elements resi- dent on different processors may incur significant com- putational overhead. Furthermore, such indicators are not affected by physical jumps at junctures or inter- faces. An element-level indicator has been derived from the higher-order element energy and applied to r and h mesh adaptation of meshes in plates and shell structures. We report on our initial experiments with a cylindrical shell that intersects with flat plates form- ing a simplified “wing-body intersection” benchmark problem.

1. Introduction

An important issue in finite element analysis is to establish the quality of the solution obtained in terms of discretization error, and proceed to improve it if necessary. Among users, common practice to deal with this matter exploits convergence arguments, which im- plicitly assume that refining the mesh improves the so- lution. Once a solution is obtained for a given mesh, a more refined mesh is used and if changes remain within certain limits acceptable to the analyst, the new solution is accepted as converged. Another com- mon procedure has been to quantify stress jumps be- tween elements as a way to ascertain the quality of a displacement-based solution. These two approaches have obvious physical appeal to engineers.

* Graduate Research Assistant ** *** Professor of Physics Copyright@ 1993 by C. A. Felippa Published by the America1 Institute of Aeronaut*ics and Astronautics, Inc. with permission.

Professor of Aerospace Engineering, Member AIAA

Up to recently most of these “convergence studies” have been carried out under user control, with perhaps some help from mesh generators. In this respect finite elements have lagged behind computational fluid dy- namics (CFD), where automatic error estimation and adaptation of body-fitted structured finite difference grids was well developed by the mid 1970s. Progress was slowed down by the essentially arbitrary and un- structured nature of finite element meshes. Treat- ment of such meshes demands mathematical methods of a different nature than the classical finite-difference truncation error measures, or Fourier analysis.

In general any adaptive mesh refinement process consists of two steps that are applied recursively:

Prediction: assessing the discretization error in- troduced by replacing the mathematical model by the finite element model. Correction: defining procedures by which the dis- crete model can be automatically adjusted to re- duce the discretization error. These corrective procedures are presently classified as follows: h adaptation: elements in regions of high errors are subdivided thus introducing additional nodes. The element type is not affected. p adaptation: elements in regions of high errors are refined by injecting polynomial shape func- tions of higher degree. The number of elements does not change. r adaptation: nodes are relocated toward higher error regions. The number and type of elements is not affected. Combinations of these adaptation procedures, such as hp or hr , are also in use.

Both (P) and (C) are based on a posteriori error assessment from one or more computed solutions. In the sequel we use the following terminology:

1. An error indicator is an assessment of local dis- cretization error that provides sufficient informa- tion to carry out t,he model correction phase.

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2. An error estimator is an approximation of the local discretization error that asymptotically ap- proaches some actual error measure as the mesh size is reduced.

An error estimator is also an error indicator but the converse is not necessarily true.

An error indicator or estimator is said to be ele- ment level if i t can be calculated without information from neighboring (adjacent) elements. This property is highly desirable in massively parallel computations in which individual elements or element groups are stored on different processing units. Element level estima- tion then reduces processor communication overhead, especially if neighboring elements are not necessarily assigned to neighboring processors.

As to the choice between correction methods, we decided to emphasize r adaptation in this research, supplemented by h adaptation. Again the decision is driven by envisioned applications to massively paral- lel processing of dynamic and nonlinear problems. On such computers r adaptation is easy and inexpensive because no modifications to the basic data structures are required. On the other hand, h adaptation involves moderate changes in run data structures, and thus it is better applied less frequently, essentially after r adap- tation “runs out of steam.”

The p adaptation, which changes element types, re- quires drastic changes to data structures. Thus it is the least suitable for massively parallel processing, and is consequently left out of the present research. Further- more an underlying rationale for the p adaptation is to keep the number of elements small. But on massively parallel SIMD computers the number of elements is less important than their “variety” because many elements of the same type can be formed simultaneously.

2. Historical Background

The importance of developing reliable and econom- ical finite element error estimates has been recognized by a large number of researchers since the late 1970s. Substantial progress in this direction has been made over the past decade. The following survey highlights key sources and advances in what is presently an active area of research. Bounding the Energy Error. The first attempt to es- tablish a posteriori error bounds in problems of struc- tural mechanics was made in the early 1960s by Fraeijs DeVeubeke1i2; follow up work may be found in the Memorial Volume? DeVeubeke advocated solving the structural problem twice, once with conforming ele- ments and once with equilibrium elements, to obtain lower and upper bounds II- and II+, respectively, to the actual potential energy II. The difference

may serve as an overall measure of discretization error because AII ---t 0 as both meshes are refined. Bounds on pointwise errors can be derived through application of dummy forces and displacements and Castigliano’s theorem. Each such estimation requires, however, the complete solution of two linear problems. Despite its elegance, t,his approach did not attract attention from finite element developers and users. Its key drawbacks are: 1. Two meshes have to be constructed. Because con-

forming and equilibrium elements have very differ- ent freedom configuration the two meshes do not have the same connectivity. Equilibrium elements have never become popular because they are difficult to construct as well as expensive in terms of degrees of freedom and con- nectivity when used within a stiffness-based pro- gram. Similarly, purely conforming, exactly inte- grated elements are rarely used because they are too stmiff. The computation of local error estimators, which are needed to drive mesh refinement, is cumber- some and expensive. Even if they had been-eco- nomical to obtain, adaptation would involve si- multaneously adapting two separate finite element models while satisfying locality constraints. This leads to significant implementation difficulties.

2.

3.

Potential Energy Estimators. A second idea emerged in the early 1970: node relocation based on energy minimization. Consider a mesh of conforming finite elements with visible degrees of freedom collected in vector v. The potential energy

II(v) = U ( V ) - W(v) = $vTKv - pTv, (2)

is considered also a function of the node locations: II(v,x). In the linear case, II =

x, the solution v = )-lp(x). This can be

to x to get an “optimal mesh” with the given topol- ogy and freedom configuration. This approach is now called r mesh adaptation, or the T method for short. Further historical discussion of this topic is given in Section 3. Local Estimators. The idea leading to the selfadaptive h method, or h adaptation for short, was apparently first proposed in two papers by BabuSka4 and Sewel15 that appeared on the same MAFELAP Proceedings volume. The article by BabuSka was, however, more specifically oriented to applications and thus exerted bigger influence. These first error estimators for h adaptat,ion were of residual type. For the prediction phase many error indicators and error measures have been developed over the past 15 years. The most successful ones are based on measuring discontinuities

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of derived field quantities such as stresses or strains across interelement boundaries. Other measures are based on interpolating residual estimates considering element patches. In 1986 Zienkiewicz and Zhu‘ pro- posed an error estimator based on the smoothing of the stress jumps at element nodes. This has been widely adopted.

By now there is a large and rapidly growing lit- erature on FEM local error estimators. The surveys by BabuEka7, Zienkiewicz et al .8~g~1011 and the recent book by Szab6 and Babuska” may be recommended.

In summary, it can be stated that present estima- tors work well for conforming elements, homogeneous domains and continuum-type linear problems. The reliance on jump conditions or the smoothing of the jumps brings difficulties, however, when such jumps are a natural part of the problem its in junctures, ma- terial interfaces, localization bands or shocks. More- over, conforming elements are not necessarily the best performers in structural problems.

3. Node Relocation bv Function Minimization

A general formulation of the r adaptation process for the design of a finite element grid is

minG(x), X (3)

where G represents the function that we want to mini- mize and x is the vector of node coordinates, on which some motion constraints discussed later apply. G can stand for the global discretization error, the potential energy in displacement-based finite elements, or any other global error measure. The dependence of G on the solution v is eliminated by assuming satisfaction of the finite element equations for fixed x:

(x) v(x> = P(X> (4) is the master stiffness matrix, v the vector of

visible nodal degrees of freedom and p the associated node forces. Should this not be the case, (4) has to be adjoined as a constraint to problem ( 3 ) . Energy Minimization. Theoretically G can be any function that produces a “nice” mesh. For structured, body-fitted meshes of the variety common in compu- tational fluid dynamics, a great variety of choices re- lated to desirable grid properties have been proposed. See for example Section 4.3 of Thompson, Warsi and Astin13. For the unstructured meshes typical of finite element models, it is convenient to relate G more di- rectly to properties of the physical solution.

Historically important choices for G(x) in meshes of conforming, displacement-based finite elements are the internal (strain) energy U and the total potential energy II:

U(X) = ~ v ( ~ ) ~ K ( ~ ) v ( x ) , T ( 5 ) II(x) = U(x) - p(x) v(x) = -$p(x>Tv(x),

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the last relation following from (4). Thus minimizing U or II with respect to x is equivalent to maximizing the external energy pTv. In other words, the nodes should be redistributed to maximize the mesh compliance.

O1iveirax4 was the first to propose that the nodes pertaining to a conforming (energy bounding) finite el- ement discretization be distributed so as to minimize II. He observed that nodes of such “energy optimized grids” would lie along iso-energetic contours, that is, lines or surfaces of equal energy density (this is easily seen by thinking of II(x) or U(x) as node-collocated particle potentials and dII/ax = 0 as particle equi- librium conditions). McNeice and Marcal15 applied a similar criterion to the manual improvement of one- and two-dimensional meshes. Nonlinear Programming. F e l i p ~ a l ~ s ~ ~ studied the min- imization of 11 under the constraints ( 5 ) in the context of nonlinear programming. Numerical experiments showed that the procedure, although successful in the sense of demonstrating a systematic improvement of the discret.e solution, required an excessive amount of work, est,imated to grow at N,” to N:, N, being the number of visible degrees of freedom. To reduce this work a local process was advocated in Reference17 in- volving node relaxation sweeps in which each node n is released in turn to follow the energy gradient

obtained by considering only the “patch” of elements that surround node n. That proposal, however, was not tested.

The use of the total internal (or potential) en- ergy minimization as distribution criterion has the im- portant practical advantage that no error calculations (with respect to the unknown exact solution) are nec- essary at all. Given two meshes characterized by node locations x1 and x2, if U(x1) > U(x2) one can ascer- tain that mesh 2 is better in an overall energy sense. But herein lies a key weakness: such a statement as- sumes that conforming and exactly integrated finite el- ements are used. Although that assumption was nat- ural at the time these early publications appeared, it would be now indefensible because high-performance elements do not satisfy those assumptions.

Curiously the same objection can be leveled against most h and p error indica.tors developed over the past decade: they are applicable to elements that are not necessarily those used in practice.

ode Adaptation by quidistribution

The minimization condition (3) can be relaxed by considering instead a minimax formulation:

min{ max c e } , X e = 1 ... N , (7)

where ce measures the "error amount" or error measure assigned to the eth element and Ne is the total number of elements in the discrete model. It is clear that a possible solution of (7) is obtained if we make

ce = constant for e = 1,. . . , N,. (8) This expresses the heuristic optimality condition of equidistribution of the error. This principle has been applied extensively in fluid dynamics13 for mesh de- sign. Equidistribution also forms the basis of several heuristic optimality criteria; see for example Berke and Venkay yal' I

Fig. 1 "Error attraction" forces acting on individual node 7 ~ . Although drawn in two dimensions for clarity, the concept extends to three dimensions with some safeguards - see text.

The Attraction Forces. With condition (6) in mind we may think of a potential problem in which the er- ror measure acts as a mass at the element centroid attracting the surrounding nodes; see Fig. 1. When the error measure inside each element is the same a state of equilibrium is reached in which nodes do not move while elements approach equilateral shapes. This idea was proposed by Diaz, Kikuchi and Taylorlg for solid mechanics. A similar procedure has been used in CFD computations," although in finite difference dis- cretizations the range of error forces is extended over all node pairs because the localizing concept of element is lacking.

In Fig. 1 we show the attraction force F,, associated with an individual node n and an element e connected to that node. This force points toward the centroid of element e . We can postulate that the components of this force are proportional to the error and that it de- cays to zero when the node n approaches the centroid c of the element. A monomial form that satisfies this condition is

F z e = we(Ax - Axce)', F Z e = w ~ ( A z - AzCe)P,

F y e = we(AY - Ayce)',

(9)

where A denotes deviation with respect to the coor- dinates xn, yn and zn of node n, exponent p 2 1 can be related to the convergence rate of the element, and we is proportional to the element error estimator ee.

x, B y and Az from the condition that the summation of error forces over a node vanishes:

e = l e = l e = l

where e ranges over the number of elements ne that meets at n. If p > 1 the increments has to be obtained by solving the system of nonlinear equations (10). But if p = 1, system (9) is linear and its solution is directly given by

where is understood to range over ne. Equations (1 1) were apparently first proposed by Winslow in 196721 for the solution of the Poisson equation and extended in 1981 to general mesh generation." They are periodically rediscovered in the finite element and finite difference literature. Relaxation Sweeps. The method is applied by sweep- ing over the nodes in sequence. Each node location is recomputed by relaxation, that is, assuming that neighboring nodes (those connected by virtue of be- longing to the same element) do not move. It is also assumed that the error does not change during a com- plete sweep. This has the advantage that relocation can start from any node and that symmetric meshes in symmetric problems will remain so. A potential draw- back is that the motion of one node does not account for eventual motions of their neighbors. This can be a source of oscillatory mesh movement and instabilityz1. A way of overcoming the problem is to introduce an underrelaxation factor p 5 1 that reduces the displace- ment of the node. In this way the following iteration formula is obtained:

where k is a relaxation sweep counter. The number of steps q during which it is assumed that the element error (and hence w;) does not change depends on the cost of error recomputation. As a rule of thumb we suggest that it be recomputed when pq 2 1.1. This recursion dlows a feedback on neighboring node move- ment and attenuates oscillations. Practical Considerations. From Winslow's work it is clear that (11) represents an approximation to the dis- crete La.placian operator. Because of this relation, node relocation by this method yields reasonably in- expensive adaptivity with low element distortion. The

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nodes should not move when all the elements reach the same error level. For triangles this can only be accom- plished if the mesh pattern allows them to reach equi- lateral shapes. If not we will always compute a nodal motion. Nevertheless, the node configuration eventu- ally reaches an equilibrium position provided the solu- tion is bounded.

An interesting property is that application of (12) without constraining boundary nodes to remain on the boundary will shrink the problem domain. (This may be easily visualized in the limit case where all nodes are allowed to move: because the error forces are attrac- tive, a simply connected domain will eventually shrink to a point.) In other words, the method is always trying to “pull in” the boundary nodes. This prop- erty represents a computational advantage because, provided that the boundary is properly constrained, it is not necessary to check whether the interior nodes in the domain will “escape” the boundary.

A desirable feature is that boundary nodes be al- lowed to move. But their motion must be constrained so that they remain on the surface or curve that defines the boundary.

Because the traction force decays as a node moves towards the element centroid, it is unlikely that af- ter relocation two nodes share the same coordinates within machine precision thus collapsing an element side. Nevertheless, elements can become distorted and penalties must be implemented in order to restrain nodes attached to potentially unacceptable distorted elements.

The r adaptation does have a drawback in the treatment of problems with fixed concentrated inte- rior loads because the load must be kept from being “dragged” by the moving mesh. The simplest rem- edy is to constrain the relocation of the node where the load is applied. More sophisticated schemes can be of course be devised, such as tracking the nearest elements to the load position, and lumping it on the re- spective nodes. Node movement also requires some po- sition dependent properties, such as thickness or elastic coefficients, to be recomputed or e ~ t r a p o l a t e d . ~ ~ This set of requirements is collectively known as the mesh transfer problem.

Treatment of Boundary Nodes. In a three-dimensional setting a geometrical boundary will be represented by a surface or a curve. With a view towards the treatment of r-adaptation on shells we consider here the general case of a surface. Its parametric representation in a Cartesian system (2, y, z ) may be written as

where r and t are surface coordinates. Solving (10) for a boundary node yields a new position x,, y, , z,

that generally falls outside the original surface thus failing to satisfy (13). From (11) we can argue that if the surface is convex the new position will be bounded by the original surface and a plane joining the three furthest centroids.

To preserve the geometric representation of the sur- face we apply the following “radial return” algorithm. Suppose that (12) gibes a node position ( zP , yp, z p ) . Then we look for the point on the surface that mini- mizes the squared-distance to that point

R = 4 (Axsp2 + Aysp2 + A z g p 2 ) ,

where Axsp = x, - xp , AYap = y., - yp, and in which the factor $ is included for convenience. The stationary condition for R gives

(14)

On linearizing this system of non-linear equations we obtain the following Newton-Raphson iterative scheme to solve for the increments in the parameters r and t :

The iterat,ive process is continued until the L2 norm of the residual is less than a prescribed tolerance:

- < to!. The primary variables in this iter- ation process are the parameters (r, t ) associated with a point on the surface of coordinates (x8, yJ, zs). The iteration is initialized assuming that the nearest point on the surface to (xp, ypj zp) is the node we want to relocate, that is PO. It means that in order to relocate a node with this algorithm we need its Cartesian coor- dinates and the pair ( r , t ) and, at the end of the itera- tion, we have to update both. This can be a problem when general surfaces without parametric representa- tion are discretized. The problem can be avoided if we automatically generate a local parametric represen- tation of the surface from the geometrical location of the surrounding nodes. This possibility, although more general, adds the cost of generating the local surface to that of the iterative process. I t has been our ex- perience with cylindrical a,nd spherical shells that the solution of (15) requires from two to four iterations.

5.

As noted in the Introduction, we have undertaken a different approach to error estimation. Felippa and Militel10”~~~ introduced a general parametrized form of the stress-strain-displacement functional of linear hyperelasticity. A general expression for the functional

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is:

where U is the generalized strain energy stored in the body volume, P is a forcing potential that embodies all other actions such as loading and boundary condi- tions, a, p and y are free parameters further discussed below, and 6, & and 6 are the varied displacements, stress, and strain fields, respectively. (A superposed tilde expresses that the symbol is subject to indepen- dent variation.) The variation HI = 0 generates all governing equations of elasticity for arbitrary a , /3 and y. All elasticity functionals with varied displacements can be obtained by appropriate specialization of these parameters .

As to the forcing potential P , three forms labelled as Pc, Pd and Pt have been studied in conjunction with parametrized Pc is the conventional forcing potential, which is used in non- hybrid finite element discretizations. Potentials P d and Pt , called displacement-generalized and traction- generalized, respectively, are useful in the construction of high-performance hybrid elements with additional boundary fields. Of these the displacement-generalized potential P d has proven to be the most useful one in the construction of high-performance element^.^' This potential depends on three varied fields

Pd = P d ( i i , zr, A), (18)

where a is a boundary displacement field defined only at element interfaces. A common property of P", P d and Pt is that they do not directly depend on the free parameters.

Now suppose that (17) is used to construct a fi- nite element discretization, while keeping one or more parameters free. (Practically this means that such pa- rameters are kept as arguments of the element stiffness subroutines.) The discrete approximants ii, 6- and i5 obtained for a given mesh will then be generally func- tions of the free parameter(s). An important property of (17) is that U takes the s a m e value if the varied fields ii, 8, 6 are set to the exact solution fields u, u, E , independently of a , P and y. Hence, we may expect that the difference between two values of U ob- tained for two different sets of parameters may provide a measure on how far we are from the converged so- lution. That is, the difference may be adopted as an error indicator. The most interesting feature of this indicator is that i t naturally provides an e l e m e n t level measure, as the following development shows.

To tackle the general case first, suppose that the two sets of parameters are (all PI, 71) and ((YO, PO, yo). The corresponding approximate values for dis- placements, strains and stresses obtained with a given

mesh are ( 6 1 , 6-1, 61) and (GO, G O , G O ) , respectively. Denote by Uf = U e ( 6 1 , 6 - ~ , 6 ~ , a ~ , P ~ , y ~ ) and U$ = Ue(i io , Go, 6 0 , ao, P o , yo) the value of the generalized strain energy evaluated over the elh element of that mesh. Then the element error indicator is defined as the difference

E e = IU,e - u;1, (19) This definition apparently requires that the problem be solved twice for a given mesh. I t will be seen, how- ever, that consideration of the structure of the stiffness matrices and their dependence on the free parameters allows the use of only one solution.

6.

As of now the three-parameter functional (17) has not been used for finite element analysis in its full generality. Work to date has been restricted to one- p a r a m e t e r specializations of (17). Two such function- als, in conjunction with the displacement-generalized forcing potential P = P d , have proven to be practically important: 1. The functional II, = U, - P d of the energy-

orthogonal Free Formulation (FF). This is a vari- ationally based form of the FF of Bergan and Nygbrd3', which was developed in subsequent p u b l i ~ a t i o n s . ~ ~ ~ ~ ~ ~ ~ ~ The only free parameter is 7. The functional 11, = U, - P d of the Assumed Natural Deviatoric Strain (ANDES) formulation. This is a variant of the Assumed Natural Strain (AN§) formulation, a name coined by Park and Stanley.35 The only free parameter is a.

Beltr6n and have successfully tested the application of the scaled Free Formulation func- tional II, for element-level error estimation on plane stress and plate bending problems. In what follows we concentrate on elements based on the ANDES func- tional II, = U, - p d .

The generalized strain energy of the ANDES func- tional may be written in matrix form as

2.

u , = $ 4 ( ; } T [ In this expression ue = stresses derived from the

tively, in which E is a matrix of elas-

derived from the independent stresses and displace- ments, respectively. Finally, I denotes the identity ma- trix of appropriate order. We collectively call ue, u', e" and e" the derived f ields.

I t may be verifiedz4 that for a = 0 and a = 1 (20) reduces to the generalized strain energy expressions

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for the stress-displacement Hellinger-Reissner and Hu- Washizu functionals, respectively. Hence IT, =: U, - P may be viewed as a linear interpolation between those two functionals. Elements based on U, - P d will be called ‘ANDES elements’ for short.

e be the stiffness matrix of an ANDES el- ement, ve the visible element degrees of freedom (those degrees of freedom in common with other elements, also called the connectors) and p the corresponding element node forces. Then the element stiffness equa- tions decompose as

i are called the basic and higher order stiff- ness matrices, respectively. The basic stiffness ma- trix, which is usually rank deficient, is constructed for consistency. The higher order stiffness matrix is con- structed for stability and (in more recent work) accu- racy. A decomposition of this nature, which also holds at the assembly level, was first obtained by Bergan and N ~ g b r d ~ ~ in their derivation of the Free Formulation.

One- Solution Error

With a view towards reducing the need for two so- lutions, the element error indicator is chosen as the difference between the element internal energies for a1 = a and ao:

Next we assume, without proof, that for two different parametrizations and for a ‘good’ mesh the following property holds: 5, N 50, Go N Go, and 8, N 8 0 ; but that 8, # eO, # e:, 80 # e: # e:. In other words, cor- responding fields in two different parametrizations con- verge toward each other faster than do different fields for each parametrization. If this assumption holds we can approximate the error indicator (22) as

(23)

Through algebraic manipulations this expression can be transformed31 to

where vk denote the element degrees of freedom calcu- lated for the a solution. Physically (24) is the higher order energy (HOE) absorbed by the element. Since

for ANDES elements (as well as for FF elements) the higher order stiffness matrix i is always available in separate form, this estimator is readily calculated on an element by element manner.

It should be notices that this error indicator is re- lated to that heuristically proposed by Melosh and Marcal= in the context of the SED (Strain Energy Density) method.

roperties of the

The error indicator inherits the properties of the higher order stiffness. By construction the higher order stiffness verifies

;v;, = 0 ( 2 5 )

where v,, are the nodal displacements associated with a rigid body motion or a constant strain state. Actu- ally v,, expands the null space of can be decomposed as ve = V& + projection of ve on the null space of

From these considerations we conclude that (24) automatically “filters out” the contributions to the dis- placement, field from rigid body motions and constant strain stat,es. Thus, in problems whose analytical so- lution consists of uniform strain states the error esti- mator vanishes over each element, in accordance with the fact that any mesh should solve those cases ex- actly. This statement may be ext,ended to cases solved by constant strain sta.tes that jump at material or geo- metric interfaces, if such interfaces are modeled exactly by the finite element mesh. Other error indicators in present use do not verify this property.

9.

The node relocation method based on the error in- dicator (24) has been extensively tested on a set of problems involving in transversely loaded thin plates and thin shells. The first set of experiments involved problems with analytical stress solutions so that a com- parison with the exact error was possible. These re- sults have been reported and discussed by Militello and F e l i ~ p a . ~ ~

Here we describe initial experiments with the ge- ometrically more complex problem shown in Fig. 2. Two flat plates intersect with a cylindrical shell. The motivation is to study an idealized system that con- tains some of the geometric characteristics of a wing- fuselage intersection without the additional fabrication complications. The behavior of the HOE error indica- tor in the vicinity of the cylindrical-shell/plate junc- ture (CPJ) is of particular interest because such junc- tures are the source of difficulties in more conventional error indicators that rely on stress smoothness assump- tions.

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E = 15 x 106 v = 0.3 t, = t, = 3 r, = 150

600

Fig. 2 Benchmark problem featuring cylinder-plate junctures (CPJ). The cylindrical shell is supported by end diaphrams, with fixed degrees of freedom as indicated.

Fig. 2 gives geometric, material, loads and B.C. data for the benchmark CPJ problem. Although no exact solution is available, it is expected that a refined model using high order curved shell elements can even- tually provide a “baseline” solution against the error estimators can be compared. Such a comparison, how- ever, was not completed as of this writing.

For the results reported herein only one quarter of the benchmark problem was considered by taking ad- vantage of the two symmetry planes. (This could be further reduced to one eighth by noticing the presence of an antisymmetry plane but the antisymmetry would disappear for some envisioned load cases.) The quar- ter portion was modeled by three-node, flat triangular shell elements with 18 degrees of freedom. This shell element combines the bending stiffness of the high- performance Kirchhoff element AQR constructed by Militello and Felippa3’ using the ANDES formulation, with the membrane stiffness of a high-performance 9- dof triangle with three corner drilling freedom^.^^^^^^^^ Each component contributes 9 freedoms. It should be noted that the membrane element has been derived using both the so-called Extended Free FormulationB (EFF) and with the ANDES formulation.

Four regular meshes containing 20 to 1280 trian- gles were generated as departure points for T refine- ment processes. Figs. 3 and 4 show results for the intermediate meshes containing 80 and 320 triangles, respectively. These figures show the initial (left plot) and final (right plot) finite-element meshes before and after T adaptation. The original color plots, produced

on a Crimson SGI workstation, depict the predicted er- ror distribution on a palette of red for lowest through white for highest. That distribution is taken from element-level results and averaged over nodes to obtain a continuous color rendition. Although the black-and- white display unfortunately masks much of the distri- bution details, key trends can still be discerned in the figures. As can be observed the models correctly pre- dict that: 1. The highest error occurs near the juncture. Fur-

thermore the finer initial meshes (320 and 1280 triangles) show that the error increases towards the juncture end (where shell theory predicts that a singularity occurs). The error localization in the general vicinity of the juncture becomes more pronounced as the initial mesh is refined.

3. The adaptation tends to further localize the peak error towards the juncture-end singularity while the rest of the structure undergoes an error equidistribution process.

Obviously T adaptation is relatively powerless to cut down on the error near the singularity. An h pro- cess should be triggered at this point and interweaved with further T adaptation cycles. This was not done because our triangle-splitting h process still displays logic problems in understanding junctures where three or more shell elements meet. The preliminary results are nonetheless encouraging since they agree with the expected physics.

2.

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Fig. 3 Results for 80-triangle model of CPJ problem. Left plot shows HOE error indicator for initial regular mesh; right plot after r adaptation.

Fig. 4 Results for 320-triangle model of CPJ problem.

IO. Conclusions

We have developed an element-level error indica- tor based on parametrized variational principles. Un- der certain assumptions the indicator reduces to the higher-order energy (HOE) absorbed by the element. The HOE indicator has been used to drive the node re- location process ( r adaptation) on finite element mod- els of thin plates and shells.

Previously reported results on simple problems

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with analytical solutions43 show that the indicator was effective as a m e s h adaptat ion driver although con- nection with the a.ctua1 error was erratic. The initial experiments on the CPJ problem suggest that driving effectiveness is unaffected by the presence of junctures. This is an important result because conventional error indicators based on interelement smoothing are known to run into difficult.ies a t junctures unless helped by substructure-partitioning logic. No such complications arise with element-level indicators.

1

Further planned studies on the benchmark CPJ

Comparison of error predictions with those ob- tained by solution-smoothing shell error estima- tors developed by the Lockheed PARL Although several analyses of the CPJ problem have been carried out using 16-node shell ele- ments, firm conclusions have not been reached as this paper is being prepared.

Assessing error prediction effectiveness by com- paring with a refined finite element solution pro- vided by the high order shell elements noted above.

problem include:

1.

2.

3. Conducting a n rh adaptation. Finally, it should be noted that conclusions regard-

ing the effectiveness of the HOE error indicator in this problem class has bearing on full-vehicle aeroelastic simulations planned by Charbel Farhat’s on massively parallel computers such as the KSR-1, CM- 5 and Intel Paragon. This is because the facet shell element chosen for the benchmark problem has been frequently used to model complete aircraft structures. Its main modeling strength is the presence of the mem- brane “drilling” freedom, which circumvents the “five versus six freedoms” dilemma associated with inter- secting stiffened shell structures.

Acknowledgements

This work has been supported by NASA Langley Research Center under Grant NASI-756, monitored by Dr. J. Housner. We thank Bryan Hurlbut, Ishak Levit and Gary Stanley of the Structures Laboratory at Lockheed PARL for assistance in providing comparison solutions that will be evaluated and discussed at the SDM meeting.

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