Energy Based Adaptive Strategy for Plates and Laminates

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Energy Based Adaptive Strategy for Plates andLaminatesA. Rajagopal a & S. M. Sivakumar ba Department of Applied Mechanics , University of Erlangen , Germanyb Department of Applied Mechanics , IIT Madras , Chennai, IndiaPublished online: 22 Apr 2009.

To cite this article: A. Rajagopal & S. M. Sivakumar (2009) Energy Based Adaptive Strategy for Plates and Laminates,International Journal for Computational Methods in Engineering Science and Mechanics, 10:3, 209-223, DOI:10.1080/15502280902795078

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International Journal for Computational Methods in Engineering Science and Mechanics, 10:209–223, 2009Copyright c© Taylor & Francis Group, LLCISSN: 1550–2287 print / 1550–2295 onlineDOI: 10.1080/15502280902795078

Energy Based Adaptive Strategy for Plates and Laminates

A. Rajagopal1 and S. M. Sivakumar2

1Department of Applied Mechanics, University of Erlangen, Germany2Department of Applied Mechanics, IIT Madras, Chennai, India

The objective of this work is the development of a numericalsolution strategy for energy–based mesh optimization based on acombined refinement strategy for laminated composite plates. Infinite element computations that rely on the principle of minimumpotetnial energy, the variational principle itself provides the basisof r-adaptive methods. The numerical solution can be improvedby further minimizing the discrete potetnial energy with respect tomaterial node point positions. A new adaptive scheme has been pro-posed and formulated for adaptive finite element analysis of lam-inates and plates. It involves a combination of the configurationalforce based r− adaption and mesh enrichment by h−refinement.These configurational forces are conjugate to the nodal motionand vanish when the potential energy is a minimum or at equilib-rium. These forces are evaluated for laminates and plates by con-sidering the weak form of the material force equilibrium. Theseforces assembled at nodes in a finite element discretization actas error indicators for r− adaption. The h− refinement is basedon a modified patch recovery based estimator based on quanti-ties of interest, enhanced by strain energy density ratios. Numer-ical study confirms that the proposed combined r − h adaption ismore efficient than a purely h−adaptive approach and more flex-ible than a purely r−adaptive approach with better convergencecharacteristics.

1. INTRODUCTIONThere has been a considerable focus on theoretical and com-

putational aspects of mesh adaptive techniques such as h−, p−,r− and s−versions [1, 2–8, 9–12] or any combination of these[13–17] for optimizing a spatial discretization. In application ofadaptive techniques for plate bending problems, the use of bestguessed stress error estimators [3, 18–19] and residual type es-timators [21], together with use of transition based quadrilateral

The authors wish to thank Prof. J. N. Reddy, Texas A and M Uni-versity, USA, for the technical input by giving the code for laminateanalysis. The authors wish to thank ARDB, Delhi, for the financialsupport offered for this work.

Address correspondence to A. Rajagopal, Department of Ap-plied Mechanics, University of Erlangen, Erlangen 91058, Germany.E-mail:[email protected]

elements [20, 28] or a combination of triangular and quadri-lateral elements, have been reported in the literature. Adaptiveanalysis of laminated composites requires consideration of bothquantitative errors due to discretization and qualitative errorsdue to mathematical modeling. Qualitative errors in terms ofparasitic shear terms [23] have been identified to occur in lami-nated plate finite element models. Quantitative errors are com-puted based on energy norm of stress resultants [22] similar toisotropic plates and shells [3, 28]. Such procedures require alocal quality smoothing technique for recovery of derivatives[24], an adaptive procedure based on local quantities of interest[24], which may be hierarchical [29] and which involves pre-diction of the desired optimal mesh sizes in the neighborhoodof the region of interest and away from the region of interest[25] based on a priori or a posteriori estimate of the error inthe quantity of interest, together with a region by region refine-ment procedure. It is reported that plate bending elements sufferfrom shear locking and edge effects [26]. Unless an appropri-ate adaptive strategy is used, such phenomena can deterioratean adaptive solution by slowing down the global convergenceand the boundary layer effects can often go uncaptured [27].There have been few works on adaptive finite element analy-sis of plates that use a combined refinement strategy. A moredetailed bibliography on adaptive finite element techniques hasbeen given in [30]. Earlier works on r − h combined adaptivestrategy [13, 14, 16] for plane problems were based on subdo-main division for either r− or h− on the divided domain withno refinement on the interface of these domains. The subdo-main divisions are based on desired element size and these arenot based on error estimators. The interface between domainsposes topological constraints on h− adaptive remeshing as wellas on the r− adaptive meshing. There has been report on use ofr − h − p refinement strategies [15]. Recently, material forceshave been used in an r-h adaptive scheme [47], where in mag-nitude of material forces are used to check discretization andnew mesh sizes are assigned through refinement factors. Theserefinement factors are obtained from the discrete material forceby a linear projection. There has been some recent remarkablework on energy-based r−adaptivity [55], where the distortion

209

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210 A. RAJAGOPAL AND S. M. SIVAKUMAR

FIG. 1. Flow chart for optimal mesh adaption procedure.

in elements are included as additional constraints to energy op-timization a solution strategy.

The focus of this paper is on a-posteriori error estimationand adaptive mesh refinement technique applied to compositelaminates and plates using combined r − h adaptive strategy.It involves a combination of the configurational force–basedr− adaption with weighted laplacian smoothing and mesh en-richment by h−refinement (see Figure (1)). The configurationaldriving force is evaluated by considering the weak form of thematerial force balance for laminates and plates, and is discussedin the next section together with the discussion on node reloca-tion techniques. These driving forces assembled at nodes act asan indicator for r− adaption. Mesh adaption by node relocationis performed until a specified level of convergence in potentialenergy is obtained. The h− enrichment strategy is based on amodified weighted average of energy norm of error enhancedby energy density ratios and evaluated using supercovergentestimators. The energy density based estimator shows betterconvergence characteristics for h− refinement. Mesh adaptionby node relocation results in an increase in flexibility of the sys-tem with decrease in potential energy. The amount of decrease isdependent on order of integration used in evaluation of the shearterms. To overcome distortion effects induced due to relocationa successive h− refinement is done after node relocation. Thecombined refinement strategy shows better convergence char-acteristics. The last section discusses the numerical examplesfor these strategies.

2. MATERIAL FORCES FOR r- ADAPTION OF PLATEBENDING PROBLEMSIn the r− adaption procedure based on configurational equi-

librium, mesh optimality is accomplished by minimizing the

potential energy [32, 33]. The potential energy of the systemis dependent on nodal coordinates in addition to the displace-ment field [34]. Configurational or material driving forces aredefined as the conjugate forces to the nodal motion with respectto the potential energy [35–37] and these forces vanish whenthe potential energy is a minimum. There has been consider-able work on the use of material forces and their equilibriumin a computational setting such as the finite element method[38–42]. There have been other novel works on use of materialforces for moving interfaces and hyperelastic fracture mechan-ics [37, 43–47]. Recently, the material forces have been usedfor adaptive finite element methods [48, 55]. In this work, aformulation of nodal errors that is consistent with material forceequilibrium has been derived for laminates and plates. In thissection the variational form of material force balance is derivedto determine the configurational forces that occur in plates andlaminates. For modeling isotropic plates using Mindlin’s the-ory, a four node isoparametric element with w, θx, θy as degreesfreedom that includes the effects of transverse shear deformation[51] has been used. For finite element analysis of compositesincorporating equivalent single layer first order shear deforma-tion theory [52], bilinear rectangular elements with five degreesof freedom per node (conforming and non conforming), whereu0, v0, w0 are the translational components and φx, φy are theshear deformation by components [53] have been used.

2.1. Material Forces for Reissner Mindilin PlatesThe Lagrangian of the plate is equal to the negative sum of the

strain energy density W and the potential of external forces V

L = −(W + V ) = L(xk, θi, θi,j , w,w,i) (1)

with

W = 1

2[Mijθj,i + Qi(θi + w,i)] (i, j, k = 1, 2) (2)

and

V = −wp (3)

where p is the uniformly distributed pressure acting over theplate. The equilibrium equations of the plate (see Figure 2) aregiven by

Mij,j − Qi = 0Qi,j + p = 0

(4)

Now considering the gradient of the Lagrangian, we can write

grad (W + V ) = (W + V ),k= Mjiθi,jk + Qi(θi,k + w,ik) − pw,k + ∂(W+V )

∂xk

(5)

The explicit derivative of (W + V ) with respect to the inde-pendent variables xk(k = 1, 2) accounts for material inho-mogeneities produced, for example, by stiffness distribution

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ENERGY BASED ADAPTIVE STRATEGY FOR PLATES AND LAMINATES 211

D(xk) and Cs(xk) smoothly or discontinuously varying withxk(k = 1, 2) (defects), and for physical inhomogeneities, pro-duced by p(xk). Integration by parts yields:

(W + V ),k = (Mjiθi,k),j − (Mji,j − Qi)θi,k

+ (Qiw,k),j − (Qj,j + p)w,k + ∂(W+V )∂xk

(6)

Using the equilibrium equations from Eq. (4) and (W + V ),k =(W + V ),j δjk we can write:

[(W + V )δjk − Mjiθi,k − Qjw,k],j = ∂(W + V )

∂xk

(7)

The left–hand side of Eq. (7) is equal to the Eshelby tensor ofplate theory and can be written as

Cjk = (W + V )δjk − Mjiθi,k − Qjw,k (8)

In the absence of inhomogeneities, the Eshelby tensor given inEq. (8) is divergence free and can be written as

Cjk,j = 0 (9)

2.2. Material Forces for Laminated Composite PlatesIn case of laminated composite plates the Lagrangian is given

by

L = −(W + V ) = L(xk, φi, θi,j , w,w,i, ui, vi) (10)

with

W = 1

2[Nijεij + Mijθj,i + Qi(φi + w,i)] (i, j, k = 1, 2)

(11)and

V = −wp (12)

where p is the uniformly distributed pressure acting over theplate. Considering the equilibrium of forces in the plane of theplate we have

Nij,j = 0 (13)

Considering the moment and transverse force equilibriumand writing it in terms of the stiffness coefficients we have

A11u,xx + 2A16u,xy + A66u,yy + A16v,xx + (A12 + A66)v,xy

+A26v,yy − B11w,xxx − 3B16w,xxy

− (B12 + 2B66)w,xyy − B26w,yyy = 0 (14)

A16u,xx + (A12 + A66)u,xy + A26u,yy + A66v,xx + 2A26v,xy

+A22v,yy − B16w,xxx − (B12 + 2B66)w,xxy

− 3B26w,xyy − B22w,yyy = 0 (15)

D11w,xxxx + 4D16w,xxxy + (D12 + 2D66)w,xxyy

+ 4D26w,xyyy + D22w,yyyy − B11u,xxx − 3B16u,xxy

− (B12 + 2B66)u,xxy − B26u,yyy − B16v,xxx

− (B12 + 2B66)v,xxy − 3B26v,xyy − B22v,yyy = q (16)

Now considering the gradient of Lagrangian we can write

grad (W + V ) = (W + V ),k= Nij εij,k + Mjiθi,jk + Qi(θi,k + w,ik)

−pw,k + ∂(W + V )

∂xk

(17)

Using the equilibrium equations and rewriting

[W + V )δjk − Nij εij − Mjiθi,k − Qjw,k],j = ∂(W + V )

∂xk

(18)

The left–hand side of Eq. (18) is equal to the Eshelby tensor ofcomposite plates and can be written as

Cjk = (W + V ) δjk − Nij εij − Mjiθi,k − Qjw,k (19)

In the absence of inhomogeneities, the Eshelby tensor given inEq. (19) is divergence free and can be written as

Cjk,j = 0 (20)

In a finite element context, it is required to compute the discreteconfigurational forces arising out of discretization. In the ab-sence of body forces the weighted residual form of the balancelaw equation using a vectorial test function η and integratingover the domain �0 is given by

∫�0

Cij,j ηi d�0 = 0 (21)

The weak form in the absence of body forces obtained by inte-grating by parts can be written as

−∫�0

Cijηi,j d�0 +∫�

Cij nj ηi d� +∫

�e �⊂�

Cij nj ηi d� = 0

(22)

As a consequence of considering stationary boundaries thetest function η vanishes on the boundaries of the domain � andhence the second term becomes zero. For a homogeneous bodyin continuous case, linear momentum balance implies materialforce balance and thus the divergence of the energy momentumtensor is zero for a homogeneous body without body forces.However, in the discrete case nodal force balance does not im-ply nodal material force balance due to the presence of nodesand hence element interface. The material force imbalance isdue to the presence of nodes (nodes can be argued to be discretedefects as they break the translational symmetry of potential en-ergy with respect to the translations in reference coordinates).

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FIG. 2. Stress resultants for a plate element.

The finite element approximation introduces spurious materialforces originating from the fact that the approximation is notsmooth with respect to strains (and stresses). Therefore in theinterior of the domain the numerical value of the nodal mate-rial forces does not vanish (see Figure 3). When material pointson the boundary are not allowed to change their position inthe reference configuration, material forces appear as reactionforces conjugates to this constraint. Thus, in a discretized formconsidering the material force equilibrium, the non-vanishingof the divergence of energy momentum tensor at the inter el-ement boundaries is taken as an error indicator. This is usedto check the discrete solution obtained through finite elementanalysis. The discrete jump in the energy momentum tensor oc-curring at the element boundaries �e (third term of the weakform equation) is the driving force used as an error measurein the node relocation process. The node relocation procedurefor configurational force method is based on minimization ofpotential energy. In literature, simple relaxation type iterativeprocedures [38, 39, 41] and improved procedures by using astandard Polak-Rebiere conjugate gradient algorithm [33, 54,

56] have been reported. A modification to this was introducedin one of the earlier works by authors [40, 41, 56] and thesame has been used for node relocation. The flow chart for noderelocation is shown in Figure 4.

3. STRESS RECOVERY AND ERROR ESTIMATIONA modified patch based stress extraction technique is used

for recovery type estimator in h− refinement. There have beenseveral strategies for obtaining optimal meshes in these cases,such as, equidistribution of errors among the elements of a pro-jected new mesh [49] and minimizing the number of elementsin projected new mesh [4–7, 50] for a specified accuracy. In thepresent work a modified adaptive strategy is used based on thecomputed values of the energy density ratios in each elementto arrive at optimal meshes. Most of the patch recovery basedestimators possess common features such as use of a discreteleast square method to compute the stress field values frombarlow points, and the extracted stress field is generally C0

continuous. All computations are local or semi local, i.e. at atime only few elements are considered for local extraction ofstresses. The problem in case of a distorted quadrilateral el-ement (which is evident in r− adaption) is that the elementcentroid does not remain super convergent point, and weightedelement patch techniques require proper choice of weightingfunctions. Unlike the earlier methods the main feature in thepresent method is that the element patch is composed of fourdiscrete node patches (Figure 5). The advantage of using such anelement patch is that since discrete node patches are combined,the assumed stress polynomial can also be formulated as bilin-ear polynomials linked by element shape functions. Secondly,the assumed stress polynomial spans the element domain; thisremoves the integration inconsistencies. Overall it is also ob-served that the use of such an element patch near the boundaryor corner of the domain also stabilizes the solution as there ismore number of stress sampling points made available for stressprojection technique.

FIG. 3. Concept of driving force at inter element boundary.

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FIG. 4. Flow chart for r- adaption based on configurational force correction.

3.1. Definition of Error EstimatorThe potential energy of the structure is given by

= 1

2

∫�

σij εij dV −∫

F+∪F−

piwid� (23)

where F+ and F− are the top and bottom faces of the plate. Thesolution to the problem uexact is the minimizer of the potential and is defined by uexact ∈ H 0(�) such that

B(uexact , v) = V (v)∀v ∈ H 0(�) (24)

FIG. 5. Node and element patches used in recovery.

For the finite element solution uexact of the Eq. (23) is such thatuh ∈ H 0

h (�),where H 0h (�) = {u = [�]U,Ui ∈ P τ

p } and P τp is

the set of globally continuous piecewise polynomials of orderp over each of the element. Thus, the above Eq. (24) can bewritten as

B(uh, vh) = V (vh)∀vh ∈ H 0h (�) (25)

FIG. 6. Laminated plate discretization considered for validation examples.

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FIG. 7. Plot of normalized deflection for various span to thickness ratios: (a) graphite epoxy and (b) glass epoxy composite.

The error in the solution is defined as e = uexact − uh. Anapproximation to the error is given as eest = uest − uh, whereuest ∈ P τ

p+k is obtained for each element τ as defined below.For each element τe let NPτ be the patch of elements in a onelayer neighborhood of τe. Over the patch NPτ we define

uest = {uest , vest , west , θxest , θyest }T = [N ]UI (26)

where UI = ∑ndof

I=1 aijpj (x, y) where aij is a vector of unde-termined coefficients pertaining to node I and pj (x, y) is thecomplete linear polynomial in two dimensions defined in termsof the centroidal coordinates of the element. This in a senserepresents a conjoint polynomial. So over each patch node, thefield is modelled as a polynomial with undetermined coefficientswhile the conjoint polynomial models the complete field overthe element patch, which is composed of four node patches. Theweighted least square functional over any node patch node maybe written as

f =M∑

k=1

(W 2

k [f (x, y) − fFE]Tk D−1 [f (x, y) − fFE]k)(27)

Instead of a discrete L2 norm functional, a discrete energyfunctional is now used, which couples some of the equation.M is the total area of the element patch in terms of number ofsuper convergent points in the element patch, fFE are the finiteelement values evaluated at the centroid and are the weightfunctions. By differentiating the above equation with respect tothe unknown coefficients, a linear set of normal equations are

obtained, which may be solved to yield the nodal parameters.Since each element is a part of more than one element patchthe f component parameters for each node are evaluated morethan once, necessitating nodal area weighted averaging. Let usconsider that our interest is in obtaining either all the stresscomponents or one of the stress components over all layersor entire thickness of the plate or one of the components (sayσxx) along one of the layers. In order to accurately obtain thepointwise information we can choose the averaged quantitiesof all or one such stress components as quantities of interest[25]. Corresponding to these averaged quantities we can definean auxiliary problem of the form, evaluate A ∈ H 0(V ) suchthat

B(A, v) = σavg

ij (v) = �(v)∀v ∈ H 0(�) (28)

Let Ah ∈ H 0h be the finite element solution for A we have

B(Ah, vh) = σavg

ij (vh) = �(vh)∀vh ∈ H 0h (�) (29)

From the orthogonality property of the finite element solutionwe have

|B(A − Ah, u − uh)| = |�(e)| (30)

or

|�(e)| ≤ |B(A−Ah, u−uh)| ≤∑

τ

|B(A−Ah, u−uh)| (31)

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FIG. 8. Plot of error in stress for various uniform meshes σxx : (a) 2 × 2; (b) 3 × 3; (c) 4 × 4; and (d) 8 × 8 meshes.

In a component form error in terms of averaged quantities forlaminates takes the form

‖

eNi‖2 = ∫

�i

(N ∗ −

N )Ti A−1(N ∗ −

N )id�i

‖

eQi‖2 = ∫

�i

(Q ∗ −

Q)Ti S−1(Q ∗ −

Q)id�i

‖

eMi‖2 = ∫

�i

(M ∗ −

M)Ti D−1(M ∗ −

M)id�i

(32)

where A, S, and D are, respectively, the extensional, thetransverse shear, and flexural rigidity. These equations providea local definition of error for laminated composite plate models.The quantities with an asterix and the quantities with a circum-flex are, respectively, the smooth solutions and the finite elementsolutions for the normal forces, transverse shear forces, and mo-ments. The effects of coupling are accounted for in the present

error estimator; further, the use of stiffness of the laminate inthe error expressions guarantees that the actual fiber orientationof each layer is accounted for. The global error for the wholediscretization provides the overall accuracy of the solution; theglobal error estimate is given by the square root of the sum ofthe squares of the local errors.

4. REFINEMENT STRATEGY FOR PLATESAND LAMINATES

The assumption in the derivation of the adaptive refinementstrategy, which essentially predicts a new element size hn froman old element size h0, is that the old element is uniformlydivided into M∗ number of new elements marked i, each of sizehn . M indicates the total number of elements in the current(old) mesh. The assumption is also consistent with the fact thatthe asymptotic convergence rate is assumed to be valid for bothundivided element e and the uniformly divided element e. Also,

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FIG. 9. (a) Final mesh obtained after adaptive h− refinement. (b) Comparisonof convergence characteristics with energy density based estimator.

the asymptotic convergence rate is assumed to be the same forboth local (element wise) and global regions. In general it is seenthat every adaptive refinement strategy is basically an equationof the form:

hn

h0= η̄e

( ‖ref‖‖E‖e

)γ

(33)

where ‖ref‖ is a reference norm, ‖E‖e is the element wiseerror in energy norm for the current (old) mesh, η̄e is a user

FIG. 10. (a) Final mesh obtained after adaptive h− refinement. (b) Compar-ison of convergence characteristics with energy density based estimator.

defined global error fraction, expressed as a percentage, and γ

is a real number. So in most cases the derivation of an adaptiverefinement strategy amounts to the derivation of the appropriateglobal reference norm. It is obvious that an upper bound of thereference norm would yield the maximum value of error fractionand minimum amount of reduction in element size. Let η̄ be thetarget global error fraction and η be the actual global relative

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FIG. 11. (a) L shaped domain problem. (b) Final mesh obtained after adaptiveh− refinement.

percentage error, then globally we get

η = ‖E‖(‖E‖2 + ‖UFE‖2)0.5

= ‖E‖‖Uest‖ (34)

η̄ = ‖E∗‖‖Uest‖ (35)

Similarly we can compute element values of ηe the local relativepercentage error and ‖UFE‖e and the element finite elementstrain energy. Hence, both the local and global relative error

FIG. 12. (a) Distribution of error in stress σxx. (b) Comparison of conver-gence characteristics with energy density based estimator.

norm percentage error energy measures can be computed. Letthe reference norm ‖ref‖ be given by the following equation

‖ref‖ =(‖E‖2 + ‖UFE‖2

)0.5

M0.5= ‖Uest‖

M0.5(36)

The reference norm can be modified accounting for the areaof the element as

‖ref‖ =(∑M

e=1 ‖Uest‖2e Ae

A

)0.5

(37)

where A and Ae are the domain and the elemental areas, re-spectively. It can be seen that for uniform meshes this referencenorm is identical to the norm that was defined in Zienkiewiczand Zhu adaptive refinement strategy. Thus the upper bound of

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FIG. 13. (a) Simply supported square plate. (b) Initial mesh and r- adapted mesh for the full plate discretization.

the (element wise) local error estimate is set as the following:

‖E‖e ≤ η̄‖ref‖ (38)

if the reference norm is based on the local energy level the usu-ally over refinement may occur [3]. So a local global energybased reference norm is usually used. The above equation usu-ally implies that the element level energy error reaches a fractionof the reference energy equally over all elements as the elementsize tends towards zero. Thus, in the limit, the optimality con-dition is reached when all the elements have equal amounts of

error in energy. The size indicator is defined as the following

ξe = ‖E‖e

η̄ ‖ref‖ (39)

where, ξe is the size indicator used to change the size of theelement as

hn = h0

ξλe

(40)

where hn and h0 are the new and old element sizes, respectively,and λ is the singularity indicator. Thus this adaptive refinement

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FIG. 14. (a) Configurational driving force in x and y direction. (b) Plot of potential energy with node relocation iterations.

strategy can also be written as

hn

h0= η̄

(‖Uest‖M0.5

) (1

‖E‖e

)(41)

In this work a better way to represent the singularity regions isdone by using the strain energy density ratios. The global finiteelement strain energy density may be defined as

Dg = ‖UFE‖A0.5

(42)

where A is the area of the domain. The local (elemental) finiteelement strain energy density is defined as

De = ‖UFE‖e

A0.5e

(43)

Thus, the modification to the target global error fraction is givenas

η̄e = η̄

[Dg

De

](44)

In physical terms, the above equation modifies the targetglobal error fraction for every element in the current mesh

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FIG. 15. (a) r-adapted mesh for quarter plate model at 650 Iterations. (b)Plot of potential energy with node relocation iterations for selective and reducedintegration rule in computation of driving force terms.

depending upon the relative measures of the local and globalstrain energy. Thus equation (41) in this case gets modified to

hn

h0= η̄

[Dg

De

](∑Me=1 ‖Uest‖2

e Ae

A

)(1

‖E‖e

)(45)

5. NUMERICAL RESULTS

5.1. Validation of EstimatorFor validation of the estimator the test problems include a

(0◦) specially orthotropic plate, (0◦/90◦) cross ply laminate,antisymmetric (45◦/−45◦) angle ply laminate, (−45◦/45◦)4 an-tisymmetric angle ply laminate and (0◦/90◦/0◦)s cross ply lam-inate. A symmetric quadrant of simply supported square platehas been considered. Figure 6 shows the simply supported lam-

inated considered, with various uniform refinements. A uni-formly distributed load of one units is considered. Two lam-inates, namely graphite epoxy and glass epoxy, are consid-ered. The normalized central deflection is computed for variousspan to thickness ratios. Figure 7(a) and Figure 7(b) show thevariation of normalized central deflection for graphite epoxyand glass epoxy laminates, for various aspect ratios. The plothas been made for classical laminate and first order theories.The effect of shear terms is more pronounced at lower ra-tios. In subsequnet examples the first order theory has beenused.

5.1.1. Example 1A symmetric quadrant of simply supported square plate (as

shown in Figure 6, subjected to a uniformly distributed loadof one units and with a span to thickness ratio of 10, has beenconsidered. The specified permissible error in energy norm wastaken as three percent for adapative analysis. Uniform meshrefinement, adaptive mesh refinement and refinement strategybased on strain energy density ratios are considered for erroranalysis. The distribution of error in stress σxx for various uni-form mesh refinements is shown in Figure 8. It is seen that withrefinement the errors tend to propagate near to the supports,needing more refinement there. The final mesh obtained afteradpative refinement is as shown in Figure 9(a). The convergencecharacteristics are shown in Figure 9(b). It is seen that the errorestimator enhanced by energy density ratios and using patchrecovery, show higher estimates of error and better convergencecharacteristics. In the conventional adpaptive refinement, theadaptive refinement and error indicators are mainly governedby the number of degrees of freedom, the order of approxima-tion and a localization factor λ, which is the same throughout theproblem domain. Also, the stress recovery by nodal averagingis approximate and has inaccuracies. Inclusion of strain energydensity ratios helps to avoid the specification of the localizationfactor λ, thereby making it a more local estimate.

5.1.2. Example 2: 3 Layer 0/90/0 Cross Ply LaminateA symmetric quadrant of simply supported square plate (as

shown in Figure 6), subjected to a uniformly distributed loadof one units and with a span to thickness ratio of 10, has beenconsidered. The plate is a 3 layer (0◦/90◦/0◦) cross ply laminateof dimensions (200 mm × 200 mm). The material propertiesare taken corresponding to those of graphite epoxy as shownin Figure 6. The specified permissible error in energy normwas taken as three percent for adaptive analysis. Uniform meshrefinement, adaptive mesh refinement and refinement strategybased on strain energy density ratios are considered for erroranalysis. The effect of various refinement strategies on relativeerror norm percentage is studied. The final h− adaptive mesh isas shown in Figure 10(a). The convergence characteristics plotof relative error norm percentage for various refinement levels isshown in Figure 10(b). As in the previous case the energy densitybased estimator indicates better convergence charactersitics.

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FIG. 16. (a) h- adapted mesh for quarter plate model. (b) r − h adapted mesh.

5.1.3. Example 3: L–Shaped DomainA symmetric portion, L-shaped domain made of cross ply

laminate with uniformly distributed load with symmetric bound-ary condtions as shown in Figure 11(a) is considered. The plateis subjected to a uniformly distributed load of one unit. A spanto thickness ratio of 10, is considerd. The material properties aretaken corresponding to those of graphite epoxy as shown in Fig-ure 6. The specified permissible error in energy norm was takenas three percent for adaptive analysis. Conventional adaptivemesh refinement, with nodal averaging and refinement strategybased on strain energy density ratios, based on patch recoveryestimates, is considered for error analysis. The error distributionin stress is as shown in Figure 12(a). The adaptive final meshis shown in Figure 11(b). The convergence characteristics areshown in Figure 12(b). The energy density based estimator giveshigher estimates of error and better convergence characteristics.

5.2. Combined r-h Adaptive StrategyA simply supported orthotropic square plate of dimensions

2 m × 2 m as shown in Figure 13(a) is considered. The plate isanalyzed using first order shear deformation theory. The mate-rial properties of the plate are Young’s modulus E = 100 GPaand poisson’s ratio ν = 0.30. Reduced and full integration

FIG. 17. (a) Plot of potential energy with node relocation iteration for variousrefinement levels. (b) Comparison of convergence characterstics of combinedrefinement strategy.

schemes are considered for shear terms. The initial mesh for afull plate is as shown in Figure 13(b). Mesh adaption by noderelocation is performed. The distribution of the driving forcesalong the x− direction and y− direction is shown in Figure14(a). It is observed that the configuration driving forces aremore pronounced at the boundaries. These forces act as errorindicators for r− adaption. The final r−adapted mesh is shownin Figure 13(b). There is a substantial decrease in potential en-ergy with node relocation iterations as seen in Figure 14(b). Thisis associated with extensive distortion in the elements; now asymmetric quadrant of plate is considered, as such the noderelocation iterations are stopped for a user defined number ofiterations. The final adapted mesh for the quadrant of plate isshown in Figure 15(a). The order of integration used for shearterms greatly influences the flexibility of the system as is seen

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from Figure 15(b). The potential energy decreases when shearterms are selectively integrated. The h− adaptive meshes for thequadrant of the plate are shown in Figure 16(a). The r − h adap-tive meshes obtained in the sequence of r− followed by h−insuccession are shown in Figure 16(b). A plot of the potentialenergy with node relocation iterations for various refinementlevels is plotted as shown in Figure 17(a). It is seen that thereis a substantial decrease in the potential energy of the systemwith node relocation iterations and with refinement. A plot ofthe relative error norm percentages for various refinement levelsis shown in Figure 17(b).

The error norm percentages are computed for patch recov-ery based estimators and compared with estimators that useconventional nodal averaging based smoothing technique. It isobserved that the combined refinement strategy gives higherestimates of error and better convergence characteristics whenpatch recovery based estimators are used. In the conventionaladaptive refinement, the adaptive refinement and error indica-tors are mainly governed by the number of degrees of freedom,the order of approximation and a localization factor λ, which isthe same throughout the problem domain. Also, the stress re-covery by nodal averaging is approximate and has inaccuracies.Inclusion of strain energy density ratios helps to avoid the spec-ification of the localization factor λ, thereby making it a morelocal estimate. Furthermore, in a combined refinement startegysuch as r −h refinement, there is a minimization of the potentialenergy with respect to the nodal positions together with refine-ment. The r adaption in a sense causes the roughening of finiteelement solution, resulting in a requirement of the higher ordercontinuity for next refinement with every step of mesh adaption.Thus, unlike in conventional adaptive refinement, wherein theorder of recovery and continuity is fixed with every refinement,here the order of continuity is possibly changing with node re-locations and refinement, which is reflected in the convergencecharacteristics.

6. CONCLUSIONSIn this work an r− adaption procedure, namely the configura-

tional force method, has been studied in detail for plate bendingapplications. The driving force has been derived for a Reiss-ner Mindlin isotropic plate and for laminated composite plates.The finite element formulations for plates incorporating variousshear deformation theories and their numerical implementationhas been discussed. Polak Rebiere conjugate gradient algorithmhas been introduced in order to increase the convergence rate ofthe node relocation technique in configurational force method.The r− adaption results in an optimal configuration of a giveninitial mesh by improving the solution. The degeneracy andincrease in error owing to r−adaption is overcome by mesh en-richment through h−refinement. The h− refinement based onweighted average recovery technique enhanced by energy den-sity ratios performs well. The combined r − h adaptivity helpsin obtaining an optimal mesh with specified degree of accuracy

and shows better convergence characteristics. The adaptivityprocedure carried out in a successive manner of r− followedby h− in cycles shows better characteristics than a completer−adaption followed by a complete h− adaption and hence isdesirable for obtaining optimal mesh for a specified accuracy.The order of integration of shear terms affects the degree offlexibility of the system. It is known that bilinear elements areassociated with zero energy mode mechanisms when reducedintegration is used for shear terms. Thus there is a possibilitythat the mesh adaption by node relocation is affected by thesezero energy modes; this forms the scope for a further work.

REFERENCES1. I. Babuska, W.C. Rheinbold, A-Posteriori Error Estimators for the Finite

Element Method, Int. J. Numerical Methods in Engineering, vol. 12, pp.1597–1615, 1978.

2. O.C. Zienkiewicz, J.Z. Zhu, A Simple Error Estimator and Adaptive Proce-dure for Practical Engineering Analysis, Intl Journal for Numerical Methodsin Engineering, vol. 24, pp. 337–357, 1987.

3. O.C. Zienkiewicz, J.Z. Zhu, Error Estimates and Adaptive Refinement forPlate Bending Problems, International Journal of Numerical Methods inEngineering, vol. 28, pp. 2839–2853, 1989.

4. O.C. Zienkiewicz, J.Z. Zhu, Super Convergence Recovery Techniques andA-Posteriori Error Estimators, International Journal of Numerical Methodsin Engineering, vol. 30, pp. 1321–1339, 1990.

5. O.C. Zienkiewicz, J.Z. Zhu, Adaptivity and Mesh Generation, Interna-tional Journal of Numerical Methods in Engineering, vol. 32, pp. 783–810,1991.

6. O.C. Zienkiewicz, J.Z. Zhu, The Super Convergent Patch Recovery and A-posteriori Error Estimates. Part I: The Recovery Technique, InternationalJournal of Numerical Methods in Engineering, vol. 33, pp. 1331–1364,1992.

7. O.C. Zienkiewicz, J.Z. Zhu, The Super Convergent Patch Recovery andA-posteriori Error Estimates. Part 2: Error Estimates and Adaptivity, Inter-national Journal of Numerical Methods in Engineering, vol. 33, pp. 1331–1364, 1992.

8. O.C. Zienkiewicz, The Background of Error Estimation and Adaptivity inFinite Elements, Computer Methods in Applied Mechanics and Engineer-ing, vol. 195, pp. 207–213, 2006.

9. J. Fish, R. Guttal, The p-version of the Finite Element Method for ShellAnalysis, Computational Mechanics, vol. 16, pp. 1–13, 1995.

10. J. Fish, The s-version of the Finite Element Method, Computers and Struc-tures, vol. 43(3), pp. 539–547, 1992.

11. J. Fish, R. Guttal, The s- version of the Finite Element Method for LaminatedComposite, International Journal of Numerical Methods in Engineering,vol. 39(21), pp. 3641–3662, 1998.

12. J. Fish, S. Markolefas, Adaptive Local-Global Refinement Strategy Basedon Interior Estimates of the h-method, International Journal of NumericalMethods in Engineering, vol. 37, pp. 827–838, 1994.

13. H. Askes, A. Rodriguez-Farran, An r-h Adaptive Strategy Based on DomainSubdivision and Error Assessment, Civil-Comp Press, Edinburgh, pp. 95–102, 2000.

14. H. Askes, A. Rodriguez-Farran, A Combined r-h Adaptive Scheme Basedon Domain Subdivision - Formulation and Linear Examples, InternationalJournal for Numerical Methods in Engineering, vol. 51(3), pp. 253–373,2001.

15. G.K. Madan, R.L. Huston, Finite Element Meshes Refinement Criteria forStress Analysis, Computers and Structures, vol. 34(2), pp. 251–255, 1990.

16. Z. Uthman, H. Askes, r-adaptive and h-adaptive Mesh Optimisation Basedon Error Assessment, 13th UK National ACME Conference, ed. R.S.Crouch, University of Sheffield, pp. 127–130, 2005.

Dow

nloa

ded

by [

Indi

an I

nstit

ute

of T

echn

olog

y H

yder

abad

] at

05:

19 0

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ovem

ber

2013


17. Jung-Ho-Cheng, Adaptive Grid Optimization for Structural Analysis -Geometry Based Approach, Computer Methods in Applied Mechanics andEngineering, vol. 107, pp. 1–22, 1993.

18. A. Selman, E. Hinton, A.W. Sibai, Edge Effects in Mindlin-Reissner PlatesUsing Adaptive Mesh Refinement, Engineering Computations, vol. 7, pp.217–226, 1990.

19. S.M. Yunus, T.P. Pawalak, M.J. Wheeler, Application of the ZZ ErrorEstimator for Plate and Shell Analysis. International Journal of NumericalMethods in Engineering, vol. 29, pp. 1281–1298, 1990.

20. C.K. Choi, Y.M. Park, An Adaptive h-refinement Using Transition Elementfor Plate Bending Problems, International Journal of Numerical Methodsin Engineering, vol. 35, pp. 145–163, 1992.

21. S.H. Lee, T. Blacker, T. Belytschko, Derivative Recovery Techniques for C0Plate Bending Problems, Engineering Computations, vol. 11, pp. 495–512,1994.

22. J.E. Abdallah Filho, J.O. Dow, An Error Analysis Approach for LaminatedComposite Plate Finite Element Models, Computers and Structures, vol.52, pp. 611–616, 1994.

23. J.E. Abdallah Filho, J.O. Dow, Qualitative Errors in Laminated CompositePlate Models, International Journal of Numerical Methods in Engineering,vol. 37, pp. 1215–1230, 1994.

24. P.M. Mohite, C.S. Upadhyay, Local Quality of Smoothening Based a-posteriori Error Estimators for Laminated Composite Plates Under Trans-verse Loading, Computers and Structures, vol. 80, pp. 1477–1488, 2002.

25. P.M. Mohite, C.S. Upadhyay, Focussed Adaptivity for Laminated Plates,Computers and Structures, vol. 81, pp. 287–298, 2003.

26. C.S. Krishnamoorthy, S. Mukherjee, Adaptive Finite Element Analysis withQuadrilateral Elements Using A New h-refinement Strategy, SADHANA,vol. 21(5), pp. 623–652, 1996.

27. Y.C. Liu, H.A. Elmaraghy, Assessment of Discretized Errors and AdaptiveRefinement with Quadrilateral Finite Elements, International Journal ofNumerical Methods in Engineering, vol. 33, pp. 781–798, 1992.

28. J.S. Sandhu, H. Liebowitz, Mesh Adaptation Using a Four Noded Quadri-lateral Plate Bending Element, Engineering Fracture Mechanics, vol. 50,pp. 737–758, 1995.

29. Jaroslav Mackerle, Error Analysis, Adaptive Techniques and Finite andBoundary elements- A Bibliography (1992–1993), Finite Elements in Anal-ysis and Design, vol. 17, pp. 231–246, 1994.

30. P. Thoutireddy, Variational Arbitrary Lagrangian-Eulerian Method, PhDThesis, Center for Advanced Computing Research, California Institute ofTechnology, Pasadena, CA, 2003.

31. P. Thoutireddy, J.F. Molinari, E.A. Repetto, M. Ortiz, A Variational r-Adaption and Shape-Optimization for Finite-deformation Elasticity, Inter-national Journal for Numerical Methods in Engineering, vol. 61(1), pp.1–21, 2004.

32. M. Braun, Configurational Forces Induced by Finite Element Discretization,Proc. Estonian Acad. Sci. Phys. Math, vol. 46(1/2), pp. 24–31, 1997.

33. J.D. Eshelby, The Elastic Energy-momentum Tensor, Journal of Elasticity,vol. 5(3–4), pp. 321–335, 1975.

34. G.A. Maugin, Material Forces Concepts and Applications, Applied Me-chanics Review, vol. 48(5), pp. 213–245, 1995.

35. P. Steinmann, Application of Material Forces to Hyper Elastostatic FractureMechanics Part- I Continuum mechanical setting, International Journal ofSolids and Structures, vol. 37, pp. 7371–7391, 2000.

36. R. Muller, G.A. Maugin, On Material Forces and Finite Element Discretiza-tion, Computational Mechanics, vol. 29, pp. 52–60, 2002.

37. R. Muller, S. Kolling, D. Gross, On Configurational Forces in the Con-text of the Finite Element Method, Intl Journal for Numerical Methods inEngineering, vol. 53, pp. 1557–1574, 2002.

38. A. Rajagopal, S.M. Sivakumar, Optimality of Finite Element Grids Basedon Material Forces and Error Assessment, Computer Assisted Mechanicsand Engineering Sciences, vol. 12, pp. 1–21, 2006.

39. A. Rajagopal, R. Gangadharan, S.M. Sivakumar, Performance Evaluationof Configurational Force and Spring Analogy Based Mesh OptimizationSchemes, International Journal of Computational Methods in EngineeringScience and Mechanics, vol. 7, pp. 241–262, 2006.

40. R. Gangadharan, A. Rajagopal, S.M. Sivakumar, An r-h Adaptive Strat-egy Based Material Forces and Error Assessment, Journal of ComputersMaterials and Continua, vol. 1(3), pp. 229–244, 2004.

41. P. Steinmann, E. Kuhl, Material Forces in Open System Mechanics, Com-puter Methods in Applied Mechanics and Engineering, vol. 193, pp. 2357–2381, 2004.

42. P. Steinmann, F. Ackermann, J. Barth, Application of Material Forces toHyper Elastostatic Fracture Mechanics. Part-II. Computational Setting,International Journal of Solids and Structures, vol. 38, pp. 5509–5526,2001.

43. P. Steinmann, E. Kuhl, H. Askes, On Convexity Conditions in Spatial andMaterial Settings of Hyperelasticity. XXI ICTAM, pp. 15–21, August 2004,Warsaw, Poland.

44. M.E. Gurtin, P. PodioGuidugli, On Configurational Inertial Forces at aPhase Interface, Journal of Elasticity, vol. 44, pp. 255–269, 1996.

45. R. Kienzler, and G. Herrmann, Mechanics in Material Space with Applica-tions to Defect and Fracture Mechanics, Springer Verlag, NY, 2000.

46. R. Muller, D. Gross, G.A Maugin, Use of Material Forces in AdaptiveFinite Element Methods, Computational Mechanics, vol. 33, pp. 421–434,2004.

47. F.J. Fuenmayor, J.L. Oliver, Criteria to Achieve Nearly Optimal Meshes inthe h− adaptive Finite Element Method, International Journal of NumericalMethods in Engineering, 39, vol. pp. 4039–4061, 1996.

48. P. Coorevits, P. Ladeveze, J.P. Pelle, An Automatic Procedure with Con-trol of Accuracy for Finite Element Analysis in 2D Elasticity, ComputerMethods in Applied Mechanics and Engineering, vol. 121, pp. 91–120,1995.

49. R.D. Cook, S.D. Malkus, M.E. Plesha, Concepts and applications of FiniteElement Analysis, John Wiley and Sons, 3rd ed. New York, 1988.

50. N.D. Phan, J.N. Reddy, Analysis of Laminated Composite Plates Using aHigher Order Shear Deformation Theory, International Journal of Numeri-cal Methods in Engineering, vol. 21, pp. 2201–2219, 1985.

51. J.N. Reddy, Mechanics of Laminated Composites Plates and Shells, CRCPress, NY, 2003.

52. J.R. Shewchuk, Report—An Introduction to Conjugate Gradient AlgorithmWithout the Agonizing Pain, Dept of Computer Science, Carnegie MellonUniversity, USA, 1994.

53. T. Blacker, T. Belytschko, Super Convergent Patch Recovery with Equi-librium and Conjoint Interpolant Enhancements, International Journal ofNumerical Methods in Engineering, vol. 37, pp. 517–536, 1994.

54. Michael Scherer, Ralf Denzer, Paul Steinmann, Energy-based r-adaptivity:A Solution Strategy and Applications to Fracture Mechanics, Int J Fract,DOI 10.1007/s10704-007-9143-9, 2007. Available online.

55. A. Rajagopal, S.M. Sivakumar, A combined r-h Adaptive Strategy Based onMaterial Forces and Error Assessment for Plane Problems and BimaterialInterfaces, Computational Mechanics, vol. 41(1), pp. 49–72, 2007.

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Energy Based Adaptive Strategy for Plates and Laminates

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