stresses and error indicators Muci-Kuchler, K.H/% Miranda … · 2014-05-13 · Kuchler and...

stresses and error indicators

Muci-Kuchler, K.H/% Miranda-Valenzuela, J.C. ,

Soriano-Soriano, S.^University of Detroit Mercy, Detroit, MI 48219-0900, USA.mucikh@udmercy. eduWessex Institute of Technology, Ashnrst, Southampton SO 40 7AA, UK.

jmiranda@wessex. etc. ukTWec Gmem/EWr/c, o m, 7V.I. 66600, M&Y7CO

[email protected]

Abstract

In this work, a new global reanalysis technique for the efficient computation ofstresses and error indicators in two-dimensional elastostatic problems is presented.In the context of the Boundary Element Method, the global reanalysis techniquecan be viewed as a post-processing activity that is carried out once an analysisusing Lagrangian elements has been performed. To do the reanalysis, thefunctional representation for the displacements is changed from Lagrangian toHermite, introducing the nodal values of the tangential derivatives of thosequantities as additional degrees of freedom. Next, assuming that the nodal valuesof the displacements and the tractions remain practically unchanged from the onesobtained in the analysis using Lagrangian elements, the Tangent DerivativeBoundary Integral Equations are collocated at each functional node in order todetermine the additional degrees of freedom that were introduced. In this fashion, asecond system of equations is generated and, once it is solved, the nodal values ofthe tangential derivatives of the displacements are obtained. This approach givesmore accurate results for the stresses at the nodes since it avoids the need todifferentiate the shape functions in order to obtain the normal strain in thetangential direction. When compared with the use of Hermite elements, the globalreanalysis technique has the attractive that the user does not have to give as inputdata the additional information required by this type of elements. Anotherimportant feature of the proposed approach is that an efficient error indicator forthe values of the stresses can also be obtained.

1 Introduction

Nowadays, most of the analyses that are carried out to solve elastostatic problemsusing the Boundary Element Method (BEM) employ Boundary Integrals Equations(BIEs) that only involve the displacements and the tractions. Thus, whenLagrangian elements are used in conjunction with those BIEs, the stresses on the

Transactions on Modelling and Simulation vol 24, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X

718 Boundary Elements

boundary are not immediately obtained from the solution of the system ofequations generated by the method. Therefore, additional computations must becarried out in order to determine the value of the stresses. The approach that iscommonly employed for that purpose uses Hooke's law together with the tractionsand the tangential derivatives of the displacements [1]. Since it is necessary todifferentiate the functional representation for the displacements to obtain thetangential derivatives of those quantities, usually the numerical solution for thestresses will be less accurate than the one for the displacements and the tractions.Furthermore, in general the stresses will not be continuous across elementboundaries because Langrangian elements only provide C° continuity for thedisplacements at those locations.

Various alternatives have been proposed to overcome the difficulties mentionedabove. The use of BIEs that explicitly involve in their formulation the tangentialderivatives of the displacements is probably the most popular one. Gosh et al. [2]recast the Conventional Boundary Integral Equations (CBIEs) in terms of thetractions and the tangential derivatives of the displacements. This scheme providessatisfactory results but no longer involves the displacement function itself, whichpresents uniqueness difficulties, especially on multiple connected regions.

In 1986, Watson [3] proposed the use of Hermitian interpolation functions toapproximate the values of the displacements and the tractions inside an element.Since Hermite elements provide C* continuity for the displacements across elementboundaries, this ensures that the stress will be continuous at those locations. Muci-Kuchler and Rudolphi [4] proposed a similar approach but used a different methodfor the regularisation of the Hypersingular Boundary Integral Equations (HBIEs).In addition, they implemented the Hermite elements in a non-isoparametricfashion, making their use more attractive.

Another approach that could be used would employ the ideas presented byGray and San Soucie [5]. In this case, the functional representation for thedisplacements is the same as the one employed for Hermite elements, but thetractions are approximated using Lagrangian shape functions. Under thesecircumstances, the nodal values of the tangential derivatives of the displacementscan be considered as additional unknowns and HBIEs are used to generate theadditional equations required to obtain a solution.

Matsumoto et al. [6] presented another alternative for the computation of thestresses on the boundary based on performing two consecutive analyses. The firstanalysis solves for the displacements and the tractions using the CBIEs. For thesecond analysis, the components of the gradient of the displacement vector areapproximated inside an element using the nodal values of those quantities andLagrangian shape functions. To solve for the new unknowns, Regularised BIEs forDisplacement Gradients are employed. Although this approach provides betterresults for the stresses, its high computational cost may make it unattractive.

Besides the previous alternatives, the use of spline elements [7] and the use ofadditional high-order shape functions [8] may also help to improve the accuracy inthe computation of the stresses.

For any of the methods described above, one way to increase the quality of thesolution is to selectively improve the discretisation of the boundary using adaptivemeshing techniques. In late years the number of references regarding these


B oundary Elements 719

techniques has increased significantly and most of the literature corresponding tothis topic has been devoted to the formulation of new and more reliable errorindicators. A comprehensive list of contributions up to 1994 is provided inreferences [9] and [10]. More recently, other options for the computation of errorindicators have appeared. In what follows, a brief explanation of some of them ispresented.

Charafi et al. [11] used the local reanalysis technique to implement an efficient/ -hierarchical adaptive meshing technique. The authors employed a localenrichment scheme to obtain an improved solution for each element of the mesh.The improved solution was compared with the original one to obtain a measure ofthe error in the element.

Guiggiani [12] used the direct differentiation approach to perform sensitivityanalysis of approximate boundary element solutions with respect to the position ofthe collocation points. A similar approach was used by Paulino et al. [13], whoproposed the evaluation of the residual of HBIEs as an error indicator. Liang et al.[14], used a modified version of the error indicator proposed by Paulino,combining the residuals of the CBIEs and HBIEs for the computation of errorindicators. In another approach, Paulino et al. [15], employed the rate of change ofthe tractions and generalised displacements with respect to the positions of middlenodes of quadratic boundary elements as error indicators.

Muci-Kuchler et al. [16] and Muci-Kuchler and Miranda-Valenzuela [17],presented a formulation for the computation of a simple error indicator for Hermiteelements. This error indicator was based on the possibility of obtaining twodifferent numerical solutions from just one analysis with Hermite elements. Thedifference of the two solutions inside an element was used as a measure of theerror for that particular element.

In this work, a new global reanalysis technique, which uses the TangentDerivative Boundary Integral Equations (TDBIEs) for the efficient computation ofstresses and error indicators in two-dimensional elastostatic problems is presented.The global reanalysis technique proposed here can be viewed as a post-processingactivity which uses as a starting point the results provided by an analysis carriedout employing Lagrangian elements. To perform the reanalysis, the functionalrepresentation for the displacements is changed from Lagrangian to Hermite,introducing the tangential derivatives of those quantities as additional degrees offreedom at the functional nodes. Then, assuming that the nodal values of thedisplacements and the tractions remain practically unchanged from the onesobtained in the analysis using Lagrangian elements, the TDBIEs are collocated ateach functional node in order to determine the new degrees of freedom that wereintroduced. In this fashion, a second system of equations is generated and, once itis solved, the nodal values of the tangential derivatives of the displacements areobtained. Hence, the nodal values of the stresses can be easily computed and amore accurate functional representation for the displacements is available.Furthermore, an error indicator for each element can be determined by comparingthe numerical solution for the stresses provided by the first analysis with the oneobtained using the results of the reanalysis.



2 Boundary Elements Formulation

In tliis section, a review of the boundary integral equations employed in this workis presented. Also, a briefing on how stresses are usually computed at a point onthe boundary is given.

2.1 The Conventional Boundary Integral Equations

Consider the two-dimensional elastostatic problem without body forces. On adomain bounded by S , the CBIEs in their completely regularised form are givenby [4]:

where the range of indices goes from 1 to 2 and the summation convention is

employed. Here, x represents the field point, | is the source point, and %,• and r,

are the displacement and traction components. The kernels Uy- and 7 are the

standard displacement and tractions kernels which can be found in severalreferences including [1].

2.2 Tangent Derivative Boundary Integral Equations

The TDBIEs are obtained through the differentiation of the displacementrepresentation integral with respect to the source point coordinates £. These

equations in their completely regularised form are given by [4]:

J<

(2)

where £• are the components of the tangent unit vector at point | and, for plain

strain, the kernels Vj, Wf, W , and Y,j are given by [4]:


Boundary Elements 721

In tlie above expressions, // is the shear modulus, v is the Poisson's ratio and r

is the distance between 3? and £ . s; and %, are tlie components of the normal and

tangent unit vectors to S at x and %// are the components of the unit outward

normal vector at the source point | .

2.3 Computation of stresses

Consider a model where an approximation for the displacements and tractionsinside each element is available from the BEM solution. The first step in thecomputation of the stresses is the evaluation of the in-plane components of thestress tensor in terms of a local coordinate system. For plain strain, the normal andshear stresses at a boundary point x can be obtained as:

(5)

Although the normal stress <J(»)(«) and the shear stress cr(n)(s) art found in a

straightforward fashion, the computation of the normal stress in the tangentialdirection 0Yr)(j) requires the prior calculation of the normal strain in the tangential

direction £(*)(*) which is given by (duj/ds)sj , where dnj/ds are the tangential

derivatives of the displacement components.If the values of 8*, / ds at the functional nodes are not directly available from

the solution of the system of equations generated by the BEM, they have to becomputed through the differentiation of the functional representation for thedisplacements employed for the element. In the case of Lagrangian elements,which only offer C° inter-element continuity, this differentiation will lead todiscontinuities in the values for the stresses across element boundaries. This, ingeneral, will degrade the overall quality of the numerical solution for the stresses.

3 Efficient evaluation of stresses

The coincident collocation of the CBIEs and TDBIEs to formulate Henniteboundary elements has proved to give very accurate results for the stresses [4].Even so, this type of elements has failed to attract the interest of researchers and



users of the method. Researchers may feel discouraged due to the complexity oftheir implementation and users may find that it is cumbersome for them to inputmore information than the one that is usually needed.

In this work, a new scheme that provides better results for the stresses than theLagrangian elements and avoids the necessity to provide additional informationduring the pre-processing stage is presented. The scheme is based on performingtwo consecutive analyses. The first one uses the CBIEs and Lagraiigian elements toobtain the values of displacements and tractions at the functional nodes. Thesecond (which constitutes the global reanalysis), uses Hermite shape functions forthe functional representation of the displacements and the TDBIEs to obtain thevalues of the tangential derivatives of the displacements at the same locations.

3.1 Global reanalysis

Once a first analysis with Lagrangian elements has been performed and the valuesof displacements and tractions are available, the next step is to modify the way inwhich the displacements are approximated inside the elements. The functionalrepresentation for those quantities will be changed from Lagrangian to Hermite,introducing the nodal values of the tangential derivatives of the displacements asadditional degrees of freedom.

With the proposed modification, the displacements are now approximated asfollows:

where HJ^ is the /* shape function associated with the i^ functional node,

*M is the j^ component of the displacement vector at node n and duy I ds is

its tangential derivative.If it is assumed that the nodal values of the displacements and the tractions

remain practically the same as the ones obtained in the first analysis despite thischange in the functional representation for the displacements, then the only newunknowns are the values of &, / ds at the functional nodes. Thus, an additional set

of equations is required to determine those degrees of freedom. These equationscan be generated by collocating the TDBIEs at each one of the functional nodes.

It is important to recall that the TDBIEs cannot be collocated at points on theboundary where the unit outward normal vector is not unique or where the tractionvector presents discontinuities. Thus, discontinuous or partially discontinuouselements are required in those cases and they must be employed in both analyses,even though the CBIEs could be collocated at those locations.

3.2 Comparison of computational cost

It is interesting to present a comparison of the computational cost involved in thesolution of the system(s) of equations generated by some of the different



alternatives that make use of two different sets of integral equations to determinethe values of the stresses on the boundary. If Gaussian elimination is employed tosolve a system of equations, the total number of operations required would be ofthe order of M*, where M is the number of equations in the system. Table 1 showsthe comparison between four different methods in terms of the total number offunctional nodes present in the mesh, which is denoted by N. As can be seen fromthe table, for meshes with the same number of nodes, the methodology presentedhere requires less operations to solve the system(s) of equations than the Hermiteelements and the method proposed by Matsumoto et al.

Table 1: Comparison of the approximate computational cost for different methods.

Method

Lagrangian

Hermite

Matsumoto et al

Global reanalysis

System of Eqs.First analysis

2Nx2N

4Nx4N

2Nx2N

2Nx2N

System of Eqs.Second Analysis

None

None

4Nx4N

2Nx2N

Approximatenumber of operations

8N^

64N^

72N

16N^

4 Computation of the error indicator

In numerical methods, it is common practice to use the difference between twoconsecutive solutions to measure the error in the most accurate one. Severalauthors have successfully applied this procedure in the BEM [9]. Here a similaridea is used since two numerical solutions for the stresses are available: one fromthe analysis using Lagrangian elements and another from the global reanalysis.Thus, it is possible to estimate the error in the latter by comparing the two.

To do so, it is useful to estimate the error in a quantity that has a clear physicalmeaning. In this regard, the Von Mises stress presents itself as a good choice sinceit combines all the components of the stress tensor in a single quantity. Hence, inthe L2 norm, the error indicator for any element V of the mesh can be defined as:

1/2(7)

where cr& and cr( are the Von Mises stresses obtained from the globalixCj JL.reanalysis and the conventional analysis, respectively, and S® is tlie part of theboundary corresponding to element" e".

5 Numerical example.

A sample problem is presented in order to show the advantages of the globalreanalysis technique in comparison to the use of the conventional Lagrangian



elements. The problem of an infinite plate with a circular hole subject to an uni-axial traction in the x± direction is considered a good choice since an analytical

solution is readily available and the problem has moderate stress gradients. Asshown in Figure 1, due to the symmetry of the problem, only one quarter of theplate needs to be modelled.

A model with 12 quadratic Lagrangian elements is adequate to demonstrate theincreased accuracy in the solution for the stresses that is obtained when the globalreanalysis technique is employed. Although the global reanalysis technique can beapplied to any Lagrangian element, the use of quadratic elements was motivated bythe great popularity that these elements have among users of the BEM.

Also, in order to do a fair evaluation of the quality of the solutions obtainedwith the global reanalysis technique and with Lagrangian elements only, acomparison based on similar numbers of operations to solve the system(s) ofequations is presented. For this purpose, a second model with 18 quadraticelements is considered. Using the formulas given in Table 1, the solution of thismodel using the standard approach has a computational cost that is close to the onerequired by the first model using the global reanalysis technique. Figure 2 showsthe meshes employed together with the approximate number of floating pointoperations required to solve the system(s) of equations that are generated.

Figure 3 shows the results for the stress 0^2 along the boundary xj = 0 before

and after the global reanalysis technique is applied to the model consisting of 12elements. The first graph compares the behaviour of both numerical solutionsagainst the analytical one. From this plot, it is clear that the application of theglobal reanalysis technique has a major impact on the accuracy of the solution.This improvement can be appreciated more easily using the second graph whichpresents the absolute value of the true error for both solutions.

Reference [4] shows a similar plot where the authors compare the performanceof Hermite and quadratic elements using models with a similar number of degreesof freedom. In general, the use of Hermite elements provides better results than theapproach presented here although their use is a little bit more complicated.

Figure 1: Boundary Value Problem and modelled region.



-4-

12 Elements390K operations

18 Elements373K operations

Figure 2: Analysed meshes, (a) Mesh for global reanalysis and (b) mesh forstandard analysis. Only the functional nodes are shown.

(NC/3

30

20

10

n

H _

:.-;•' :

# Lagrangiann Reanalysis

; Analytic' , , , , i , , , , i , , ,

\ #

\o

Absolute

true error in S22

3

LA

O

L# Lagrangian

a Reanalysis

- :•-••-

#",f, ,,§,,, I , n ,

#

* o

, , , n i , , , ,

1 2 3 4

A re Length

1 2 3 4 5

A re Length

Figure 3: Comparison between the analysis with Lagrangian elements and theglobal reanalysis technique for the mesh with 12 elements.

Figure 4 compares the solution for the model with 12 elements using the globalreanalysis technique, against the solution for the model with 18 elements using thestandard approach. As stated before, both solutions have a similar computationalcost. From the plots presented for 022 along the boundary x\ = 0 , it is clear that

the global reanalysis technique still provides a better accuracy than the one that thequadratic elements alone can offer. This feature makes the technique presentedhere even more appealing.

The problem shown in Figure 1 was also used to test the reliability of the errorindicator to lead adaptive processes. As improvement criteria, all the elements in agiven mesh that had a value for the error indicator larger than 0.65 times themaximum value of the error indicator for that mesh, were refined into two elementsof equal size. The adaptive process was stopped when all the elements of a meshhad a value for the error indicator smaller than 5.0 (5% the applied load). Figure 5shows the three meshes that took the adaptive process to complete.



Computing the real error in the solution for the Von Mises stress on each

element by E = (() (a - dS fi*, where o*$ is the analytical solution

for the Von Mises stress, it is possible to compare the values for the real andpredicted errors.

The first plot in Figure 6 shows a comparison between the values for £*> and

fiW on the elements of the first mesh. The second graph presents the convergenceto the analytical solution achieved by the adaptive process by plotting on each step

the maximum value of ffl and the maximum value of E\*'. As can be seen, thepredicted and true errors have a very good agreement.

Conclusions

A new technique for the computation of stresses and error indicators in theBoundary Element Method has been presented. This global reanalysis techniqueuses the TDBIEs in a post-processing fashion to improve the solution for thestresses when an analysis with Lagrangian elements has already been performed.Its main attractive is that it increases the overall accuracy of the numerical solutionfor the stresses at a reasonable computational cost. Furthermore, this approachavoids the inconveniences related to the use of Hermite elements since it is notnecessary to specify the tangential derivatives of the boundary conditions as part ofthe input data. Although the methodology is less accurate than the use of Hermiteelements, its performance is superior than the one corresponding to the standardapproach and it is computationally cheaper than other proposals of the same typepresented by other authors. In addition, an efficient and reliable error indicator canbe computed from the difference between the original and the global reanalysissolutions, which adds another advantage to the proposed method.

30,

20

10

0

1 [r,,•••••

D Reanalysis

Analytic

# Lagrangian, , , ,

13

q#

\l

1 2 3 4

A re Lengt h

Absolute

true error in S22

O LA

G)

L

D Reanalysis

# Lagrangian 1

; •

" ; o

• •i , O , , i § . , , , , § , , , , n . , , , ,

1 2 3 4 5

Arc Length

Figure 4: Comparison between the analysis with Lagrangian elements and theglobal reanalysis technique using approximately the same number of operations.


<UOq-,oS


-^ e ^

16 Elements148 DOF

bo*ot oot

Figure 5: Adaptive process. Only functional nodes are shown.

12

10

8

6

4

2

0

D Predicted

HReal

MlJLnJkJi1,

,

1 3 5 7 9

Element

110 120 130 140 150

Degrees of freedom

Figure 6: Comaprison of the estimated and true errors for the initial mesh andconvergence to the analytical solution.

Acknowledgements

The first author dedicates this paper to the memory of Prof. Gundo Nariboli, whointroduced him to the beautiful field of continuum mechanics. J.C. Miranda-Valenzuela acknowledges the financial support provided by CONACyT, theMexican Council for Science and Technology, for the realisation of this work.

References

[1] Becker, A. A. The Boundary Element Method in Enginnering - A completecourse. Mc-Graw Hill International, 1992.

[2] Gosh, N., Rajiyah, H., Gosh, S. and Mukerjee, H. A new boundary element



method formulation for linear elasticity. J. Appl Mech. 53, 69-76, 1986.

[3] Watson, J.O. Hermitian cubic and singular elements for plane strain.Developments in Boundary Element Methods-4 Chap. 1, Ed. Banerjee, P. K.and Watson, J.O., Elsevier Applied Science Publishers, London, 1-28, 1986.

[4] Muci-Kuchler, K. H. and Rudolphi, T.J. Coincident collocation ofdisplacement and tangent derivative boundary integral equations inelasticity. IJNME36, 2837-49, 1993.

[5] Gray, L. J. and San Soucie, C. A Hermite interpolation algorithm forhypersingular boundary integrals. IJNME36, 2357-67, 1993.

[6] Matsumoto T., Tanaka M. and Hirata, H. Boundary stress calculation usingregularized Boundary Integral Equation for displacement gradients. IJNME36, 783-97, 1993.

[7] Cabral, U.S., Wrobel, L.C. and Brebbia, C.A. A BEM formulation using B-spline: I-uniform blending functions. Eng. Anal. 7, 136-44, 1990.

[8] Zhao, Z. Interelement stress evaluation by boundary elements. IJNME, 39,2399-415, 1996.

[9] Liapis, S., A review of error estimation and adaptivity in the boundaryelement method. Eng. Anal, 14, 315-23, 1994.

[10] Kita, E. and Kamiya, N., Recent studies on adaptive boundary elementmethods. Adv. Eng. Software, 19, 21-32, 1994.

[11] Charafi, A., Neves, A.C. and Wrobel, L.C. h-Hierarchical adaptive boundaryelement method using local reanalysis. IJNME, 38, 2185-207, 1995.

[12] Guiggiani, M., Sensitivity analysis for boundary element error estimationand mesh refinement. IJNME., 39, 2907-20, 1996.

[13] Paulino, G. H., Gray, L. J. and Zarikian, V., Hypersingular residuals - a newapproach for error estimation in the boundary element method. IJNME, 39,2005-29, 1996.

[14] Liang, M.T., Chen, J.T. Yang, S.S., Error estimation for boundary elementmethod, Eng. Anal., 23, 257-65, 1999.

[15] Paulino, G. H., Shi, F., Mukherjee, S. and Ramesh, P. Nodal sensitivities aserror estimates in computational mechanics. Ada Mech., 121, 191-213,1997.

[16] Muci-Kuchler, K.H., Miranda-Valenzuela, J.C. and Rudolphi, T.J. A newerror indicator for adaptive meshing with Hermite Boundary Elements.Comp. Meth. App. Mech. Eng, 173, 419-31, 1999.

[17] Muci-Kuchler, K.H. and Miranda-Valenzuela, J.C. A New Error IndicatorBased on Stresses for Adaptive Meshing with Hermite Boundary Elements,in Boundary Elements XX (Ed. Kassab, A. J.; Chopra, M. and Brebbia, C.A.) 279 - 288, Orlando, Florida, USA, 1998.


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