Analytical response sensitivity computation using hybrid finite elements

Analytical response sensitivity computation using hybrid®nite elements

P.C. Pandey*, P. Bakshi

Department of Civil Engineering, Indian Institute of Science, Bangalore, 560 012, India

Received 25 May 1997; accepted 18 November 1998

Abstract

This paper presents analytical sensitivity computation using hybrid ®nite elements. Expressions have been derived

for analytical response sensitivities using two-dimensional hybrid elements developed on the basis of the modi®edHu±Washizu variational principle. Computational algorithms have been formulated for sensitivity computation andthe same have been implemented using a partial symbolic computational scheme, veri®cation being done withstandard bench-mark examples. Illustrations of the response sensitivity computation with respect to sizing variables

have been presented. The improvement in the accuracy of sensitivity values using hybrid elements as compared withconventional displacement based elements is examined. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Hybrid ®nite elements; Modi®ed Hellinger±Reissner variational principle; Response sensitivity; Symbolic computations

1. Introduction

Sensitivity analysis has evolved as a major area of

research in structural analyses holding out immense

promise for widespread applications. The past three

decades have witnessed a spurt of research activities in

the computational aspects of sensitivity. Theoretical

bases were constructed and ®nite element codes were

formulated for computation of sensitivity for static re-

sponse, transient response and buckling and eigenvalue

problems. Noteworthy contributions have been made

by Dems and MroÂ z [1±3], Choi and his co-workers

[4,5], Haftka and Barthelemy [6,7], Arora and Haug [8],

Rajan and Belegundu [9,10] and Wang et al. [11]. Non-

linear sensitivity has also been approached by a number

of researchers [12±15]. In accordance with theoretical

developments of computational formulations on one

hand, and emergence of computer hardware and soft-

ware on the other, the integration of the two has also

emerged as an important area of study [16±18]. Recent

reviews of the developments in this ®eld [19±21] provide

a better perspective in this ®eld of study.

However, e�cient implementation of computational

procedures into ®nite element programs is still attract-

ing investigators searching for improved accuracy. It is

generally pointed out that any scheme adopted for the

purpose should be acceptably accurate and computa-

tionally e�cient. This implies that the ease-of-im-

plementation and performance of the algorithm are the

governing factors in the satisfactory performance of a

particular scheme of sensitivity computation. Several

algorithms using ®nite elements as well as boundary el-

ements have been used as computational tools.

However, the extent of compromise between ease of

implementation and accuracy of results is found to be

Computers and Structures 71 (1999) 525±534

0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00293-4

* Corresponding author. Tel.: +91-80-309-2667; fax: +91-

80-334-1683.

E-mail address: [email protected] (P.C. Pandey)

sensitive to a particular approach adopted. The ®nite

elements used for sensitivity computation, to date,have been the conventional displacement elements

based on the principle of minimum potential energy.

These elements, although remarkably simplistic, per-

form poorly in several situations. Inaccuracies havebeen observed [22], such as poor performance in con-

strained media problems, loss of accuracy in calculat-

ing derived ®eld variables (including derivatives of

primary ®eld variables) and slow convergence for pro-blems with high gradients. These elements are highly

sensitive to mesh distortion, and exhibit severe locking

when applied to problems involving incompressible

materials. Moreover, for elements with arbitrary geo-metry, it is di�cult, if not impossible, to construct in-

terpolation functions for the displacements in an

element to ful®ll the interelement compatibility con-

ditions, especially in C 1 elements. Most importantly,these elements yield inferior stress values except at the

gauss points. All these defects that are inherent in the

displacement-based elements get re¯ected on the accu-

racy of the sensitivity values computed using them.

Hence, a better option is envisaged in using hybrid

®nite elements for sensitivity computations which is

pursued in this paper.

The hybrid ®nite element method, since its inception

by Pian [23] in 1964, has been the subject of rigorous

research. It has presently evolved as one of the most

competitive ways of devising ®nite element models for

the analysis of solid continuum. The variational and

the computational aspects of hybrid elements have

been critically examined over the years leading to a

wide spectrum of application in structural mechanics.

The use of multiple independent ®eld variables in el-

ement formulations created a rich arena of theoretical

considerations leading to better ®nite element approxi-

mations. This yielded elements with improved conver-

gence behavior, better stress prediction, avoidance of

locking in constrained media problems and the capa-

Nomenclature

n number of element degrees of freedomm number of design variablesnb number of stress parameters

na number of strain parametersnl number of internal displacement parametersu displacement vector of size 2�1= uq+ uluq the compatible part of uul the incompatible part of uss the stress vector

ssh the stress vector with the constant terms excludedEE the strain vector[A ] elasticity matrix[K ] sti�ness matrix of size n�n

{F } external load vector of size n�1{q } vector of nodal displacements of size n�1{b } vector of design variables of size m�1

{b } vector of stress parameters of size nb�1{a } vector of strain parameters of size na�1{l } vector of internal displacement parameters of size nl� 1

P interpolation matrix for stress of size 3�nbN interpolation matrix for strain of size 3�naF interpolation matrix for compatible displacements of size 2�n

M interpolation matrix for incompatible displacements of size 2�nlGq fvPTBqdVGl fvPT

hBldVH fvNTANdV

W fvPTNdVBq DTF

Bl DTM

V elemental volumeD di�erential operator of equilibrium

P.C. Pandey, P. Bakshi / Computers and Structures 71 (1999) 525±534526

bility to represent traction-free edge conditions and

singular stress ®elds [22].

The advantages of the hybrid ®nite element method

over the displacement-based ®nite element methods

prompted the possible merging of these element tech-

nologies. It was expected that this merger would lead

to more accurate displacements and stress derivatives

at a lesser cost. For problems where the accuracy of

stress sensitivity is crucial, it would be desirable to

adopt a ®nite element which can yield superior stress

to the displacement-based elements, and hence superior

sensitivities. Thus, the argument goes in favour of the

hybrid elements. Sensitivity computation using hybrid

and mixed elements have not been addressed in the lit-

erature so far, mainly due to lower popularity of such

elements in existing commercial packages of ®nite el-

ement programs. This lack of acceptance may largely

be attributed to the ignorance of the industry about

the bene®ts of the hybrid ®nite elements in selective

applications. Also, most of these elements are only

recently being extensively researched and made more

widely available for practical use. Unmistakably, these

elements hold out great promise in ®nite element

analysis, and it is envisaged that the present lacuna

between research and practice will be reduced in time

to come. As pointed out by Zienkiewicz [24]: ``Much

further research will elucidate the advantages of some

of the forms discovered and we expect the use of such

developments to increase in the future''.

The present work focuses on the sensitivity analysis

using hybrid stress method based on the Hellinger±

Reissner principle. Analytical sensitivity derivatives

have been explicitly derived here for the four and eight

node elements, based on this functional. The im-

plementation has been done partially using symbolic

algebra. The results are compared to the sensitivity de-

rivatives obtained by using conventional displacement-

based ®nite element method. The application of the

technique is demonstrated in the case of static analysis

Fig. 1. Classi®cations of sensitivity analysis.

P.C. Pandey, P. Bakshi / Computers and Structures 71 (1999) 525±534 527

of two-dimensional problems of elasticity with respect

to sizing variables.

2. Response sensitivity

2.1. General de®nition and classi®cation

In analysis problems, there is a need to study thee�ect of the variation of design variables or variationof some parameters that characterizes (globally or

locally) the response, such as stress or displacementstates. Corresponding variations in response caused bydesign perturbation constitute the study matter of de-

sign sensitivity analysis. The sensitivity analysis maythus be de®ned as the methods for obtaining the ratesof change of response quantities due to variations in

design variables. In a structural sense, response quan-tities are simply the derived quantities like displace-ment, stress, eigenvalue/eigenvector, buckling load,natural frequency and interlaminar stresses, and failure

indices for composites. On the other hand, the designvariables may be de®ned as those components of astructure that de®ne the topology and geometry of the

system being analysed.From the point of view of response quantities, sensi-

tivity derivatives can be classi®ed into several classes.

If the e�ect of change of displacements or strains on abehavioural function are to be examined with respectto any design variable, the corresponding terms are dis-

placement sensitivity or strain sensitivity, respectively.

Similarly, considering stresses, one can obtain the

stress sensitivity. On similar lines, the eigenvalue and

eigenvector sensitivities, buckling load sensitivity, fre-

quency response sensitivity and so on, have been

de®ned in the literature [19,20]. Another classi®cation

of the sensitivity methods can be made with respect to

design variables which are being varied in order to

study the corresponding variations in response quan-

tities. Thus, there are the sizing sensitivity and the

shape sensitivity according as the design variables are,

respectively, sizing variables and shape variables.

Finally, sensitivity gradients may be categorized

depending upon the methods used to compute the

same.

From the formulation point of view, sensitivity

analysis of structural response has followed two paral-

lel paths. The ®rst path involves the application of sen-

sitivity techniques to an already discretized system and

is known as the discrete approach. The other path is

concerned with the di�erentiation of the continuum

equations which were then discretized. This was termed

the variational method. The various types of sensitivity

derivatives and their analysis techniques are brie¯y

summarized in Fig. 1. They have been reviewed

[20,21,26] in detail elsewhere.

The displacement sensitivities for discrete static sys-

tems can be obtained from the static equation of equi-

librium, as follows:

Table 1

The hybrid elements used in the paperÐelement 1

Aspects Description

Name PS4

Variational principle PHR ��V

hÿ 1

2sssTSsss� sssTÿDuq

�ÿ �DTsss�Tuli

dV

Modi®ed Hellinger±Reissner principle [33]

Independent variables ss=Pbu=uq+uluq=Fq

ul=MlInterpolation functions

P �24 1 0 0 x 0 0 Z 0 00 1 0 0 x 0 0 Z 00 0 1 0 0 x 0 0 Z

35Fi � 1

4 �1� xix��1� ZiZ�, i � 1,NN

M �24ÿ1ÿ x2

� ÿ1ÿ Z2

�0 0

0 0ÿ1ÿ x2

� ÿ1ÿ Z2

�35


�K�b��fqg � �F�b� �1�

where [K ] is the sti�ness matrix of size n�n, {q } is the

displacement vector of size n� 1, {F } is the externalload vector of size n�1, {b } is the vector of designvariables of size m� 1.

Di�erentiating Eq. (1), we get

�K ��@q

@bi

��@F

@bi

�ÿ�@ �K �@bi

�fqg �2�

The right-hand side is known as the ``Pseudo Load''

vector, denoted P�. Once P� is calculated, the displace-ment sensitivity, @q/@bi can be easily obtained by con-ventional ®nite element analysis. Having obtained the

displacement derivatives, the stress and strain sensi-tivities can be obtained easily.In hybrid ®nite element formulation, besides interpo-

lating the stresses in terms of unknown coe�cients, b,the element displacements, u, are separated into twoparts: the compatible part uq which is expressed in

terms of the nodal displacements, q, and the additional

part ul, which is expressed in terms of the internal dis-

placement parameters, l. Here, the equilibrium of the

stresses is coerced within each element only in an aver-

age sense. Thus, a purely stress-based approach was

formally derived. Current variational bases for ®nite el-

ement formulations are embedded in the most general

Hu±Washizu functional in which displacements, stres-

ses and strains are all assumed as independent quan-

tities. Based on a variational functional introduced by

Chien [27], proposing the separation of the stress

terms, also, Chen and Cheung [25,28±31] presented a

series of element formulations, with all three quan-

titiesÐdisplacements, stresses and strains indepen-

dently interpolated.

The various elements that have been developed in

the literature and used in this paper have been listed in

Tables 1±4, along with the variational principles upon

which they are formulated, the independent ®elds

assumed and the respective references.

Table 2

The hybrid elements used in this paperÐelement 2

Aspects Description

Name RGH4

Variational principle PG ��V � 12EEETAEEEÿ sssT�EEEÿDuq� � sssT

h �Dul�� dVFunctional proposed by Pian and Tong [34,28]

Independent variables ss=Pbssh=PhbEE=Nau=uq+uluq=Fq


P �24 1 0 0 x 0 0 Z 0 00 1 0 0 x 0 0 Z 00 0 1 0 0 x 0 0 Z

35

Ph �24 0 0 0 x 0 0 Z 0 00 0 0 0 x 0 0 Z 00 0 0 0 0 x 0 0 Z

35

N �24 1 0 0 x=jJj 0 0 Z=jJj 0 00 1 0 0 x=jJj 0 0 Z=jJj 00 0 1 0 0 x=jJj 0 0 Z=jJjB

35Fi � 1

4 �1� xix��1� ZiZ�, i � 1,NN

M �24ÿ1ÿ x2

� ÿ1ÿ Z2

�0 0

0 0ÿ1ÿ x2

� ÿ1ÿ Z2

�35


3. Analytical sensitivity using hybrid elements

Considering the modi®ed functional proposed by Cheung and Chen [25], as

PG �Xe

�V e

�1

2EEETAEEEÿ sssTÿEEEÿDuq

�� sssTh �Dul�

�dV �3�

where u=uq+ ul, uq=the compatible part of the displacement vector u, ul=the incompatible part of the displace-

ment vector, EE=the strain vector, ss=the stress vector, ssh=the stress vector with the constant terms excluded,and A=elasticity matrix.Admissible variations of Eq. (3) and static condensation of the parameters corresponding to the internal displace-

ments at the element level lead to the formulation of the sti�ness matrix of a new hybrid element as follows:

K � G TqW

ÿTHW ÿ1hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq �4�

In order to calculate the displacement sensitivity, from Eq. (2), one needs to calculate the pseudo load vector, whichcomprises the derivatives of the load vector as well as the sti�ness matrix, with respect to some design variables.Thus, di�erentiation of the hybrid sti�ness matrix, K, with respect to any design parameter, b, would yield

@K

@b� @G T

q

@b�W ÿTHW ÿ1 �

hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq � G T

q

@


�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq � G T

q�W ÿTHW ÿ1 �

� @

@b

hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq � G T

q�W ÿTHW ÿ1 �

�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

i@

@b

�5�

Table 3


Aspects Description

Name RGH8

Variational principle PG ��V � 12EEETAEEEÿ sssT�EEEÿDuq� � sssT

h �Dul�� dVFunctional proposed by Pian and Tong [34,28]

Independent variables ss=Pbssh=PhbEE=Nau=uq+uluq=Fq

ul=MlInterpolation functions P � �I3 xI3 ZI3 xZI3 �1ÿ 3x2�I3 �1ÿ 3Z2�I3�

Ph � �0I3 xI3 ZI3 xZI3 �1ÿ 3x2�I3 �1ÿ 3Z2�I3�N � �I3 xI3=jJj ZI3=jJj xZI3=jJj �1ÿ 3x2�I3=jJj �1ÿ 3Z2�I3=jJj�Fi � 1

4 �1� xix��1� ZiZ�, i � 1,NN

M ��x3 Z3 0 00 0 x3 Z3

�

I3 �24 1 0 00 1 00 0 1

35


From the above expression, it is clear that, in order to obtain the derivative of the sti�ness matrix, it would berequired to evaluate the derivatives of the matrices H, Wÿ1, Gq and Gl, with respect to the design variable b.The load vector is, generally, not a function of either shape or sizing variables, except in the case of directed

loads. Hence, assuming the derivative of the load vector to be zero, the displacement derivative with respect to thedesign variable, b, can be obtained by substituting Eq. (5) into Eq. (2).Having obtained the displacement derivatives, the derivatives of the strain and the stress parameters, a and b can

be easily obtained. From the formulation of the element [25], the dependence of the stress and the strain parameters

on the displacement parameters can be given as:

b �W ÿTHW ÿ1hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq �6�

and

a �W ÿ1hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq �7�

Di�erentiating Eqs. (6) and (7), with respect to b,

Table 4


Aspects Description

Name HS4

Variational principle P � �V � 12EEETAEEEÿ sssT�EEEÿDuq�� dV

Functional proposed by Chen and Cheung [31]

Independent variables ss=PQbssh=PhQbEE=NQau=uq+uluq=Fq


P �24 1 0 0 x 0 0 Z 0 00 1 0 0 x 0 0 Z 00 0 1 0 0 x 0 0 Z

35

Ph �24 0 0 0 x 0 0 Z 0 00 0 0 0 x 0 0 Z 00 0 0 0 0 x 0 0 Z

35

N �24 1 0 0 x=jJj 0 0 Z=jJj 0 00 1 0 0 x=jJj 0 0 Z=jJj 00 0 1 0 0 x=jJj 0 0 Z=jJjB

35Fi � 1

4 �1� xix��1� ZiZ�, i � 1,NN

M �24ÿ1ÿ x2

� ÿ1ÿ Z2

�0 0

0 0ÿ1ÿ x2

� ÿ1ÿ Z2

�35

ÅQ � Aÿ1

Q

Q: �)� �

V

PThDM dV

�b� � 0; b � Qb�


@b@b� @


hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq� �W ÿTHW ÿ1 �

� @

@b

hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq� �W ÿTHW ÿ1 �

�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

i@Gq

@bq� �W ÿTHW ÿ1 �

�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq@q

@b

�8�

@a@b� @W ÿ1

@b

hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq�W ÿ1

� @

@b

hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGqq�W ÿ1

�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

i@Gq

@bq�W ÿ1

�hIÿ Gl

ÿG T

lWÿTHW ÿ1Gl

�ÿ1G T

lWÿTHW ÿ1

iGq@q

@b

�9�

With the derivatives of the stress and strain parameters, from Eqs. (8) and (9), the stress and strain sensitivities can

be obtained using the relations from Tables 1±4. Thus,

@s@b� @P

@bb� P

@b@b

�10�

and

@ E@b� @N

@ba� N

@a@b

�11�

4. Implementation and veri®cation

The above equations for sensitivities can be greatly

simpli®ed, if the design parameters of interest are theintrinsic variables, such as the cross-sectional dimen-sions or material properties of the element. When the

design variables are de®ned by the size of the struc-tural components or their mechanical properties, di�er-entiation of the above matrices is straightforward.

Here, the expressions from Eqs. (4)±(11) have beenworked out analytically, and ®nal matrix operationshave been executed by using the symbolic software

package, Mathematica [32].Structural sensitivity problems were addressed, for

numerical veri®cation, using the conventional displace-ment-based elements as well as the hybrid elements, as

listed below:

1. PS4 : this is the four-node hybrid isoparametric el-

ement developed by Pian and Sumihara [33].2. H4 : this is the four-node hybrid isoparametric element

developed by Chen and Cheung [31] based on a con-

stant parameter, a. Here, a has been assumed as 0.4.3. RGH4 : this is the re®ned plane four-node hybrid

isoparametric element developed by Cheung andChen [25].

4. RGH8 : this is the re®ned plane eight-node hybridisoparametric element that was proposed by Cheungand Chen [25].

5. Q4 : the standard four-node isoparametric element.6. Q8 : these are the standard isoparametric eight-node

elements.

A comparative study has been carried out between thetwo formulations, hybrid as well as displacement-based, with respect to accuracy, cost e�ectiveness and

robustness, using identical mesh.

4.1. Illustration

1. A straight cantilever beam (Fig. 2) of nearly incom-pressible material and under plane strain conditionis considered, which is acted upon by end shear [23].

The sensitivity of displacement and stress are com-puted with respect to thickness. Normalized deriva-tives of the tip displacements at the point A of thebeam (Fig. 2a) and normalized stress derivatives at

the point B (Fig. 2b) are compared for each el-ement, with di�erent values of Poisson's ratio. Sincestresses in displacement-based elements are most

accurate at the gauss points, stress derivatives havebeen computed at those points for the hybrid el-ements also. A constant mesh is adopted here for all

the elements for e�ective comparison. The normali-zations are achieved with respect to the derivatives


computed from the elasticity theory. The accuracy

in results re¯ected from the graphs is in favour of

the hybrid elements. Although the eight node quad-

rilateral element is of comparable accuracy to the

four-node hybrid elements, its performance is far in-

ferior to the eight node hybrid element. Similarly, the

four-node quadrilateral element exhibits high error

compared to their four-node hybrid counterparts. The

error exhibited by the displacement-based elements

may be attributed to the fact that for higher values of

Poisson's ratio, they tent to ``lock'' [25]. However,

with the re®nement of the mesh, although the accuracy

of all the elements is enhanced, the relative perform-

ance of the same was found to remain unchanged.

Hence, the accuracy of the lower order hybrid el-

ements for displacement as well as stress derivatives

can be observed to be comparable to the accuracy of

the higher order displacement-based elements, at any

value of Poisson's ratio.

2. A curved cantilever loaded with unit force at the free

end (Fig. 3), with inner radius=80 units, outer

radius=100 units, arc=908, thickness=20 units,

Young's modulus, E=1500 units, Poisson's ratio,

m=0.25, mesh=5�1 is used as a test for accuracy

[25]. The accuracy of the di�erent models would re¯ect

the performance of the elements in bending. The nor-

malized displacement derivatives at point A and the

normalized stress derivatives at point B have been

taken for comparison for di�erent element types

(Table 5). The sensitivity derivatives have been com-

puted with respect to the thickness of the beam. From

the classical theory of elasticity for curved beams, the

expressions for displacements and stresses have been

taken and di�erentiated algebraically to obtain the de-

rivatives. With respect to these derivatives, the nor-

malization of the derivatives obtained from the

sensitivity analysis has been done. It can be concluded

from the tabulated values that both the four node and

the eight node hybrid elements have better accuracy in

sensitivity values in bending.

From the above illustrations, it can be concluded that

the hybrid element method could be a better option

for sensitivity analysis against the traditional displace-

ment-based ®nite element methods, especially for stress

and strain derivatives. It has been demonstrated here

that the hybrid method is highly suitable for deriving

stress and displacement sensitivities with respect to de-

sign variables for static structural systems. Since the

sti�ness matrix can be derived explicitly, without intro-

ducing any numerical integration, the formulation is

free from errors, which is re¯ected here in the results.

Again, numerical integration not being necessary at

any step in the formulation made here, the accuracy is

high and the computational cost is reduced consider-

ably. Due to these two advantages of improved accu-

racy and reduced cost, the alternate method of analysis

using hybrid elements would hold better promise in

sensitivity analysis.

Fig. 2. Cantilever beam with end shear (plane strain con-

dition). (a) Displacement derivatives. (b) Stress derivatives.

Fig. 3. Sensitivity analysis of curved cantilever beam (discre-

tized structure).

Table 5

The normalized displacement and stress derivatives for the

curved cantilever beam

Model

Normalized results PS4 H4 RGH4 RGH8 Q4 Q8

dDA 0.951 0.955 0.955 0.993 0.516 0.963

dsB 0.982 0.848 0.982 1.110 0.680 1.152


5. Conclusions

Response sensitivity using the hybrid elementmethod has been presented. Sensitivity expressions arederived explicitly and implemented, using symbolic

algebra partially. The results of the numerical exper-imentation are encouraging and demonstrate the su-perior performance of hybrid elements in sensitivity

computations. Here, attention has been focused mainlyon sizing sensitivity. Shape sensitivity using hybrid el-ements requires further investigation. Further investi-

gation is also warranted to compute sensitivities ofother derived quantities, like eigenvalue/eigenvectorsfor both shape and sizing variables.

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Analytical response sensitivity computation using hybrid finite elements