Finite Element Method - Mixed formulatinon and constraints - in complete ( hybrid ) field methods, buondary - Trefftz methods_13

Finite Element Method - Mixed formulatinon and constraints - in complete ( hybrid ) field methods, buondary - Trefftz methods_13 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

13

Mixed formulation and constraints

methods, boundary/Trefftz methods

In the previous two chapters we have assumed in the mixed approximation that all the variables were defined and approximated in the same manner throughout the domain

of the analysis This process can, however, be conveniently abandoned on occasion with different formulations adopted in different subdomains and with some variables being only approximated on surfaces joining such subdomains In this part we shall

discuss such incomplete or partial jield approximations which include various so-

called hybrid formulations

In all the examples given here we shall consider elastic solid body approximations only, but extension to the heat transfer or other field problems, etc., can be readily made as a simple exercise following the procedures outlined

13.2 Interface traction link of two (or more) irreducible form subdomains

One of the most obvious and frequently encountered examples of an ‘incomplete field’ approximation is the subdivision of a problem into two (or more) subdomains in each

of which an irreducible (displacement) formulation is used Independently approxi- mated Lagrange multipliers (tractions) are used on the interface to join the subdomains,

as in Fig 13.1(a)

In this problem we formulate the approximation in domain R1 in terms of displace- ments u1 and the interface tractions t’ = 1 With the weak form using the standard

virtual work expression [see Eqs (1 1.22)-( 1 1.24)] we have

in which as usual we assume that the satisfaction of the prescribed displacement on

rul is implied by the approximation for ul Similarly in domain R2 we can write,

now putting the interface traction as t2 = -1 to ensure equilibrium between the

Interface traction link of two (or more) irreducible form subdomains 347

Fig 13.1 Linking of two (or more) domains by traction variables defined only on the interfaces (a) Variables

in each domain are displacements u (internal irreducible form) (b) Variables in each domain are displacements

and stresses c-u (mixed form)

two domains,

jaz S(Su2)TD2Su2dR + h, S u 2 T I d r - In2 Su2Tbdf2 - fr; Su2*idr = 0 (13.2)

The two subdomain equations are completed by a weak statement of displacement

continuity on the interface between the two domains, i.e.,

jr, SIT(u2 - ul) d r = 0 (13.3)

Discretization of displacements in each domain and of the tractions I on the

interface yields the final system of equations Thus putting the independent

approximations as

u 1 = N , I ~ 1 (13.4)

we have

[ t 2 : ;I{;}={;} (13.7a)

348 Mixed formulation and constraints

where

K2 = jnz B2TD2B2 dR

(13.7b)

We note that in the derivation of the above matrices the shape function NA and

hence h itself are only specified along the interface line - hence complying with our definition of partial field approximation

The formulation just outlined can obviously be extended to many subdomains and

in many cases of practical analysis is useful in ensuring a better matrix conditioning and allowing the solution to be obtained with reduced computational effort.' The variables u' and u2, etc., appear as internal variables within each subdomain

(or superelement) and can be eliminated locally providing the matrices K' and K2

are non-singular Such non-singularity presupposes, however, that each of the sub- domains has enough prescribed displacements to eliminate rigid body modes If this is not the case partial elimination is always possible, retaining the rigid body modes until the complete solution is achieved

The process described here is very similar to that introduced by Kron2 at a very early

date and, more recently, used by Farhat et d 3 in the FETI method which uses the process

on many individual element partitions as a means of iteratively solving large problems The formulation just used can, of course, be applied to a single field displacement for- mulation in which we are required to specify the displacement on the boundaries in a weak sense (rather than imposing these directly on the displacement shape functions) This problem can be approached directly or can be derived simply via the first equation of (13.7a) in which we put u2 = U , the specified displacement on I?,

Now the equation system is simply

where

f - - - jr, N I u d r

(13.8)

(13.9)

This formulation is often convenient for imposing a prescribed displacement on a displacement element field when the boundary values cannot fit the shape function field

Interface traction link of two or more mixed form subdomains 349

We have approached the above formulation directly via weak forms or weighted

residuals Of course, a variational principle could be given here simply as the minimi-

zation of total potential energy (see Chapter 2) subject to a Lagrange multiplier 1

imposing subdomain continuity The stationarity of

(Su)TD(Su)dR- UTbdR- UTidI'+ LT(u' - u 2 ) d r (13.10)

SQ h., h., would result in the equation set (13.1)-( 13.3) The formulation is, of course, subject to

limitations imposed by the stability and consistency conditions of the mixed patch test

for selection of the appropriate number of 1 variables

subdomains

The problem discussed in the previous section could of course be tackled by assuming

a mixed type of two-field approximation (cr/u) in each subdomain, as illustrated in

Fig 13.1(b)

Now in each subdomain variables u and cr will appear, but the linking will be

carried out again with the interface traction 1

We now have, using the formulation of Sec 1 1.4.2 for domain R1 [see Eqs (1 1.29)

and (1 1.22)],

- S u l ] d R = O (13.11a)

and for domain R2 similarly

- S u 2 ] d R = 0 (13.12a)

S(Su2)To2dR + Su2T1dr - Su2TbdR - 1 Su21idr = 0 (13.12b)

With interface tractions in equilibrium the restoration of continuity demands that

On discretization we now have

u I = N,iU'

c 1 = N , I ~ '

u 2 = N,2U2

G 2 = N,zti2

1 = NxL

350 Mixed formulation and constraints

with A, C, f , , and f2 defined similarly to Eq (11.32) with appropriate subdomain

subscripts and Q' and Q2 given as in (13.7b)

All the remarks made in the previous section apply here once again - though use of the above form does not appear frequently

1 3.4 Interface displacement 'frame'

In the preceding examples we have used traction as the interface variable linking two

or more subdomains Due to lack of rigid body constraints the elimination of local subdomain displacements has generally been impossible For this and other reasons

it is convenient to accomplish the linking of subdomains via a displacement field

defined only on the interface [Fig 13.2(a)] and to eliminate all the interior variables

so that this linking can be accomplished via a standard stiffness matrix procedure using only the interface variables

The displacement frame can be made to surround the subdomain completely and if

all internal variables are eliminated will yield a stiffness matrix of a new 'element'

Fig 13.2 Interface displacement field specified on a 'frame' linking subdomains: (a) two-domain link; (b) a

'superelement' (hybrid) which can be linked to many other similar elements

Interface displacement 'frame' 351

which can be used directly in coupling with any other element with similar displace-

ment assumptions on the interface, irrespective of the procedure used for deriving

such an element [Fig 13.2(b)]

In all the examples of this section we shall approximate the frame displacements as

v = N , i on (13.15) and consider the 'nodal forces' contributed by a single subdomain R' to the 'nodes' on

this frame Using virtual work (or weak) statements we have with discretization

( 13.16)

where t are the tractions the interior exerts on the imaginary frame and q' are the

nodal forces developed The balance of the nodal forces contributed by each sub-

domain now provides the weak condition for traction continuity

As finally the tractions t can be expressed in terms of the frame parameters V only,

we shall arrive at

q' = K ' t + fh (13.17) where K' is the stiffness matrix of the subdomain R' and f i its internally contributed

'forces'

From this point onwards the standard assembly procedures are valid and the sub-

domain can be treated as a standard element which can be assembled with others by

ensuring that

i

where the sum includes all subdomains (elements!) We thus have only to consider a

single subdomain in what follows

We shall assume as in Sec 13.3 that in each subdomain, now labelled e for generality,

the stresses be and displacements ue are independently approximated The equations

(1 3.1 1) are rewritten adding to the first the weak statement of displacement continuity

We now have in place of (1 3.1 1 a) and (1 3.13) (dropping superscripts)

SoT(D-'c - Su) dR -

- 1, SuT(STo + b) dR +

StT(u - v) d r = 0 (13.19) Equation (1 3.1 1 b) will be rewritten as the weighted statement of the equilibrium

relation, i.e.,

SuT(t - i) d r = 0

or, after integration by parts

352 Mixed formulation and constraints

In the above, t are the tractions corresponding to the stress field IJ [see Eq (1 1.30)]:

t = G o (13.21)

In what follows rip, i.e, the boundary with prescribed tractions, will generally be taken

as zero

On approximating Eqs (13.19), (13.20) and (13.16) with

u = N , u o = N $ and v=N,V

we can write, using Galerkin weighting and limiting the variables to the 'element' e,

A' C' Q'

where

(1 3.22a)

(13.22b)

Elimination of 6' and u from the above yields the stiffness matrix of the element and the internally contributed force [see Eq (13.17)]

Once again we can note that the simple stability criteria discussed in Chapter 11 will help in choosing the number of IJ, u, and v parameters As the final stiffness matrix of

an element should be singular for three rigid body displacements we must have [by

Eq (ll.lS)]

nu 3 nu + n, - 3 (13.23)

in two-dimensional applications

Various alternative variational forms of the above formulation exist A particularly

useful one is developed by Pian et al.4>5 In this the full mixed representation can be written completely in terms of a single variational principle (for zero body forces) and no boundary of type r r present:

II, = - j&crD-'crdR - jo(STa)TuI dR + j aTSvdR (13.24)

In the above it is assumed that the compatible field of v is speciJied throughout the

element domain and not only on its interfaces and uI stands for an incompatible

field defined only inside the element d0main.t

R

t In this form, of course, the element could well fit into Chapter 1 1 and the subdivision of hybrid and mixed forms is not unique here

Interface displacement 'frame' 353

We note that in the present definition

To show the validity of this variational principle, which is convenient as no inter-

face integrals need to be evaluated, we shall derive the weak statement corresponding

to Eqs (13.19) and (13.20) using the condition (13.25)

We can now write in place of (1 3.19) (noting that for interelement compatibility we

have to ensure that uI = 0 on the interfaces)

After use of Green's theorem the above becomes simply

6 a T ( D - ' ~ - SV) dR + ( S T S ~ ) T ~ I d r = 0

SI,

(13.26)

(13.27)

In place of (13.20) we write (in the absence of body forces b and boundary r,)

(1 3.28)

and again after use of Green's theorem

/ 6uTSTodR - 1 S(Sv)TodR = 0 (if 6v = 0 on r,) (13.29)

These equations are precisely the variations of the functional (13.24)

Of course, the procedure developed in this section can be applied to other mixed or

irreducible representations with 'frame' links Tong and Pian6.' developed several

alternative element forms by using this procedure

In this form we shall assume a priori that the stress field expansion is such that

and that the equilibrium equations are identically satisfied Thus

STo = 0; SToo = b in R and G a = 0; Goo = t on rf

In the absence of

Chapter 1 1, Sec 1 1.7)

Eq (13.20) is identically satisfied and we write (13.19) as (see

6aT(D-'aT - Su) dR +

(13.31)

On discretization, noting that the field u does not enter the problem

o = N g 6 v = N , i

354 Mixed formulation and constraints

we have, on including Eq (1 3.16)

where

Q' = / (GN,)TN,dr

r l e

and

F

f' - N,Goodr

2 - J,.

Here elimination of 5 is simple and we can write directly

K'V = q' - fz - QeT(A')-'f' and K' = QeT(A')-'Qe (13.33)

In Sec 11.7 we have discussed the possible equilibration fields and have indicated the difficulties in choosing such fields for a finite element, subdivided, field In the present case, on the other hand, the situation is quite simple as the parameters describing the equilibrating stresses inside the element can be chosen arbitrarily in

a polynomial expression

For instance, if we use a simple polynomial expression in two dimensions:

0, = "0 + a1x + "2.Y

a y = Po + P l X + P 2 Y (13.34)

Txy = Yo + Y l X + Y2Y

we note that to satisfy the equilibrium we require

(13.35)

and this simply means

7 2 = -"1

71 = - 0 2

Thus a linear expansion in terms of 9 - 2 = 7 independent parameters is easily achieved Similar expansions can of course be used with higher order terms

1 nu 2 n, - 3 is needed to preserve stability

2 By the principle of limitation, the accuracy of this approximation cannot be better than that achieved by a simple displacement formulation with compatible expan- sion of v throughout the element, providing similar polynomial expressions arise in stress component variations

It is interesting to observe that:

linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 355

However, in practice two advantages of such elements, known as hybrid-stress

elements, are obtained In the first place it is not necessary to construct compatible

displacement fields throughout the element (a point useful in their application to,

say, a plate bending problem) In the second for distorted (isoparametric) elements

it is easy to use stress fields varying with the global coordinates and thus achieve

higher order accuracy

The first use of such elements was made by Pian* and many successful variants are

in use t ~ d a y ~ - ~ ~

13.5 Linking of boundary (or Trefftz)-type solution by the

We have already referred to boundary (Trefftz)-type solutions23 earlier (Chapter 3)

Here the chosen displacement/stress fields are such that a priori the homogeneous

equations of equilibrium and constitutive relation are satisfied indentically in the

domain under consideration (and indeed on occasion some prescribed boundary

traction or displacement conditions)

Thus in Eqs (13.19) and (13.20) the subdomain (element e ) Re integral terms

disappear and, as the internal St and Su variations are linked, we combine all into a

single statement (in the absence of body force terms) as

h T ( t - i ) d r = 0 (13.36)

This coupled with the boundary statement (1 3.16) provides the means of devising

stiffness matrix statements of such subdomains

For instance, if we express the approximate fields as

u = N i (13.37) implying

o = D(SN)a and t = Go = GD(SN)a

we can write in place of (13.22)

(13.38)

where

Q' = 1 [GD(SN)ITNu d r (13.39)

r l e

In Eqs (13.38) and (13.39) we have omitted the domain integral of the particular

solution oo corresponding to the body forces b but have allowed a portion of the