Convergence Criteria Download

8/11/2019 Convergence Criteria Download

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CRITERIA FOR SELECTION OF FEM MODELS.

Prof. P. C.Vasani ,Applied Mechanics Department, L. D. College of Engineering ,Ahmedabad- 380015

Ph.(079) 7486320 [R]

E-mail:[email protected]

1. Criteria for Convergence.

The finite element method provides a numerical solution to a

complex problem. It may, therefore, be expected that the solution must

converge to the exact solution under certain circumstances. It can be shown

that the displacement formulation of the method leads to an upper bound tothe actual stiffness of the structure. Hence the sequence of successively finer

meshes is expected to converge to the exact solution if assumed element

displacement fields satisfy certain criteria. These are,

1. The displacement field within an element must be continuous. In

other words, it does not yield a discontinuous value of function

but rather a smooth variation of function, and the variation do notinvolve openings, overlap, or jumps, which are inherently

continuous. This condition can easily be satisfied by choosing

polynomials for the displacement model. The function w is

indeed continuous if for example it is expressed as,

W = C1 + C2 x + C3 x2+C4 x

3+.

2. The approximate function should provide inter-element

compatibility up to a degree required by the problem.

For instance, for the column problem involving axialdeformations, it is necessary to ensure inter-element compatibility

at least for displacements of adjacent nodes. That is theapproximate function should be such that the nodal displacements

between adjacent nodes are the same.

Inter-element compatibility must be enforced for displacements

and their derivatives up to the order n-1, where n is the highest

order derivative in the energy function.

The highest order of derivative for a column element, for

example, is 1. Hence the inter-element compatibility should

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include order of v at least upto 0 (zero), that is, displacement v, In

general, the formulation should provide inter-element

compatibility upto order n-1, where n is the highest orderderivative in the energy function. Approximate functions thatsatisfy the conditions of compatibility can be called conformable

(compatible or conforming elements).

In contrast to the plane deformations, for realistic approximationof the physical conditions in the case of bending, it is necessary to

satisfy inter-element compatibility with respect to both thedisplacements and slopes, i.e. first derivative (gradient) of

displacement.

As a consequence, it becomes necessary to use higher order

approximation for the displacement in the case of bending, since itis necessary to provide for inter-element compatibility for slopes

also, we can add slope at the node as an additional unknown. This

leads to two primary unknowns, displacement w and slope

= dw/dx, at each node, hence, for an element there are a total offour degrees of freedom: w1, 1 at node 1 and w2 , 2 at node 2.The approximate function is conformable since it provides for

inter-element compatibility upto n-1 = 2-1 = 1 derivative of w,that is, for both w and its first derivative, where n=2 is the highest

order of derivative in the potential energy function.

The displacement function for a beam element thus can be written

as,

W = C1 + C2 x +C3 x2+C4 x

3

3. The displacement model must include the rigid bodydisplacements of the element.

A rigid body displacement is the most elementary deformation

an element may undergo. Hence when the nodes are given such

displacement corresponding to a rigid motion, the element

should not experience any strain and hence leads to zero nodalforces. The constant terms in the polynomials used for the

displacement models would usually ensure this condition. For

instance the constant term C1 provides for a rigid body



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1

X | Y

X2 XY Y

2

X3 X

2Y | XY

2 Y

3

X4

X3Y X

2Y

2 XY

3 Y

4

X5 X

4Y X

3Y | X

2Y

3 XY

4 Y

5

Symmetry

Axis

For example, if we would like to construct a polynomial with four terms,

invariance is achieved by selecting

U = C1 + C2 X + C3 Y + C4 XY

Thus, geometric invariance can be achieved by selecting the correspondingorder of terms on either side of the axis of symmetry.

2. FINITE ELEMENT ANALYSIS OF A CONTINUUM.

Conventional engineering structures can be visualized as an

assemblage of structural elements interconnected at a discrete number of

nodal points. If the force-displacement relationships for the individualelements are known it is possible, by using various well-known techniques

of structural analysis, to derive the properties and study the behavior of the

assembled structure.

In elastic continuum the true number of interconnection points is

infinite, and here lies the biggest difficulty of its numerical solution. In many

phases of engineering the solution of stress and strain distributions in elasticcontinua is required. Special cases of such problems may range from two-



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dimensional plane stress or strain distributions, axisymmetrical solids, plates

bending, and shell, to fully three-dimensional solids. In all cases the number

of interconnections between any !finite element" isolated by some imaginaryboundaries and the neighboring elements is infinite. It is therefore difficultto see at first glance how such problems may be discretized in the same

manner as is done for simpler structures.

The difficulty can be overcome (and approximation made) in the

following manner.

(a) The continuum is separated by imaginary lines or surfaces into a numberof !finite elements".

(b) The elements are assumed to be interconnected at a discrete number of

nodal points situated on their boundaries. The displacements of thesenodal points will be the basic unknown parameters of the problem, just as

in the simple structural analysis.(c) A function (or functions) is chosen to define uniquely the state of

displacement within each !finite element" in terms of its nodal

displacements.

(d) The displacement functions now define uniquely the state strain withinan element in terms of the nodal displacements. These strains, together

with any initial strains and the elastic properties of the material will

define the state of stress throughout the element and, hence, also on itsboundaries.

(e) A system of forces concentrated at the nodes and equilibrating the

boundary stresses and any distributed loads is determined, resulting instiffness relationship same as for simpler structure

Once this stage has been reached the solution procedure can follow

the standard structural routine.

So far, the process described is justified only intuitively, but what in

fact has been suggested is equivalent to the minimization of the totalpotential energy of the system in terms of a prescribed displacement field. If

this displacement field is defined in a suitable way, then convergence to thecorrect result must occur. The process is then equivalent to the well-known

Ritz procedure. The derivation of characteristics of !finite element" of acontinuum in detail is beyond the scope of this article.



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3. FINITE ELEMENT ANALYSIS OF AXISYMMETRIC SOLIDS.

The problem stress distribution in bodies of revolution (axi-symmetric solids) under axisymmetric loading is of considerable practical

interest. The mathematical problems presented are very similar to those of

plane stress and plane strain as, once again, the situation is two-dimensional.By symmetry, the two components of displacements in any plane

sectioning the body along its axis of symmetry define completely the state ofstrain and, therefore, the state of stress. Such a cross-section is shown in

Figure above. The volume of material associated with an !element" is equal

to of a body of revolution indicated on fig, and all integration have to be

referred to this.

In plane stress or strain problems it is being shown that internal

work is associated with three strain components in the co-ordinate plane, thestress component normal to this plane not being involved due to zero values

of either the stress or the strain.

In the axisymmetrical situation any radial displacement

automatically induces a strain in the circumferential direction, and as the

stresses in this direction are certainly non-zero, this fourth component ofstrain and of the associated stress has to be considered. Here lies the

essential difference in the treatment of the axisymmetric situation.

The simplest examples are a circular cylinder loaded by a uniform

internal or external pressure, a circular footing resting on a soil mass,pressure vessels, rotating wheels, flywheels etc. (See Fig. below).The

deformation being symmetrical with respect to the y-axis, the stress

components are independent of the angle and all derivatives with respect

Axisymmetric element

r

12

3

z



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AXISYMMETRIC PROBLEMS.

To vanish. The components w, x, y, x and y are zero. The strain-displacement relations are given by

x= u ; = u ; y= v ; xy= u + vx x y y x

Thus, the constitutive relation is,

y,v

x,u

X

Y

xyx

= E_____

(1+)(1-2)

(1-) 0 x(1-) 0 y

Symmetric (1-) 0 (1-2)/2 xy



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The stiffness matrix :

The stiffness matrix of the any axisymmetric element can now be computedaccording to the general relationship. Remembering that the volume integral

has to be taken over the whole ring of material we have,

[K

e

] = 2 [B]T

[D] [B] r drdz

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