4.15 Adaptive mesh refinement

Back

The plate in Fig. 4.52 was analyzed with bilinear elements, with an attempt

to improve the results by adaptively refining the mesh in three steps. At the

start the mesh consisted of 160 elements, and the final mesh consisted of 1231

elements. At that stage in the analysis, the energy error was about the same

in most of the elements.

The thickness of the plate was t = 0.25 m, and the material properties

were E = 3.4 · 104 MN/m2, ν = 0.167.

The energy norm (squared) of the FE solution is the strain energy product

between the stresses and the strains,

||uh||2

E =

_

Ω

σh • εh dΩ , (4.117)

and the energy norm squared of the error e = u − uh is

||eh||2

E =

_

Ω

(σ − σh) • (ε − εh) dΩ . (4.118)

Because the exact solution u is unknown the energy norm is replaced—as

discussed in Sect. 1.31, p. 147—by the following estimate [132],

||eh||2

E

≤ η2 =

_

i

η2

i =

_

i

0.42 h2

λ + 5μ

||ri||20

+

1.22 h

λ + 5μ

||ji

||20

(4.119)

where ri = p−ph are the residual forces within an element Ωi, and where ji

are the jumps of the traction vector × 0.5 on the edges of the element Ωi:

||ri||20

=

_

Ωi

(p − ph)2 dΩ ||ji

||20

= 0.5

_

Γi

t2

Δ ds . (4.120)

384 4 Plane problems

P = 10 kN

A

B

C

D

1

1

1.5 m

2.0 m 2.0 m 2.0 m 2.0 m

2.5 m 1.0 m

a - a

b - b

d - d

1.0 m

Fig. 4.52. Adaptive refinement of a shear wall: a) original mesh, b) final mesh

At the start of the adaptive refinement the relative error is

ηrel := η _

||uh||2

E + η2

=

%

1.78

25 + 1.78

· 100% = 25.78% (4.121)

and then it slowly decreases as can be seen in Table 4.9.

Theoretically the mesh should be refined further near the supports and

the base of the single force—if this makes sense. Close to these critical points,

the grey is a very dark grey.

4.15 Adaptive mesh refinement 385

Table 4.9. Adaptive refinement

Step ||uh||2

E

||eh||2

E ηrel Nodes Elements d.o.f.

0 2.50E+06 1.78E+05 25.78% 199 160 388

1 2.77E+06 1.59E+05 23.30% 407 328 802

2 3.01E+06 1.49E+05 21.68% 821 685 1630

3 3.25E+06 1.43E+05 20.53% 1442 1231 2872

The form of the resulting mesh essentially depends on the percentage of

elements that are refined. Usually only the first 20, 30 or 50% of the elements

that exceed the critical value are refined. The lower the percentage the better

the refinement process will concentrate on the truly singular points. Here only

the first 50% were refined. The relative error ηrel dropped from about 25% to

20.53% which is not a great gain.

In a second analysis only the first 30% were refined and at the end the

estimated relative error was η = 20.87%. This level was reached with 781

elements, while the 50% strategy required 1,231 elements—a larger effort for

practically the same result. This observation indicates that the two hot spots,

the point support and the single force, dominate the error.

The corner points of the openings, where the stresses oscillated, were also

critical points (see Table 4.10), while the results at the more backward interior

point C, for example, were stable. The refinement of the mesh hardly had any

effect on the stresses at that point. The same holds for the section 1 − 1; see

Table 4.12.

Table 4.10. Stresses (kN/m2) at selected nodes at the beginning (0) and after

three refinements (3) and BE stresses

Nodes A B C D

σ(0)

xx −39.87 32.26 16.23 7.43

σ(3)

xx −106.96 90.44 16.76 9.36

σBE

xx

−76.814 68.94 16.41 11.84

σ(0)

yy −54.33 22.88 −1.66 −12.32

σ(3)

yy −127.28 79.52 −1.24 −15.39

σBE

yy

−99.95 56.96 −0.91 −16.71

To confirm that stress resultants are more stable than isolated stresses, the

stresses were integrated along three sections; see Table 4.11. These resultant

forces also vary, but they are more stable than the stresses at the isolated

points.

Hence the presence of singular points does not necessarily imply that the

whole solution is worthless. Rather the results show that in regions where the

386 4 Plane problems

Table 4.11. Integral of the stresses σyy (kN/m) along three sections

Cross section a-a b-b d-d

Step 0 −26.80 5.12 −11.63

Step 3 −32.1 9.85−

12.64

BE −29.32 6.92 −13.48

solution is smooth, the stresses are stable, although this doesn’t necessarily

mean that they are accurate! It is difficult to judge how large the influence

of the corner singularities is, that is how large the pollution error is, but we

think that in structural analysis—considering the error margins we are accustomed

to—pollution is a secondary effect. In standard situations, modeling

errors will probably have a more negative impact on the accuracy of, say, the

support reactions or the bending moments than an unresolved singularity on

the boundary.

Table 4.12. Stress distribution (kN/m2) in section 1-1

Level y = 2.5 y = 2.0 y = 1.5 y = 1.0 y = 0.5 y = 0.0

σ(0)

xx 7.86 13.09 18.11 23.43 28.96 34.44

σ(3)

xx 8.72 13.67 18.38 23.36 28.54 33.64

σ(BE)

xx 8.9 13.62 18.18 22.97 27.91 32.69

Навигация