Threepoint bending test
The first example is concerned with simple twodimensional analysis of a simply
supported concrete beam loaded by a concentrated force
(Fig. 1a). Due to symmetry, the centerline of the process zone
is straight and the failure mechanism corresponds to pure ModeI fracture.
The constitutive properties are set to: Young's modulus
GPa, Poisson's ratio , tensile strength MPa,
strain controling the softening
, and nonlocal interaction radius
mm.
The mesh contains 696 nodes and 1298 constantstrain elements (Fig. 1b).
The formation of the process zone and failure of the specimen are simulated
in 15 incremental steps, controled by imposed increments of the
displacement under the applied load. The loaddisplacement diagram
is shown in Fig. 3.
Three solution strategies are compared:
 SSM10: secant stiffness matrix updated after each 10 equilibrium iterations.
 SSM5: secant stiffness matrix updated after each 5 equilibrium iterations.
 TSM: fully consistent nonlocal tangent stiffness matrix updated
after each equilibrium iteration.
The rates of convergence of the global equilibrium iteration procedure are
illustrated in Fig. 2, which shows the dependence of the error
(norm of residual forces) on the number of iterations. As expected, the secant
stiffness leads only to a linear convergence rate, i.e., the logarithm of the
error is approximately a linear function of the
iteration number, .
With increasing , the ratio
tends to where is the slope of the straight line in the
semilogarithmic convergence plot in Fig. 2b.
The steps A, B and C correspond to peak, postpeak and fully damaged
regime, as indicated in the loaddisplacement diagram in Fig. 3.
For the tangent stiffness, the convergence rate is quadratic,
i.e., the convergence curves are approximately parabolic
(Fig. 2a). The residual is then
reduced to the level of machine precision within a few iterations,
typically 3 to 6.
The only exception is step 5 (around the peak of the loaddisplacement curve),
in which the initial prediction is not sufficiently close to the exact solution,
and the quadratic rate of convergence is attained only after iteration number 7.
Figure 1:
Threepoint bending test: (a) specimen geometry and loading,
(b) finite element mesh

Figure 2:
Threepoint bending test: convergence characteristics

Figure 3:
Threepoint bending test: load displacement diagram

The relative efficiency of the strategies using the secant or tangent stiffness
matrices strongly depends on the prescribed convergence tolerance.
For a relative tolerance
, the number of iterations per step
using SSM10 does not exceed 68; see Table 1. With TSM, the
number of iterations per step is at most 8, but each iteration is much more
costly. The secant stiffness matrix is symmetric and has a smaller average bandwidth
than the nonsymmetric tangent stiffness matrix. Moreover, the rate of convergence
is not substantially reduced if the secant stiffness
is assembled and factorized only once per every 10 iterations,
which results into additional savings. The tangent stiffness must be updated
in every iteration, otherwise the quadratic rate of convergence would be lost.
As shown in Table 2, the total time needed for the complete analysis using
SSM10 is more than three times shorter than that needed by TSM with a direct solver.
For this small problem, an iterative solver would not be
advantageous. The assembly procedure takes 0.02
seconds for SSM10 approach, while for TSM approach it takes between
0.02 to 1.2 seconds. The factorization time for SSM10 approach is
constant, equal to 0.34 seconds, while for TSM approach it varies between
0.75 and 1.06 seconds.
The relative efficiency of SSM and TSM changes dramatically if a strict convergence
criterion is required. For a relative tolerance
, the number
of iterations per step
using SSM10 becomes excessive, and no improvement is obtained if the stiffness
matrix is updated after every 5 iterations; see Table 3. On the other
hand, TSM typically converges within 3 to 6 iterations, and only in one step it
needs 10 iterations. As a result, the total analysis with TSM is now more than
3 times faster than with SSM10; see Table 2.
Table 1:
Threepoint bending test: convergence and times per
iteration (``nite'' is the number of equilibrium iterations) for tolerance

SSM10 
TSM 
step 
nite 
time[s] 
nite 
time[s] 
1 
1 
0.30 
1 
0.60 
2 
1 
0.30 
1 
0.60 
3 
14 
1.61 
3 
3.66 
4 
34 
3.65 
3 
6.94 
5 
46 
4.78 
8 
17.35 
6 
36 
3.75 
4 
10.45 
7 
27 
2.78 
4 
11.30 
8 
23 
2.42 
3 
9.37 
9 
29 
2.93 
4 
12.01 
10 
63 
6.25 
3 
9.82 
11 
68 
6.75 
2 
7.53 
12 
55 
5.55 
2 
7.62 
13 
46 
4.57 
2 
7.64 
14 
19 
1.96 
2 
7.74 
15 
15 
1.62 
2 
7.69 

Table 2:
Threepoint bending test: total analysis times for different strategies
and different convergence tolerances
tolerance 
strategy 
total time 

SSM10 
00m:50s 

TSM 
02m:01s 

SSM10 
14m:33s 

SSM5 
15m:03s 

TSM 
03m:52s 

Table 3:
Threepoint bending test: convergence and times per
iteration (``nite'' is the number of equilibrium iterations) for tolerance

SSM10 
SSM5 
TSM 
step 
nite 
time[s] 
nite 
time[s] 
nite 
time[s] 
1 
1 
0.32 
1 
0.32 
1 
0.60 
2 
1 
0.32 
1 
0.32 
1 
0.60 
3 
46 
4.98 
44 
4.82 
4 
4.73 
4 
110 
11.38 
108 
11.63 
4 
8.81 
5 
191 
19.11 
187 
19.27 
10 
21.45 
6 
426 
41.79 
423 
42.90 
6 
14.85 
7 
696 
67.34 
694 
69.91 
6 
15.91 
8 
807 
77.91 
805 
80.28 
5 
14.21 
9 
732 
70.49 
730 
72.68 
6 
16.85 
10 
813 
77.86 
811 
80.84 
5 
14.69 
11 
897 
85.79 
895 
88.55 
4 
12.58 
12 
989 
94.26 
986 
97.84 
4 
12.97 
13 
1053 
98.27 
1051 
104.07 
4 
12.87 
14 
1153 
110.18 
1151 
113.60 
3 
10.33 
15 
1156 
112.30 
1155 
115.68 
3 
10.43 

This example indicates that for a low accuracy the strategy
using a secant stiffness matrix is numerically
more efficient. For a high accuracy,
the results illustrate the potential benefits of the tangent
stiffness matrix.
The overhead of storing the whole matrix instead of its
symmetric part and of the more timeconsuming solution procedure
is more than compensated by
very fast convergence.
This page is part of the OOFEM project documentation (www.oofem.org)
(c) 2016 Borek Patzak, info(at)oofem(dot)org