## Three-point bending test

The first example is concerned with simple two-dimensional 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 Mode-I 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 constant-strain 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 load-displacement diagram is shown in Fig. 3. Three solution strategies are compared:

• SSM-10: secant stiffness matrix updated after each 10 equilibrium iterations.
• SSM-5: 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, post-peak and fully damaged regime, as indicated in the load-displacement 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 load-displacement 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.   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 SSM-10 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 SSM-10 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 SSM-10 approach, while for TSM approach it takes between 0.02 to 1.2 seconds. The factorization time for SSM-10 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 SSM-10 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 SSM-10; see Table 2.

Table 1: Three-point bending test: convergence and times per iteration (nite'' is the number of equilibrium iterations) for tolerance SSM-10 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: Three-point bending test: total analysis times for different strategies and different convergence tolerances
 tolerance strategy total time SSM-10 00m:50s TSM 02m:01s SSM-10 14m:33s SSM-5 15m:03s TSM 03m:52s

Table 3: Three-point bending test: convergence and times per iteration (nite'' is the number of equilibrium iterations) for tolerance SSM-10 SSM-5 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 time-consuming solution procedure is more than compensated by very fast convergence.