9. Verification of the consistency of homogenization: tangent op-
11.2 REV size and shape
11.2.2 The periodic frame effect
For the series of 100-grain REVs, the orientation-dependent stress response to bi-axial loading is averaged over the 100 unique realizations. This gives statistically representative stress-strain curves for REVs with 100 grains at different orienta-tions θI. Figure 11.8-left shows these curves for the REVs with 100% sphericity
11.2 REV size and shape 135
optimization. Two types of curves can be destinguished based on their soften-ing rate. These two types of softensoften-ing response correspond to the two types of damage patterns in the interfaces; the stronger softening corresponds to a single localization path in the REV and the weaker softening to a double localization path. To demonstrate the relation between the REV orientation θI and the post-peak response, a crosssection of the stress-strain curves is taken at εa = −0.025.
Figure 11.8-right shows the crosssections for the averaged stress-strain curves of 0% and 100% optimization. It is clear that there is a strong influence of the ori-entation of the REV and more precicely the oriori-entation of the periodic boundary conditions in the response, with peaks in a 45◦ interval. Consequence is that the REV boundary orientations remain influencing the post-peak behaviour and presents preferential orientation of localization through the softening response.
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005 00
5 10 15 20 25
εa[−]
nominal stress [MPa] cross-section
−900 −60 −30 0 30 60 90
5 10 15 20 25
orientation θI [°]
nominal stress [MPa]
η = 0 η = 1
Fig. 11.8: Left: 100-test averaged stress-strain response for different loading orienta-tions of 100-grains REVs with grain sphericity optimization η = 1. Right:
orientation-dependent 100-test averaged response at −2.5% axial strain for η = 1 and η = 0.
12. PERFORMANCE AND COMPUTATIONAL EFFICIENCY
Compared to the classical FE methods with macroscale phenomenological con-stitutive laws, the FE2 method is computationally expensive. Although the rel-ative cost of the microscale computations goes down significantly with increasing numbers of degrees of freedom at the macro scale, the major part of the compu-tation time is spent on the micro scale compucompu-tations.
The possible gain in computational efficiency is therefore an important ar-gument in the choice for computational homogenization by condensation, as it makes the additional four computations needed for the finite difference approx-imation of the consistent tangent stiffness matrix by numerical perturbation ob-solete. Without these four additional computations, the theoretical reduction in computation time through the introduction of the condensation is a factor five (for mechanical problems) or eight (for hydromechanical coupling). However, the routine for homogenization by condensation requires some matrix operations as well and a certain overhead of initialization of the problem is required. This will reduce the factor 5 (or 8) to a theoretical upper limit of efficiency which, especially for relatively small microstructures, is difficult to obtain.
For the computational homogenization by static condensation (CHSC) to be effective, both its overall and its absolute efficiency has to be competitive with the method of numerical perturbation (NP). This requires in the first place a convergence of the macroscale Newton-Raphson scheme to be of the same quality as obtained by NP. Secondly, the computational effort required for the condens-ation needs to be much smaller than the computcondens-ations it is replacing. In this chapter the performance of the developed methods are discussed with respect to computational effort and quality of convergence in doublescale computations.
12.1 Doublescale convergence studies
To assess the convergence of doublescale computations with hydromechanical coupling, several loading steps of the computation of Test B in Section 10.4 are repeated. Starting from the same state of deformation, identical loading steps are applied using either NP or CHSC for obtaining the consistent tangent stiffness matrices. Three loading steps are evaluated; a loading step in the elastic domain (εa = −0.25 %), a loading step just after the peak (εa = −1.00 %) and one at the end of the snap-through (εa = −1.20 %). Figure 12.1 shows the graphs for force convergence (left) and displacement convergence (right) as evaluated by the macroscale finite element program Lagamine, with;
FNORM the norm of the out-of-balance forces (on the DOFs to be solved for), RNORM the norm of the reaction forces (the prescribed DOFs),
UNORM the norm of the iterative updates of the DOFs to be solved for, DNORM the norm of the loading step updates of the DOFs to be solved for.
The ratios FNORM/RNORM and UNORM/DNORM can be compared with con-vergence criteria for the forces (combining nodal (double) forces and nodal fluid mass balance) and displacements (DOFs ui νij and p). Details of the normaliza-tion of the different terms are not discussed here, as only a comparison between NP and CHSC is made.
1 2 3 4 5 6 7 8 9 10 11 12
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
log(FNORM/RNORM)
iteration
CHSC | εa=0.0025 CHSC | εa=0.0100 CHSC | εa=0.0120 NP | εa=0.0025 NP | εa=0.0100 NP | εa=0.0120
1 2 3 4 5 6 7 8 9 10 11 12
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100
iteration
log(UNORM/DNORM)
CHSC | εa=0.0025 CHSC | εa=0.0100 CHSC | εa=0.0120 NP | εa=0.0025 NP | εa=0.0100 NP | εa=0.0120
Fig. 12.1: Convergence graphs for computational homogenization by static condensa-tion (CHSC) and numerical perturbacondensa-tion (NP) for the HM-coupled biaxial compression test D = 1 × 10−11in Section 10.4
The convergence graphs in Figure 12.1 demonstrate that the convergence of the macroscale Newton-Raphson scheme obtained when using CHSC is of the same quality as when NP is used. This holds for the elastic domain as well as the softening domain. However, it is not guaranteed that this result always holds and the Newton-Raphson algorithm does not necessarily leads to convergence and stiffness matrix obtained by a numerical approximation such as the numerical perturbation might sometimes better deal with the material non-linearities to be taken into account around the test configuration.
For the simulation of the biaxial compression with fluid in Section 10.3, the convergence profiles for forces and displacements are generated using numerical perturbation and condensation for obtaining the tangent operators. The com-parison at different states of deformation (linear domain, at peak response and after snapthrough) are given in Figure 12.2. The convergence criteria present the combined normalized errors for nodal forces and double forces.