PSI - Issue 71

Manan Ghosh et al. / Procedia Structural Integrity 71 (2025) 445–452 ) 0, −∑ ∫ 0 ( 1 ∫ 0

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, ( )=∑ ∫ 0, ( 1 ∫ =0 (8) the cycle scale weak form, where

0 ) 0

, are the coefficients of the nodal residual vector. , is solved at discrete cycle points using an iterative scheme such as Quasi-Newton for the nodal displacement coefficients, , such that ( )= ( , ) = ∑ ( ) ( ) , In Eq. 8, are the components of Cauchy stress tensor at τ k s in a cycle (N). To obtain at every integration point in a domain, the values of ( , = 0) = , 0 , ( , = 0) = 0 ∧ ( , = 0) = 0 . These variables form the internal state at the coarse scale and separate cycle scale rate equations are defined as

, 0 ( , ) = ( , , ) − , 0 ( , ) 0 ( , ) = ( , , ) − 0 ( , ) 0 ( , ) = ( , , ) − 0 ( , )

0 ( , ) = ( , , ) − 0 ( , ) (9) to obtain their coarse scale evolution. These equations are integrated in the cycle scale with jumps in the order of multiple cycles (∆N) using a backward difference formula. β -Titanium alloy with a BCC crystal structure is considered in this study, which results in 106 coarse and fine scale internal variables consisting of nine components of , 0 , 48 slip system resistances 0 , and 48 slip system back stresses 0 , along with 0 . Since, the implicit integration of Eqs. 9 involves 106 variables and requires inversion of a 106 × 106 matrix (Jacobian), which can impact the simulation speed negatively, a two-level procedure is followed. In this iterative process, , 0 , 0 and 0 are updated using the Newton – Raphson method, followed by an implicit update of 0 , until convergence is achieved. The initial guess for internal variables is made using a second-order explicit method (Chakraborty and Ghosh (2013)). A cycle step in the multi-time scale method involves fine-scale integration over one time period of loading to derive the coarse-scale evolution rate. The backward Euler method is used to numerically integrate at discrete time points within a cycle. An optimal time step, ∆τ, depending on convergence, is used. Typically, near the peak and valley in a cycle, smaller time steps are required. The method uses varying maximum resolutions for different segments, with the total set of coefficients given by = = 1 (10) where I is the total set of coefficients, are the number of segments and repesents the set of coefficients in each segment. Both single time and multi-time scale CPFEM formulation are implemented in a parallel Message Passing Interface (MPI) environment. Domain decomposition is done based on load balancing, considering the number of elements and nodal displacement degrees of freedom per processor. A parallel direct sparse solver, SuperLU, factorizes the global stiffness matrix. Quasi-Newton method is used in both the methods to enhance solution efficiency.