An efficient implementation of the high-fidelity generalized method of cells for complex microstructures

Highlights

• High-fidelity generalized method of cells (HFGMC) is computationally expensive.

• The use of reformulation and sparse matrices improved efficiency.

• High-resolution analyses are now possible using HFGMC.

• High-resolution micromechanical analysis is necessary for complex microstructures.

Abstract

The high-fidelity generalized method of cells (HFGMC) enables micromechanical analysis of heterogeneous materials with high accuracy but does so at the cost of computational efficiency. In this paper, an implementation of the triply periodic HFGMC is developed to enable high-resolution simulations of materials with complex microstructures at a significantly reduced computational cost. This paper describes efficient reformulation and develops low-cost algorithms to reduce overall computation time and memory required to analyze complex 3D microstructures. The low-cost algorithms exploit the sparsity of the data by storing and performing calculations on only the non-zero values. The Parallel Direct Sparse Solver (PARADISO) subroutine is used to execute the most computationally intensive processes in parallel on multiple cores. Simulations of two selected test cases demonstrate the validity, computational efficiency, and value of the developed implementation. The results indicate that the savings in computation time and required memory are substantial and more than 100 times in some cases. In addition, parallel processing further reduces the computation time. The efficiency achieved through this work makes the high-resolution simulation of complex microstructures using HFGMC for the prediction of accurate local stress/strain fields computationally feasible.

Introduction

At the lowest length scales, most materials are heterogeneous in nature and consist of multiple constituents or phases. The individual behaviors of the material constituents at the microscale govern the overall material behavior, and micromechanics-based analysis and predictive techniques, which allow researchers to predict macro- and even structural scale response based on microstructural architecture and constituent properties, have become increasingly popular material analysis and design tools.

The wide array of methods for micromechanical analysis can be broadly classified into analytical methods such as the Mori-Tanaka method [1], and numerical methods such as the finite element analysis (FEA) [2]. The group of related methods for materials with periodically repeating microstructures, namely, the method of cells (MOC) [3], generalized method of cells (GMC) [4] and the high-fidelity generalized method of cells (HFGMC) [5] based on the theories developed by Aboudi et al. [6], are considered semi-analytical and promise the computational efficiency of analytical methods while retaining the fidelity of fully numerical methods. MOC, GMC, and HFGMC satisfy the laws of continuum mechanics in a volume averaged sense within the discretized volumes (subcells) and in a surface averaged sense between subcells. These methods were implemented by researchers at the NASA Glenn Research Center in the MAC/GMC micromechanics code [7] and others [8] and have seen wide use in materials ranging from composite materials [9] to metals [10]. These have been used to model simple linear elastic material behavior as well as more complex behaviors such as plasticity [11], thermal expansion [12], and damage [13], [14]. Their use as a multiscale method in synergy with FEA to enable structural scale analysis [15], [16], or recursively within themselves to enable analyses of additional length scales in woven or braided composites [14], [17] has also seen much success.

The purpose of micromechanical analyses is two-fold: (1) the prediction of effective global behavior (homogenization), and (2) the recovery of local stress and strain fields (localization). While MOC and GMC have demonstrated the capability to predict the effective macroscopic properties with acceptable accuracy for most cases of linear elastic material behavior, their localization capabilities are poor [18]. The MOC and GMC derive from essentially the same micromechanical formulations, but GMC extends the discretization capabilities of MOC from eight subcells (four subcells in the doubly periodic version) to an arbitrary number of subcells. This makes GMC capable of resolving the geometry of more complex multiphase microstructures. However, because GMC assumes a first-order displacement field, it is ambiguous to the actual architecture of the microstructure. This makes GMC suitable only for ordered microstructures and prohibits analysis of more complex microstructures or effects of architectural stochasticity [19]. The first-order assumption also results in a lack of coupling between local normal and shear stresses. The lack of shear coupling capability implies that local fields, particularly under non-linear regimes such as inelasticity and damage, are often not captured well under multiaxial loading, and the resulting predicted homogenized material response can have significant error [20].

HFGMC overcomes GMC’s limitations in localization using higher-order (second-order) displacement fields. The localization accuracy of HFGMC with cuboid subcells is found to be comparable to FEA [20], and the accuracy further increases with the use of arbitrarily shaped subcells [21]. However, because of the higher-order formulation, the number of linear equations in the triply periodic formulation of the HFGMC increases from six for each subcell in GMC to 21 for HFGMC. As a consequence, the computation time increases by approximately (21/6)³$\approx$ 43 times, and computer memory increases by (21/6)²$\approx$ 12.3 times [6]. The substantial increase in computation time and memory required implies that HFGMC achieves accuracy in localization at the expense of computational efficiency. As a result of the increase in memory requirements, the highest mesh resolution possible in a computer with 1 GB of memory would be 1,000 subcells [22]. While a direct comparison is not reasonable, most FEA codes could simulate larger mesh configurations on a computer with identical specifications.

Efforts have been made to address the high computational expense of the HFGMC, with one approach being to reconsider the HFGMC formulation [23], [24]. By using displacement continuity conditions between subcells in a surface-averaged sense, the reformulation reduces the number of equations per subcell in a triply-periodic formulation from 21 in the original formulation [5] to 9. The reformulation results in the number of equations being reduced by (21/9) = 2.33 times. As a result, computation time is reduced by a maximum of (21/9)³ $\approx$ 12 times, and the maximum number of subcells feasible increases from 1,000 to 1,728 on a computer with 1 GB of memory. While the improvement in computational efficiency is large, the limit on the maximum number of subcells is still a hindrance. Moreover, to the authors’ knowledge, the efficient reformulation for the triply periodic version of the HFGMC has not yet been published in detail. Another approach taken for the doubly periodic HFGMC to achieve computational efficiency is using order-reduction approaches such as proper orthogonal decomposition (POD) [25], [26]. In POD, the system of equations is reduced to a smaller set of equations. The use of POD in HFGMC resulted in a reduction of the computational time by 70% [26]. However, such methods are not exact and require expert use to ensure that no errors are introduced from order reduction. More importantly, the POD approach would not result in computation time savings when only a single timestep simulation is needed or when the material stiffness changes with timesteps (as in a progressive damage model). This is a significant limitation because damage in composite materials using micromechanical models is often modeled as progressive damage [13], [27].

An often-overlooked approach to improving computational efficiency is the numerical implementation itself. One aspect of the numerical implementation that has been critical in making large simulations using FEA feasible is the use of sparse matrix representation and storage and associated algorithms for the solution of a system of linear equations represented by sparse matrices, i.e., a sparse solver, [28]. The stiffness matrices in FEA are highly sparse, with relatively few constituents/elements being non-zero. Most implementations of FEA use sparse matrix storage formats such as banded, skyline, and compressed sparse row (CSR) to store only the non-zero elements [29]. Sparse matrix storage is coupled with solvers that perform computations only on non-zero values, resulting in significant savings in both memory and computation time. The governing matrices in semi-analytical methods are also highly sparse. The sparsity of the matrices has been exploited in the doubly periodic version of the GMC [30], and combined with the efficient GMC reformulation, it has facilitated the analysis of microstructures at a high spatial resolution previously unattainable because of the associated computational cost [31]. In HFGMC, the sparsity of the matrices has been recognized [22], [32]. The sparseness in the triply periodic HFGMC ‘stiffness’ matrix is 99.5% and only increases for configurations with a larger number of subcells. For 1,728 subcells, the sparsity increases to 99.98%. In the reformulated HFGMC, the sparseness is marginally less. The sparseness for 64 and 1,728 subcells in the reformulated HFGMC is 98.2% and 99.90%, respectively. Exploiting this high degree of sparsity has been attempted, but the details of the implementation, as well as the resulting computational efficiency, have not been discussed [33], [34], [35]. In one implementation of HFGMC, the entire matrix was assembled and then reduced to a sparse matrix and afterward passed on to a sparse solver [35]. Since the assembly of the full matrix is the bottleneck for memory use, it can be expected that this implementation would not result in appreciable memory savings, and limitation for the maximum number of subcells in a configuration remains the same. Additionally, to the authors’ knowledge, sparse matrix implementations in HFGMC have been limited to the doubly periodic case. Therefore, in addition to sparse matrix solution algorithms, sparse matrix assembly procedures and algorithms for full 3D triply periodic microstructures are critical to fully exploit the sparse nature of the matrices in HFGMC. This will enable increased computational efficiency and facilitate improved high-fidelity material analysis.

A final approach to reduce computation time (but not necessarily increase computational efficiency) is to take advantage of the architecture of the computing machines. Implementation of the algorithms to execute processes in parallel on multiple cores/processors has the potential to reduce computation time significantly. Approaches to parallel processing can be classified into shared memory parallelism (SMP) and distributed memory parallelism (DMP) [36]. In SMP, each processor operates on data/variables stored in a shared memory directory that is accessible to all cores. In DMP, each core operates on a separate memory, and communication is required between those memories. The exchange of data between memories results in performance penalties and communication lag. Shared memory parallelism does not involve any data transfer and is more suitable for execution on a single multicore machine. Most commercial FEA codes are capable of parallel processing on multicore systems, and many micromechanics tools, for instance, SwiftComp [37], are also capable of parallel processing. Parallel processing algorithms have been developed for GMC [38] but not for HFGMC.

Among micromechanics analysis methods and tools, computational efficiency is a key aspect that directs the choice of the method [39]. The fidelity of HFGMC in a wide range of scenarios, materials, and behaviors has been proven, but the high computational cost has remained a persistent barrier to its more widespread use. This paper uses a combination of approaches to develop a computationally efficient implementation of the triply periodic HFGMC, including efficient reformulation, sparse matrix representation, and parallel processing. The validity of the resulting implementation is verified using linear elastic test cases from a benchmarking work for micromechanics analysis tools by Sertse et. al.[39], referred to as the benchmarking paper henceforth. The resulting significant improvements in computational efficiency from the proposed implementation are presented and discussed. The value of the developed efficient implementation of the triply periodic HFGMC in simulating the behavior of materials with complex microstructures is shown. Additionally, the use of the efficient HFGMC implementation to predict inelastic and damage behaviors and pathways to achieve further computational efficiency are discussed.