An efficient monolithic solution scheme for FE2 problems

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Abstract

The FE2 method is a very flexible but computationally expensive tool for multiscale simulations. In conventional implementations, the microscopic displacements are iteratively solved for within each macroscopic iteration loop, although the macroscopic strains imposed as boundary conditions at the micro-scale only represent estimates. In order to reduce the number of expensive micro-scale iterations, the present contribution presents a monolithic FE2 scheme, for which the displacements at the micro-scale and at the macro-scale are solved for in a common Newton–Raphson loop. In this case, the linear system of equations within each iteration is solved by static condensation, so that only very limited modifications to the conventional, staggered scheme are necessary. The proposed monolithic FE2 algorithm is implemented into the commercial FE code Abaqus. Benchmark examples demonstrate that the monolithic scheme saves up to  60% of computational costs.

Introduction

Most engineering materials are inhomogeneous at a characteristic length-scale. The properties of this so-called micro-scale, consisting of certain dissimilar constituents, determine the behavior at the macro-scale. The macro-scale, in turn, is usually the actual scale of interest (for engineering structures). While it is generally possible to directly resolve the microstructure in structural computations, it is often not favorable, since this results in models far too complex to be dealing with large structures [1]. A common way to reduce the model size by means of multi-scale approaches. The mutual connection between the micro- and macro-scales is established through localization and homogenization processes. In this scale transition, heterogeneous microstructure is assigned to a homogeneous material with effective properties (homogenization), while the inverse procedure is called localization [2]. A comprehensive overview on the theoretical background of different multi-scale approaches can, for instance, be found in [1], [3], [4], [5].

The use of finite element analysis (FEA) at both length scales is referred to as FE2 modeling [6] and illustrated in Fig. 1. In linear problems, a sequential solution procedure is possible, since the effective (macro-)stiffness of the microstructure is representative of all deformation states and can therefore be calculated beforehand. For many nonlinear problems, a concurrent solution scheme is necessary, in which a micro-scale model has to be solved at each integration point of the macroscopic mesh [3], [4], [7]. While the implementation of the sequential scheme is comparatively simple, the concurrent approach is challenging and computationally expensive, but still quite universally applied. To name a few examples, it has been used in numerical multi-scale analysis of fiber-reinforced composites [4], [6], [8], woven composites [9], [10], biomechanics [11], elastic–plastic matrix-inclusion problems [4], [12], or the growth of microvoids [12], [13] and texture formation in polycrystalline metals [14], and in the design optimization of microstructures [15]. Following much research attention over the last two decades, the FE2 method is no longer restricted to conventional mechanical problems. Extensions of the method to the realm of multi-physics and generalized continua have been proposed by many groups, e.g. for thermo-mechanical [16], [17], electro-mechanical [18], [19], magneto-mechanical [20], micromorphic [21], [22] or even three-field problems [23], [24] — see also the references therein.

Several implementations can be found in literature. The first FE2-program was programmed by Feyel [6]. He used the ZeBuLoN code for both scales in the way that the code is reentrant, which means that it is able to call itself. Kouznetsova et al. [13] presented an implementation in 2001, where the macroscopic model is computed by a MATLAB FE-code and the microscopic problems are solved by the commercial package MARC. Yuan and Fish [9], Tchalla et al. [25] and Tikarrouchine et al. [8], [10] utilize Abaqus for both scales with the aid of a python script and a procedure UMAT to let the Abaqus macro program calls itself at each integration point. Naturally this listing is not complete.

The issue of high computational costs has already been addressed in some of the mentioned publications. Yuan and Fish [9] specify the number of linear solution operations of the microscopic problem with NcellsnImacroImicro where Ncells is the number of macroscopic integration points, n the number of load increments and Imacro and Imicro the average number of iterations on the macro- and micro-scale. As this number can become quite large, even for small problems, the necessity to substantially reduce the computational effort is evident. They name coarse graining or model reduction and parallel computation as the main options to lower computational costs [9]. The need for parallel computation is also mentioned in other publications, cf. [6], [8]. Highly parallelizing implementations of FE2 have been presented in [26], [27].

Some publications suggest algorithmic modifications to the standard FE2 solution procedure. Feyel for example mentions the use of a Quasi-Newton algorithm, so that the macroscopic material tangent does not have to be recalculated in every Newton step [6]. Temizer and Wriggers [28] investigate condensation and perturbation procedures in the context of computing the macroscopic tangent. Nezamabadi et al. [29] combine the multi-scale finite element procedure with the asymptotic numerical method, which can be efficient for certain types of problems, e.g. in buckling analysis. A completely different approach described in the literature is to employ a fast Fourier transformation method to solve the micro-problem efficiently, while using finite element analysis for the macro-problem, cf. [30], [31], [32]. This spectral approach is, however, limited to voxalized microstructures and its computational costs increase with the contrast between the moduli of the constituents. The idea of solving micromechanical problems by means of Fourier transformations was first developed by Moulinec and Suquet [33]. An overview over recent applications of this method can be found in [34]. Another approach proposed in several publications towards computationally efficient multi-scale simulations is a sequential solution procedure, which is capable of describing nonlinear and irreversible effects with the aid of neural networks (NN), see [31], [35], [36], [37]. The efficiency of this approach arises from the fact that the constitutive relation at the macroscopic integration point is provided by a trained neural network instead of through concurrent microscopic finite element simulations. Disadvantages are the lower flexibility and the costly training process associated with the NN-approach.

The fundamental equation of the FEM is the following general equilibrium relation, to be fulfilled at both the macro- and micro-levels Rˆ̲(Uˆ̲,uˆ̲)=Fˆ̲int(Uˆ̲,uˆ̲)=α=1nαwαB̲̲αTΣ̲α(uˆ̲α,H̲α=B̲̲αUˆ̲)=0̲,rˆ̲α(uˆ̲α,H̲α=B̲̲αUˆ̲)=0̲. Here small symbols denote microscopic quantities, while large symbols represent a macroscopic quantity. Rˆ̲,rˆ̲ are the residuals, Uˆ̲,uˆ̲ represent nodal displacements and the index α denotes macroscopic integration points. For simplicity, and without the loss of the general applicability of the presented algorithmic concept, only internal forces Fˆ̲int have been considered in Eq. (1). It can be seen that (1), (2) are coupled through the macroscopic stress Σ̲α and displacement gradient H̲α.

The previously cited works all imply a staggered solution scheme, as illustrated in Fig. 2(a). In this context the term staggered indicates that a microscopic Newton loop is nested into the macroscopic iterative Newton process. The micro-scale problems receive the macroscopic displacement gradient H̲αK for the (periodic) boundary condition and iterate until convergence is reached, despite the fact that H̲αK is only an estimate for H̲αi+1 in the Kth Newton–Raphson step. Recently, Tan et al. [38] proposed the “Direct FE2” method where the macroscopic and microscopic nodes and elements are implemented in a single global FE model, coupled by multi-point constraints. Consequently, the macroscopic and all microscopic problems are solved in a common Newton–Raphson loop in this monolithic scheme as shown Fig. 2(b). This method has the advantage that no costly microscopic iterations are necessary and all features of existing FE codes can be used at both scales. However, the global system of equations comprises the macroscopic nodal displacements Uˆ̲ and the microscopic displacements uˆ̲α of all macroscopic Gauss points α and can thus become exceedingly large even for medium-sized problems [27]. Interestingly, similar concepts can be found in phase-field fracture modeling approaches, in which efficient monolithic schemes have been implemented and successfully shown to be capable of saving computing time, when compared to the more conventional staggered, algorithmically decoupled approach [39], [40]. A more general discussion on this kind of algorithm and generally on iterations at the material level can be found in [41], [42].

In the present article, a new algorithmic strategy is proposed for solving FE2 problems more efficiently in a monolithic way, in which static condensation [11], [28] is employed to avoid prohibitively large systems of equations.

Section snippets

Homogenization

The microscopic FEA can be viewed as a complex material routine in the overall workflow of a macroscopic FEM simulation. It replaces the phenomenological material description used in classical FEM. It gives the relation between stress Σ̲α and displacement gradient H̲α1 in dependence of the load history encountered by the micro model, which is

Implementation

The monolithic as well as the staggered FE2 method have been implemented into the commercial FE code Abaqus. Existing Abaqus-FE2 implementations are build on the idea to use Abaqus both on the macro- and microscopic level, see [8], [9], [25]. The advantage is, that Abaqus is already well tested and has a lot of built-in features, which can be used for the micro and macro model. However it has the big disadvantage, that the Abaqus code is not reentrant and has to be restarted via a python

Benchmarking

To evaluate differences in the computational effort, benchmarks between the staggered and the monolithic (w & w/o stored factorized stiffness matrix) algorithm are performed, by computing model problems with highly nonlinear character. Thereby the computing time strongly depends on the nonlinearity of the problem chosen. In linear problems the staggered and monolithic algorithm are equivalent. To get comparable results, all examples were calculated on the same computer with one processor (

Conclusions

The computational costs of FE2 simulations are generally very high. Currently this fact, and the lack of implementations in commercial FE software, often hinder their application in actual engineering problems. Hence, a monolithic solution strategy to FE2 problems has been outlined in the present contribution, which solves the macro- and the micro-scale problems in a common Newton–Raphson iteration loop. It was shown that the microscopic degrees of freedom can be removed from the global system

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank Vincent Tan and Karthikayen Raju for providing the Direct FE2 implementation (Example 4) and for many fruitful discussions. Furthermore, the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the SFB 920 “Multi-Functional Filters for Metal Melt Filtration — A Contribution towards Zero Defect Materials” – project ID 169148856 – is gratefully acknowledged.

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