# Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

- Annika Radermacher
^{1}, - Brett A. Bednarcyk
^{2}, - Bertram Stier
^{1}, - Jaan Simon
^{1}, - Lei Zhou
^{1}and - Stefanie Reese
^{1}Email author

**3**:29

https://doi.org/10.1186/s40323-016-0082-8

© The Author(s) 2016

**Received: **25 March 2016

**Accepted: **24 October 2016

**Published: **15 November 2016

## Abstract

Many applications are based on the use of materials with heterogeneous microstructure. Prominent examples are fiber-reinforced composites, multi-phase steels or soft tissue to name only a few. The modeling of structures composed of such materials is suitably carried out at different scales. At the micro scale, the detailed microstructure is taken into account, whereas the modeling at the macro scale serves to include sophisticated structural geometries with complex boundary conditions. The procedure is crucially based on an intelligent bridging between the scales. One of the methods derived for this purpose is the meanwhile well established FE\(^2\) method which, however, leads to a very high computational effort. Unfortunately, this impedes the use of the FE\(^2\) method and similar methodologies for practically relevant problems as they occur e.g. in production or medical technology. The goal of the present paper is to significantly improve computational efficiency by using model reduction. The suggested procedure is very generally applicable. It holds for large deformations as well as for all relevant types of inelasticity. An important merit of the work is the computation of the consistent tangent operator based on the reduced stiffness matrix of the microstructure. In this way a very fast (in most cases quadratic) convergence within the Newton iteration at macro level is achieved.

## Keywords

## Background

Many applications are based on the use of materials with heterogeneous microstructure. Prominent examples are fiber-reinforced composites, multi-phase steels or soft tissue to name only a few. The modeling of structures composed of such materials is suitably carried out at different scales. At the micro scale, the detailed microstructure is taken into account, whereas the modeling at the macro scale serves to include sophisticated structural geometries with complex boundary conditions. The procedure is crucially based on an intelligent bridging between the scales. This challenge, commonly denoted as multiscale modeling, has been in the center of international research for several years.

Among the methods which have been established for this purpose are the so-called FE\(^2\) method (see e.g [1, 2]) and the multiscale LATIN method (see e.g. [3, 4]). A disadvantage of the FE\(^2\) method is the relatively high computational effort. The main reason for that is the fact that in every evaluation point of the macroscopic structure a boundary value problem has to be solved. This impedes the use of FE\(^2\) for practically relevant problems as they occur e.g. in production or medical technology.

In the recent years, several methodologies have been developed to overcome this problem. One possibility is to take advantage of parallel computing e.g. like Feyel and Chaboche [1]. Another approach is to work with very efficient semi-analytical methods at the micro scale, for example, the generalized method of cells (GMC). This method has been developed, extended and widely used in the field of composites by Aboudi [5], Aboudi and Arnold [6]. In the GMC, the underlying repeating composite unit cell is discretized into sub-cells and the microscopic field equations of each sub-cell are explicitly coupled with the macroscopic constitutive equations without using finite elements. Michel and Suquet [7] embed the so-called nonuniform transformation field analysis (NTFA), introduced by the same authors in [8] and based on the early work of Dvorak and Benveniste [9], into a macroscopic finite element analysis for elastoplastic behavior with nonlinear isotropic hardening. The idea of the method is to create an effective constitutive relation for the nonlinear microstructure based on a reduced number of internal variables with inelastic modes to be defined by precomputed inelastic strain fields. The method is steadily further developed. Fritzen and Böhlke [10] show that the reduced basis order-reduction techniques applied in the NTFA can be very much improved by micromechanical considerations. Their method captures the anisotropic transient response of viscoelastic composites with a high gain in CPU time effort. Using a mixed variational framework, Fritzen and Leuschner [11] extend the previous concept to generalized standard materials. Further, consistency of the method is improved in the sense that an increasing number of basis functions leads to a monotonic increase of accuracy. In a very recent paper, Fritzen et al. [12] raise the speed-up of the method enormously by incorporating a parallel GPU implementation. The homogenization method of Fritzen and Leuschner [11] is additionally extended by nonuniform hardening modes which can account for locally varying hardening states. Largenton et al. [13] investigate several versions of the NTFA and apply their method to viscoelastic composites in the presence of aging and swelling. Another variant of the transformation field analysis is found in the paper of Oskay and Fish [14] and the follow-up work of Sparks and Oskay [15], where the evolution of failure is modeled by means of the eigen deformation.

Among the most common strategies to achieve model reduction in general are reduced basis (RB), proper orthogonal decomposition (POD) and proper generalized decomposition (PGD) techniques. Whereas the NTFA is either combined with RB, POD or mixtures of the two, there is a small group of authors who manage to incorporate the PGD into a multiscale framework. A good overview of the PGD is given by Chinesta et al. [16]. The procedure presented in Ladeveze et al. [17] is crucially based on the LATIN method which splits the entire problem into a macro problem defined over the whole structure and coarse sub-intervals which are treated along with linear micro problems. The PGD is applied on the latter which results into significant computational savings. In Ladeveze et al. [17] a viscoelastic fiber-reinforced structure is investigated. In a recent paper (Cremonesi et al. [18]), the method has been further refined by the derivation of the so-called homogenized operator and the extension of the PGD representation to the interface macro displacements. The, according to the knowledge of the authors, only combination of the FE\(^2\) method with PGD is found in Halabi et al. [19], where a two-dimensional linear-elastic structure of simple geometry is investigated. Both the reduction in CPU time effort as well as the accuracy of the results show the high potential of this method.

One of the first papers on a “reduced” multiscale methodology involves the reduced-basis approach but is restricted to linear elasticity [20]. The very interesting question of error estimation has been investigated by Kerfriden et al. [21] in the context of projection-based reduced order modeling. By means of the derivation of upper and lower error bounds, the authors enable adaptive computations of random linear-elastic composites. The coupling of a reduced basis method to multiscale finite element methods for elliptic problems with highly oscillating coefficients can be found in Hesthaven et al. [22]. A finite element-based heterogeneous multiscale method applied to crack domains has been published in Abdulle and Bai [23].

Concerning nonlinear constitutive modeling, the work of Goury et al. [24] should be mentioned, where a FE\(^2\) framework with damage is set up and the representative volume element (RVE) is modeled by POD. The investigation is restricted to a RVE with prescribed macroscopic strains. Highly interesting is the incorporation of the gappy method which allows to evaluate the internal force vector in only a small set of points of the structure. The selection of the points (“controlled” elements) is performed by means of the discrete empirical interpolation method. The example shows that—as expected—the maximum damage values are found between the inclusions. These are also the regions where the controlled elements are placed. A further development of the so-called gappy POD is found in the paper of Miled et al. [25] who couple the approach with the a priori hyperreduction method of Ryckelynck [26] to come to adaptive strategies for the computation of viscoelastic-viscoplastic composites. Closely related to hyperreduction is the method of empirical interpolation used by Hernandez et al. [27]. The latter authors apply POD at the level of their RVE which is assumed to behave elasto-plastically. An important merit of the paper is the development of a new interpolatory integration method which overcomes the well-known problem of ill-posedness when replacing the non-affine term only by POD terms. In the work of Monteiro et al. [28] nonlinearly electrical as well as nonlinearly thermal transient conduction problems are solved by means of a POD-reduced multiscale approach.

According to the knowledge of the authors there are hardly any contributions, where reduced multiscale modeling is applied to hyperelastic solids undergoing large deformation. In this context, it is important to mention the work of Yvonnet and He [29], who combine the FE\(^2\) method with POD at the micro scale. The method shows very convincing results, whereby the reduction in CPU time effort leaves room for improvements. One reason for that could be the missing “analytical” (consistent) tangent which is replaced by a numerical tangent. Obviously, the computation of the latter is elaborate and leads to additional numerical effort. Very interesting is also the work of Xia and Breitkopf [30], where POD (at RVE level) and diffuse approximation techniques are combined to arrive at structural topology optimization of nonlinearly elastic structures. In order to cope with this challenging problem, the linearized strain measure is used throughout the paper.

The present work follows the line suggested by Yvonnet and He [29]. In the first part of the paper, the FE\(^2\) method is presented in its standard format using periodic boundary conditions at the micro scale (not considered by Yvonnet and He [29]). The derivation incorporates large deformations, inelasticity as well as the formulation of the consistent tangent to achieve approximately quadratic convergence in the Newton–Raphson iteration at the macro scale. In the second step, model reduction, more specifically POD, is brought into play which means to split the multiscale simulation into an offline and an online procedure. In the offline step, the so-called representative volume element or unit cell is subjected to nine characteristic deformation states additionally varied over time in order to compute a physically meaningful projection subspace. The latter is then used in the online step to significantly reduce the dimension of the equation system to be solved at the micro level. The work is crucially based on using the FE\(^2\) interface of the finite element solver FEAP [31]. In this way, a very general multiscale solver is created. An important contribution of the work is the consistent tangent operator which is modified to incorporate the process of model reduction. The example computations performed in the third part of the paper show that speed-ups of two orders of magnitude can be reached. Very good results with negligible deviations from the reference simulation are achieved by means of only a small number of modes.

## POD-based reduced multiscale method

### Classical multiscale approach based on finite elements (FE\(^2\))

Before presenting the reduced multiscale approach, attention is devoted to the classical first-order computational homogenization scheme. For detailed information please refer to e.g. Feyel [32], Kouznetsova [33], Miehe and Koch [34] and Geers et al. [2]. The local macroscopic constitutive response—for example given in terms of the first Piola-Kirchhoff stress tensor \({\mathbf P}_M\) in dependence of the macroscopic deformation gradient \({\mathbf F}_M\)—is derived from the solution of a micro-structural boundary value problem in every evaluation point of the macroscopic structure. Speaking of FE\(^2\) means that the spatial discretization at both scales is performed by means of finite elements. The volume to be considered at micro level should represent the usually heterogeneous microstructure sufficiently well. If this is the case, for a clear definition see the work of e.g. Kanit et al. [35], one speaks of a representative volume element (RVE). In the present work, we assume that a RVE is available. However, the present multiscale approach could be equally applied, if the conditions for the presence of a RVE were not fulfilled. Certainly then, the meaning of the results would have to be discussed with care.

In the following the three steps—macro-to-micro scale transition, solution of RVE boundary value problem, micro-to-macro scale transition—are briefly summarized.

#### Macro-to-micro scale transition

#### Solution of RVE boundary value problem

*R*internal variables \({\mathbf Q}_r\) (covering the elastoplastic behavior):

*n*independent degrees-of-freedom are stored:

*n*independent degrees-of-freedom included in \({\mathbf U}_i\):

*a*,

*b*running according to \(a=n_d+1,\ldots ,\bar{n}\) and \(b=1,\ldots ,n_d\), respectively. The other quantities in the equations for \(\widetilde{\mathbf K}^j\) and \(\widetilde{\mathbf G}^j\) are determined analogously.

*n*(\(n=n_b+n_f\)). The linearization (10) including the decomposition of the displacement components into \({\mathbf U}_b\) and \({\mathbf U}_f\) (11) is written as

#### Micro-to-macro scale transition

Upon the solution of the RVE problem, the macroscopic stress tensor \({\mathbf P}_M\) is obtained by computing the volume average of the micro-structural stress field \({\mathbf P}_m\). In addition, to achieve quadratic convergence at the macro level, the correct macroscopic tangent \({\mathbf C}_M\) in \(\Delta {\mathbf P}_M={\mathbf C}_M:\Delta {\mathbf F}_M\) is needed. Note that \({\mathbf C}_M\) includes in general geometric and material parts (see e.g. [39]).

*B*by using the divergence theorem and, secondly, that the divergence of \({\mathbf P}_m\) vanishes in the micro-structural equilibrium. For a detailed derivation of this step, we refer to Kouznetsova et al. [38] and Kouznetsova [33]. In the case of a finite element discretization of the RVE, the boundary integral in Eq. (15) is computed as sum of the products of the \(3 \times 1\) nodal reaction vector \({\mathbf L}_b^{A}\) and the \(3 \times 1\) position vector \({\mathbf X}^{A}\) at the \(N_b\) prescribed boundary nodes \(A \in [1,N_b]\):

*B*by the relation

### POD-based multiscale method (FEPOD)

The idea of the proposed POD-based multiscale approach is to couple a POD-reduced RVE simulation at the micro scale to a finite element computation (without any reduction) at the macro scale. Therefore, in each evaluation point of the macroscopic system, a *reduced* RVE computation is performed to define the current stress state instead of a standard finite element computation as in the classical approach.

The macro-to-micro scale transition as explained above can be used without any changes for the reduced approach. In contrast, the RVE problem as well as the macro-to-micro scale transition differ from the classical approach.

#### Solution of reduced RVE boundary value problem

The computation is split as usual into an offline and an online part, to apply POD-based model reduction at the micro scale. The offline part is used to define a suitable lower dimensional subspace. In the computationally efficient online step the reduced computation of the RVE based on this subspace is performed.

In the **offline step**, the RVE is analyzed for each independent deformation mode separately. In the case of small deformations, we use six precomputations: the three tension modes and the three shear modes. In the case of large deformations, nine precomputations are investigated corresponding to the nine entries in the deformation gradient \({\mathbf F}_M\). The subspace is computed by using the method of snapshots [40, 41]. A snapshot is one computed solution vector \({\mathbf U}_f\) of a full-dimensional microstructural FE simulation. Obviously, for each precomputation, arbitrarily many snapshots can be stored. The choice of the number of snapshots is very much dependent on the problem under investigation. In particular, the material behavior plays a major role. In the context of inelasticity, it becomes necessary to store snapshots on loading as well as on unloading paths. Also the time dependence of the material behavior is an aspect which deserves special attention.

If the number of snapshots is lower than their dimension (which is usually the case), the method of snapshots is very efficient to compute the lower-dimensional subspace. The *l* computed solution vectors \({\mathbf U}_f\) of the unknown displacement dofs in these evaluated full-field simulations according to Eq. (14) are saved in the \(n_f \times l\)-dimensional snapshot matrix \({\mathbf D}=[{\mathbf U}_f^1,\,{\mathbf U}_f^2,\, \cdots , \, {\mathbf U}_f^l]\). The computation of the lower-dimensional subspace is based on the minimization of the approximation error of the collected snapshots and their new approximation. In addition orthogonality is claimed. These conditions lead to an eigenvalue problem of the correlation matrix \(1/l \; {\mathbf D}^T \, {\mathbf D}\). The \(n_f \times m\) projection matrix \({\varvec{\Phi }}\) is then filled with the *m* eigenvectors—multiplied by \({\mathbf D}\) and normalized—corresponding to the first *m* largest eigenvalues of the correlation matrix. The offline part has to be evaluated once for each type of RVE and has to be performed before the multiscale simulation.

**online step**of the multiscale simulation, the reduced RVE is called in each material point of the macroscopic system. Therefore, the unknown displacements of the RVE boundary value problem (14) are approximated by

*m*-dimensional unknown vector \({\mathbf U}_{red}\). Typically, the reduced dimension

*m*is much smaller than the full dimension \(n_f\): \(m \ll n_f\).

*m*instead of the full dimension \(n_f\), thus solving the RVE problem is computationally much more efficient than in the classical approach. In the case of nonlinear mechanical RVE problems, it is still necessary to evaluate all local quantities (stress and material tangent) in each evaluation point of the RVE (see for a more detailed derivation of the presently used POD method in the nonlinear context [42, 43]). In order to reduce the computational effort further, extended projection-based reduction methods have been developed (e.g. [26, 27, 44–49]). The idea of such methods is in general to significantly reduce the number of evaluation points. This important step could be also integrated into the present method but goes beyond the scope of this paper.

#### Reduced micro-to-macro scale transition

Solution scheme of FEPOD

Macroscale | Microscale | ||
---|---|---|---|

Offline | Set up RVE | ||

Apply predefined idenpendent deformation states | |||

Run and save solutions for each case | |||

Calculate subspace for each RVE separately | |||

Online | 1. Set up macroscopic model | ||

Geometry, boundary conditions | Initialize RVEs | ||

FE discretization | Load subspace | ||

Set RVEs for each material | |||

2. Run simulation | |||

Loop time or load step | |||

Apply load increment | |||

Loop Newton iteration | |||

Loop elements | |||

Loop material points | |||

Calculate \({\mathbf F}_M\) | |||

Load history variables | |||

\(\Rightarrow \) | Calculate boundary conditions for RVE | ||

Solve RVE with POD reduction | |||

If convergence is reached: | |||

Calculate \({\mathbf P}_M\) | |||

Calculate \({\mathbf C}_M\) | |||

Save local history variables globally | |||

Store \({\mathbf P}_M\), \({\mathbf C}_M\) and history | \(\Leftarrow \) | ||

End (material points) | |||

End (elements) | |||

Global assembling | |||

Check convergence | |||

If converged: BREAK | |||

End (Newton iteration) | |||

Update displacements | |||

End (time/load step) |

#### Summary of FEPOD

Table 1 summarizes the presented reduced multiscale approach. In the offline step, the RVE is analyzed for prescribed independent deformation modes separately. These are standard mechanical simulations of the microstructural boundary value problem. Based on the computed solutions (snapshots) the subspace \({\varvec{\Phi }}\) is computed for the RVE. Afterwards, in the online step the multiscale simulation with reduction is done. Therefore, in a first step the macroscopic model has to be defined. The material behavior is defined by coupling RVEs to the macroscopic material constituents, which have to be initialized. Then the nonlinear mechanical simulation of the macroscopic model starts. A load increment is applied to the macroscopic structure and the nonlinear deformation response is computed in a Newton–Raphson scheme. Thereby, a nested reduced nonlinear FE simulation of the RVE problem is called in each integration point of the macroscopic model. The current macroscopic deformation gradient sets the boundary conditions for the current reduced RVE computation. The RVE problem is then calculated using POD reduction. If the RVE boundary value problem is converged, the macroscopic stress tensor as well as the material tangent are computed by means of averaging the microscopic data using the equations derived in the previous subsections. After the problem has converged on the macro scale the next load increment is applied.

## Numerical results

### Multiscale problem settings

### Linear elastic material behavior

*i*and

*j*refer to the components of the matrix. As expected, the comparison shows a perfect agreement between the unreduced and the reduced solution (effectively zero approximation errors). Of course, this can only be reached in the case of linear elasticity where the principle of superposition holds. At the RVE scale, the POD simulation runs around 60 times faster than the unreduced simulation.

Relative displacement and relative stress deviations of the multiscale problem with a linear elastic RVE

\(u_x\) | \(u_y\) | \(u_z\) | ||||
---|---|---|---|---|---|---|

Min ( | 0 | 0 | 0 | |||

Max ( | 0 | 0 | \(1.86\cdot 10^{-8}\) |

\(\sigma _{xx}\) | \(\sigma _{yy}\) | \(\sigma _{zz}\) | \(\sigma _{xy}\) | \(\sigma _{yz}\) | \(\sigma _{zx}\) | |
---|---|---|---|---|---|---|

Min ( | 0 | 0 | \(1.98\cdot 10^{-14}\) | 0 | \(4.58\cdot 10^{-9}\) | \(2.05\cdot 10^{-8}\) |

Max ( | \(9.06 \cdot 10^{-10}\) | \(8.52\cdot 10^{-10}\) | \(1.16\cdot 10^{-9}\) | \(6.17\cdot 10^{-10}\) | \(6.30\cdot 10^{-5}\) | \(1.89\cdot 10^{-3}\) |

This first example shows the validity of the presented reduced multiscale approach, especially the computation of the consistent tangent based on the reduced stiffness matrix and its implementation in the FE\(^2\) framework of the finite element solver FEAP [31], in the context of linear elasticity.

### Elastoplastic material behavior

In the next step, we investigate a metal matrix composite. The matrix is modeled as elastoplastic material with a Young’s modulus of \(E=110300\,\mathrm {N/mm^2}\) and a Poisson’s ratio of \(\nu =0.26\). The yield stress is \(\sigma _{y}=371.5\,\mathrm {N/mm^2}\) and the linear hardening modulus is \(H=28921.5\,\mathrm {N/mm^2}\). The isotropic fibers behave linearly elastically with a Young’s modulus of \(E_f=393000\,\mathrm {N/mm^2}\) and a Poisson’ ratio of \(\nu _f=0.25\). The elastoplastic behavior of the microstructure is represented in Fig. 5. The stress-strain relation of the RVE for uniaxial loading in *x*-as well as in *y*-direction is shown. The curves show a pronounced hardening effect and a high degree of plastification.

Relative displacement, accumulated plastic strain and stress deviations of the multiscale problem with a elastoplastic RVE for different numbers of POD modes at time point of maximum loading

\(u_x\) | \(u_y\) | \(u_z\) | \(\varepsilon _{pl}\) | |||
---|---|---|---|---|---|---|

\(m=6\) | \(2.8\cdot 10^{-3}\) | \(5.5\cdot 10^{-4}\) | 16.5 | \(4.8\cdot 10^{-3}\) | ||

\(m=12\) | \(5.0\cdot 10^{-4}\) | \(1.6\cdot 10^{-4}\) | 4.5 | \(7.0\cdot 10^{-4}\) | ||

\(m=24\) | \(1.5\cdot 10^{-4}\) | \(4.5\cdot 10^{-5}\) | 0.7 | \(1.5\cdot 10^{-5}\) |

\(\sigma _{xx}\) | \(\sigma _{yy}\) | \(\sigma _{zz}\) | \(\sigma _{xy}\) | \(\sigma _{xz}\) | \(\sigma _{yz}\) | |
---|---|---|---|---|---|---|

\(m=6\) | \(1.3\cdot 10^{-2}\) | \(2.5\cdot 10^{-3}\) | \(4.3\cdot 10^{-3}\) | \(1.1\cdot 10^{-2}\) | 12.8 | 1.0 |

\(m=12\) | \(1.9\cdot 10^{-3}\) | \(7.4\cdot 10^{-4}\) | \(9.5\cdot 10^{-4}\) | \(2.4\cdot 10^{-3}\) | 2.3 | 0.4 |

\(m=24\) | \(6.6\cdot 10^{-5}\) | \(2.9\cdot 10^{-4}\) | \(5.3\cdot 10^{-5}\) | \(2.2\cdot 10^{-4}\) | 0.15 | 0.19 |

Relative displacement, accumulated plastic strain and stress deviations of the multiscale problem with a elastoplastic RVE for different numbers of POD modes after unloading

\(u_x\) | \(u_y\) | \(u_z\) | \(\varepsilon _{pl}\) | |||
---|---|---|---|---|---|---|

\(m=6\) | \(2.1\cdot 10^{-1}\) | \(2.2\cdot 10^{-1}\) | 289 | \(1.1\cdot 10^{-1}\) | ||

\(m=12\) | \(3.0\cdot 10^{-2}\) | \(1.4\cdot 10^{-2}\) | 13.14 | \(2.3\cdot 10^{-2}\) | ||

\(m=24\) | \(3.0\cdot 10^{-2}\) | \(3.8\cdot 10^{-2}\) | 18.23 | \(2.7\cdot 10^{-2}\) |

\(\sigma _{xx}\) | \(\sigma _{yy}\) | \(\sigma _{zz}\) | \(\sigma _{xy}\) | \(\sigma _{xz}\) | \(\sigma _{yz}\) | |
---|---|---|---|---|---|---|

\(m=6\) | \(7.6\cdot 10^{-2}\) | \(2.1\cdot 10^{-1}\) | \(2.8\cdot 10^{-2}\) | \(6.2\cdot 10^{-2}\) | 43 | 1.99 |

\(m=12\) | \(7.0\cdot 10^{-3}\) | \(5.1\cdot 10^{-2}\) | \(3.8\cdot 10^{-2}\) | \(2.2\cdot 10^{-2}\) | 5.6 | 0.165 |

\(m=24\) | \(1.0\cdot 10^{-2}\) | \(3.9\cdot 10^{-2}\) | \(3.9\cdot 10^{-2}\) | \(2.5\cdot 10^{-2}\) | 1.28 | 0.81 |

To summarize, the second example of a metal matrix composite shows that the proposed FEPOD multiscale approach leads to good results with errors smaller \(5\%\) for the relevant data and a high speed-up of around 180 (\(m=12\)). Note also that working with the FEPOD approach does not significantly change the number of iterations per Newton step compared to the FE\(^2\) approach. Similarly to the FE\(^2\) approach, the FEPOD method shows quadratic convergence in the Newton iteration.

### Hyperelastic material behavior

The derived FEPOD method is finally applied to a hyperelastic microstructure. In this example a simplified microstructure with four cubic material domains and 384 free degrees-of-freedom is investigated. Two of them contain a rather stiff material with a Young’s modulus of \(100\,\mathrm {N/mm^2}\) and a Poisson’s ratio of 0.3. The other two are filled with a soft material (Young’s modulus equal to \(1\,\mathrm {N/mm^2}\), Poisson’s ratio equal to 0.45).

As expected, the error decreases with increasing number of POD modes. Using around 27 POD modes yield errors in the range of \(10^{-3}\), which promises good results in the reduced multiscale simulation.

Relative displacement and relative stress error of the multiscale problem with a hyperelastic RVE

| \(u_3\) | \(\sigma _{33}\) |
---|---|---|

1000 | 0.000 | 0.000 |

2000 | 0.000 | 0.008 |

3000 | 0.009 | 0.010 |

4000 | 0.028 | 0.024 |

5000 | 0.048 | 0.045 |

6000 | 0.071 | 0.064 |

## Conclusions

This paper presents a new multiscale concept FEPOD where a reduced RVE computation is embedded into a finite element-based macro-mechanical simulation. Therefore, the POD reduction is carried out for the micro-mechanical boundary value problem. After this offline step, the RVE is represented by only a few POD modes. In this way it is possible to decrease the computational demands of a multiscale simulation enormously. Furthermore, the consistent macroscopic tangent for the FEPOD has been derived.

In case of a linear elastic composite, the FEPOD approach with only six POD modes leads to the same accuracy as a classical FE\(^2\) approach. This is due to the principle of superposition which holds for linear elasticity. In case of plasticity, a few more POD modes are necessary to reach good accuracy. It is shown that also unloading phases are approximated in a very good way. In the investigated examples, speed-ups of around 250 times are possible while maintaining a high level of accuracy with errors less than \(1\%\) on the micro scale and less than \(5\%\) on the macro scale. Furthermore, a first test example concerning hyperelastic RVEs is shown. The example shows the successful application of FEPOD to large deformation. Of course, this topic has to be investigated in more detail in further research.

The computation of the reduced modes in the offline step is one important part of the proposed FEPOD approach, because the quality of the computed subspace significantly influence the accuracy of the reduced computation. In that offline step, the RVE has to be solved at least for the independent deformation states. This set-up was shown in the examples. In some cases, more than the six or nine independent deformation states have to be precomputed. This holds in particular for the case of highly nonlinear behavior where it is not possible to work with the assumption of superposition. It will be addressed in further investigations which states are most useful and how these states can be chosen. Also updating strategies of the modes on the fly during a FEPOD simulation should be further explored.

In summary, the proposed FEPOD method is able to increase the computational efficiency of a finite element based multiscale simulation significantly while maintaining a high level of accuracy. Therefore it is a promising approach for future research in multiscale modeling of not only fiber-reinforced composites but also other heterogeneous materials.

### Acknowledgements

The presented work was mainly done during a research stay founded by a Heinrich Hertz scholarship at the NASA Glenn Research Center at Lewis Field in Cleveland, Ohio. The authors thank the Ministry for Innovation, Science, Research and Technology, which awarded the scholarship, as well as the head of the Mechanics and Life Prediction Branch of NASA Glenn Research Center Dr. S. M. Arnold and his colleagues. Further, the financial support of project RE1057/32-1 within the priority program SPP 1542 funded by the German Science Foundation (DFG) is gratefully acknowledged.

### Competing interests

The authors declare that they have no competing interests.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Feyel F, Chaboche J-L. Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput Methods Appl Mech Eng. 2000;183(3–4):309–30. doi:10.1016/S0045-7825(99)00224-8.View ArticleMATHGoogle Scholar
- Geers MGD, Kouznetsova VG, Brekelmans WAM. Multi-scale computational homogenization: trends and challenges. J Comput Appl Math. 2010;234(7):2175–82. doi:10.1016/j.cam.2009.08.077. Fourth International Conference on Advanced Computational Methods in Engineering (ACOMEN 2008).
- Ladevèze P, Loiseau O, Dureisseix D. A micro-macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Methods Eng. 2001;52(1–2):121–38. doi:10.1002/nme.274.View ArticleGoogle Scholar
- Ladevèze P, Nouy A. On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput Methods Appl Mech Eng. 2003;192(28–30):3061–87. Multiscale Computational Mechanics for Materials and Structures.Google Scholar
- Aboudi J. Micromechanical analysis of thermo-inelastic multiphase short-fiber composites. Compos Eng. 1995;5:839–50.View ArticleGoogle Scholar
- Aboudi J, Arnold SM, Bednarcyk BA. Micromechanics of composite materials: a unified multiscale analysis approach. Amsterdam: Elsevier; 2013.Google Scholar
- Michel JC, Suquet P. Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput Methods Appl Mech Eng. 2004;193(48–51):5477–502. doi:10.1016/j.cma.2003.12.071. Advances in Computational Plasticity.
- Michel JC, Suquet P. Nonuniform transformation field analysis. Int J Solids Struct. 2003;40(25):6937–55. doi:10.1016/S0020-7683(03)00346-9 (Special issue in Honor of George J. Dvorak).MathSciNetView ArticleMATHGoogle Scholar
- Dvorak GJ, Benveniste Y. On transformation strains and uniform fields in multiphase elastic media. The Royal Society; 1992. pp. 291–310.Google Scholar
- Fritzen F, Böhlke T. Reduced basis homogenization of viscoelastic composites. Compos Sci Technol. 2013;76:84–91. doi:10.1016/j.compscitech.2012.12.012.View ArticleGoogle Scholar
- Fritzen F, Leuschner M. Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng. 2013;260:143–54. doi:10.1016/j.cma.2013.03.007.MathSciNetView ArticleMATHGoogle Scholar
- Fritzen F, Hodapp M, Leuschner M. GPU accelerated computational homogenization based on a variational approach in a reduced basis framework. Comput Methods Appl Mech Eng. 2014;278:186–217. doi:10.1016/j.cma.2014.05.006.MathSciNetView ArticleGoogle Scholar
- Largenton R, Michel J-C, Suquet P. Extension of the nonuniform transformation field analysis to linear viscoelastic composites in the presence of aging and swelling. Mech Mater. 2014;73:76–100. doi:10.1016/j.mechmat.2014.02.004.View ArticleGoogle Scholar
- Oskay C, Fish J. Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Comput Methods Appl Mech Eng. 2007;196(7):1216–43. doi:10.1016/j.cma.2006.08.015.MathSciNetView ArticleMATHGoogle Scholar
- Sparks P, Oskay C. Identification of optimal reduced order homogenization models for failure of heterogeneous materials. Int J Multiscale Comput Eng. 2013;11(3):185–200.View ArticleGoogle Scholar
- Chinesta F, Ammar A, Cueto E. Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng. 2010;17:327–50. doi:10.1007/s11831-010-9049-y.MathSciNetView ArticleMATHGoogle Scholar
- Ladevèze P, Passieux JC, Néron D. The multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng. 2010;199(21—-22):1287–96.MathSciNetView ArticleMATHGoogle Scholar
- Cremonesi M, Néron D, Guidault P-A, Ladevèze P. A pgd-based homogenization technique for the resolution of nonlinear multiscale problems. Comput Methods Appl Mech Eng. 2013;267:275–92. doi:10.1016/j.cma.2013.08.009.MathSciNetView ArticleMATHGoogle Scholar
- El Halabi F, González D, Chico A, Doblaré M. Fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition. Comput Methods Appl Mech Eng. 2013;257:183–202. doi:10.1016/j.cma.2013.01.011.View ArticleMATHGoogle Scholar
- Boyaval S. Reduced-basis approach for homogenization beyond the periodic setting. Multiscale Model Simul. 2008;7(1):466–94. doi:10.1137/070688791.
- Kerfriden P, Ródenas JJ, Bordas SP-A. Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error. Int J Numer Methods Eng. 2014;97(6):395–422. doi:10.1002/nme.4588.MathSciNetView ArticleGoogle Scholar
- Hesthaven JS, Zhang S, Zhu X. Reduced basis multiscale finite element methods for elliptic problems. Multiscale Model Simul. 2015;13(1):316–37. doi:10.1137/140955070.
- Abdulle A, Bai Y. Adaptive reduced basis finite element heterogeneous multiscale method. Comput Methods Appl Mech Eng. 2013;257:203–20. doi:10.1016/j.cma.2013.01.002.MathSciNetView ArticleMATHGoogle Scholar
- Goury O, Kerfriden P, Bordas S. Bridging analytical and computational homogenisation for nonlinear multiscale problems: a reduced order modelling approach for a damage problem. Technical report; 2014. https://hal.archives-ouvertes.fr/hal-00994923
- Miled B, Ryckelynck D, Cantournet S. A priori hyper-reduction method for coupled viscoelastic-viscoplastic composites. Comput Struct. 2013;119:95–103. doi:10.1016/j.compstruc.2012.11.017.View ArticleGoogle Scholar
- Ryckelynck D. A priori hyperreduction method: an adaptive approach. J Comput Phys. 2005;202(1):346–66. doi:10.1016/j.jcp.2004.07.015.MathSciNetView ArticleMATHGoogle Scholar
- Hernández JA, Oliver J, Huespe AE, Caicedo MA, Cante JC. High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng. 2014;276:149–89. doi:10.1016/j.cma.2014.03.011.MathSciNetView ArticleGoogle Scholar
- Monteiro E, Yvonnet J, He QC. Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Comput Mater Sci. 2008;42(4):704–12. doi:10.1016/j.commatsci.2007.11.001.View ArticleGoogle Scholar
- Yvonnet J, He Q-C. The reduced model multiscale method (r3m) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys. 2007;223(1):341–68. doi:10.1016/j.jcp.2006.09.019.MathSciNetView ArticleMATHGoogle Scholar
- Xia L, Breitkopf P. A reduced multiscale model for nonlinear structural topology optimization. Comput Methods Appl Mech Eng. 2014;280:117–34. doi:10.1016/j.cma.2014.07.024.MathSciNetView ArticleGoogle Scholar
- Taylor RL. Feap—a finite element analysis program, Version 8.3 User manual. University of California at Berkeley; 2011. www.ce.berkeley.edu/projects/feap. Accessed 05 Apr 2011.
- Feyel F. Multiscale FE2 elastoviscoplastic analysis of composite structures. Comput Mater Sci. 1999;16(1–4):344–54. doi:10.1016/S0927-0256(99)00077-4.View ArticleGoogle Scholar
- Kouznetsova VG. Computational homogenization for the multi-scale analysis of multi-phase materials. PhD thesis, Technische Universiteit Eindhoven; 2002.Google Scholar
- Miehe C, Koch A. Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech. 2002;72(4):300–17. doi:10.1007/s00419-002-0212-2.View ArticleMATHGoogle Scholar
- Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct. 2003;40(13–14):3647–79. doi:10.1016/S0020-7683(03)00143-4.View ArticleMATHGoogle Scholar
- van der Sluis O, Schreurs PJG, Brekelmans WAM, Meijer HEH. Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling. Mechan Mater. 2000;32(8):449–62.View ArticleGoogle Scholar
- Terada K, Hori M, Kyoya T, Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approach. Int J Solids Struct. 2000;37(16):2285–311.View ArticleMATHGoogle Scholar
- Kouznetsova V, Brekelmans WAM, Baaijens FPT. An approach to micro-macro modeling of heterogeneous materials. Comput Mech. 2001;27:37–48.View ArticleMATHGoogle Scholar
- Reese S, Wriggers P. A finite element method for stability problems in finite elasticity. Int J Numer Methods Eng. 1995;38(7):1171–200.MathSciNetView ArticleMATHGoogle Scholar
- Breuer KS, Sirovich L. The use of the karhunen-loéve procedure for the calculation of linear eigenfunctions. J Comput Phys. 1991;96(2):277–96. doi:10.1016/0021-9991(91)90237-F.MathSciNetView ArticleMATHGoogle Scholar
- Sirovich L. Chaotic dynamics of coherent structures. Phys D Nonlinear Phenom. 1989;37(1–3):126–45. doi:10.1016/0167-2789(89)90123-1.MathSciNetView ArticleGoogle Scholar
- Radermacher A, Reese S. A comparison of projection-based model reduction concepts in the context of nonlinear biomechanics. Arch Appl Mech. 2013;83(8):1193–213. doi:10.1007/s00419-013-0742-9.View ArticleGoogle Scholar
- Radermacher A, Reese S. Proper orthogonal decomposition-based model reduction for non-linear biomechanical analysis. Int J Mater Eng Innov. 2013;4(2):149–65. doi:10.1504/IJMATEI.2013.054393.View ArticleGoogle Scholar
- Astrid P, Weiland S, Willcox K, Backx T. Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control. 2008;53:2237–51. doi:10.1109/TAC.2008.2006102.MathSciNetView ArticleGoogle Scholar
- Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737–64. doi:10.1137/090766498.MathSciNetView ArticleMATHGoogle Scholar
- Grepl MA. Model order reduction of parametrized nonlinear reaction-diffusion systems. Comput Chem Eng. 2012;43:33–44. doi:10.1016/j.compchemeng.2012.03.013.View ArticleGoogle Scholar
- Nguyen NC, Patera AT, Peraire J. A ‘best points’ interpolation method for efficient approximation of parametrized functions. Int J Numer Methods Eng. 2008;73(4):521–43. doi:10.1002/nme.2086.MathSciNetView ArticleMATHGoogle Scholar
- Carlberg K, Bou-Mosleh C, Farhat C. Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Int J Numer Methods Eng. 2011;86(2):155–81. doi:10.1002/nme.3050.MathSciNetView ArticleMATHGoogle Scholar
- Farhat C, Avery P, Chapman T, Cortial J. Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int J Numer Methods Eng. 2014;98(9):625–62. doi:10.1002/nme.4668.MathSciNetView ArticleGoogle Scholar
- Stier B, Bednarcyk BA, Simon JW, Reese S. Investigation of micro-scale architectural effects on damage of composites. NASA/TM–2015-218740; National Technical Information Service; 2015.Google Scholar
- Stier B, Simon JW, Reese S. Numerical and experimental investigation of the structural behavior of a carbon fiber reinforced ankle-foot orthosis. Med Eng Phys. 2015;37:505–11.View ArticleGoogle Scholar