Uncertainty quantification in multiscale simulation of woven fiber composites

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Abstract

Woven fiber composites have been increasingly employed as light-weight materials in aerospace, construction, and transportation industries due to their superior properties. These materials possess a hierarchical structure that necessitates the use of multiscale simulations in their modeling. To account for the inherent uncertainty in materials, such simulations must be integrated with statistical uncertainty quantification (UQ) and propagation (UP) methods. However, limited advancement has been made in this regard due to the significant computational costs and complexities in modeling spatially correlated structural variations coupled at different scales. In this work, a non-intrusive approach is proposed for multiscale UQ and UP to address these limitations. We introduce the top-down sampling method that allows to model non-stationary and continuous (but not differentiable) spatial variations of uncertainty sources by creating nested random fields (RFs) where the hyperparameters of an ensemble of RFs is characterized by yet another RF. We employ multi-response Gaussian RFs in top-down sampling and leverage statistical techniques (such as metamodeling and dimensionality reduction) to address the considerable computational costs of multiscale simulations. We apply our approach to quantify the uncertainty in a cured woven composite due to spatial variations of yarn angle, fiber volume fraction, and fiber misalignment angle. Our results indicate that, even in linear analysis, the effect of uncertainty sources on the material’s response could be significant.

Introduction

It is widely accepted that materials are heterogeneous and possess a hierarchical structure where the coarse-scale behavior is greatly affected by the fine-scale details (i.e., the microstructure). Because the traditional one-scale continuum mechanics does not suffice to investigate the effect of microstructure on materials’ properties, significant effort has been devoted to the development of multiscale computational models. These models have provided the means to study the effect of microstructure on many phenomena including damage evolution [[1], [2]], fracture initiation [3], and strain localization [4].

Uncertainty is inevitably introduced in materials’ behavior starting from the design and constituent selection stages, through the manufacturing processes, and finally during operation. For this reason, ever-growing research [[5], [6], [7]] is being conducted to rigorously couple computational models with statistical uncertainty quantification (UQ) and propagation (UP) methods to provide probabilistic predictions that are in line with the observed stochasticity in materials.

UQ and UP are actively pursued in various fields of science and engineering [[6], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]]. They are, however, seldom applied to multiscale simulations due to the significant computational costs and complexities. Our goal is to devise a non-intrusive UQ and UP approach that characterizes the uncertainties via random fields (RFs) and is applicable to multiscale simulations where multiple uncertainty sources (including spatial microstructural variations) arise from different length-scales. We are particularly interested in a non-intrusive approach because not only they are more general, but also opening multiscale computer models (i.e., changing the formulations) to directly introduce uncertainty sources into them requires considerable effort.

We take woven fiber composites as our motivating example. Such materials have been increasingly used in aerospace, construction, and transportation industries due to their superior properties including high strength-to-weight ratio, non-corrosive behavior [24], enhanced dimensional stability [25], and high impact resistance [26]. Woven fiber composites possess, as illustrated in Fig. 1, a hierarchical structure that spans multiple length-scales. Macroscale is at the highest length-scale where the overall mechanical performance under some loading conditions is evaluated and characteristics such as fiber and yarn (aka tow or bundle) volume fractions, effective properties, and part geometry are of interest. The individual yarns and their architecture (i.e., their dimension and relative spatial arrangement, see Fig. 2 (a) and (b)) in the laminates are modeled in the mesoscale. The fibers (and their relative position within the yarns) and the matrix belong to the microscale. Finally, the constituent properties and the interaction between them (e.g., the interphase) are modeled in the nanoscale.

Multiple uncertainty sources that must be considered in the computational models are introduced at each of these length-scales [24]. For instance, during the preforming process, the high pressure and flow of the resin or draping change the local architecture of the fibers [27], see Fig. 2 (c). Additionally, processing variations and material imperfections cause the fiber volume fraction to spatially vary over the part; especially along the yarn path where there is compact contact [[28], [29]]. These macroscopic uncertainties are manifestations of a multitude of uncertainty sources that exist at the finer scales where, due to the delicacy of materials, the number and dimensionality of the uncertainty sources increase [10]. Moving down the scale ladder in Fig. 1, one can observe that the uncertainty sources are of different nature (e.g., morphological, geometrical, or property related), and spatially (within and across scales) correlated. These features render the UQ of a macroscopic quantity of interest extremely challenging. It is further noted that macroscopic uncertainty depends heavily on the quantity of interest. For example, the homogenized response in linear analyses and damage are clearly not equally sensitive to spatial microstructural variations.

To illustrate the various uncertainty sources along with the abovementioned challenges associated with UQ and UP tasks in multiscale materials we narrow our focus, as depicted in Fig. 3, to two scales from the composite in Fig. 1. In the finite element (FE) mesh of the mesostructure in Fig. 3, there are over 10 000 integration points (IP’s, aka Gauss or material points). Each of these points can be thought of as a realization of a sufficiently large structure (aka representative volume element, RVE) at the next finer scale, i.e., the microscale. A widely-recognized method for modeling the behavior of such a hierarchical material is through computational homogenization. This method involves the solution of partial differential equations (PDEs) at both scales where the boundary conditions of the microscale PDEs are determined at the mesoscale. Once the microscale PDEs are solved, the homogenized stresses and tangent moduli are passed up to be used in the PDEs at the mesoscale. If the FE method is used in both scales to solve the boundary value problems, this method is referred to as FE2 [[32], [33]]. Since the FE2 method is generally very expensive, many recent works have focused on more efficient methods [[34], [35], [36], [37]].

In a deterministic simulation, all the parameters (and their evolution) at the IPs in both scales (related to, e.g., the material properties, the morphology, or the applied load) are known a priori. In the presence of uncertainty, however, such parameters may only be characterized statistically. For instance, in the mesoscopic RVE in Fig. 3, the dimensions and relative arrangement of the yarns, the fiber volume fraction, and material properties (such as Young’s modulus and Poisson ratio) might vary spatially due to multiple factors such as manufacturing inconsistencies. These spatial variations across the IPs of the mesostructure imply that the corresponding microstructures are not identical. More importantly, these mesoscale variations are spatially correlated. For instance, the data reported in [[27], [28], [38]] demonstrate that in unidirectionally reinforced fiber composites (with either metallic or polymeric matrix) not only the fiber misalignment angles and volume fraction change spatially, but also these changes are negatively correlated (i.e., high volume fraction implies small misalignment angle and vice versa). Similarly, some of the properties and parameters at the microstructures might be spatially inhomogeneous. Hence, to quantify the mesoscale uncertainty, the microscale stochasticity must be quantified, propagated, and finally coupled with uncertainty sources at the mesoscale.

One way to quantify the uncertainty of an effective coarse-scale property (e.g., shear or Young’s moduli), is via Monte Carlo (MC) sampling where many simulations are conducted to find the distributional characteristics. However, MC sampling is practically infeasible in multiscale simulations due to (i) the curse of dimensionality that arises from the number of IPs (i.e., degrees of freedom), (ii) the nested nature of the uncertainty sources, and (iii) the high cost of multiscale simulations. For example, in the two-scale structure in Fig. 3, a single MC sample would consist of:

  • 1.

    Assigning correlated parameters (morphological, geometrical, property related, and so on) to all the IPs of the mesostructure.

  • 2.

    Generating microstructures where each one is consistent with the assigned parameters to the corresponding IP in the mesoscale.

  • 3.

    Conducting a multiscale simulation via, e.g., the FE2 method.

As the number of scales and uncertainty sources increase (see Fig. 1), the computational costs of UQ and UP analyses rapidly increase. More efficient sampling techniques (such as quasi MC, importance sampling, subset simulation, line sampling, and variants of Latin hypercube sampling [[39], [40], [41]]) suffer from the same issue. Since the use of multiscale materials in science and engineering is rapidly growing, there is a clear need [[6], [42]] for efficient methods and frameworks to quantify the coarse-scale uncertainty of such materials as a function of the fine-scale uncertainties. We believe that our approach paves the path to this end by leveraging the physics of the problem at hand in addition to the statistical UQ and UP methods. With our approach, fewer simulations could be used to determine the distributional characteristics.

There has been some work focused on UQ and UP in fibrous composites. Komeili and Milani [24] conducted a two-level factorial design at the mesoscale to illustrate the sensitivity of orthogonal woven fabrics to the yarn geometry (including yarn spacing, width, and height) and material properties. They concluded that, based on the applied load, these parameters could have a significant effect on the effective response (which was taken as the reaction force). A similar sensitivity study based on Sobol’s indices was conducted in [12] to demonstrate that the friction coefficient and yarn height significantly affect the macroscale mechanical response of interest in dry woven fabrics. In [[12], [24]], yarn properties do not spatially vary and fibers are well aligned. Savvas et al. [[43], [44]] studied the required RVE size as a function of spatial variations of volume fraction and fiber architecture. They concluded that the RVE size must increase at higher volume fractions and, additionally, fiber orientation is more important than waviness in determining the mesoscale RVE size. In another work [17] they also illustrated that geometrical characteristics (i.e., the shape and arrangement of the fibers) and the material properties (Young’s moduli of the constituents) affect the homogenized response (average axial and shear stiffness) in unidirectional (UD) composites quite significantly (with the former being more important). The variations were shown to decrease as the number of fibers and RVE size increased. Vanaerschot et al. [27] also studied the variability in composite materials’ properties and concluded that the stiffness in the mesoscale RVE is affected by the load orientation [38] and, additionally, it significantly decreases as the fiber misalignment (see Fig. 2 (c)) increases. In a series of works, Hsiao and Daniel [[45], [46], [47]] experimentally and theoretically investigated the effect of fiber waviness in UD composites. They demonstrated that under uniaxial compression loading the stiffness and strength decrease.

The focus of most prior works on UQ and UP in multiscale materials, as briefly reviewed above in the case of woven composites, has not been placed on rigorously modeling the uncertainty sources and statistically propagating their effects across multiple scales. For instance, modeling spatial variations via RFs, connecting them across different spatial scales, and investigating stochastic simulations are often neglected. On the contrary, as we review in Section 2, significant progress has been made in rigorous UQ and UP analyses in single-scale materials. Our goal is to exploit the strength of these two bodies of literature to devise a flexible approach for statistical uncertainty quantification and propagation in multiscale materials. To this end, we introduce the top-down sampling method that allows to model non-stationary and C0 (i.e., continuous but not differentiable) RFs at fine length-scales (i.e., mesoscale and microscale) with a stationary and differentiable RF at the macroscale. We motivate the use of multi-response Gaussian processes to quantify the RFs and conduct sensitivity analyses for dimensionality reduction. The resulting approach is non-intrusive (in that the computational models need not be adapted to account for the uncertainties) and can leverage statistical techniques (such as metamodeling and dimensionality reduction) to address the considerable computational costs of multiscale simulations.

The rest of the paper is organized as follows. In Section 2, we review the methods and tools for single-scale UQ to motivate the need for quantifying the spatial variations via RFs. Our proposed approach is introduced in Section 3 which, in Section 4, is employed to quantify the macroscale uncertainty in a multiscale cured woven composite sample. We conclude the paper and provide future directions in Section 5.

Section snippets

Methodological base for uncertainty quantification and propagation

As opposed to multiscale materials such as fibrous composites, there are many works focused on UQ and UP in single-scale materials. Broadly speaking, there are three major methods for UQ and UP in computational mechanics’ literature which use MC sampling, the perturbation approach, or the spectral stochastic approach. These three methods start with the random field representation of uncertainty sources. Next, the characterized field is discretized to assign random (but correlated) values to the

The proposed approach for multiscale UQ and UP

As illustrated in Fig. 5, our approach for multiscale UQ and UP has two main stages: Intra-scale uncertainty quantification and inter-scale uncertainty propagation. We start by identifying the uncertainty sources at each scale and modeling them via RFs where one RF is associated with each structure realization (see Section 3.2 for the details). We employ RFs with sensible (i.e., physically interpretable) parameters for three main reasons i to easily couple the uncertainty sources across the

Application of the proposed approach in cured woven fiber composites

As argued in Section 1, woven fiber composites possess a hierarchical structure where multiple sources of uncertainty exist across the scales. In this section, we follow the steps of our approach to quantify the macroscale uncertainty in the elastic response of a cured woven composite as a function of spatial variations in the fiber volume fraction (microscale and mesoscale), yarn angle, and fiber misalignment angles (mesoscale). As illustrated in Fig. 6 (a), the structure is composed of four

Conclusion and Future Works

Limited advancement has been made in integrating statistical uncertainty quantification (UQ) and propagation (UP) methods with multiscale simulations. Herein, we have introduced an approach for multiscale UQ and UP that models the uncertainty sources with multi-response Gaussian processes (MRGPs) and employs statistical techniques (such as metamodeling and sensitivity analyses) to address the computational costs. Our choice of random field enables manageable uncertainty quantification by

Acknowledgment

This work was sub-contracted from Ford Motor Company which has received the award from US Department of Energy (Award Number: DE-EE0006867).

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