On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties

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ScienceDirect Comput. Methods Appl. Mech. Engrg. 350 (2019) 432–461 www.elsevier.com/locate/cma

On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties Dmytro Pivovarov ∗, Kai Willner, Paul Steinmann Chair of Applied Mechanics, University of Erlangen—Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany Received 2 August 2018; received in revised form 17 January 2019; accepted 18 February 2019 Available online 16 March 2019

Highlights      

A unified formulation for fuzzy, stochastic, and fuzzy–stochastic FEM is presented. The inner product of fuzzy variables is discussed. We introduce fuzzy–stochastic FEM with local bases in non-deterministic dimensions. Incorporation of the XFEM into the fuzzy–stochastic FEM is proposed. A novel fuzzy FEM output representation is proposed. A severe dimension reduction for imprecise probabilities is proposed.

Abstract In this work we review a unified formulation for spectral fuzzy, spectral stochastic, and spectral fuzzy–stochastic FEM. We propose some modifications for the fuzzy and fuzzy–stochastic FEM involving local bases in the non-deterministic dimensions and the incorporation of extended FEM into the non-deterministic FEM. These modifications were previously used for stochastic FEM only. We discuss advantages of the proposed techniques for problems with uncertainties in the geometry and demonstrate their application to computational homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. We address also some theoretical aspects of fuzzy FEM, that have not been tackled in the literature, namely the inner product of fuzzy variables. We address also another important problem, which becomes often the target of criticism of fuzzy approach. The membership function of the fuzzy input has often some degree of arbitrariness: it is known only approximately or it is constructed based on some assumptions. Here we propose a fuzzy FEM representation, which requires only the modal value and support of the fuzzy input parameters and is completely independent of the membership function’s shape. We also discuss the case of imprecise probabilities which allows for severe dimension reduction of the non-deterministic problem if the accurate spectral simulation technique is used. Application of dimension reduction is also demonstrated on the example of computational homogenization. c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Stochastic FEM; Fuzzy FEM; eXtended FEM; Polymorphic uncertainties; Imprecise probability; Random geometry

∗

Corresponding author. E-mail address: [email protected] (D. Pivovarov).

https://doi.org/10.1016/j.cma.2019.02.024 c 2019 Elsevier B.V. All rights reserved. 0045-7825/⃝

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1. Introduction 1.1. Classes of uncertainties Uncertainties in real systems and specifically in material’s microstructure result from different sources. Classically they are divided in two big groups: aleatoric and epistemic. Aleatoric uncertainty results from the natural variability of process realizations. This interpretation of aleatoric uncertainty is very close to the frequentist interpretation of probability, thus in the case of sufficient experimental data available, this uncertainty can be described using probabilistic methods. Aleatoric uncertainties are non-reducible by increasing the experimental effort. No further experimental study will reduce the range of variation. On the contrary, epistemic uncertainties are reducible, they result from insufficient knowledge, inaccurate measurements, round-off errors, noise in experimental data, imperfection of the theoretical model used to describe the real process, etc. It is well appreciated that the possibility theory [1] is an appropriate tool for problems involving epistemic uncertainty [2–5]. By definition, epistemic uncertainties do not possesses any known probability distribution function. In some cases only the interval of possible parameter values is available. Obviously, interval analysis is the appropriate tool here. In case when more information is available, the more precise but also computationally more expensive fuzzy analysis is involved. Let us consider, e.g. some nonlinear physical process, which is studied experimentally. In order to use empirical data in simulations it is fitted with some function. Usually we want to keep the fitting function as simple as possible, i.e. we want to avoid overfitting, and further investments in experiments (in order to reduce noise) are not reasonable. Due to the fact that the fitting curve never coincides with experimental data, we want to know, how the changes in the function parameters (different fits) may influence the results. In this case we consider iteratively increasing intervals of confidence for the fitting parameters. The smallest interval of confidence (i.e. the case of highest confidence) is just a point — the crisp or modal (main) value obtained from fitting. The largest interval of confidence (i.e. the case of least confidence ) is the smallest interval covering all experimental data. The set of nested and ordered intervals results naturally in a fuzzy description [2,3]. Another example of epistemic uncertainty, where fuzziness appears, is the case of linguistic variables (no objective distribution available) and approximate reasoning [6,7] — a common subject in the fuzzy logic community. Stochastic analysis has proven to be very efficient for aleatoric uncertainties with precisely known probabilistic input [8]. Probability based optimal design and risk assessments should be preferred in many cases [9–11], especially for large systems including many components. However in the case when no probabilistic input is given or the data is insufficient, the possibilistic treatment (fuzzy analysis) of aleatoric uncertainties is involved [8,12]. The basic idea is as follows: if the input data is not sufficient for a precise estimation of the input’s probability distribution (imprecise probabilities), a family of probability distributions is considered with upper and lower bounds that are commonly presented by possibility and necessity functions [2,13–18]. Note that the possibility and necessity functions are not the most general representations of probability bounds [19] but the simplest one [14], thus the most widely used. Possibilistic treatment is in this case a coarser tool than probabilistic methods due to the extra uncertainties hidden in the spread between possibility and necessity functions. Another approach to simulate imprecise probabilities is to consider the entire family of probability distributions and parametrize it by considering the parameter values to be also uncertain. This approach results in some combination of probabilistic methods augmented with fuzzy or interval arithmetic [4,20–26]. This hybrid simulation approach results in a much higher precision compared to the upper–lower bound estimation in the possibilistic modeling [23], however it requires much more computations due to the increased problem’s dimensionality and also needs additional input information. So, e.g. the principal distribution model and the possible parameter variation should be known. Remark. The transformation between probabilistic and possibilistic descriptions is often addressed in the literature. This transformation is possible, but it results in information loss or insertion, wherein the inserted information is always arbitrary to some degree [2,27]. From an engineering point of view both methods differ drastically [5]. The required input and output quantities are completely different in stochastic and fuzzy simulations. It is demonstrated in the literature through numerous examples that probabilistic and possibilistic approaches may result in very different conclusions regarding the system’s failure and optimal design [9–11].

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Imprecise probabilities are only one case of the more general class of polymorphic uncertainties. Polymorphic uncertainties are the combination of both epistemic and aleatoric uncertainties. Any system involving separately epistemic and aleatoric uncertainty in its different components represents also the case of polymorphic uncertainty. 1.2. Simulation techniques Simulation techniques used for both the stochastic and the fuzzy problem may be classified in two big groups: spectral and sampling based. Spectral techniques for stochastic problems, namely the stochastic Galerkin based FEM (SFEM), was proposed around 1991 [28]. On the contrary, the spectral version of fuzzy FEM (FFEM) was proposed only very recently in 2014 [29]. Further developments of spectral FFEM are presented in [30]. For recent developments in spectral SFEM we refer to [31–38]. A classical numerical approach for uncertainty propagation is the sampling based approach. For stochastic problems the basic method is the Monte-Carlo simulation. Further developments are the non-intrusive polynomial chaos and stochastic collocation method [39–42] which are sometimes denoted as pseudo-spectral, because they implicitly utilize the spectral expansion. Sampling methods for fuzzy problems are the general transformation method [43–45], sparse grids [46] and the optimization method [47]. In contrast to stochastic problems the sampling is here used to find the output’s global maximum and minimum on nested intervals, thus presenting a completely different mathematical task. In this contribution we define SFEM exclusively as spectral Galerkin-type techniques, other methods often referred to as SFEM like, e.g. perturbation methods [48–50], are not considered here. In order to avoid ambiguity, sampling based techniques are also denoted separately. After the first introduction of spectral SFEM a number of modifications were proposed, namely the stochastic local FEM (SL-FEM) and the stochastic extended FEM (SX-FEM) [35,37,51–56]. Application of these methods to the computational homogenization of heterogeneous materials with random microstructure [57] demonstrated their huge potential and flexibility. To our knowledge corresponding generalization of the spectral fuzzy FEM, namely the fuzzy local FEM (FL-FEM) and the fuzzy extended FEM (FX-FEM), were not introduced yet. Thus their formulation is one of the central goals of this paper. In the case of imprecise probabilities and more generally in the case of polymorphic uncertainties the mixed fuzzy–stochastic or interval–stochastic approach is used. In most applications some sampling based methods were utilized. Recently a fully spectral formulation of fuzzy–stochastic problem was introduced [58]. However, still many questions remain open, like, e.g. the definition of an inner product for the fuzzy parameters, the interpretation of the fuzzy output, and the advisability of spectral methods for hybrid problems. The second part of this paper is addressing these challenging questions. The well-known disadvantage of spectral methods is the “curse of dimensionality”, which is also the case for pseudo-spectral methods [59]. This problem may partially be solved by using order reduction and/or hyperreduction techniques [60–62]. A novel approach to avoid the exponential growth of the problem complexity is proposed in [59]. In the case of imprecise probabilities a drastic reduction of computational effort may be achieved using the fact that spectral methods provide actually much more information than normally used in uncertainty analysis. The advantage of local basis formulations (SL-FEM, FL-FEM) for reduced modeling is also discussed. In this work we consider models with polymorphic uncertainties and sufficient input information to utilize a precise fuzzy–stochastic description. We review generalized spectral modeling techniques for stochastic, fuzzy, and general hybrid interval–stochastic and fuzzy–stochastic problems (Section 3). We also discuss some aspects of the spectral fuzzy FEM that have not been tackled in the literature, namely the inner product definition for fuzzy variables (Section 3.3). Section 4 introduces novel contributions to the spectral non-deterministic FEM. Sections 4.1 and 4.2 represent modifications involving local and discontinuous basis functions. Next some theoretical aspects of spectral fuzzy analysis are discussed, namely the novel formulation void of any assumptions on the shape of the input’s membership function (Section 4.3). The novel dimension reduction for imprecise probabilities within the spectral FEM framework is discussed in Section 4.4. Application of the proposed techniques is demonstrated based on examples of computational homogenization with geometrical uncertainties in the microstructure. Numerical examples are presented in Section 5. Finally, Section 6 concludes the paper.

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2. Notation In this work we distinguish between deterministic and random variables, vectors and tensors, matrices and operators. We use the following notation:  Second order tensors and vectors are denoted by bold (e.g. F) and bold italic (e.g. x) scripts, respectively.  Random variables, second order tensors and vectors are represented [63,64] as functions of the elementary event ω, e.g. g(ω), F(ω), θ (ω).  A random field is any function of the spatial coordinates x and the elementary event ω (e.g. G(x, ω)).  Fuzzy numbers, vectors, and matrices are represented as function of the elementary event ω, ˜ like, e.g. χ (ω). ˜  Capital calligraphic letters are used for the domains of functions and sets (e.g. D, S, F).  Bold calligraphic letters denote function spaces like, e.g. the Hilbert space H.  Differential operators are denoted by capital upright letters, e.g. D(x, ω).  In particular Div and Grad denote divergence and gradient operators applied in the reference configuration of a geometrically nonlinear continuous body. 3. Non-deterministic FEM 3.1. Physical, probability/stochastic, and possibility/fuzzy spaces For problems involving uncertainties the corresponding modifications of FEM are proposed in the literature, namely the stochastic FEM (SFEM) for aleatoric uncertainty [28] and the fuzzy FEM (FFEM) for epistemic uncertainty [29]. To set the stage, deterministic FEM is defined in the physical space E, i.e. the Euclidean space with coordinates x ∈ R3 augmented by a corresponding algebra. The solution of deterministic FEM belongs to the physical Hilbert space of functions H defined over the physical domain D ⊂ E. In the probabilistic and possibilistic approaches additional spaces should be introduced. The probability space is denoted as (Ω , F, P), where Ω is the sample space or the space of elementary events ω, F is the corresponding σ -algebra, and P is the probability measure assigned to the elements of F [63,64]. In the most general case the elementary events ω are not numerical and not ordered. If we order them and assign them numerical values, we end up with more convenient random variables (RVs). By introducing the vector of independent random variables θ (ω) : Ω → Rn we map the probability space to the stochastic space S with corresponding Borel σ -algebra. In analogy to the physical space the vector of basis random variables θ (ω) can be represented as coordinates in the stochastic space. Similarly to the physical space the choice of the coordinates is not unique. Basis transformation in the stochastic space is similar to the transformation of a curvilinear coordinate system in physical space. Correspondingly, we introduce the stochastic Hilbert space of functions Q defined over the stochastic domain S ⊂ S. ˜ Π ), where Ω˜ is In order to distinguish from the probability space we denote the possibility space as (Ω˜ , F, ˜ the sample space, F represents a corresponding σ -algebra, and Π is the non-additive possibility measure [1]. For detailed explanations on the meaning of Ω˜ please see [12]. The fuzzy space S˜ is the space of fuzzy variables with the basis set of independent fuzzy variables denoted as χ (ω) ˜ : Ω˜ → Rm . The corresponding fuzzy Hilbert space of ˜ defined over the fuzzy domain S˜ ⊂ S˜ is introduced. functions Q The solution of the stochastic FEM belongs to the physical–stochastic product space H × Q of functions defined over the physical–stochastic product domain D × S. Elements of H × Q are called random fields. The solution ˜ of the fuzzy FEM is called the fuzzy field and represents an element of the physical–fuzzy product space H × Q ˜ For polymorphic uncertainty problems the of functions defined over the physical–fuzzy product domain D × S. ˜ of functions defined solution of the fuzzy–stochastic FEM (FS-FEM) belongs to the product space H × Q × Q ˜ The tensor product space and corresponding domains are depicted in Fig. 1. over the product domain D × S × S. A random variable θ (ω) is characterized by its probability density function (pdf) f θ and its cumulative distribution function (cdf) Fθ , such that ∫ θ Fθ (θ ) = P(t ≤ θ) = f θ (t)dt. (1) −∞

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Fig. 1. Physical, stochastic, and fuzzy domains D, S, and S˜ and tensor product domains.

Fig. 2. The membership function of a symmetric triangular fuzzy number with modal value m x decomposed into 6 α-cuts.

The pdf is a function, whose value at any given point represents a relative likelihood. On the contrary, the possibility density function represents the degree of confidence [12], feasibility, or plausibility [16]. For historical reasons the possibility density function of a fuzzy variable χ(ω) ˜ is called the membership function µχ . The analogies to the cumulative distribution function are the possibility function M(χ) and the necessity function N (χ) Mχ (χ ) = Π (x ≤ χ) = sup(µχ (x)), x≤χ

Nχ (χ ) = 1 − Π (x ≥ χ) = infx≥χ (1 − µχ (x)).

(2)

The main goal of fuzzy arithmetic is to reconstruct the membership function for the quantity of interest from the known membership function of the input variable. The classical approach is the application of fuzzy arithmetical operations derived from Zadeh’s extension principle. Since fuzzy sets were introduced in 1965 [65] the original idea was adopted to a large number of uncertainty-propagation problems including also engineering applications. The classical approach has proven to be inefficient due to the high complexity and loss of accuracy for algebraic division [66]. An alternative approach is to decompose the fuzzy variable into α-cuts, i.e. subintervals containing all points that have a membership degree greater or equal α (Fig. 2). Thus the fuzzy problem is then converted into a set of nested interval problems, whereby the global min and max values of the output on every subinterval (α-cut) are evaluated. The joint probability density function for the vector of independent random variables θ is obtained as the product of individual pdfs. On the contrary, the membership function of the vector of independent fuzzy variables (Fig. 3) is obtained as the minimum of individuals membership functions: µχ (χ ) = min(µχ1 (χ1 ), . . . , µχm (χm )).

(3)

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Fig. 3. Examples of the joint probability density function f θ and the joint possibility density function µχ for two independent variables.

The important difference between possibilistic and probabilistic approaches is that the probabilistic approach is based on the evaluation of weighted averages, like the mean and standard deviation, estimations, etc. This approach considers the analysis of the averaged scenario in contrast to the possibilistic approach that considers the worstcase scenario. This difference results from the non-additive formulation of the possibility measure. For a deeper discussion on possibilistic and probabilistic decision making see [8]. 3.2. Spectral formulation of non-deterministic FEM In this section we exploit the general Galerkin-type FEM. Different versions of FEM like the deterministic FEM, SFEM, FFEM, and FS-FEM can be considered as special cases of the Galerkin method with ansatz functions defined ˜ or in only one of its subspaces. For example, the deterministic FEM in the generalized product space H × Q × Q utilizes shape functions entirely belonging to the space H. An advantage of this approach is the fact that the nondeterministic FEM may be viewed and analyzed as the common deterministic FEM, however, in an n-dimensional product space [54,57]. Let us introduce a general fuzzy–stochastic differential operator D(x, ω, ω): ˜ D(x, ω, ω) ˜ y(x, ω, ω) ˜ = f (x, ω, ω). ˜

(4)

where f (x, ω, ω) ˜ is the non-deterministic loading and y(x, ω, ω) ˜ is the unknown non-deterministic solution function. Stochastic, fuzzy, and deterministic differential operators are obtained as particular cases of (4). Next we evaluate Galerkin projections of the differential operator (4) and the unknown function y(x, ω, ω) ˜ onto some basis ϕ(x, ω, ω). ˜ y(x, ω, ω) ˜ =

N ∑

yi ϕi (x, ω, ω), ˜

(5)

i=1

⟨[ ] ⟩ D(x, ω, ω) ˜ y(x, ω, ω) ˜ − f (x, ω, ω) ˜ ϕi (x, ω, ω) ˜ = 0, ∀i = 1, ... , N .

(6)

where N is the number of basis functions, ⟨ ⟩ denotes the inner product in the considered Hilbert space of functions. A detailed discussion on the inner product in general product spaces is presented in the sequel. Here we consider the following expression: ∫ ∫ ∫ ⟨g1 (x, ω, ω)g ˜ 2 (x, ω, ω)⟩ ˜ := g1 (x, θ , χ )g2 (x, θ , χ ) f θ dχ dθ dx, (7) D

S

S˜

where f θ is the joint probability density function. For a geometrically nonlinear elastic problem (in the sense of continuum mechanics) with nonlinear constitutive equations the differential operator in (4) reads explicitly as ( ) D(x, ω, ω) ˜ y(x, ω, ω) ˜ := − Div P F(x, ω, ω) ˜ , F = Grad y(x, ω, ω), ˜ f (x, ω, ω) ˜ := f (x, ω, ω), ˜

(8)

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where y(x, ω, ω) ˜ corresponds to the non-deterministic deformation map describing the position of material points in the spatial configuration, f (x, ω, ω) ˜ denotes the non-deterministic body force density, P and F represent the Piola stress tensor and the deformation gradient tensor, respectively. Please note that the Grad and Div operators involve differentiation only with respect to the physical coordinates x in the material configuration. Thus, after integration by parts the expressions (6) and (8) read R = F ext − F int → 0, F iint = ⟨P · Grad ϕi (x, ω, ω)⟩ ˜ , ext F i = ⟨ f (x, ω, ω)ϕ ˜ i (x, ω, ω)⟩ ˜ ,

(9)

˜ Further, Newton iterations may be used in where R is the residual and ⟨ ⟩ is the inner product in H × Q × Q. order to find the solution. Note that the basis functions are expressed in the n-dimensional product space with generalized coordinates {x, θ (ω), χ (ω)}, ˜ but the differential operator involves derivatives with respect to the physical coordinates x only. Observe that the model exhibits no deformation in non-deterministic directions. In fact spectral SFEM and FFEM return a continuous response surface for the quantity of interest in contrast to sampling techniques, which provide only discrete output. In SFEM the response surface obtained is used to evaluate integrals in stochastic dimensions and, e.g. estimate the mean value and the standard deviation for the quantity of interest (or any other probabilistic moments, skewness, kurtosis, etc.). In FFEM the response surface is used in order to find the minimum and maximum values of the quantity of interest on different subintervals (α-cuts), i.e. the response surface is further used as surrogate model. Due to the continuity of the obtained solutions spectral methods provide extra information which may be used for deeper problem analyses and further problem simplifications (Section 4.4). 3.3. Discussion I: inner product in general product space The main property of a Hilbert space is the existence of an inner product, which is a key component in projection based methods (6). The inner product of two elements in the physical Hilbert space of functions H is defined as integral over the physical domain D. ∫ ⟨g1 (x), g2 (x)⟩ := g1 (x)g2 (x)dx. (10) D

A similar inner product of two elements in the stochastic Hilbert space Q (called expectation), i.e. of two functions over Ω , is represented as the Lebesgue integral: ∫ ⟨g1 (ω), g2 (ω)⟩ := g1 (ω)g2 (ω)dP, (11) Ω

or more conveniently as the Riemann integral over the stochastic domain S ∫ ⟨g1 (ω), g2 (ω)⟩ := g1 (θ )g2 (θ) f θ dθ ,

(12)

S

where the weight function f θ is the joint probability density function of the basis random variables θi (ω). However, to our knowledge there is no well-known and widely accepted definition for the inner product of fuzzy numbers. The spectral fuzzy FEM proposed in [29] used the Galerkin projection defined in the same manner as for functions in H (10). This approach was successfully applied to a number of problems [30,58]. But the mathematical meaning, limitations and the efficiency of this implementation was never discussed. Thus, firstly, we discuss possible definitions of the inner product for fuzzy variables available in the literature. Next we demonstrate that the chosen definition of the inner product is valid for interval variables but not for fuzzy numbers, thus reducing fuzzy FEM to interval FEM. Finally, we explain the high efficiency and optimality of the approach in [29]. Obvious candidates for the inner product of fuzzy numbers are the Choquet integral, defuzzification operators, and different generalizations of expectation. One of the restrictions is that the inner product applied to the crisp numbers (fuzzy numbers with membership functions presented by a Kronecker delta) should coincide with their algebraic product. Due to the specificity of the possibility measure, which is not additive, the linearity property

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˜ is not necessarily satisfied. Nonlinear Galerkin projections are sophisticated, thus changing the entire of ⟨ ⟩ in Q framework. The Choquet integral is a nonlinear and non-additive generalization of the expectation. If the probability measure is considered, the Choquet integral reduces to (11). If applied to the possibility and necessity measure, the Choquet integral yields upper and lower expected values [12,16]. A further modification of the Choquet integral is the generalized fuzzy integral [67]. It may be very useful in the context of the non-deterministic Galerkin method, however, many mathematical aspects still must be addressed. There are also other generalizations of the expectation for fuzzy numbers proposed in the literature [68,69]. Another candidate for the projection operator ⟨ ⟩ is the defuzzification procedure which returns a scalar value if applied to the fuzzy number (or/and any function of fuzzy numbers). There are many different versions of the defuzzification procedure [70,71]. For an overview of the most widely used techniques see also [3,4]. Different defuzzification algorithms yield different versions of FFEM. The simplest algorithm is the center of mass defuzzification resulting in an expression analogously to the inner product in stochastic space. ∫ ∫ ⟨g1 (ω), ˜ g2 (ω)⟩ ˜ := g1 (χ )g2 (χ )µχ dχ / µχ dχ , (13) S˜

S˜

where the ˜ The constant ∫ weight function µχ is the joint possibility density function of the basis fuzzy variables χi (ω). term 1/ S˜ µχ dχ in Eq. (13) after substitution into (6) vanishes thus resulting in an expression identical to (12). Note that µχ evaluated as (3) is only a C 0 -continuous function, thus resulting in a quite complicated n-dimensional Gauss integration scheme. The inner product for interval variables is obtained by reducing the expression for the expectation (12) or the expression for defuzzification (13) to the case of a uniform weight function, thus neither the membership function nor the pdf is involved. ∫ ⟨g1 (ω), ˜ g2 (ω)⟩ ˜ := ⟨g1 (χ )g2 (χ )⟩ = g1 (χ )g2 (χ )dχ . (14) S˜

The application of this expression in [29] was motivated by convenience. The basis fuzzy number was transformed in [29] into an interval variable by performing singular mapping, thereby resulting in an information loss (of all information contained in the membership function). This simplification allows the use of well-known orthogonal Legendre polynomials and also avoids any complicated integration procedure. In order to explain the efficiency and correctness of this approach let us remember the fact that the fuzzy problem is decomposed into a set of nested interval problems. Thus, intuitively we suppose that interval approach could be used. However only one interval problem is considered instead of the entire set. This reduction of computational costs is possible due to the continuity of the spectral solution, which provides more information than actually used during the postprocessing stage, thus compensating the information loss due to the singular mapping. Another argument is as follows. Let us analyze the influence of the weight factor onto the method’s accuracy. Introducing the weights we enforce a more precise function evaluation at regions with higher weight factors. In the case of global basis used in stochastic domain (SG-FEM) this procedure strongly improves the convergence. In the case of local basis (SL-FEM) the weight distribution is irrelevant, if a sufficient number of elements is used. In FFEM we would like to obtain higher accuracy near global minima and maxima, which are, however, unknown. The FFEM design based on (13) results in superior accuracy for the center of the interval, where the probability of finding the global minimum or maximum is the lowest, and in a deteriorating accuracy for the ends of the interval, where the global minimum and maximum are located in the case of a monotonic response function. Thus the weight function in form of the joint membership function renders an a priori deteriorating accuracy for the considered problem. 3.4. Discussion II: spectral and sampling-based methods The spectral methods benefit from the solution of the entire continuum of possible/probable configurations in one computation. They are formulated in enlarged product spaces and result in extremely large stiffness matrices. As long as the stiffness matrix inversion is feasible, the spectral methods are advantageous due to the solution’s continuity and fast evaluation possibilities.

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The robustness of sampling methods depends strongly on the sampling strategy. Thus the brute-force MC simulation requires usually an extremely large number of samples but demonstrates convergence only weakly depending on the number of non-deterministic dimensions. Optimization algorithms used to evaluate min and max bounds in fuzzy arithmetic demonstrate also a polynomial increase in computational time, thus being only very efficient for large number of non-deterministic dimensions. These sampling methods are extremely slow for a small number of non-deterministic dimensions but advantageous otherwise. Grid based sampling methods use a predefined number of samples chosen based on some distribution properties. The well-known stochastic collocation method [39], the non-intrusive polynomial chaos, and the extended transformation method for fuzzy problems [43] provide the response surface in a manner similar to spectral methods but also share their main disadvantage: exponential growth of problem complexity, i.e. the “curse of dimensionality”. Moreover sampling based fuzzy techniques depend strongly on the number of α-cuts in contrast to the spectral FFEM. Application of sparse grids and sparse polynomial chaos may strongly accelerate spectral and pseudo-spectral methods, however at the cost of accuracy loss. Isoparametric n-dimensional serendipity finite elements [57] utilized in the n-dimensional product space may be also advantageous due to the fact that they use an incomplete set of polynomials similar to sparse grids and sparse polynomial chaos. Further reduction of computational costs is achieved by applying domain decomposition [72,73], reduced order modeling [74,75] and hyperreduction techniques [76]. In some cases hybrid sampling-spectral methods provide an optimal balance between the size of the stiffness matrix and the number of samples, thus resulting in an optimal solution speed. As a conclusion, a unique recommendation, which method (spectral, pseudo-spectral, fully sampling based) should be applied, does not exist. At least in the nonlinear case the choice depends strongly on the problem at hand [36]. The reduction of the model uncertainties is critically important for real applications. This may be performed by introducing new basis random variables corresponding to the principal system’s components, or reduction of the two uncertainties into one as demonstrated in Section 4.4. 4. Novel approaches for fuzzy–stochastic FEM 4.1. Proposal I: local basis in non-deterministic dimensions In the current work we introduce fuzzy local FEM (FL-FEM) similarly to the stochastic local FEM (SL-FEM) studied earlier [54,57]. Application of a local basis in non-deterministic dimensions results in a unified treatment of all problem dimensions, thus allowing a simple and straight forward implementation of advanced FEM techniques, like, e.g. the isoparametric concept, adaptive meshing, generation of conforming meshes, and the use of serendipity finite elements, which possesses some properties of sparse polynomial chaos. Local bases allow also a simple treatment of problems possessing geometric uncertainties and random boundaries, when the solution is discontinuous ˜ The fuzzy–stochastic local FEM (FSL-FEM) presenting the generalization of this concept in the space E × S × S. for the case of polymorphic uncertainties is derived elementwise as follows: ⎡ ⎤ ⎡ ⎤ x x ⎢ ⎥ ⎢ i⎥ ⎢ ⎥ ∑⎢ ⎥ ⎢ θ(ω) ⎥ ⎢θi ⎥ ⎢ ⎥= ⎢ ⎥ Ni (ξ ), (15) ⎢ ⎥ ⎢ ⎥ ⎢ χ (ω) ⎥ ˜ i ⎢χ i ⎥ ⎣ ⎦ ⎣ ⎦ y(x, ω, ω) ˜ yi with dim(ξ ) = dim(x) + dim(θ (ω)) + dim(χ (ω)), ˜ ξ ∈ [−1, 1]n , where ξ are local (isoparametric) coordinates. The corresponding inner product reads elementwise ∫ ⟨g1 (x, ω, ω)g ˜ 2 (x, ω, ω)⟩ ˜ := g1 (ξ )g2 (ξ ) det Jdξ , [−1, 1]n

⏐ ⏐ ⏐ ,χ ) ⏐ is the Jacobian. where det J = ⏐ ∂(x,θ ∂(ξ ) ⏐

(16)

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Fig. 4. Logarithm of the absolute error in stress values plotted versus the stochastic coordinate θ (ω) for SG-FEM (left) and for SL-FEM (right). In the case of global basis, error values are minimal near the modal value of f θ , i.e. the weight factor in (7).

Note that the (global) weight factor f θ (7) is purposely missing in this expression due to the local basis representation. By introducing the weights in classical stochastic global FEM (SG-FEM) we enforce a more precise function approximation of (5) near the weight function’s modal value and accuracy loss at the ends of the interval, where the weight factor is close to zero. A typical error distribution of the SG-FEM is depicted in Fig. 4 left. This error distribution was obtained for a problem studied in [77] (using SG-FEM) and in [57] (using SL-FEM), which represents a rectangular material cell with centered circular inclusion of random radius, where the radius is expressed as a mapping of a truncated normal RV θ (ω). A similar example is also presented in Section 5.4, however, for a different radii distribution rule. In the case of SG-FEM an implementation of the weight factor increases the accuracy of the mean value and standard deviation evaluations for the obtained solution and improves the convergence. However, this is different for SL-FEM. Due to the fact that the residual is evaluated locally on elements (Fig. 4 right) the increase of the element number in the non-deterministic dimensions neglects the influence of the global weight factor. Thus, the weight factor can be skipped without loss of accuracy. Important advantages of local bases are: (a) the unified treatment of all problem dimensions; (b) high sparsity of the resulting stiffness matrix; (c) absence of the global weight factor in (16); (d) simple and straight-forward application of advanced FEM techniques, like extended bases (Section 4.2); (e) stronger reduction of computational costs in case when discrete empirical interpolation or similar hyperreduction techniques are applied. 4.2. Proposal II: extended basis in non-deterministic dimensions It is sometimes difficult to generate an efficient mesh for models with random boundaries, e.g. a model involving random geometry or random material interfaces. Thereby the natural extension of the FEM is the incorporation of the extended FEM (XFEM) technique into the general FS-FEM framework (Section 3.2). The deterministic XFEM technique was initially developed in [78] for cracks propagation through the underlying discretization. Simultaneously, partition of unity method and generalized finite element method (GFEM) were introduced in [79–82] representing different enrichment strategies. The XFEM and GFEM are basically identical methods [83] with the difference that the first developments on the GFEM involved global enrichments. In contrast the XFEM technique considers only local enrichment of the finite element basis with some set of non-smooth and/or discontinuous functions. Local enrichment means here that only elements containing the discontinuities, high gradients and/or cracks are enriched. The approximation space for this elements includes additional basis functions (17). The Stochastic eXtended FEM (SX-FEM) was introduced in [37] and was used in a number of applications [35,55,56]. Accuracy of the SL-FEM and SX-FEM was analyzed previously in [57,62]. Here we propose a formulation for the fuzzy extended FEM (FX-FEM) and fuzzy–stochastic extended FEM (FSX-FEM) both utilizing the isoparametric concept and local bases in deterministic and non-deterministic dimensions (Section 4.1). Thus the element-wise solution for enriched elements expands as e

y (ξ ) =

J ∑ j=1

y j N j (ξ ) +

J ∑

a j F(ξ )N j (ξ ),

j=1

where F(ξ ) is the enrichment function. Depending on the problem two and more enrichments may be used.

(17)

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In this work we apply FS-FEM to the computational homogenization of heterogeneous materials with uncertainties in the microstructure. Therefore, here the main reason for using XFEM is the matrix–inclusion interface modeled as n − 1-dimensional hypersurface in the general product space E × S˜ × S. Thus we introduce the corresponding enrichment function proposed in [84]. F(ξ ) =

J ∑ j=1

|z j |N j (ξ ) − |

J ∑

z j N j (ξ )|,

(18)

j=1

where z j are the nodal values of level-set functions. Note that this is not a unique way to define the enrichment function. The combination of the Heaviside and/or ramp functions is an alternative approach [35,78]. However, the formulation (18) proved to be an ideal choice for weak discontinuities (C 0 -continuous displacements and C −1 -continuous material properties) due to the fact that there is no need for a special treatment of so-called blending elements. The concept of level-set functions is not a strictly necessary part of the XFEM [85], but it is a common and highly efficient complement. Furthermore, the level-set concept is widely used as an independent technique in order to indicate and describe surfaces and boundaries. The zero level-set of a level-set function coincides with the discontinuity or phase interfaces in heterogeneous materials like, e.g. cracks, matrix–inclusion or matrix–void interfaces. Positive and negative values are used to indicate different phases. Numerical examples of solutions obtained using the isoparametric FL-FEM and FX-FEM are presented in Section 5 for the case of geometrical uncertainties. Furthermore, the implementation of FSX-FEM is trivial and therefore not demonstrated here. 4.3. Proposal III: independence on the particular form of the membership function Results obtained using fuzzy arithmetic depend strongly on the considered shape of the membership function. If the fuzzy number decomposition is used, the problem is turned into a set of nested interval problems, wherein the input’s membership function plays a critical role in definition of α-cuts. The sequence of α-cuts represents a set of nested intervals, for which the possibility measure is defined. This fact is highlighted particularly when the fuzzy probability is described as the set of nested p-boxes [4]. The quantity of interest is then the output’s min and max bounds for different α-cuts, i.e. bounding curves plotted versus the number α : µ(x ∈ cutα ) ≥ α. If the input’s membership function is updated, the set of nested intervals, for which the possibility measure is defined, must be redrawn, and all evaluations must be repeated. On the other hand, the shape of the membership function is usually arbitrary to some degree. E.g. the main or modal value (the most expected/important value) must always be given, otherwise the interval approach is preferred. The modal value may result from the current model fitting, may represent the actual measured value, or may correspond to the experts’ highest confidence. But the determination of the membership function is not obvious. In most cases the fuzzy number is just assumed to be triangular, which is actually a very good assumption [2,86]. The fuzzy approach is often criticized due to the certain degree of arbitrariness in this assumption: what is the advantage of the fuzzy approach over stochastic modeling, if the possibility and probability distributions are both unknown and must be constructed based on subjective feelings and assumptions? On the one hand, the role of the exact shape of the membership function is less critical in contrast to probabilistic models [8]. On the other hand, the influence of the possibility distribution assumption is in many cases unclear and this criticism must be addressed more rigorously. Thus we may conclude that the output’s bounds derived independently of the input’s membership function would be advantageous in many engineering applications. Let us consider as an example some arbitrary asymmetric fuzzy number x(ω) ˜ (Fig. 5) and some arbitrary mapping function f (x) such that y(ω) ˜ = f (x(ω)) ˜ is the fuzzy output. The standard procedure is as follows. For every α-cut the min and max bounds of the output are evaluated and plotted versus the corresponding value α (Fig. 6). The obtained plot demonstrates in fact the reconstruction of the output’s membership function. Our proposal is to evaluate the min and max bounds for the right and left branches of the fuzzy number independently (Fig. 7). Firstly we follow the mapping function from the center (main value) to the left and plot the min and max bounds as functions of the distance to the center (Fig. 7 left). From the fuzzy arithmetical point

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Fig. 5. Asymmetric fuzzy number x(ω) ˜ and arbitrary mapping function f (x).

Fig. 6. The upper and lower bounds of y = f (x) computed for different α-cuts and plotted versus the value α. This plot depicts the reconstructed membership function of y(ω). ˜ Note that the evaluated bounds depend strongly on the membership function assigned to the input variable x(ω). ˜

Fig. 7. Separate evaluation of the min and max bounds for the left and right branches of the fuzzy number.

of view this means that we consider the left branch of a fuzzy number as a new fuzzy number and propagate it independently. Similarly we obtain the min and max bounds for the right branch of a fuzzy number (Fig. 7 right). The final plot (Fig. 8) is called the two-sided diagram and demonstrates the evolution of the min and max bounds while moving from the modal value to the ends of an interval. This representation is not much more expensive, since new optimization problems are formulated on smaller intervals, is independent on the input’s membership function, and requires only knowledge about the fuzzy number’s modal value and support. Note, that the modal value should be necessarily known. The two-sided diagram may be used to reconstruct the output’s membership function for any given input’s membership function with same support and modal value. Let us consider any arbitrary interval within the support which includes the modal value. The min and max bounds at the considered interval are obtained by simply choosing the lowest and the highest values at the ends of the interval in the two-sided diagram (Fig. 8). Note that we may

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Fig. 8. Two-sided diagram demonstrating min and max bounds.

Fig. 9. Multivariate fuzzy variable with non-convex support (left) and corresponding two-sided diagram (right).

work only with intervals, which include the modal value, because the possibility measure must be defined on the set of nested intervals which tends to the modal value. In the case of multivariate fuzzy numbers the changes in the framework are more significant. The determination of the two-sided diagram for a multivariate case is depicted in Fig. 9. The colored area and the lines intersection point represent the support and the modal value of the multivariate fuzzy variable, respectively. Colors on the surface (Fig. 9 left) correspond to the values of the mapping function f (x). In this case the two-sided diagram (Fig. 9 right) is obtained by performing a min and max search along optimization directions (black dashed lines) and interpolating in between. The min and max bounds for some arbitrary α-cut (as the one depicted in Fig. 9 left) are obtained by choosing the lowest and the highest values on the boundary of the α-cut in the two-sided diagram. Note that in the case of the transformation method or the spectral approach, the response surface is obtained. Correspondingly the min and max search is performed using the response surface, thus again no costly additional simulations are involved. 4.4. Proposal IV: Dimension reduction for polymorphic uncertainties In this section we consider the very common case of polymorphic uncertainties described by fuzzy probabilities. The fuzzy probability distribution [4] or the imprecise probability [22] is represented as a combination of nondeterministic variables including fuzzy, interval, and random variables. This is a special case of polymorphic uncertainties, where system states are encoded by uncertain parameters through some function F : S × S˜ → R ˜ Every point in (see example below) defined over the n-dimensional region S × S˜ in the product space S × S. S × S˜ captures some unique parameter combination. Every function value encodes some unique system state. Obviously, if the function is continuous, the function values repeat along isolines. The space of the function values is one-dimensional. By replacing the function F of a set of non-deterministic variables by a single non-deterministic variable, which takes the same values as the function F, we may convert the problem to a problem of much smaller

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Fig. 10. Histogram of inclusion diameter distribution and the obtained cumulative probability density (Courtesy of Bastian Walter, Chair of Applied Mechanics, University of Erlangen–Nuremberg).

Fig. 11. Log-normal fitting of an experimental curve from Fig. 10 representing the cdf of the inclusions radii.

dimension. Important is however that we should be able to decompose the obtained solution of the reduced problem into the solution of the initial n-dimensional problem. Let us demonstrate this on an example. Common examples of polymorphic uncertainty are the fuzzy normal and log-normal variables obtained from fitting empirical data. Here experimental data (Fig. 10) is fitted using a truncated [77] log-normal random variable F(ω) = exp(m + sθ (ω)), where θ (ω) ∈ [−3, 3] is the truncated Gaussian variable. However, due to the noise in experimental data the distribution parameters m and s are considered to be fuzzy numbers, thus resulting in a fuzzy probability distribution. F(ω, ω) ˜ = exp(m(ω) ˜ + s(ω)θ(ω)). ˜

(19)

Fig. 11 demonstrates an experimental cdf, its least-squares fitting and the error bounds representing the upper and lower limits of the fuzzy distribution. The fuzzy-random variable F(ω, ω) ˜ is expressed in terms of three independent non-deterministic variables. Therefore the possible states described by F(ω, ω) ˜ cover a 3-dimensional region in the fuzzy–stochastic product ˜ However, all these states may be covered by one single random variable F(ω): ¯ space S × S. ¯ F(ω) = exp(m¯ + s¯ θ (ω)), ln(F2 ) + ln(F1 ) , (20) 2 ln(F2 ) − ln(F1 ) s¯ = , 6 where [F1 , F2 ] = supp(F(ω, ω)) ˜ is the support of the fuzzy-random variable. Thus performing the replacement of ¯ the fuzzy-random variable F(ω, ω) ˜ by the corresponding random variable F(ω) with the same support we transform m¯ =

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Fig. 12. Scheme of propagation of polymorphic uncertainty in example (19) using FEM.

the initial multidimensional problem into an equivalent one-dimensional problem, which covers the same manifold of possible states. Schematically, the transformation of a fuzzy log-normal variable defining a family of curves to ¯ a single equivalent log-normal variable is denoted as F(ω, ω) ˜ ⇒ F(ω). Please note that only the spectral version of SFEM should be applied to the equivalent problem. In contrast to sampling techniques, spectral methods are faster for low-dimensional problems and control the approximation error between the integration points. The solution continuity and error control is what we denoted earlier as extra information which is normally not used in mean value and standard deviation estimations. This properties make, however, spectral methods optimal for the solution of equivalent problems. We would like to demonstrate the reduction procedure by two schemes. The scheme in Fig. 12 shows graphically the structure of FEM based propagation of polymorphic uncertainty. Without loss of generality we consider again an example of a fuzzy-random radius r = F(ω, ω) ˜ (19). Input data is presented usually by a family of random variables r (ω) with corresponding probability densities fr (Fig. 12 right). For convenience we represent all random variables in this family as a mapping (19) of the basis random variable θ (ω) with predefined pdf f θ . This mapping is depicted as the set of curves F(ω, ω). ˜ The quantity of interest is, e.g, the mechanical stress σ . Following the standard procedure we need to evaluate the response curves σ (r ) for every curve r (ω) out of the family F(ω, ω). ˜ Thereby, a set of FEM solutions (using either SFEM, or MC, SC, etc.) is required. But the dependency σ (r ) is the same for all curves, since it depicts the same physical process and does not depend on the input’s pdf, thus, all curves σ (r ) overlap. The output quantities are however the stress pdf f σ (Fig. 12 bottom left), which must be evaluated accounting for inputs pdf fr , and/or stress curves representing stresses as mapping of basis RV σ (θ ). The family of curves σ (θ ) are obtained simply by substituting r (θ) into σ (r ). Note that output quantities are evaluated at the stage of postprocessing. ¯ The idea is to perform the costly and expensive FEM simulation only once for an equivalent curve F(ω) (Fig. 13) and to obtain one single response curve σ (r ) covering the entire support of F(ω, ω). ˜ Here, it is important that the spectral SFEM or a similar method reconstructing the entire response curve is used to solve the equivalent problem. Next the obtained response curve σ (r ) may be used to extract the partial solutions for every RV from the family F(ω, ω) ˜ at the stage of postprocessing (simply by substituting different input curves r (ω) into the solution σ (r )). Note that the equivalent curve may be even a straight line with corresponding support. For numerical reasons this curve is chosen (20) as similar as possible to the input curves in order to achieve the optimal density of integration (interpolation) points.

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Fig. 13. Scheme of propagation of polymorphic uncertainty replaced by an equivalent problem (20) in example (19).

Remark. In this simple example we transformed the family of RVs into one equivalent RV. In case, when the SL-FEM or the FL-FEM is used, i.e a local basis in non-deterministic dimensions is utilized, the solver does not distinguish between fuzzy, interval, and stochastic variables. Thus, an equivalent variable may be an interval variable. In case, when a global basis in non-deterministic dimensions is utilized, the stochastic treatment of an equivalent variable is preferred (if possible), because the probability density function can be used as a weight function to enforce higher solution accuracy. Formally the dimension reduction can be applied in a very general setting, but if the non-deterministic variables are combined into expressions, which do not include any physical variables x. Thus the reduction cannot be applied for fuzzy-random fields, but only to fuzzy-random variables. D(x, θ (ω), χ (ω)) ˜ = D(x, F1 (ω, ω) ˜ . . . FN (ω, ω)) ˜ ¯ ¯ ⇒ D(x, F1 (ω) . . . FN (ω)),

(21)

N < dim θ + dim χ , where Fi (ω, ω) ˜ are independent fuzzy-random variables containing the entire model uncertainty. Please note that here Fi (ω, ω) ˜ are arbitrary expressions containing arbitrary numbers of random and/or fuzzy variables. 5. Application to computational homogenization 5.1. Representative volume element In the following examples we consider homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. The macroscopic material properties are obtained from the response of a representative volume element (RVE) of the microstructure. Following our previous works [57,77] we consider a rectangular RVE with the size 2a, where a = 1, with one elliptic inclusion possessing uncertainty in the inclusion’s geometry. Due to the periodic boundary conditions the inclusions position inside the RVE does not influence the homogenized stress values, thus for the sake of simplicity we consider only a model with the inclusion in the center of the RVE. Due to the uncertainties considered the RVE possesses also additional non-deterministic dimensions. For the sake of demonstration only one non-deterministic dimension θ (ω) or χ (ω) ˜ is depicted in the schematic Figs. 14 and 25. The non-deterministic dimension captures the evolution of the microstructure by varying the non-deterministic parameter, so every orthogonal section of the RVE corresponds to some deterministic sample.

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Fig. 14. The fuzzy representative volume element for a 2D problem considering the fuzziness in the inclusion’s orientation.

We model an inclusion as a jump in elastic properties (C −1 -continuity), whereby the displacements are C -continuous. We assume for simplicity a constant Poisson’s ratio ν = 0.3. Then only the shear modulus is a (fuzzy) random field and is given as 0

[ ] 1 G(x, ω, ω) ˜ = G m + [G i − G m ] 1 − sign z(x, ω, ω) ˜ , (22) 2 where G m and G i are the shear moduli of the matrix and the inclusion, respectively, z is a cone-like level-set function [77], which indicates whether the material point with coordinates x belongs to the matrix or to the inclusion (z < 0: inclusion, z > 0: matrix). In order to verify our model, we use the values of material parameters from [87]: G m = 8, G i = 80. For the sake of demonstration the Piola stress tensor P(x, ω, ω) ˜ is given as the first derivative of the Neo-Hookean energy potential Ψ (F(x, ω, ω)). ˜ Ψ (F) =

1 1 Λ [F : F − 3 − 2 ln J ] + λ ln2 J, 2 2

∂Ψ = P = ΛF + λ0 F−t , ∂F where Λ and λ are Lam´e parameters, with Λ the shear modulus and λ is related to the Poisson’s ratio ν = λ0 = [λ ln J − Λ]; J = det F is the Jacobian determinant; and F−t denotes the transposed inverse of F. The stress derivative with respect to the deformation gradient is explicitly defined as: ∂P ∂ 2Ψ = = λF−t ⊗ F−t − λ0 F−t ⊗F−1 + ΛI⊗I, ∂F ∂F2

(23) (24) λ ; 2[λ+Λ]

(25)

where I is the identity tensor; the symbols ⊗ and ⊗ denote the non-standard tensor products of two second order tensors A and B represented component-wise as follows: [A⊗B]i jkl = [A]ik [B] jl and [A⊗B]i jkl = [A]il [B] jk . For the sake of demonstration all simulations were performed with periodic boundary conditions with the macroscopic deformation gradient F¯ applied as: ⎡ ⎤ 1.1 0 0 ⎢ ⎥ ⎢ ⎥ F¯ = ˆ⎢ 0 1 0⎥ . ⎣ ⎦ 0 0 1

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5.2. Example I: fuzzy local and fuzzy extended bases In this example we consider only epistemic uncertainties and demonstrate the efficiency of the local approximation. The problem is solved using FL-FEM and FX-FEM techniques. In contrast to classical fuzzy FEM with global basis functions in fuzzy domain (FG-FEM) both novel methods can be applied to problems with discontinuous fuzzy fields (22). We consider heterogeneous materials with periodic microstructure. A soft matrix material is reinforced with stiff elliptic inclusions. The resulting macroscopic material properties are orthotropic. Induced orthotropy is often required in order to achieve optimal material strength and hardness in the loading directions. Due to the elliptic shape of the inclusions, the microstructural response is also orthotropic. However, the inclusion demonstrates some fluctuations from the preferred directions. If no statistical analysis of particle angles is available, the inclusion’s orientation must be considered fuzzy and may be modeled as a symmetric triangular fuzzy number with modal value representing the preferred direction. The corresponding fuzzy RVE is schematically depicted in Fig. 14. For the sake of demonstration we consider the preferred inclusion’s orientation as ϕ¯ = π/12 and the variation of the fuzzy parameter as ∆ϕ = π/9. One possible choice of the level-set function for the considered problem reads: [√ ] √ X 12 X 22 + 2 −1 , z(x, ω) ˜ = r1 r2 r12 r2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ (26) X1 x1 cos(ϕ(ω)) ˜ sin(ϕ(ω)) ˜ ⎣ ⎦=⎣ ⎦ · ⎣ ⎦. X2 − sin(ϕ(ω)) ˜ cos(ϕ(ω)) ˜ x2 ϕ(ω) ˜ = ϕ¯ + ∆ϕ χ (ω), ˜

(27)

where r1 = 0.5, r2 = 0.32 and χ(ω) ˜ is a symmetric triangular basis FV with modal value χ¯ = 0 and support [−1, 1]. For a detailed discussion on level-set functions please see [77]. The reference solution is obtained using the optimization based sampling method, wherein fuzzy numbers are decomposed in 11 α-cuts (22 optimization problems). Every sample represents a deterministic solution. The total number of deterministic samples used by an optimization algorithm reaches 286 in this particular example. The FL-FEM is used to obtain the response surface for the homogenized stresses, which is further used to reconstruct the membership function of the homogenized stress. The number of element layers in the non-deterministic dimension ˜ The is 12. Quadratic n-dimensional serendipity finite elements are used to discretize the product domain D × S. mesh in the physical domain D is depicted in Fig. 15 at ϕ¯ = π/12. Note that the homogenized stress is represented by a non-monotonic nonlinear function of the basis fuzzy variable χ(ω) ˜ with global maximum at χ = −0.75 (Fig. 16). Comparison of the sampling FFEM and FL-FEM is presented in Fig. 16. The considered problem is solved also using the FX-FEM. The reference membership function and the membership function obtained from FX-FEM are depicted in Fig. 17. The error Ri is estimated for every α-cut and for the left and right branches of the membership function separately. Ri = σ F F E M (µ = αi ) − σ F L−F E M (µ = αi ).

(28)

Fig. 18 represents the error estimation (28) obtained for the FL-FEM. Please note the very good agreement between the two methods. A deteriorating accuracy is, however, demonstrated by the FX-FEM solution (Fig. 19). Analogous results are also obtained for SL-FEM and SX-FEM in [57]. In the presented example the FL-FEM solution required 291 s, in contrast the sampling based FFEM that needed 894 s, i.e. FL-FEM was 3 times faster than sampling based FFEM. 5.3. Example II: fuzzy–stochastic local basis In this example we consider not only a variable particle orientation but also a variable particle radii ratio with fixed particle area. The particle shapes are normally studied before manufacturing, thus the probability distribution for the particle radii ratio ρ(ω) must be known. In this example the particle orientation represents the epistemic

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Fig. 15. Discretization of the physical domain D at ϕ¯ = π/12. Quadratic serendipity elements are used.

Fig. 16. Reconstruction of the membership function of the homogenized stress. Comparison of the optimization based sampling FFEM used as reference solution and the isoparametric FL-FEM.

Fig. 17. Reconstruction of the membership function of the homogenized stress. Comparison of the optimization based sampling FFEM used as reference solution and the FX-FEM.

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Fig. 18. Absolute error of the FL-FEM compared to the optimization based sampling FFEM.

Fig. 19. Absolute error of the FX-FEM compared to the optimization based sampling FFEM.

uncertainty, whereas the particle radii ratio displays the aleatoric uncertainty. The problem is polymorphic, but dimension reduction cannot be applied due to the fact that both uncertainties are connected through the fuzzy-random field z(x, ω, ω). ˜ The expression for the corresponding level-set function (30) is presented in [77]. ϕ(ω) ˜ = ϕ¯ + ∆ϕ χ (ω), ˜ ρ(ω) = exp (ρ¯ + ∆ρ θ (ω)) , A r1 (ω)2 = ρ(ω) , π A r2 (ω)2 = , ρ(ω) π ⎤ √ ⎡√ X 12 X 22 A⎣ z(x, ω, ω) ˜ = + − 1⎦ , π r1 (ω)2 r2 (ω)2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X1 cos(ϕ(ω)) ˜ sin(ϕ(ω)) ˜ x ⎣ ⎦=⎣ ⎦ · ⎣ 1⎦ . X2 − sin(ϕ(ω)) ˜ cos(ϕ(ω)) ˜ x2

(29)

(30)

where χ(ω) ˜ is a symmetric triangular basis FV with modal value χ¯ = 0 and support [−1, 1]; θ (ω) is a basis truncated Gaussian RV; supp(ρ) = [1.1, 2], therefore ρ¯ = [ln 2 + ln 1.1]/2 and ∆ρ = [ln 2 − ln 1.1]/6; A = 0.16π is the particle area.

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Fig. 20. Response surface for homogenized stresses representing a nonlinear and non-monotonic function of the basis FV χ (ω) ˜ and RV θ (ω).

Fig. 21. Dependency of the homogenized stresses on the fuzzy angle. The stress is plotted versus the basis FV χ (ω). ˜ Different colors correspond to different radii ratio ρ = exp (ρ¯ + ∆ρ θ).

For the sake of demonstration we choose ϕ¯ = π/12 as in the previous example, but increase the angle variation to ∆ϕ = π/6, so the maximum stress value must be reached at χ = −0.5. The response surface for the homogenized stresses (Fig. 20) for the considered problem is generated using FSL-FEM with 6 element layers in each non-deterministic dimension and the same discretization of the physical domain D as in the previous example (Fig. 15). Figs. 21 and 22 represent dependencies of the homogenized stress on the fuzzy angle and random radii ratio, respectively. The two-sided diagram, which represents independently the max and min bounds for the right and left sides of the fuzzy number, is depicted in Fig. 23. Note that this representation is independent of the considered shape of the input’s membership function and requires only knowledge of its modal value and support. For every fixed value of the basis RV θ (ω) the homogenized stress is fuzzy and may be represented by upper and lower bounding curves. The evolution of the spread between the upper and lower bounds by moving from the fuzzy number’s modal value χ = 0 to the ends of the interval demonstrates the system’s sensitivity to the fuzzy input. For every value of the basis RV θ (ω) and every given subinterval L α ∈ supp(χ), which includes the modal value L α ∋ 0, the upper and lower bounds on L α may be easily found from the two-sided diagram by picking the lowest and the highest values at the ends of L α .

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Fig. 22. Dependency of the homogenized stresses on the random radii ratio. The stress is plotted versus the basis RV θ (ω). Different colors correspond to different radii ratio ϕ = ϕ¯ + ∆ϕ χ.

Fig. 23. Two-sided diagram demonstrating the evolution of upper and lower bounds of the homogenized fuzzy stress for every possible value of the basis RV θ (ω).

5.4. Example III: dimension reduction In this section we apply dimension reduction to an example studied in [88]. In this example we consider a material with circular inclusions of different size. We simulate the model with a fuzzy-random inclusion’s radius pn (ω, ω) ˜ which possesses the same probability distribution as the probability distribution of the function describing the ratio between the particle radii and the size of corresponding Voronoi cells in Fig. 24. By using this sophisticated normalized particle radii distribution we capture the effect of different distances between particles in real composites as depicted in Fig. 24. The evaluation of the normalized particle distribution is, however, beyond the scope of the current paper. Here we mention only that the actual distribution is smooth, continuous, and may be fitted well using low-order polynomials. The corresponding level-set function is presented as follows ⎡√ ⎤ x12 x22 z(x, ω) ˜ = pn (ω, ω) ˜ ⎣ + − 1⎦ , (31) pn (ω, ω) ˜ 2 pn (ω, ω) ˜ 2 with pn (ω, ω) ˜ =a1 H1 (θ (ω)) + a2 (ω)H ˜ 2 (θ (ω)) +a3 H3 (θ (ω)) + a4 (ω)H ˜ 4 (θ (ω)).

(32)

where Hi (θ) are cubic Hermite splines. Thus the normalized particle radius is approximated as a cubic function of the basis RV θ(ω). Four distribution parameters are obtained from the experimental data fitting. The two crisp

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Fig. 24. Modeling of non-overlapping randomly distributed circular inclusions based on experimental data. Material cells associated with inclusions are depicted in orange . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 25. The stochastic representative volume element for a 2D problem considering a random radius distribution.

parameters a1 = 0.0920 and a3 = 0.8980 may be obtained exactly. The other two parameters cannot be estimated exactly thus becoming fuzzy variables. They are modeled as asymmetric triangular fuzzy numbers a2 (ω) ˜ with modal value a¯ 2 = 0.0729 defined in the range [0.0108, 0.1340] and a4 (ω) ˜ with modal value a¯ 4 = 0.2542 defined in the range [0.2328, 0.2892]. Schematically the stochastic RVE is presented in Fig. 25. For an optimal speed the reference solution is obtained by using the hybrid sampling-spectral fuzzy–stochastic FEM with five α-cuts. Sampling is applied to the fuzzy variables, while the spectral method is used for the stochastic part of the problem. The samples are generated using the extended transformation method [3], so every sample

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Fig. 26. Sampling strategy for the model involving a cubic random variable with two triangular fuzzy parameters.

Fig. 27. Stochastic homogenized stress curves plotted versus the basis RV θ (ω) ∈ [−3, 3]. The different curve’s colors correspond to the samples in Fig. 26. Reference solution.

corresponds to some stochastic problem and is solved using the SL-FEM with 12 element layers in the stochastic dimension. The sampling strategy in the space (a2 (ω), ˜ a4 (ω)) ˜ is presented in Fig. 26. In total 41 samples are used. Quadratic finite elements are considered. The discretization of the physical domain D is topologically the same as depicted in Fig. 15. The quantity of interest is presented by the first component of the homogenized stress tensor σ11 plotted versus the basis RV θ (ω) (Fig. 27). The different curves correspond to the colors of the different samples in Fig. 26. Note that the stress changes caused by the fuzzy number variation are in this example in particularly much smaller than the stress variation caused by the stochastic part of the problem. The small influence of the epistemic (reducible) uncertainty means that the model is stable and the fitting procedure is accurate. Due to the small stress spread in Fig. 27 the stress curves are plotted also in the form of the deviation from the modal stress curve σ11 − σ¯ 11 = σ11 (a2 , a4 ) − σ11 (a¯ 2 , a¯ 4 ). Thus we extract the variation caused by the stochastic parameter and look closely on the variation caused by the epistemic uncertainty (Fig. 28). Note that the influence of the fuzzy parameter variation is different for different values of the random variable θ , thus producing a complex α net of overlapping curves. Fig. 29 demonstrates the upper and lower bounds of the stress curves sup{σ11 } − σ¯ 11 and α inf{σ11 } − σ¯ 11 for all α-cuts. An equivalent problem is obtained by fixing the parameter values a2 = 0.0729 and a4 = 0.2542. The solution of the equivalent stochastic problem is obtained using 12 element layers in the stochastic dimension and the same

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Fig. 28. Deviation of the homogenized stresses from the modal values σ11 − σ¯ 11 = σ11 (a2 , a4 ) − σ11 (a¯ 2 , a¯ 4 ) plotted versus the basis random variable θ (ω). Reference solution.

α α } − σ¯ 11 . Reference } − σ¯ 11 and inf{σ11 Fig. 29. Stress boundaries for each α-cut plotted in form of deviation from the modal value sup{σ11 solution.

Fig. 30. Deviation of the homogenized stresses from the modal values σ11 − σ¯ 11 = σ11 (a2 , a4 ) − σ11 (a¯ 2 , a¯ 4 ) plotted versus the basis random variable θ (ω). Data extracted from the equivalent model.

discretization in the physical domain. The equivalent problem is 41 times faster for this particular example due to the relatively small number of α-cuts and therefore small number of samples. For a more precise simulation the gain in speed will increase even more. Figs. 30 and 31 demonstrate the stress curves extracted from the solution of the equivalent problem and the corresponding upper and lower bounds for different α-cuts. The error of the equivalent model compared to the full solution is estimated for every stress curve independently as the expectation of the least-square difference between the reference solution and the equivalent model data (33).

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α α } − σ¯ 11 . Data extracted } − σ¯ 11 and inf{σ11 Fig. 31. Stress boundaries for each α-cut plotted in form of deviation from the modal value sup{σ11 from the equivalent model.

Fig. 32. Least-square error of the equivalent model compared to the reference solution. The error is estimated as (33) for every sample. Colors and marker radii represent error values . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 32 demonstrates a diagram of the error distribution. Circle centers represent the samples in space (a2 , a4 ), circle sizes and colors denote the error values. Note that the maximum error ∥Ri ∥2 value is less than 9e-5, thus stating a high accuracy of the equivalent model solution. √∫ 2 ref eqv ∥Ri ∥2 = |σi (θ ) − σi (θ )| f θ dθ . (33) S

6. Conclusions In this contribution we review the general description of the non-deterministic Galerkin-based spectral FEM formulated in an enlarged physical–fuzzy–stochastic product space. We discuss the definition of the inner product of fuzzy numbers, indirectly applied in the spectral fuzzy FEM proposed in the literature. We list different candidates for the inner product of fuzzy numbers and demonstrate that the formulation proposed in the literature reduces the fuzzy FEM to the interval FEM. This approach is, however, highly efficient despite of some information loss due to the fact that the fuzzy problem is anyway decomposed into a set of interval problems for practical applications. Only one of the sets of interval problems is, however, solved. Extraction of the entire information from one single submodel is possible due to the properties of spectral methods, namely the solution’s continuity and the error control over the entire interval. We address the common criticism regarding the arbitrariness in the membership function determination and propose a fuzzy FEM formulation which is void of any assumptions on the input’s membership function. This

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formulation requires only the precise knowledge of the fuzzy parameter support and the modal value. However, if the information regarding the support and the modal value is not available, the interval approach should be preferred anyway. We propose a dimension reduction for the case of imprecise probabilities using an equivalent model. Due to the continuity of the spectral solution and the robustness of spectral methods for low dimensional problems, the spectral fuzzy–stochastic FEM is the perfect tool for the simulation of the equivalent model. We introduce also the modification of the fuzzy and the fuzzy–stochastic FEM involving local bases and n-dimensional isoparametric serendipity finite elements. The advantages of local bases are:     

good description of the model’s complex boundaries, simplified representation of the inner product in the stochastic space, uniform representation of the Galerkin projections in stochastic and fuzzy spaces, high sparsity of the stiffness matrix, higher accuracy for equivalent problems due to the local residual minimization.

The incorporation of XFEM into the fuzzy–stochastic FEM with local bases is presented for the case of a complex fuzzy-random geometry where the mesh generation may become infeasible. XFEM results, however, in more complicated element integration and additional DOFs. Applications of local bases and extended bases are demonstrated for examples of computational homogenization of materials with epistemic (fuzzy problem) and polymorphic (fuzzy–stochastic problem) uncertainties in the microstructure. Acknowledgments The support of this work by the Deutsche Forschungs-Gemeinschaft (DFG), Germany through the Priority Program SPP1886 is gratefully acknowledged. References [1] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1) (1978) 3–28. [2] D. Dubois, L. Foulloy, G. Mauris, H. Prade, Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities, Reliab. Comput. 10 (4) (2004) 273–297. [3] M. Hanss, Applied Fuzzy Arithmetic, Springer Berlin Heidelberg, 2005. [4] B. Moller, M. Beer, Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics, first ed., Springer-Verlag Berlin Heidelberg, 2004. [5] D.J. Segalman, M.R. Brake, L.A. Bergman, A.F. Vakakis, K. Willner, Epistemic and aleatoric uncertainty in modeling, in: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 8: 22nd Reliability, Stress Analysis, and Failure Prevention Conference; 25th Conference on Mechanical Vibration and Noise, 2013. [6] L.A. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (3) (1975) 407–428. [7] L.A. Zadeh, Fuzzy Logic = Computing with Words, Physica-Verlag HD, Heidelberg, 1999, pp. 3–23. [8] S.Q. Chen, Comparing Probabilistic and Fuzzy Set Approaches for Designing in the Presence of Uncertainty (Ph.D. thesis), Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 2000. [9] S. Chen, E. Nikolaidis, H.H. Cudney, R. Rosca, R.T. Haftka, Comparison of probabilistic and fuzzy set methods for designing under uncertainty, in: 40th Structures, Structural Dynamics, and Materials Conference and Exhibit, Structures, Structural Dynamics, and Materials and Co-located Conferences, 1999. [10] G. Maglaras, E. Nikolaidis, R.T. Haftka, H.H. Cudney, Analytical-experimental comparison of probabilistic methods and fuzzy set based methods for designing under uncertainty, Struct. Optim. 13 (2) (1997) 69–80. [11] E. Nikolaidis, S. Chen, H. Cudney, R.T. Haftka, R. Rosca, Comparison of probability and possibility for design against catastrophic failure under uncertainty, J. Mech. Des. 126 (2003) 386–394. [12] N.-P. Walz, Fuzzy Arithmetical Methods for Possibilistic Uncertainty Analysis (Ph.D. thesis), Institut fuer Technische und Numerische Mechanik der Universitaet Stuttgart, 2016. [13] G. De Cooman, Generalized possibility and necessity measures on fields of sets, in: Proceedings of the International ICSC Symposium on Fuzzy Logic (ISFL’95), 1995, pp. A91–A98. [14] D. Dubois, Possibility theory and statistical reasoning, Comput. Statist. Data Anal. 51 (1) (2006) 47–69, the Fuzzy Approach to Statistical Analysis. [15] D. Dubois, Fuzzy measures on finite scales as families of possibility measures, in: Europian Society of Fuzzy Logic and Technology (EUSFLAT-LFA), 2011, pp. 822–829. [16] D. Dubois, H. Prade, Possibility theory and its applications: Where do we stand? in: J. Kacprzyk, W. Pedrycz (Eds.), Springer Handbook of Computational Intelligence, Springer Berlin Heidelberg, 2015, pp. 31–60. [17] M. Inuiguchi, S. Greco, R. Slowinski, T. Tanino, Possibility and necessity measure specification using modifiers for decision making under fuzziness, Fuzzy Sets and Systems 137 (1) (2003) 151–175, preference Modelling and Applications.

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