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In order to assess earth structures for diverse applications, commercial finite ..... [3] W. Ehlers and J. Bluhm: Foundations of multiphasic and porous materials, ...
PAMM · Proc. Appl. Math. Mech. 17, 59 – 62 (2017) / DOI 10.1002/pamm.201710018

Polymorphic uncertainty quantification for stability analysis of fluid saturated soil and earth structures Carla Henning1,∗ and Tim Ricken1,∗∗ 1

TU Dortmund University, Chair of Mechanics, Structural Analysis, and Dynamics

Nowadays, numerical simulations enable the description of mechanical problems in many application fields, e.g. in soil or solid mechanics. During the process of physical and computational modeling, a lot of theoretical model approaches and geometrical approximations are sources of errors. These can be distinguished into aleatoric (e.g. model parameters) and epistemic (e.g. numerical approximation) uncertainties. In order to get access to a risk assessment, these uncertainties and errors must be captured and quantified. For this aim a new priority program SPP 1886 has been installed by the DFG which focuses on the so called polymorphic uncertainty quantification. In our subproject, which is part of the SPP 1886 (sp12), the focus is driven on quantification and assessment of polymorphic uncertainties in computational simulations of earth structures, especially for fluid-saturated soils. To describe the strongly coupled solid-fluid response behavior, the theory of porous media (TPM) will be used and prepared within the framework of the finite element method (FEM) for the numerical solution of initial and boundary value problems [2, 3]. To capture the impacts of different uncertainties on computational results, two promising approaches of analytical and stochastic sensitivity analysis will enhance the deterministic structural analysis [6–8]. A simple consolidation problem already provided a high sensitivity in the computational results towards variation of material parameters and initial values. The variational and probabilistic sensitivity analyses enable to quantify these sensitivities. The variational sensitivities are used as a tool for optimization procedures and capture the impact of different parameters as continuous functions. An advantage is the accurate approximation of the solution space and the efficient computation time, a disadvantage lies in the analytical derivation and algorithmic implementation. In the probabilistic sensitivity analysis from the field of statistics, the expense only increases proportionally to the problems dimension. Instead of a constant value, the model parameters are defined as probability distribution, which provides random values. Thus, a set of solution data is built up by several cycles of the simulation. Different approaches of the Bayes statistics will enable to receive accurate information with just a few simulations. The overall objective is the development of more efficient methods and tools for the sizing of earth structures in the long-run. c 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

In order to assess earth structures for diverse applications, commercial finite element programs are available, which perform deterministic structural analyses. However, a closer look at the individual process steps points out different sources of uncertainties. Here, two main categories my be distinguished: aleatoric uncertainties UA , which represent natural scattering, and epistemic Uε , which cover lacks of knowledge.

Fig. 1: a) general BVP b) pore fluid pressure λ over the height of the soil column of a reference BVP and c) for varying BVPs at t = 10 s ∗

∗∗

Corresponding author: e-mail [email protected], phone +49 231 755 7264, fax +49 231 755 2532 e-mail [email protected], phone +49 231 755 5840, fax +49 231 755 2532

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DFG Priority Programme 7: Turbulent superstructures

By means of a simple consolidation problem, see fig. 1 a), the system sensitivity due to changes in the model set up is illustrated. For a crisp boundary value problem, the decrease of pore fluid pressure λ over the height is evaluated for several time steps in fig. 1 b), which provides the reference solution. Fig. 1 c) compares the pore fluid pressure at time t = 10 [sec] of the referential calculation with further simulations, where small changes, e.g. one varying material parameter, result in large deviations. Uncertain initial values (IV), material parameter (MP), outer geometry (OG) and inner geometry (IG) result in different extents of sensitivity and can be uniquely assigned to aleatoric uncertainties UA , i.e. UA ∈ {MP, IV, OG, IG}. Here, uncertainty in the numerical solution (NS) is represented by a too little number of finite elements and belongs to the epistemic uncertainty class UE . In addition the model accuracy (MA), which is not considered in the example, also can be classified as aleatoric uncertainty, so that UE ∈ {NS, MA}. The chosen boundary conditions (BC) fall into both categories. The boundary conditions represent epistemic uncertainties as they form a part of the model. When the boundary conditions are specified, the characteristics and behavior of the soil exhibit variation which is aleatoric. The endeavor to quantify solution sensitivities in general to ensure the reliability of numerical results currently is very high, thus, there are accesses from different fields of research. In the following the soil structure is captured by the Theory of Porous Media (see, e.g. B OER [2], E HLERS & B LUHM [3], R ICKEN ET AL . [4]), in the framework of Finite Element Method (FEM). For such a complex numerical model, the variational sensitivities can be developed with the help of the local convective kinematic (see, e.g. BARTHOLD [6] and S TIEGHAN [7]).

2

Multiphase model

The load bearing mechanism of fully or partially saturated soils depends on the coupled interaction between the porous solid skeleton and saturating liquid and/or gas phase on the micro scale. Mathematical models for the description of soils are mainly based on multiphase homogenization approaches since a direct representation of the pore soil structure, e.g. via a finite element discretization, is impractical due to computational limitations. The true structure of fully saturated soil consists of grains and surrounding water, see fig. 2. The two phases are homogenized over a certain domain and represented by solid S and fluid F. Both phases are available at each geometrical point and required to fulfill the saturation condition nS + nF = 1, where nα denotes the volume fraction of a phase α with α ∈ {S, F}. In addition, some further requirements to the phase interaction and the relation between real density ραR and partial density ρα , with ρα = nα ραR , the Theory of Mixtures provides the basis for the Theory of Porous Media (TPM). In order to reach a stable equilibrium state, the minimum of the potential function Π can be identified by its variations with respect to the chosen field quantities. Here, uS denotes the solid displacement vector and λ the pore fluid pressure, which are strongly coupled by the balance Fig. 2: Homogenisation of the constituents equations. For the numerical solution in the framework of the finite element method (FEM), see section 3, the standard Galerkin procedure is used to build the necessary weak (variational) forms. Derived by local requirements to momentum and mass, here, the state equations result in two weak forms, given in eq. 1 and 2. The balance equation of momentum for the mixture, which considers a variation in the displacement vector δuS , is given by:

δuS Π = RMo (uS , λ, δuS ) =

Z

{(SSE −λ I)·δε−(ρS0S +ρF 0F ) b·δuS

B0S

} dV0S −

Z

{(SSE −λ I) N·δuS } dA0S = 0 , (1)

∂B0S

and the balance of mass for the mixture, which depends on the variation of lambda δλ:

δλ Π = RMa (uS , λ, δλ) =

Z

B0S

{tr(ε˙ S ) δλ − nF wFS0 · GradT δλ} dV0S −

Z

{nF wFS0 · N δλ } dA0S = 0 . (2)

∂B0S

Herein, SSE denotes the solid extra stresses, ε the solid displacement gradient, ρα 0α the initial partial density, b the body force vector, N the surface normal vector, and wFS0 the initial seepage velocity, which represents the solid to liquid relative velocity due to independent motion functions of both phases. The residual vector R = [RMo RMa ]T contains the variations of the potential function and its zero represents the global equilibrium state of a system with given boundary conditions. For a more detailed information about the model development, the reader is referred to R ICKEN ET AL . [4]. c 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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PAMM · Proc. Appl. Math. Mech. 17 (2017)

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61

Variational sensitivity

The weak forms of balance equations in section 2 can be numerically implemented and geometrically approximated by the Finite Element Method. Depending on the boundary value problem, the specific zero of the residual R can be iteratively determined by the Newton-Raphson scheme. In fig. 3, the green line schematically displays the residual function course, depending on a field quantity u. Considering a varying input value, e.g. the second Lamé constant µS , the equilibrium state, and hence the determined field quantity u, changes. The black dashed line R(u, µS ), which is located on the u-µS -level, represents the equilibrium state for an arbitrary Lamé constant µS . Because of the high nonlinearity of the real problem, the analytical derivation of the residual for varying equilibrium states causes problems. The local convective kinematic, see section 3.2, enables the derivation of variational sensitivities of the actual equilibrium state with respect to arbitrary values, e.g. material parameter m or reference geometry X. The Fig. 3: 1D visualization of equilibrium state and variational sensitivity tangent is given by δm Ru=un (µS ), which is represented by the black solid line, and its slope serves as a measure for the system sensitivity to the shear modulus µS . The example simplicity just enables a visualization. This methodology can be used for multi-dimensional problems likewise. On this basis one can decide upon further investigations and actions regarding the simulation reliability. The implementation is effected prior the discretization scheme which is an essential advantage in view of the algorithmic conversion. To increase the convenience of the algorithmic implementation, use will be made of the local-convective description, cf. BARTHOLD [5]. The resulting local-convective sensitivity analysis has already been applied on a similar description in the framework of the Theory of Porous Media by S TIEGHAN [7]. As an analytical method the variational sensitivity analysis is essentially more efficient in contrast to empirical or probabilistic approaches. The main steps are briefly discussed in the following. 3.1

Variation of the residual

As mentioned before, the residual R is the first variation of the potential Π with respect to the set of field quantities u, with u ∈ {uS , λ}. In order to use the Newton-Raphson scheme, the zero of the residual tangent is required and reads as: δu [Π(u)] + δu2 [Π(u)] = R(δu, u) + δu [R(δu, u)] = R(δu, u) +

∂R(δu, u) ∆u = 0 . ∂u

(3)

If one considers a variation of further parameters in the residual function, e.g. the geometry X and material parameters m, the total variation of R additively extends to: δ[R(δu, u, X, m)] = δu [R(δu, u, X, m)] + δX [R(δu, u, X, m)] + δm [R(δu, u, X, m)] =

∂R(δu, u, X, m) ∂R(δu, u, X, m) ∂R(δu, u, X, m) ∆u + ∆X + ∆m ∂u ∂X ∂m

(4)

and finally leads to the discretized form ≈

E [

e=1

{δuT Ke ∆u + δuT Se ∆x + δuT He ∆m} ,

(5)

where the matrices Ke , Se and He denote the tangential element stiffness, geometry sensitivity and material sensitivity matrices. After assembling, they contain the information about the system sensitivity. The possibly very extensive data can be analyzed with the help of the singular value decomposition (SVD). In order to receive this residual derivations, a reformulation with the help of the local convective description of the TPM is required to insulate material and geometric quantities. www.gamm-proceedings.com

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DFG Priority Programme 7: Turbulent superstructures

3.2

Local convective kinematic

Usually, kinematic relations are defined in the euclidean space E3 , mapping a reference configuration at the initial time t = t0 , which contains the geometry with specific material properties, on a current configuration with t = t0 + ∆t. In order to derive the variational sensitivities, this reference configuration has to be divided, see fig. 4. Integer numbers Θα in Z3 only contain the material information without any geometric assignments. Only after mapping through the function of representation κ0α , the reference configuration point Xα can be determined by Xα = κ0α (Θα ). The separation enables a pull-back transformation of the residual R into the geometry-free space Z3 by Θα = κ−1 0α (Xα ). Furthermore, xα is a material point of the current configuration and χtα and ϕXα Fig. 4: Local convective kinematic TPM, see BARTHOLD [6] the functions of representation of the respective configurations. Thus, the deformation gradient FXα can be expressed by FXα = FΘα K−1 , where F denotes the local deformation gradient, and KΘα the local geomeΘα Θα try gradient. This approach is the basis for the development of variational sensitivities in general.

4

Conclusion and further work

The variational sensitivity analyses enables to quantify different kinds of sensitivities. It is a tool for structural optimization procedures, which captures the impact of different parameters as continuous functions. One advantage is the accurate approximation of the solution space and the efficient computation time, a disadvantage lies in the analytical derivation and algorithmic implementation. Another approach is given by the probabilistic sensitivity analysis from the field of statistics, which is also part of our subproject of the SPP1886. Its expense only increases proportionally to the problems dimension. Instead of a constant value, the model parameters are defined as probability distribution, which provides random values. Thus a set of solution data is built up by several cycles of the simulation. By means of this data base, the probabilistic sensitivity analysis can be applied independently from the underlying model. Different approaches of the Bayes statistics will enable to receive accurate information with just a few simulations. In detail, the Gaussian process regression will be used to create a meta model. By means of the resulting posteriori-distribution, the sensitivity analysis will be performed, depending on the choice of the a-priori-distribution. Both approaches will be pursued and advantages and disadvantages will be captured, compared, and combined, so more complex problems can be handled in the long run through further development of the models. Acknowledgements The authors gratefully acknowledge the financial support of the project (Grant No.: RI 1202/6-1) by the German Research Foundation (DFG).

References [1] R. M. Bowen: Theory of mixtures, in A.C. Eringen, editor, Continuum Physics, volume III, pages 1-127, Academic Press, New York, 1976. [2] R. De Boer: Theory of Porous Media, Springer, New York, 2008. [3] W. Ehlers and J. Bluhm: Foundations of multiphasic and porous materials, Porous Media: Theory, Experiments and Numerical Applications, pages 3-86., Springer-Verlag, 2002. [4] T. Ricken, A. Schwarz, J. Bluhm: A triphasic model of transversely isotropic biological tissue with applications to stress and biologically induced growth Computational Materials Science 39(1):124-136, 2007. [5] F.-J. Barthold: Zur Kontinuumsmechanik inverser Geometrieprobleme Number 44-2002 in Braunschweiger Schriften zur Mechanik, Technische Universität Braunschweig, 2002. Habilitation. [6] F.-J. Barthold: Theorie und Numerik zur Berechnung und Optimierung von Strukturen aus isotropen, hyperelastischen Materialien Technische Universität Braunschweig, Forschungs- und Seminararbeit aus dem Bereich Mechanik der Universität Hannover, BerichtNr. F 93/2, 1993. [7] J. Stieghan: Variationelle Sensitivitätsanalyse in der Theorie poröser Medien Technische Universität Carolo-Wilhelmina zu Braunschweig, 2008. [8] J. E. Oakley and A. O’Hagan: Probabilistic sensitivity analysis of complex models: a Bayesian approach, J. R. Statist. Soc. B, 66(3):751769, 2004. c 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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