Global sensitivity analysis of certain and uncertain factors for a circular tunnel under seismic action

Abstract

There are many certain and uncertain design factors which have unrevealed rational effects on the generation of tensile damage and the stability of the circular tunnels during seismic actions. In this research paper, we have dedicated three certain and four uncertain design factors to quantify their rational effects using numerical simulations and the Sobol’s sensitivity indices. Main effects and interaction effects between the design factors have been determined supporting on variance-based global sensitivity analysis. The results detected that the concrete modulus of elasticity for the tunnel lining has the greatest effect on the tensile damage generation in the tunnel lining during the seismic action. In the other direction, the interactions between the concrete density and both of concrete modulus of elasticity and tunnel diameter have appreciable effects on the tensile damage. Furthermore, the tunnel diameter has the deciding effect on the stability of the tunnel structure. While the interaction between the tunnel diameter and concrete density has appreciable effect on the stability process. It is worthy to mention that Sobol’s sensitivity indices manifested strong efficiency in detecting the roles of each design factor in cooperation with the numerical simulations explaining the responses of the circular tunnel during seismic actions.

Keywords

shear waves / Sobol’s sensitivity indices / maximum principal stress / maximum overall displacement / tensile damage

Introduction

The underground structures such as circular tunnels respond to dynamic loads imposed by seismic actions in different way compared to that of above ground structures, where these tunnels do not undergo resonance with the surrounding ground but their responses are dependent on the response of the surrounding ground. The effect of seismic loads on tunnels is described through the imposed deformations from the surrounding media. The design of tunnels is conducted to accommodate these deformations. The loads imposed by the earthquakes are generating shear waves vertically which are predominant and cause ovaling deformations in circular tunnels and as a result the distortion of the tunnel lining cross section [¹,²].

Many researchers have studied the effect of earthquakes in circular tunnels and tried to obtain data about the behavior and responses of the structural system using finite element analysis in addition to stochastic methods. Nariman et al. [³] have performed numerical analysis and simulation of a circular tunnel with uncertain three factors related to the surrounding soil during earthquakes. They quantified the uncertainty of those factors which are related to the responses of the tunnel therough meta-models using an experimental method. Nariman et al. [⁴] studied the structural optimization of four certain factors related to the tunnel lining properties subjected to a ground motion through using coupled XFEM-BCQO approach to analyze the predicted damage in the tunnel lining wher they obtained the optimized values of the factors and predicted the strat of cracks and spalling through using XFEM technique.

Mollon et al. [⁵] studied the 3D analysis and face stability of a shallow circular tunnel and proposed a reliability approach adopting the failure modes due to blow-out and the collapse of the ultimate limit state. They discovered that collapse mode of failure result in the most critical deterministic sequences against face stability where they used it in the stochastic analysis and design. They dedicated soil shear strength factors as random variables and quantified the reliability index of Hasofer-Lind in addition to the failure probability. Furthermore, they carried on sensitivity analysis on the soil shear strength factors and finally they performed a reliability-based design to determine the desired tunnel pressure aiming the probability of a target collapse failure.

Kalab and Stemon [⁶] studied the influence of earthquakes on shallow circular tunnels using three types of rock ground (soft rock, medium hard rock, and hard rock). They performed sensitivity analysis considering Wang’s methodology which is soil interaction approach. They variated and tested both the tunnel lining diameter and the Young’s modulus of elasticity of the lining, where the thrust force was calculated conditioning no-slip case. They revealed the relations between these factors in the process of the influence of the earthquake on the shallow circular tunnels.

The possible failure of tunnels due to seismic actions might result in damages or in the loss of human lifes. The evaluation of seismic safety for tunnels still a critical problem in the design of tunnel structures which led in development of effective techniques for damage and crack detection [⁷–¹⁷]. To assess the reliability and structural safety of tunnels, advanced computational methods are commonly applied which are capable of capturing the damages. These methods can be classified into continuous and discrete fracture approaches. Continuous approaches to fracture smear the crack over a certain width. They include non-local damage models [¹⁸,¹⁹] and gradient models [²⁰,²¹]. Also the introduction of a viscosity [²²–²⁶] smears the crack over a certain width. With the seminal work of Miehe et al. [²⁷], phase field approaches have become another alternative to non-local and gradient models. Instead of relating the non-local damage to internal state variables, the damage is obtained by solving a partial differential equations. Phase field approaches have meanwhile been applied to numerous interesting problems [²⁸–³⁴] including fracture in thin shells [³³]. Discrete crack approaches include meshfree methods [³⁵–⁴⁰], extended finite element methods [⁴¹–⁴³], smoothed extended finite element methods [⁴⁴–⁴⁶], phantom node methods [⁴⁷–⁵⁰], extended meshfree methods [⁵¹–⁶¹], extended IGA [⁶²–⁶⁵], cracking particle methods [⁶⁶–⁷³], the numerical manifold method [⁷⁴–⁸⁰], efficient remeshing techniques [⁸¹–⁸⁷], multiscale models for fracture [⁸⁸–¹⁰⁰], peridynamics [¹⁰¹–¹⁰³] and dual-horizon peridynamics [¹⁰⁴,¹⁰⁵]. One challenge in these methods is tracking the crack path which can become an overworking task especially in 3D and for dynamic fracture.

In this paper we are exploiting Sobol’s sensitivity indices which is a variance-based global sensitivity analysis method to quantify the rational effects of seven certain and uncertain factors in the design of circular tunnels related to the damage and stability during seismic actions. This study will reveal the importancy of each factor so that to account for each of which in the design of circular tunnels in conjunction with the numerical simulations using finite element analysis.

Finite element model

The 2D model of both the tunnel lining and the soil media generated in ABAQUS program with dimensions details (see Fig. 1) is created to simulate the seismic action for 10 s duration both in vertical and horizontal directions. The tunnel lining thickness, tunnel diameter, tunnel concrete modulus of elasticity, tunnel lining concrete density, peak ground acceleration, soil density, and soil modulus of elasticity are the selected seven certain and uncertain factors respectively to predict the responses of the tunnel regarding the damage in the tunnel lining. The dimensions of the soil media are 50 m ×100 m. The tunnel lining material is made of reinforced concrete where the reinforcement effect is neglected for simplicity. Concrete damaged plasticity model used for damage detection analysis supporting on real data in the literature, where the dilation angle is 35° and the eccentricity, f_b0/f_c0, K, and viscosity parameter equal zero for all . Mohr Coulomb plasticity model was used for the soil media to define its shear strengths at different effective stresses (see Table 1). In the initial step two predefined fields were created for the soil, geostatic stress of 20601 Pa and void ratio of 1. A geostatic step of 1 s duration was generated, where the gravity loads of the model system were assigned. Two different boundary conditions were assigned for both the base and the sides of the soil media, by preventing the displacement in x-axis direction for both of them and providing two degrees of freedom for the sides in y-direction and rotation about z-axis for both of them. A static step with duration of 1e−10 s was created for the model system. A dynamic implicit step with 10 s duration was created to simulate the ground motion due to the earthquake by assigning acceleration boundary conditions in both x and y directions with magnitude of 4.905 m/s² . A tie constraint was assigned to model the surface contact between the soil media and the tunnel lining. The soil media and the tunnel lining were modeled with standard plain strain linear with reduced integration hourglass control elements CPE4R.

The model of the circular tunnel has been prepared after performing mesh convergence analysis to obtain the most efficient model which does not changes in the outputs values when further mesh refinements are performed both for the soil and the tunne (Fig. 2). The criteria for the output is the displacement of the a point at the top point inside the tunnel due to earthquake excitation for the duration of 10 s. Uniform mesh convergence was adopted for the assembly by considering 18 seeding cases starting from 1008 to 20112 total elements. The selected total elements case was 20112 elements where the results of the displacements are constant approximately.

Sobol’s sensitivity indices

Sobol sensitivity analysis is intended to find out how much of the variability in model output is dependent upon each of the input factors, either upon a single factor or upon an interaction between different factors. The decomposition of the output variance in a Sobol sensitivity analysis employs the same principal as the classical analysis of variance in a factorial design. Sobol sensitivity analysis is not intended to identify the cause of the input variability. It only indicates what impact and to what extent it will have on model output. As a result, it cannot be used to identify the source(s) of variance. The Sobol sensitivity indices is dedicated for seven certain and uncertain factors related to the soil type, tunnel lining properties and the peak ground acceleration.The Sobol’s sensitivity indices are ratios of partial variances to total variance, and for independent factors satisfy the relationship:

(1)\mathrm{1}=\mathrm{\sum }_i{{\displaystyle s}}_i+\mathrm{\sum }_i\mathrm{\sum }_{j>\mathrm{1}}{{\displaystyle s}}_{ij}+\mathrm{\sum }_i\mathrm{\sum }_{j>\mathrm{1}}\mathrm{\sum }_{k>\mathrm{1}}{{\displaystyle s}}_{ijk}+\mathrm{\cdots .}

The Sobol’ sensitivity indices are:

First effect sensitivity index

(2){{\displaystyle s}}_i={\displaystyle \frac{{{\displaystyle V}}_i}{V}}.

Second effect sensitivity index

(3){{\displaystyle s}}_{ij}={\displaystyle \frac{{{\displaystyle V}}_{ij}}{V}}.

Total sensitivity index

(4){{\displaystyle s}}_{Ti}={{\displaystyle s}}_i+\mathrm{\sum }_{j\ne \mathrm{1}}{{\displaystyle s}}_{ij}+\mathrm{\cdots .}

The first effect index, {{\displaystyle s}}_i, is a measure for the variance contribution of the individual factor {{\displaystyle X}}_i to the total model variance. The partial variance {{\displaystyle V}}_i in Eq. (2) is given by the variance of the conditional expectation {{\displaystyle V}}_i=V\left[E\left(Y/{{\displaystyle X}}_i\right)\right] and is also called the main effect of {{\displaystyle X}}_i on Y. The impact on the model output variance of the interaction between factors {{\displaystyle X}}_i and {{\displaystyle X}}_j is given by {{\displaystyle s}}_{ij} and {{\displaystyle s}}_{Ti} is the result of the main effect of {{\displaystyle X}}_i and all its interactions with the other factors (up to the p\mathrm{t}\mathrm{h} effect) [¹⁰⁶–¹⁰⁸].

Ranges of factors

The certain and uncertain factors under study to quantify the Sobol’s sensitivity indices for the damage detection and stability of the circular tunnel under the seismic action are illustrated in the following (Tables 2 and 3) where the name and symbol of each factor would be used in the global sensitivity analysis process.

Results of sensitivity indices—certain factors

The main effects of the sensitivity indices for each certain factor in addition to their interaction effects were calculated targetting the maximum principal stress output of the tunnel lining. Also, the main and interaction effects of sensitivity indices for each certain factor were calculated considering the maximum overall displacement output of the tunnel structure, (see Table 4). Where the main effects of the uncertain factors have been calculated using variation approach, (see Figs. 3 and 4).

In relation with the maximum principal stress, the total effect sensitivity index for the design factor X₂ is 4.89, this value is bigger than the total effect sensitivity index of design factor X₁ which is 4.43, also bigger than the total effect sensitivity index of design factor X₄ which is 4.66, but very much smaller than the total effect sensitivity index of design factor X₃ which is 86.89%, this means that the tensile damage in the tunnel lining is 86.89% due to variation in the Young’s modulus of elasticity of the tunnel lining, and it is 4.89% due to the variation in the tunnel diameter, also it is 4.66% due to the variation in the concrete density of the tunnel lining, and it is 4.43% due to the variation in the tunnel lining thickness. It is worthy to mention that the tensile damage in the tunnel lining during seismic action is also related to the interactions between X₂ and X₄ and between X₃ and X₄, which are 1.15 and 2.05, respectively, while the interactions between X₁ and X_2,X₁ and X₃, X₁ and X₄, X₂ and X₃ are not so appreciable compared to the previous two inteactions, which are 0.12, 0.4, 0.54, and 0.68, respectively.

While considering the maximum overall displacement, the total effect sensitivity index for the design factor X₄ is 9.07, this value is bigger than the total effect sensitivity index of design factor X₃ which is 0.68, also bigger than the total effect sensitivity index of design factor X₁ which is 0.42, but very much smaller than the total effect sensitivity index of design factor X₂ which is 89.98%, this means that the stability of the tunnel structure due to seismic action is 89.98% due to variation in the tunnel diameter, and it is 9.07% due to the variation in concrete density of the tunnel lining, also it is 0.68% due to the variation in the Young’s modulus of elasticity of the tunnel lining, and it is 0.42% due to the variation in the tunnel lining thickness. Furthermore, we can say that the stability of the tunnel lining during seismic action is also related to the interactions between X₂ and X₄, which is 1.26, while the interactions between X₁ and X₂, X₁ and X₃, X₁ and X₄, X₂ and X₃, X₃ and X₄ do not exist, which are 0.01, 0.0, 0.0, 0.0, and 0.0, respectively.

Results of sensitivity indices—uncertain factors

In the same way, the main effects of the sensitivity indices for each uncertain factor in addition to their interaction effects were calculated targetting the maximum principal stress output of the tunnel lining. Also, the main and interaction effects of sensitivity indices for each uncertain factor were calculated considering the maximum overall displacement output of the tunnel structure, (see Table 5). Where the main effects of the uncertain factors have been calculated using variation approach, (see Figs. 5 and 6).

By considering the maximum principal stress, the total effect sensitivity index for the design factor X₅ is 42.25, this value is bigger than the total effect sensitivity index of design factor X₆ which is 31.75, also bigger than the total effect sensitivity index of design factor X₇ which is 27.73. This means that the tensile damage in the tunnel lining is 42.25% due to variation in the peak ground acceleration of the seismic action, and it is 31.75% due to the variation in the soil density, also it is 27.73% due to the variation in the soil modulus of elasticity. Furthermore, the tensile damage in the tunnel lining during seismic action is also related to a small effects of the interactions between X₅ and X₆, X₅ and X₇, X₆ and X₇, which are 0.89, 0.55, and 0.56, respectively.

In the other side related to the maximum overall displacement, the total effect sensitivity index for the design factor X₅ is 99.83, this value is so much bigger than the total effect sensitivity index of design factor X₆ which is 0.09, also so much bigger than the total effect sensitivity index of design factor X₇ which is 0.1. This an indication that the stability of the tunnel structure due to seismic action is 99.83% due to variation in peak ground acceleration, and it is 0.09% due to the variation in soil density which is not appreciable. Also it is 0.1% due to the variation in the soil modulus of elasticity. We can also detect that the stability of the tunnel lining during seismic action is not related to the interactions between X₅ and X₆ which is 0.01, and the interactions between X₅ and X₇, X₆ and X₇ which are 0.01, 0.01, and 0.0, respectively, because of the so low magnitudes of them which approach zero approximately.

Results of numerical simulations

Depending on the results of the Sobol’s sensitivity indices, the most effective design factors are considered to carry on numerical simulations for the exposure of a circular tunnel to a seismic action with a constant peak ground acceleration of 4.905 m/s² and constant design factors (tunnel thickness, tunnel diameter, tunnel concrete density, soil density and soil modulus of elasticity) regarding tensile damage and the stability of the tunnel.

Numerical simulations of tensile damage

For the tensile damage output, the variation in the concrete modulus of elasticity values is dedicated to determine the effects on the tensile damage generation in the tunnel lining where three values are dedicated (23E+ 09, 27E+ 09, and 31E+ 09 Pa).

For minimum value of the concrete modulus of elasticity the maximum principal stress value is 8.366E+ 05 Pa, and it is equal to 9.156E+ 05 Pa for the medium value of the mentioned factor. While it has a value of 9.383E+ 05 Pa for the maximum value. We discover a nonlinear relation between the variation of the concrete modulus of elasticity and the maximum principal stress which is a criteria of the tensile damage in four regions as seen in red color inside the tunnel lining at four regions.The deformations in the tunnel lining increase with the increase of this design factor where a negative maximum principal stress in the surrounding soil shown in light green color in the perpendicular positions to the red regions inside the tunnel lining, see (Figs. 7(a)–7(f)) is an indicator of the increase of the positive maximum principal stress in the tunnel lining. Another indication is the increase of the light green color area with the increase of the value of the mentioned factor.

Results of stability

While for the stability output, the variation in the tunnel diameter values is dedicated to determine its effects on the stability of the tunnel where three values are considered which are (8.0, 9.0, and 10.0 m) (see Fig. 8).

The variation in the diameter value does not result in an appreciable change in the overal displacement of the tunnel where the change starts obviouslt after eight seconds from the seismic action. It is worthy to mention that the maximum displacements don’t reach 0.2 m in all three cases. As a result we discover that in the critical case of diameter design factor the stability of the tunnel won’t change despite the fact that there is a change in the diameter of the tunnel.

Conclusions

We can conclude the following aspects supporting on the numerical simulations and the Sobol’s sensitivity indices regarding the tensile damage in the tunnel lining and the stability of the tunnel structure during the seismic action:

1) The most effective design factor which is deciding in the generation of tensile damage in the tunnel lining is the concrete modulus of elasticity of the tunnel lining. Also the soil mechanic properties (modulus of elasticity and density) have an appreciable effects on this output results. While the tunnel lining thickness, diameter and conceret density have small effects on the generation of the tensile damage which can be considered safe compared to other mentioned design factors.

2) The interaction between the design factors concrete modulus of elasticity and concrete density has the greatest effect on the generation of tensile damage, also the interaction between the tunnel diameter and the concrete density has the second rank effect on this output. While the interactions between the other design factors have an equal small effects on this output approximately.

3) The most effective design factor which is deciding in the stability of the tunnel structure is the tunnel diameter. Also the concrete density is taking part in the stability process. While the tunnel lining thickness, conceret modulus of elasticity, soil density and soil modulus of elasticty have a very small effects on the stability of the tunnel which can be regarded safe in the design process.

4) The interaction between the design factors tunnel diameter and concrete density has the greatest effect on the stability of the tunnel during seismic action which is related to the self weight of the tunnel. Also the interaction between the tunnel thickness and the tunnel diameter has a small appreciable effect on the the stability. While the interactions between the other design factors have zero effects on the stability of the tunnel.

5) The Sobol’s sensitivity indices approach manifested a great efficiency in revealing the predicted roles of each certain and uncertain design factors in the generation of tensile damage and stability of the tunnel during seismic actions.

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[63] Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchho-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[64] Jia Y, Anitescu C, Ghorashi S, Rabczuk T. Extended isogeometric analysis for material interface problems. IMA Journal of Applied Mathematics, 2015, 80(3): 608–633

[65] Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture nalysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[66] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[67] Rabczuk T, Areias P M A. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Engineering, 2006, 30(11): 1159–1172

[68] Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49

[69] Rabczuk T, Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[70] Rabczuk T, Areias P M A, Belytschko T. A simplied meshfree method for shear bands with cohesive surfaces. International Journal for Numerical Methods in Engineering, 2007, 69(5): 993–1021

[71] Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6–8): 641–658

[72] Rabczuk T, Song J H, Belytschko T. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics, 2009, 76(6): 730–741

[73] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[74] Cai Y, Zhuang X, Zhu H. A generalized and ecient method for nite cover generation in the numerical manifold method. International Journal of Computational Methods, 2013, 10(5): 1350028

[75] Liu G, Zhuang X, Cui Z. Three-dimensional slope stability analysis using independent cover based numerical manifold and vector method. Engineering Geology, 2017, 225: 83–95

[76] Nguyen B H, Zhuang X, Wriggers P, Rabczuk T, Mear M E, Tran H D. Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 132–150

[77] Nguyen B H, Tran H D, Anitescu C, Zhuang X, Rabczuk T. An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275

[78] Zhu H, Wu W, Chen J, Ma G, Liu X, Zhuang X. Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rock mass. Tunnelling and Underground Space Technology, 2016, 51: 30–40

[79] Wu W, Zhu H, Zhuang X, Ma G, Cai Y. A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis. Theoretical and Applied Fracture Mechanics, 2014, 72: 136–149

[80] Cai Y, Zhu H, Zhuang X. A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modelling. Frontiers of Structural and Civil Engineering, 2013, 7(4): 369–378

[81] Nguyen-Xuan H, Liu G R, Bordas S, Natarajan S, Rabczuk T. An adaptive singular ES-FEM for mechanics problems with singular eld of arbitrary order. Computer Methods in Applied Mechanics and Engineering, 2013, 253: 252–273

[82] Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41

[83] Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315

[84] Areias P M A, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[85] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[86] Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[87] Areias P, Rabczuk T. Finite strain fracture of plates and shells with congurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

[88] Silani M, Talebi H, Hamouda A S, Rabczuk T. Nonlocal damage modeling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23

[89] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modelling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[90] Silani M, Talebi H, Ziaei-Rad S, Hamouda A M S, Zi G, Rabczuk T. A three dimensional extended Arlequin method for dynamic fracture. Computational Materials Science, 2015, 96: 425–431

[91] Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38

[92] Talebi H, Silani M, Bordas S, Kerfriden P, Rabczuk T. A computational library for multiscale modelling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071

[93] Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541

[94] Yang S W, Budarapu P R, Mahapatra D R, Bordas S P A, Zi G, Rabczuk T. A meshless adaptive multiscale method for fracture. Computational Materials Science, 2015, 96: 382–395

[95] Budarapu P, Gracie R, Bordas S, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[96] Budarapu P R, Gracie R, Yang S W, Zhuang X, Rabczuk T. Ecient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[97] Zhuang X, Wang Q, Zhu H. Multiscale modelling of hydro-mechanical couplings in quasi-brittle materials. International Journal of Fracture, 2017, 204(1): 1–27

[98] Zhu H, Wang Q, Zhuang X. A nonlinear semi-concurrent multiscale method for fractures. International Journal of Impact Engineering, 2016, 87: 65–82

[99] Zhuang X, Wang Q, Zhu H. A 3D computational homogenization model for porous material and parameters identification. Computational Materials Science, 2015, 96: 536–548

[100] Kouznetsova V, Geers M G D, Brekelmans W A M. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering, 2002, 54(8): 1235–1260

[101] Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 2017, 225: 42–48

[102] Amani J, Oterkus E, Areias P, Zi G, Nguyen-Thoi T, Rabczuk T. A non-ordinary state-based peridynamics formulation for thermoplastic fracture. International Journal of Impact Engineering, 2016, 87: 83–94

[103] Ren H, Zhuang X, Rabczuk T. A new Peridynamic formulation with shear deformation for elastic solid. Journal of Micromechanics and Molecular Physics, 2016, 1(2): 1650009

[104] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[105] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[106] Glen G, Isaacs K. Estimating sobol sensitivity indices using correlations. Journal of Environmental Modelling and Software, 2012, 37: 157–166

[107] Nossent J, Elsen P, Bauwens W. Sobol sensitivity analysis of a complex environmental model. Journal of Environmental Modelling and Software, 2011, 26(12): 1515–1525

[108] Zhang X Y, Trame M N, Lesko L J, Schmidt S. Sobol sensitivity analysis: A tool to guide the development and evaluation of systems pharmacology models. CPT: Pharmacometrics & Systems Pharmacology, 2015, 4(2): 69–79

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