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A weighted Nitsche stabilized method for small-sliding contact on frictional surfaces
Abstract
We propose a weighted Nitsche framework for small-sliding frictional contact problems on three-dimensional interfaces. The proposed method inherits the advantages of both augmented Lagrange multiplier and penalty methods while also addressing their shortcomings. Algorithmic details of the traction update and consistent linearization in the presence of Nitsche terms are provided. Several benchmark numerical experiments are conducted and the results are compared with existing studies. The results are encouraging and indicate accurate satisfaction of the non-interpenetration constraint, stable tractions and asymptotic quadratic convergence of the Newton–Raphson method.
... Classical numerical methods, such as FEM and XFEM, treat frictional fracture surfaces as discrete discontinuities subjected to physical constraints on contact and frictional conditions (Liu and Borja, 2009). In numerical implementations, the contact surfaces are required to satisfy a set of physical constraints, such as no-penetration under compression, enforced by modern algorithms like the Lagrange multiplier method (Simo et al., 1985), penalty method (Liu and Borja, 2010), Nitsche method (Annavarapu et al., 2014(Annavarapu et al., , 2015, etc., which still pose challenges in computational contact mechanics. Some limitations still exist, including (i) the need for special ad-hoc failure criteria to determine frictional fracture initiation, and (ii) difficulties in tracking complex frictional crack growth paths. ...
... We then simulate a pre-fractured block in a plane-strain compression test, serving as the second benchmark for frictional contact problems (Annavarapu et al., 2014(Annavarapu et al., , 2015. The square pre-fractured block has the size of 1.0 m, as shown in Fig. 7(a). ...
... The diffuse interface damage zone is mathematically described by the equation = 0.2 + 0.4586 and the corresponding tilt angle is ar ct an = 0.2, as plotted in Fig. 7(b). According to Annavarapu et al. (2014Annavarapu et al. ( , 2015, we assume the identical material properties on both sides of the interface, i.e., young's modulus = 10.0 GPa, Poisson's ratio = 0.2 and mass density = 1000.0 kg/m 3 . ...
Frictional fracture phenomena in geological media are often closely related to fault instability in earthquakes and slip surface formation in geohazards. In this work, we propose a new phase-field model for capturing frictional fractures in pressure-sensitive geomaterials. Our model has three novel features: (i) a thermodynamically consistent energetic interface for contact and friction conditions; (ii) incorporation of a level set function to couple phase-field evolution and frictional-contact slips; and (iii) a transition from stored energy to yielding for describing different plastic-like frictional stick–slip fractures. Based on the energy conservation law and a variational inequality of virtual work, we formulate the governing equations for frictional fractures, including the dynamic equilibrium equation, phase-field evolution law, and most importantly, frictional interface plastic-like driving forces. We also present a robust numerical technique to handle the spatiotemporal formation and evolution of frictional fractures in rocks. We validate the model by simulating several benchmark examples. Our model is shown to reproduce both frictional stick and slip phenomena in rocks. We also apply this model to study the effect of confining pressure on frictional crack initiation and propagation in rocks, which helps us better understand the deep mechanisms of frictional fracture.
... However, it is still subjected to the stability problem (when the multiplier is discretized independently as in Alart and Curnier [14]) or requires additional iterative steps to calculate the multiplier (when the multiplier is not discretized separately as in Simo and Laursen [15]). Recent studies have thus developed alternative and more advanced methods such as the weighted Nitsche method [16][17][18] or explored diffuse approaches for algorithm-free modeling of frictional interfaces [19][20][21][22]. Yet none of the existing methods is considered optimally robust and efficient for modeling embedded interfaces with frictional contact. ...
... where k N := −∂ p N /∂u N defined in Eq. (18). Notably, when ∥u T ∥ is found to be zero (i.e. ...
We present a barrier method for treating frictional contact on interfaces embedded in finite elements. The barrier treatment has several attractive features, including: (i) it does not introduce any additional degrees of freedom or iterative steps, (ii) it is free of inter-penetration, (iii) it avoids an ill-conditioned matrix system, and (iv) it allows one to control the solution accuracy directly. We derive the contact pressure from a smooth barrier energy function that is designed to satisfy the non-penetration constraint. Likewise, we make use of a smoothed friction law in which the stick-slip transition is described by a continuous function of the slip displacement. We discretize the formulation using the extended finite element method to embed interfaces inside elements, and devise an averaged surface integration scheme that effectively provides stable solutions without traction oscillations. Subsequently, we develop a way to tailor the parameters of the barrier method to embedded interfaces, such that the method can be used without parameter tuning. We verify and investigate the proposed method through numerical examples with varied levels of complexity. The numerical results demonstrate that the proposed method is remarkably robust for challenging frictional contact problems, while requiring low cost comparable to that of the penalty method.
... [51,52]. In [53], a weighted Nitsche's method is proposed to model frictional sliding on interfaces embedded in background meshes. ...
... (43)-(45) at Gauss points. This is to circumvent the splitting of each nodal reaction into respective contributions from Gauss points, which very likely leads to 365 divergence [53]. With a relatively fine discretisation, the influence of the approximation in our approach would be insignificant. ...
Implicit boundary methods, which enrich the interpolation structure with implicit weight functions, are straightforward methods for the enforcement of Dirichlet boundary conditions. In this article, we follow the implicit boundary method that uses approximate step functions (the step boundary method) developed by Kumar et al. and provide modifications that have several advantages. Roller boundary conditions have wide practical applications in engineering, however, the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations. Thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach, which can be implemented directly in fictitious domain finite element analysis. The approach is then extended, we believe for the first time, to the nonlinear cases of frictional boundary conditions and elasto-plastic material behaviour. The proposed formulation and procedures are validated on a number of example problems that test different aspects of the method.
... Taking for example the primal unknown with enough regularity, i.e., u ∈ H 2 (Ω) so that s = 1/2, we obtain a convergence, as in Theorem 1.2 for FEM, of order O(h) (optimal) from Theorem 1. 6. In this thesis we use the semi-smooth Newton method to obtain the numerical solution of the discrete Signorini problem. ...
... Residual-based a posteriori error estimates are presented and analyzed by Chouly [39], with a saturation assumption, and by Gustafsson et al. [81], without a saturation assumption. The topic of small-sliding frictional contact in 3D has been studied by Annavarapu et al. [6], where a weighted Nitsche method is designed and tested numerically. In Hansbo et al. [83], a least-squares stabilized Augmented Lagrangian method, inspired by Nitsche's method, is introduced for unilateral contact. ...
This thesis is concerned with the devising and the analysis of hybrid discretization methods for nonlinear variational inequalities arising in computational mechanics. Salient advantages of such methods are local conservation at the cell level, robustness in different regimes and the possibility to use polygonal/polyhedral meshes with hanging nodes, which is very attractive in the context of mesh adaptation. Hybrid discretizations methods are based on discrete unknowns attached to the mesh faces. Discrete unknowns attached to the mesh cells are also used, but they can be eliminated locally by static condensation. Two main applications of hybrid discretizations methods are addressed in this thesis. The first one is the treatment using Nitsche's method of Signorini's contact problem (in the scalar-valued case) with a nonlinearity in the boundary conditions. We prove optimal error estimates leading to energy-error convergence rates of order (k+1) if face polynomials of degree k >= 0 are used. The second main application is on viscoplastic yield flows. We devise a discrete augmented Lagrangian method applied to the present hybrid discretization. We exploit the capability of hybrid methods to use polygonal meshes with hanging nodes to perform local mesh adaptation and better capture the yield surface. The accuracy and performance of the present schemes is assessed on bi-dimensional test cases including comparisons with the literature
... In the recent decades, Nitsche's method has been utilized for solving a wide range of interface problems in an efficient way [46][47][48][49][50][51][52][53][54][55]. More recently, Nitschebased methods have been developed for frictional-sliding on embedded interfaces [56,57] and small-sliding contact on frictional surfaces, including stick-slip behavior [58]. In this article, we propose a stabilized finite element method for cohesive fracture problems, which is inspired by the Nitsche's method for general boundary conditions developed by Juntunen and Stenberg [59]. ...
... The stabilization parameters β τ , β n and the weights γ 1 , γ 2 play a key role in the numerical performance of the method. This so-called weighted Nitsche method [58] is particularly advantageous for dissimilar material interfaces with large contrast in material properties or for unstructured meshes with significant variations in mesh size. For constant strain triangular (CST) and tetrahedral elements, Annavarapu et al. [46] provided estimates for the stabilization parameters using a local coercivity analysis as given by ...
We present a stabilized finite element method that generalizes Nitsche's method for enforcing stiff anisotropic cohesive laws with different normal and tangential stiffness. For smaller values of cohesive stiffness, the stabilized method resembles the standard method, wherein the traction on the crack surface is enforced as a Neumann boundary condition. Conversely, for larger values of cohesive stiffness, the stabilized method resembles Nitsche's method, wherein the cohesive law is enforced as a kinematic constraint. We present several numerical examples, in two-dimensions, to compare the performance of the stabilized and standard methods. Our results illustrate that the stabilized method enables accurate recovery of crack-face tractions for stiff isotropic and anisotropic cohesive laws; whereas, the standard method is less accurate due to spurious traction oscillations. Also, the stabilized method could mitigate spurious sensitivity of load–displacement results to displacement increment in mixed-mode fracture simulation, owing to its stability and accuracy.
... Equation (11) states that the pressure of a fluid in a fracture pf should be calculated, which presents a very complicated problem, especially for fractures that are not straight. Moreover, the fluid pressure in the fracture must be transferred into an equivalent nodal force, the linear integral form, which requires integral points on the fracture surface (Mao and Gensheng, 2014;Annavarapu et al., 2015). Therefore, this simulation set the net pressure inside the fractures at 3 MPa. ...
... In the coupling analysis, sets the time step dt as 30 s. This article is not yet considered that the problem of fracture surface contact, which could accomplish by Nitsche algorithm and Linear complementary algorithm (Mao and Gensheng, 2014;Annavarapu et al., 2015). The x-axis stress shown in Fig. 12 and the y-axis stress shown in Fig. 13. ...
Most unconventional oil and gas reservoirs contain some natural fractures, which play an essential role during reservoir reconstruction. Given the strong discontinuities in the displacement on both sides of the fracture as well as weak discontinuities in the pore fluid pressure, a novel three-dimensional seepage-stress coupling model using extended finite element method is proposed for fractured reservoirs. This directly coupled scheme for displacement field and stress field avoids the cumbersome process during calculating the fluid pressure in complicated fracture networks and translating into equivalent nodal force. Numerical examples are presented to simulate the fracture propagation pathway during the laboratory experiment on the staged synchronous fracturing of a horizontal well, revealing the deformation response and interaction mechanism between hydraulic fractures and natural fractures at orthotropic and non-orthotropic angles. The results show that due to the stress shadow effects, a non-planar fracture deflecting to wellbore would be formed during the progress of staged synchronous fracturing for a horizontal well. Moreover, the adjacent section to the intersection is opened and the others are closed for orthogonal natural fracture. In contrast, the non-orthogonal natural fracture is activated near the intersection at first and then fully opens as time increases; eventually, it is in the tensile-shear and composite state. In other words, the hydraulic fracture tends to traverse the orthogonal natural fracture and continue to propagate, as it is more easily arrested by the oblique natural fracture and transferred to propagation.
... The numerical simulations presented in this work are performed using the GEOS simulation framework. GEOS is a massively parallel multiphysics simulation framework developed at Lawrence Livermore National Laboratory, which offers coupling of the presented method with other capabilities of GEOS, including simulation of hydraulic fracture stimulation in two and three dimensions Settgast et al., 2012Settgast et al., , 2014Settgast et al., , 2017Vogler, Settgast, et al., 2017); modeling geochemical transport and reaction geothermic drawdown ; fracture shearing (Annavarapu et al., 2015), matrix flow, and heat transport (Guo et al., 2016); and simulations of immiscible fluid flow . Preliminary results of the current study, using the GEOS framework, can also be found in Vogler, Settgast, et al. (2016) and Vogler (2016). ...
... with nodal index h, spatial direction i, finite element shape function Φ, solid body density m , the Cauchy stress tensor T ij , body force (gravity) b i , and externally applied traction t i . The penalty method is used to enforce 10.1002/2017JB015057 frictional contact conditions under small-deformation assumptions (Annavarapu et al., 2015). The residual for the rate of fluid mass in a finite volume (r) is given by ...
In this work, we present the application of a fully-coupled hydro-mechanical method to investigate the effect of fracture heterogeneity on fluid flow through fractures at the laboratory scale. Experimental and numerical studies of fracture closure behavior in the presence of heterogeneous mechanical and hydraulic properties are presented. We compare the results of two sets of laboratory experiments on granodiorite specimens against numerical simulations in order to investigate the mechanical fracture closure and the hydro-mechanical effects, respectively. The model captures fracture closure behavior and predicts a non-linear increase in fluid injection pressure with loading. Results from this study indicate that the heterogeneous aperture distributions measured for experiment specimens can be used as model input for a local cubic law model in a heterogeneous fracture to capture fracture closure behavior and corresponding fluid pressure response.
... Equation (11) states that the pressure of a fluid in a fracture pf should be calculated, which presents a very complicated problem, especially for fractures that are not straight. Moreover, the fluid pressure in the fracture must be transferred into an equivalent nodal force, the linear integral form, which requires integral points on the fracture surface (Mao and Gensheng, 2014;Annavarapu et al., 2015). Therefore, this simulation set the net pressure inside the fractures at 3 MPa. ...
... In the coupling analysis, sets the time step dt as 30 s. This article is not yet considered that the problem of fracture surface contact, which could accomplish by Nitsche algorithm and Linear complementary algorithm (Mao and Gensheng, 2014;Annavarapu et al., 2015). The x-axis stress shown in Fig. 12 and the y-axis stress shown in Fig. 13. ...
Conventional stimulation methods such as matrix acidizing, acid fracturing, or proppant fracturing have resulted in products that perform poorly and/or fail within months. Other options, such as water fracs with light sand, give better results but are prohibitively expensive. Mineral composition, brittleness index, stress regime, and petrophysical properties, which are favorable for creating complex fracture networks, can be obtained by geochemical and geomechanical analysis. The extended Reshaw and Pollard criterion shows that hydraulic fractures tend to be arrested by pre-existing natural fractures, and complex fracture networks would be created during fracturing. Additionally, the critical stressed faults theory indicates that the pre-existing natural fractures tend to slip with the shear mode as the pore fluid pressure increases. Rotating disk experiments and conductivity tests with artificial sheared plates have shown that flow channels can be etched at the location of scratches on fracture surfaces. Meanwhile, the carbonate cement in natural fractures can be chelated to form wormhole likely flow channels. Complex fracture networks with sufficient acid etched conductivity can be generated by water fracs with acid. A novel and economical volume stimulation strategy known as network acid fracturing has provided Tarim Oil Company the means to develope ultra-deep, ultra-high pressure, high temperature, and ultralow permeability but fractured gas reservoirs. Post-stimulation production performances of numerous wells with network acid fracturing are comparable to those with stimulated reservoir volumes.
... Nitsche's method (e.g. Wriggers and Zavarise [22]) also sets free contact constraints; however, it is established based on a different concept in which the contact stress vector is computed in terms of the stress field of the bodies rather than the Lagrange multipliers [22,49]. Therefore, Nitsche's method does not give rise to extra degrees of freedom. ...
Large deformation problems in practical engineering are often accompanied by contact phenomena. While the conventional material point method (MPM) is efficient at solving large deformation problems, it cannot handle slip contacts. This paper presents Nitsche’s method for analysing large deformations with frictional contact via the MPM. Nitsche’s method has good features of variational consistency and no additional unknowns, and it is integrated into the MPM in a weak manner based on the principle of virtual power. Within the integrated formulation, both biased and unbiased computational schemes are derived to adapt to different forms of contact. Additionally, B-spline shape functions are employed to alleviate cell-crossing noise, and an improved particle extrapolation approach for accurate contact detection is introduced. The efficacy of the proposed Nitsche-based MPM is validated through several representative benchmarks from the literature. We further apply the proposed method to simulate the water leakage problem of the lining gasketed joint in shield tunnels. Comparison with experimental results demonstrates the applicability of the proposed method.
... The first one is sliding friction, where a set-valued force law with two degrees of freedom generates a force resisting only to sliding F T . Different numerical simulations, which include sliding friction, were given in [8,22,55]. The second kind of friction model is rolling friction, where two translation degrees of freedom are required in the formulation. ...
The general motion of a sphere in a mechanism in contact with a rigid planar surface under rolling, sliding and spinning friction is studied in the context of non-smooth contact dynamics. The equations of motion are solved by the non smooth generalized–α implicit time integration scheme, where the position and velocity level constraints are satisfied exactly without requiring to define any particular value for a penalty parameter. The geometrical properties of the spheres are described by a rigid-body formulation with translational and rotational degrees of freedom. The robustness and the performance of the proposed methodology is demonstrated by different examples, including both flexible and/or rigid elements.
... We will focus on such weak constraint enforcement strategies and develop a weighted Nitsche's approach for enforcing jump constraints across the interface. Nitsche's method has been used extensively for weakly enforcing constraints on interfaces [2][3][4][5][6][7]14,[24][25][26]29]. The method employs a stabilization parameter that should be chosen to be sufciently large as to maintain coercivity. ...
We develop a numerical strategy based on a weighted Nitsche’s approach to model a general class of interface problems with higher-order simplex elements. We focus attention on problems in which the jump in the field quantities across an interface is given. The presented method generalizes the weighted Nitsche’s approach of Annavarapu et al. (Comput. Meth. Appl. Mech. Eng. 225–228:44–54, 2012) to higher-order simplices. Specifically, for higher-order simplex elements, we derive closed-form analytical expressions for the stabilization parameter arising in Nitsche’s variational form. We also prescribe corresponding weights for the discrete fluxes in the consistency terms present in Nitsche’s variational form. The prescribed choice of weights is shown to be optimal such that it minimizes the stabilization parameter while ensuring coercivity of the bilinear form. In the presence of large contrasts in material properties and mesh sizes, the proposed weighting yields better conditioned systems than the traditional Nitsche formulation by bounding the maximum eigenvalue of the discrete system from above. Further, the geometrical representation of curved interfaces is improved through a hierarchical local renement approach. Several numerical examples are presented with quadratic triangles to demonstrate the efficacy of the presented method.
... The augmented Lagrangian method is free of the aforementioned problems of the Lagrange multiplier and the penalty methods, but it requires additional iterative steps. Recent studies have thus developed alternative and more advanced methods such as the weighted Nitsche method [12][13][14] or explored diffuse approaches for algorithm-free modeling of frictional interfaces [15][16][17][18]. Yet none of the existing methods is considered optimally robust and efficient for modeling embedded interfaces with frictional contact. ...
We present a barrier method for treating frictional contact on interfaces embedded in finite elements. The barrier treatment has several attractive features, including: (i) it does not introduce any additional degrees of freedom or iterative steps, (ii) it is free of inter-penetration, (iii) it avoids an ill-conditioned matrix system, and (iv) it allows one to control the solution accuracy directly. We derive the contact pressure from a smooth barrier energy function that is designed to satisfy the non-penetration constraint. Likewise, we make use of a smoothed friction law in which the stick-slip transition is described by a continuous function of the slip displacement. We discretize the formulation using the extended finite element method to embed interfaces inside elements, and devise an averaged surface integration scheme that effectively provides stable solutions without traction oscillations. Subsequently, we develop a way to tailor the parameters of the barrier method to embedded interfaces, such that the method can be used without parameter tuning. We verify and investigate the proposed method through numerical examples with various levels of complexity. The numerical results demonstrate that the proposed method is remarkably robust for challenging frictional contact problems, while requiring low cost comparable to that of the penalty method.
... Nitsche methods overcome the issue of numerical instability affecting penalty-like formulations, by adding consistency terms [62,63]. Nitsche's method has been extended for modeling frictionalsliding on embedded interfaces [64,65] and small-sliding contact on frictional surfaces, including stick?slip behavior [66]. Inspired by the work of [67], we recently extended the Nitsche's method to cohesive fracture problems, and developed a stabilized FEM that alleviates traction oscillations with stiff, anisotropic cohesive laws [7]. ...
We demonstrate the ability of a stabilized finite element method, inspired by the weighted Nitsche approach, to alleviate spurious traction oscillations at interlaminar interfaces in multi-ply multi-directional composite laminates. In contrast with the standard (penalty-like) method, the stabilized method allows the use of arbitrarily large values of cohesive stiffness and obviates the need for engineering approaches to estimate minimum cohesive stiffness necessary for accurate delamination analysis. This is achieved by defining a weighted interface traction in the stabilized method, which allows a gradual transition from penalty-like method for soft elastic contact to Nitsche-like method for rigid contact. We conducted several simulation studies involving constant strain patch tests and benchmark delamination tests under mode-I, mode-II and mixed-mode loadings. Our results show clear evidence of traction oscillations with the standard method with structured and perturbed finite element meshes, and that the stabilized method alleviates these oscillations, thus illustrating its robustness.
... So we have resorted to Nitsche's method, see Hansbo [36], which could be regarded as a hybrid of penalty methods and Lagrange multiplier methods, as our choice, because it provides a stabilized linear equations and is better conditioned than standard penalty methods. Moreover, we avoid saddlepoint problem complications inherent to Lagrange multiplier methods (e.g., [6,7,10,22,35]). ...
We propose a numerical method for analyzing fault slip tendency under fluid injection using the extended finite element method (XFEM) both for fluid flow and poroelasticity. The fault is modeled as a zero-thickness interface, and we use a reduced model for the fluid flow in the fault to account for its hydraulic behavior. We use the rate- and state-dependent friction model as the fault friction model, and Biot’s theory of poroelasticity to study the coupling between fluid flow and mechanical deformation in the surrounding porous media. Since a fully coupled method between fluid flow and poromechanics is computationally expensive, we have investigated the use of the so-called fixed-stress split in this context. In such a scheme, the fluid flow problem is solved firstly by freezing the total means stress field, and then the results are used to solve the mechanical problem. The fixed-stress split is unconditionally stable, consistent and more accurate for a given number of iterations compared with other type of splitting strategies. In order to verify our method, some test cases are presented. For the coupling between fluid flow and poromechanics, we consider the Terzaghi Problem and the Mandel Problem, comparing our results with those of previously published works. While, for the mechanic problem, we compare the results with those obtained using the software Pylith.
... The Nitsche's method [6] can be viewed as a variationally consistent penalty method, with the advantage that the discrete system of equations are better conditioned provided the stabilization parameters are chosen appropriately. As a result of the pioneering work of Hansbo [7], the method has become popular for a wide class of interface (contact) problems [8][9][10]. A more detailed account of the Nitsche's method and its application to various interface problems in computational mechanics can be found in the review article by [11]. ...
... A detailed discussion on the method is provided in Settgast et al. (2017). The sliding contact and crack propagation techniques used in GEOS can be found in Annavarapu et al. (2015) and Annavarapu et al. (2016). ...
Wellbore integrity is a critical component of long-term carbon storage. Depleted reservoirs that are potential CO 2 storage sites, typically contain several wells. Due to years of operations and abandonment, these wells can have cracks in the cement, cement-casing interface, and/or cement-formation interface. During CO 2 injection, changes in temperature may result in stress variations that can further damage the well, threatening its integrity. The temperature difference between the cold injected CO 2 and warm reservoir, and different thermal properties for the wellbore casing, cement, and the lithology, will stress the near wellbore environment, potentially extending pre-existing defects creating leakage pathways from the storage reservoir to the overlying strata. We have conducted a systematic numerical study to explore the role of CO 2 injection temperature, downhole effective in-situ horizontal stress, and the thermo-mechanical properties by coupling a linear elastic stress model with heat conduction. We consider conditions in non-perforated casing above the injection zone where conductive heating is dominant. The injection temperatures considered covers current industrial practice as well as sub-zero temperatures. The latter represents direct injection following ship transport, without pre-heating. In this study, we consider the connection between damage risk and the temperature difference between the injected CO 2 and the formation and downhole effective in-situ horizontal stress. The study found that the negative impacts of thermal stress in the wellbore environment are mitigated by the presence of effective in-situ horizontal stresses. Stresses normal to the well have the potential to reduce the tensile stress and stress intensity factor. In the absence of sufficient effective in-situ horizontal stress, thermal stress may cause the fracture to propagate due to high stress concentration near the fracture edges. In general, formations with large effective in-situ horizontal stress can prevent leakage paths from growing even when large temperature difference exists between the formation and the injected CO 2. Our simulations suggest that CO 2 can be injected at sub-zero temperatures, suitable for ship transport, when the downhole effective in-situ horizontal stress is greater than 10 or 12 MPa, depending on the location of the pre-existing cracks. For onshore transportation, injection of liquid CO 2 results in minimal damage, provided there is ample in-situ horizontal stress.
... Thus, the mechanical and failure behavior of the interfaces between these phases can be complex. Studies in recent years have begun incorporating phasedependent material properties into interfacial constitutive models for intergranular cracks, dislocation transmission, frictional sliding, etc. [33][34][35]. ...
The Discontinuous Element Insertion Program is a MATLAB/Octave toolbox for inserting zero-thickness interface elements into two and three dimensional finite element meshes. These interface elements, termed herein as “couplers” are used for intrinsic cohesive zone modeling and for the Discontinuous Galerkin method. The underlying algorithm is topology based and is suitable for complex, unstructured meshes of mixed-type linear and quadratic elements. Insertion is specified according to regions or subdomains within the overall analysis domain, a geometrically intuitive means to designate the coupler locations.
... Upper and lower bounds are proved under a saturation assumption, and the performance of the error estimates is investigated numerically. • The topic of small-sliding frictional contact on 3D interfaces is the object of [3], where a weighted-Nitsche method is designed, and tested numerically. • In [48] a least-square stabilized augmented lagrangian method, inspired by Nitsche's method, is described for unilateral contact. ...
We summarize recent achievements in applying Nitsche’s method to some contact and friction problems. We recall the setting of Nitsche’s method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H1(Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multi-body contact, contact in large transformations and contact in elastodynamics are presented as well.
... The numerical simulator GEOS, developed at Lawrence Livermore National Laboratory (Settgast et al. 2014(Settgast et al. , 2016, is used to simulate the injection phases. GEOS is a massively parallel multi-physics framework, which incorporates capabilities to model hydraulic fracturing , Settgast et al. 2016; seismic events (Sherman et al. 2016); fracture shearing (Annavarapu et al. 2015); thermal drawdown ; matrix flow and heat transport (Guo et al. 2016); flow in rough fractures (Vogler et al. 2016a); geochemical transport and reaction ; and simulations of immiscible fluid flow . The numerical solver employed in this study offers fully coupled hydro-mechanics (HM) and implicit time stepping. ...
... Among the various strategies for the enforcement of contact constraints, e.g., Lagrange multiplier methods [Amdouni et al. (2014)], augmented Lagrangian approaches [Zavarise and De Lorenzis (2012)] and Nitsche methods [Annavarapu et al. (2014[Annavarapu et al. ( , 2015], the penalty methods [Benson and Hallquist (1990); Zavarise (2015)] are probably the most widely used algorithms due to there simplicity. In the classical NTS contact discretization, by means of the penalty approach, the normal contact traction, p n , takes the form ...
Laminated glass is a simple sandwiched structure, which, however, is widely used in automotive and architectural industries. The well-known extrinsic cohesive zone model has proved to be a powerful numerical approach for the glass-ply crack simulations in laminated glass, where the classical node-to-segment contact treatment is usually employed. However, an unphysical phenomenon, named contact force jump, may arise during crack simulations, thereby resulting in numerical instabilities or incorrect simulation results. To address this issue, we develop an improved node-to-segment contact formulation to make the contact interactions in extrinsic crack simulations more stable. The main idea is to keep the number of contact constraints constant before and after cracking. The improved contact formulation is very simple and easy to implement, which, however, is effective to address the unphysical phenomenon. Three numerical examples, including the impact crack simulations of a laminated glass beam, are performed to validate the accuracy and effectiveness of the improved contact formulation.
... GEOS is a massively parallel, multi-physics numerical framework, which was designed mainly to enable simulations of subsurface hydraulic reservoir stimulations. It has been used to simulate hydraulic fracturing [Settgast et al., 2017, Vogler et al., 2017a, reactive transport , drawdown in geothermal systems [Fu et al., 2016], immiscible fluid flow , shearing of fractures using FEM [Annavarapu et al., 2015], crack growth using XFEM methods [Annavarapu et al., 2016], and hydro-mechanically coupled flow in rough fractures [Vogler et al., 2016a[Vogler et al., , 2018. GEOS was chosen for DS as it offers an internal mesh generator, which greatly simplifies running simulations on a wide range of DFNs. ...
To enable fast uncertainty quantification of fluid flow in a discrete fracture network (DFN), we present two approaches to quickly compute fluid flow in DFNs using combinatorial optimization algorithms. Specifically, the presented Hanan Shortest Path Maxflow (HSPM) and Intersection Shortest Path Maxflow (ISPM) methods translate DFN geometries and properties to a graph on which a max flow algorithm computes a combinatorial flow, from which an overall fluid flow rate is estimated using a shortest path decomposition of this flow. The two approaches are assessed by comparing their predictions with results from explicit numerical simulations of simple test cases as well as stochastic DFN realizations covering a range of fracture densities. Both methods have a high accuracy and very low computational cost, which can facilitate much-needed in-depth analyses of the propagation of uncertainty in fracture and fracture-network properties to fluid flow rates.
... In [18] a stabilized augmented Lagrange formulation is developed for frictionless contact. A stabilized formulation based on the Nitsche method is presented in [4,5] for small sliding contact in 2D and 3D respectively. In both formulations the stabilizing term involves the finite element tractions. ...
This paper proposes a method of solving 3D large deformation frictional contact problems with the Cartesian Grid Finite Element Method. A stabilized augmented Lagrangian contact formulation is developed using a smooth stress field as stabilizing term, calculated by Zienckiewicz and Zhu Superconvergent Patch Recovery. The parametric definition of the CAD surfaces (usually NURBS) is considered in the definition of the contact kinematics in order to obtain an enhanced measure of the contact gap. The numerical examples show the performance of the method.
... Equation (9) states that the fluid pressure inside the fracture, , should be calculated, which presents a very complicated problem, especially for fractures that are not straight. Moreover, the fluid pressure inside the fracture must be transferred into an equivalent nodal force using the linear integral form, which requires integral points on the fracture surface [23]. We set the net pressure inside the fractures at 3 MPa. ...
There has been a growing consensus that preexisting natural fractures play an important role during stimulation. A novel fully coupled hydromechanical model using extended finite element method is proposed. This directly coupled scheme avoids the cumbersome process during calculating the fluid pressure in complicated fracture networks and translating into an equivalent nodal force. Numerical examples are presented to simulate the hydraulic fracture propagation paths for simultaneous multifracture treatments with properly using the stress shadow effects for horizontal wells and to reveal the deformation response and interaction mechanism between hydraulic induced fracture and nonintersected natural fractures at orthotropic and nonorthotropic angles. With the stress shadow effects, the induced hydraulic flexural fracture deflecting to wellbore rather than transverse fracture would be formed during the progress of simultaneous fracturing for a horizontal well. The coupled hydromechanical simulation reveals that the adjacent section to the intersection is opened and the others are closed for orthogonal natural fracture, while the nonorthogonal natural fracture is activated near the intersection firstly and along the whole section with increasing perturbed stresses. The results imply that the induced hydraulic fracture tends to cross orthotropic natural fracture, while it is prior to being arrested by the nonorthotropic natural fracture.
... The numerical simulator GEOS, developed at Lawrence Livermore National Laboratory (Settgast et al. 2014(Settgast et al. , 2016, is used to simulate the injection phases. GEOS is a massively parallel multi-physics framework, which incorporates capabilities to model hydraulic fracturing , Settgast et al. 2016; seismic events (Sherman et al. 2016); fracture shearing (Annavarapu et al. 2015); thermal drawdown ; matrix flow and heat transport (Guo et al. 2016); flow in rough fractures (Vogler et al. 2016a); geochemical transport and reaction ; and simulations of immiscible fluid flow . The numerical solver employed in this study offers fully coupled hydro-mechanics (HM) and implicit time stepping. ...
In-situ hydraulic fracturing has been performed on the decameter scale in the Deep Underground rock Laboratory (DUG Lab) at the Grimsel Test Site (GTS) in Switzerland in order to measure the minimum principal stress magnitude and orientation. Conducted tests were performed in a number of boreholes, with 3–4 packer intervals in each borehole subjected to repeated injection. During each test, fluid injection pressure, injection flow rate and microseismic events were recorded amongst others. Fully coupled 3D simulations have been performed with the LLNL's GEOS simulation framework. The methods applied in the simulation of the experiments address physical processes such as rock deformation/stress, LEFM fracture mechanics, fluid flow in the fracture and matrix, and the generation of micro-seismic events. This allows to estimate the distance of fracture penetration during the injection phase and correlate the simulated injection pressure with experimental data during injection, as well as post shut-in. Additionally, the extent of the fracture resulting from simulations of fracture propagation and microseismic events are compared with the spatial distribution of the microseismic events recorded in the experiment.
... • The topic of small-sliding frictional contact on 3D interfaces is the object of [3], where a weighted-Nitsche method is designed, and tested numerically. ...
We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are recalled: consistency, well-posedness, fully optimal convergence in H 1 (Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multi-body contact, contact in large transformations and contact in elastodynamics are presented as well.
... For the local approach, we use ε = 1 × 10 −8 . The algorithm has been implemented in GEOS: a flexible multi-scale, multi-physics simulation environment developed at Lawrence Livermore National Laboratory [42,43,44]. ...
We develop a local, implicit crack tracking approach to propagate embedded failure surfaces in three-dimensions. We build on the global crack-tracking strategy of Oliver et al. (Int J. Numer. Anal. Meth. Geomech., 2004; 28:609–632) that tracks all potential failure surfaces in a problem at once by solving a Laplace equation with anisotropic conductivity. We discuss important modifications to this algorithm with a particular emphasis on the effect of the Dirichlet boundary conditions for the Laplace equation on the resultant crack path. Algorithmic and implementational details of the proposed method are provided. Finally, several three-dimensional benchmark problems are studied and results are compared with available literature. The results indicate that the proposed method addresses pathological cases, exhibits better behavior in the presence of closely interacting fractures, and provides a viable strategy to robustly evolve embedded failure surfaces in 3D.
... It is shown that damping caused by contact interfaces can account for 90% of the total [7] in the whole structure. Consequently, efforts to study the nonlinear mechanisms of jointed structures with contact interfaces is of great value [8,9]. ...
... This example demonstrates the accuracy of the stress fault model. Note that no numerical stability issues on the contact tractions on the fault surface such as alluded to in Annavarapu et al. (2015) are encountered in our procedure. This may be due to the fact that we use a penalty formulation (see Section 4.1), and do not attempt to analytically derive the tangent operators arising from the Coulomb stick/slip logic on the interface. ...
Faults are geological entities with thicknesses several orders of magnitude smaller than the grid blocks typically used to discretize reservoir and/or over-under-burden geological formations. Introducing faults in a complex reservoir and/or geomechanical mesh therefore poses significant meshing difficulties. In this paper, we consider the strong-coupling of solid displacement and fluid pressure in a three-dimensional poro-mechanical (reservoir-geomechanical) model. We introduce faults in the mesh without meshing them explicitly, by using the extended finite element method (X-FEM) in which the nodes whose basis function support intersects the fault are enriched within the framework of partition of unity. For the geomechanics, the fault is treated as an internal displacement discontinuity that allows slipping to occur using a Mohr-Coulomb type criterion. For the reservoir, the fault is either an internal fluid flow conduit that allows fluid flow in the fault as well as to enter/leave the fault or is a barrier to flow (sealing fault). For internal fluid flow conduits, the continuous fluid pressure approximation admits a discontinuity in its normal derivative across the fault, whereas for an impermeable fault, the pressure approximation is discontinuous across the fault. Equal-order displacement and pressure approximations are used. Two- and three-dimensional benchmark computations are presented to verify the accuracy of the approach, and simulations are presented that reveal the influence of the rate of loading on the activation of faults.
Fracture propagation in layered media is investigated using an adaptive phase-field method. We focus on the interplay between cracks and interfaces, considering both perfectly and imperfectly bonded interfaces. For perfectly bonded interfaces, three-layer models are analyzed to study the effects of mechanical property mismatches, layer thickness, and confinement pressure on crack growth. Results reveal that critical energy release rate mismatch significantly influences the crack geometry, leading to single through-going fractures, middle layer fragmentation, or delamination. There is an inverse relationship between layer thickness and fragmentation, and between confinement pressure and delamination. For imperfectly bonded interfaces, a phase-field method incorporating an interface energy term is introduced and validated with benchmark examples. This model is used to study the combined effects of mechanical property mismatch and interface strength on crack growth. Our findings demonstrate that the interface strength strongly influences the dominant failure mechanism, with high strength favoring mechanical property mismatch-driven fracture and low strength leading to interfacial failure. Finally, the robustness of the proposed method is illustrated through a complex seven-layer model. This study provides valuable insights into the various factors influencing macroscopic failure mechanisms in layered materials.
The transition zone is a typical interface with enough thickness and high heterogeneity between the sandstone and coal and has a great effect on the fracture propagation in coal measure strata. In the study, the true tri-axial experiments were conducted with the combination of sandstone, coal and limestone outcrops. The effect of cementing strength and thickness of the transition zone was mainly studied. Three types of hydraulic fracture initiation positions were used to simulate three fracturing situations in the field. The results showed that the effect of the transition zone on the fracture vertical extension could be concluded: the fracture was easy to cross or slip with low cementing strength and turned inside the transition zone with high cementing strength; the fracture tended to branch or be blocked by the thick transition zone. The horizontal flow through face cleats and natural fracture activation in coal contributed to a more developed fracture network. The characters of pumping curve pressure “increasing – fluctuation – dropping”, and the “fluctuation pressure more than initiation pressure” indicated the interface crossing. The results of experiments agreed with the previous numerical simulation and helped to optimize the fracturing design in layered coal strata.
To enable fast uncertainty quantification of fluid flow in a discrete fracture network (DFN), we present two approaches to quickly compute fluid flow in DFNs using combinatorial optimization algorithms. Specifically, the presented Hanan Shortest Path Maxflow (HSPM) and Intersection Shortest Path Maxflow (ISPM) methods translate DFN geometries and properties to a graph on which a max flow algorithm computes a combinatorial flow, from which an overall fluid flow rate is estimated using a shortest path decomposition of this flow. The two approaches are assessed by comparing their predictions with results from explicit numerical simulations of simple test cases as well as stochastic DFN realizations covering a range of fracture densities. Both methods have a high accuracy and very low computational cost, which can facilitate much-needed in-depth analyses of the propagation of uncertainty in fracture and fracture-network properties to fluid flow rates.
In this thesis, we present and study a new formulation of frictional contact between two elastic bodiesbased on Nitsche's method. This method aims to treat the interface conditions in a weak sense, thanksto a consistent additional term stabilized with the parameter gamma. At first, we introduce the study carriedout in the small strain framwork for an unbiased version of the method. The non-distinction between amaster surface and a slave one will allow the method to be more generic and directly applicable to theself-contact problem. The restrictive framework of small strain allowed us to obtain theoretical results onthe consistency and convergence of the method. Then, we present the extension of the Nitsche methodto the large strain case more relevant for industrial applications and situations of self-contact. ThisNitsche's method is formulated for an hyper-elastic material and declines in the two versions: biased andunbiased. We describe a class of methods through a generalisation parameter theta . Particular variantshave different properties from a numerical point of view, in terms of accuracy and robustness. To provethe accuracy of the method for large deformations, we provide several academic and industrial tests. Wealso study the inuence of numerical quadrature on the accuarcy and convergence of the method. This study covers a comparison of several integration rules proposed in the literature for other integral methods.
Les assemblages sont des éléments critiques pour les structures industrielles. De fortes non-linéarités de type contact frottant, ainsi que des précharges mal maîtrisées rendent complexe tout dimensionnement précis. Présents en très grand nombre sur les structures industrielles (quelques millions pour un A380), cela implique de rafiner les modèles localement et donc de gérer des problèmes numé-riques de très grandes tailles. Les nombreuses interfaces de contact frottant sont des sources de difficultés de convergence pour les simulations numériques. Il est donc nécessaire de faire appel à des méthodes robustes. Il s’agit d’utiliser des méthodes itératives de décomposition de domaine, permettant de gérer des modèles numériques extrêmement grands, couplées à des techniques adaptées afin de prendre en compte les non-linéarités de contact aux interfaces entre sous-domaines. Ces méthodes de décomposition de domaine restent encore très peu utilisées dans un cadre industriel. Des développements internes aux codes éléments finis sont souvent nécessaires et freinent ce transfert du monde académique au monde industriel.Nous proposons, dans ces travaux de thèse, une mise-en-oeuvre non intrusive de ces méthodes de décomposition de domaine : c’est-à-dire sans développement au sein du code source. En particulier, nous nous intéressons à la méthode Latin dont la philosophie est particulièrement adaptée aux problèmes non linéaires. La structure est décomposée en sous-domaines reliés entre eux au travers d’interfaces. Avec la méthode Latin, les non-linéarités sont résolues séparément des aspects linéaires. La résolution est basée sur un schéma itératif à deux directions de recherche qui font dialoguer les problèmes linéaires globaux etles problèmes locaux non linéaires.Au cours de ces années de thèse, nous avons développé un outil totalement non intrusif sous Code_Aster permettant de résoudre par une technique de décomposition de domaine mixte des problèmes d’assemblage. Les difficultés posées par le caractère mixte de la méthode Latin sont résolues par l’introduction d’une direction de recherche non locale. Des conditions de Robin sur les interfaces des sous-domaines sont alors prises en compte simplement sans modifier les sources de Code_Aster. Nous avons proposé une réécriture algébrique de l’approche multi-échelle assurant l’extensibilité de la méthode. Nous nous sommes aussi intéressés à coupler la méthode Latin en décomposition de domaine à un algorithme de Krylov. Appliqué uniquement à un problème sous-structuré avec interfaces parfaites, ce couplage permet d’accélérer la convergence. Des structures préchargées avec de nombreuses interfaces de contact frottant ont été traitées. Des simulations qui n’auraient pu être menées par un calcul direct sous Code_Aster ont été réalisées via cette stratégie de décomposition de domaine non intrusive.
A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is established both for two and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two and three-dimensional numerical experiments illustrate the theory.
We propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method introduced previously for frictionless unilateral contact. Both cases of unilateral and bilateral contact with friction are taken into account, with emphasis on frictional unilateral contact for the numerical analysis. We manage to prove theoretically the fully optimal convergence rate of the method in the H-1(Omega)-norm which is O(h(1/2+nu)) when the solution lies in H3/2+nu(Omega), 0 < nu <= k - 1/2, in two dimensions and three dimensions, for Lagrange piecewise linear (k = 1) and quadratic (k = 2) finite elements. No additional assumption on the friction set is needed to obtain this proof.
We introduce a Nitsche-based formulation for the finite element discretization of the uni-lateral contact problem in linear elasticity. It features a weak treatment of the non-linear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the H 1 (Ω)-norm for linear finite elements in two dimensions, which is O(h 1 2 +ν) when the solution lies in H 3 2 +ν (Ω), 0 < ν ≤ 1/2. An interest of the formulation is that, conversely to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf–sup condition needs to be satisfied.
Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that. for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29 from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.
Taking into account arbitrary crack geometries, crack closure generally occurs independently of the load case. As the standard eXtended Finite Element Method (XFEM) does not prevent unphysical crack face penetration in this case, a formulation allowing for crack face contact is proposed in terms of a penalty formulation for normal contact. The discretization is developed for non-planar cracks intersecting hexahedral elements in an arbitrary manner. Typical problems of many crack face contact implementations within the XFEM, like locking or the introduction of additional degrees of freedom, are avoided by projecting the contact contribution onto the hexahedral element nodes. The method is tested by means of suitable numerical examples, finally presenting an application in form of a multiscale setup with arbitrarily arranged micro cracks in the vicinity of a macro crack front.
The extended finite element method (X-FEM) has
been developed to minimize requirements on the mesh in a
problem with a displacement discontinuity. We present the
development carried out to take advantage of the X-FEM
approach in simplifying the meshing of complex 3D networks
of discontinuities with junctions. Contact with large
sliding along the branched discontinuities is discussed. Solutions
are proposed and discussed to solve some matrix conditioning
issues. Several examples are presented in this paper
in order to prove the efficiency of the proposed approach.
In this work, we propose a novel weighting for the interfacial consistency terms arising in a Nitsche variational form. We demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method. Consequently, we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings; ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of (a) elements with arbitrarily small volume fractions, (b) large material heterogeneities and (c) both large heterogeneities as well as arbitrarily small elements. We then highlight the accuracy and efficiency of the proposed formulation through numerical examples, focusing particular attention on interfacial quantities of interest.
An interface finite element for three-dimensional problems based on the penalty method is presented. The proposed element can model joints/interfaces between solid finite elements and also includes the propagation of damage in pure mode I, pure mode II and mixed mode considering a softening relationship between the stresses and relative displacements. Two different contact conditions are considered: point-to-point constraint for closed points (not satisfying the failure criterion) and point-to-surface constraint for opened points. The performance of the element is tested under mode I, mode II and mixed mode loading conditions.
Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright © 2009 John Wiley & Sons, Ltd.
In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact
model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence
analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence
of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results
and show the capabilities of the method.
Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh-dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials.
In this paper a finite element formulation for frictionless contact problems with non-matching meshes in the contact interface
is presented. It is based on a non-standard variational formulation due to Nitsche and leads to a matrix formulation in the
primary variables. The method modifies the unconstrained functional by adding extra terms and a stabilization which is related
to the classical penalty method. These new terms are characterized by the presence of contact forces that are computed from
the stresses in the continuum elements. They can be seen as a sort of Lagrangian-type contributions. Due to the computation
of the contact forces from the continuum elements, some additional degrees-of-freedom are involved in the stiffness matrix
parts related to contact. These degrees-of-freedom are associated with nodes not belonging to the contact surfaces.
This book presents the fundamentals of nonlinear mechanics within a modern computational approach based mainly on finite element methods. Both material and geometric nonlinearities are treated. The topics build up from the mechanics of finite deformation of solid bodies through to nonlinear structural behaviour including buckling, bifurcation and snap-through. The principles are illustrated with a series of solved problems. This book serves as a text book for a second year graduate course and as a reference for practitioners using nonlinear analysis in engineering and design.
We investigate a finite element method for frictional sliding along embedded interfaces within a weighted Nitsche framework. For such problems, the proposed Nitsche stabilized approach combines the attractive features of two traditionally used approaches: viz. penalty methods and augmented Lagrange multiplier methods. In contrast to an augmented Lagrange multiplier method, the proposed approach is primal; this allows us to eliminate an outer augmentation loop as well as additional degrees of freedom. At the same time, in contrast to the penalty method, the proposed method is variationally consistent; this results in a stronger enforcement of the non-interpenetrability constraint. The method parameter arising in the proposed stabilized formulation is defined analytically, for lower order elements, through numerical analysis. This provides the proposed approach with greater robustness over both traditional penalty and augmented Lagrangian frameworks. Through this analytical estimate, we also demonstrate that the proposed choice of weights, in the weighted Nitsche framework, is indeed the optimal one. We validate the proposed approach through several benchmark numerical experiments.
This paper presents a grain level finite element model to simulate the cracking behavior of the intermetallic
compound (IMC) layer in solder joints. The grain microstructure of the IMC layer is explicitly
included in the model by Voronoi tessellations. Cohesive interface elements with a coupled cohesive
law are embedded along the grain boundaries to simulate microcrack initiation, propagation and coalescence
in the IMC layer. A model with a Weibull distributed grain interfacial strength is adopted to account
for randomly distributed grain boundary defects. The average thickness of the IMC layer and the wavelength
and the roughness of the waved solder/IMC interface are used to characterize the IMC microstructure.
Using the numerical approach developed, the effects of the grain shape, the randomly distributed
grain boundary defects, the thickness of the IMC layer and the morphology of the solder/IMC interface
on the microcrack patterns and on the overall response of solder joints are investigated. The results indicate
that the overall mechanical strength is not sensitive to the grain shape, but the microcrack pattern
and the crack path depend heavily on the grain shape. In the model containing randomly distributed
grain boundary defects, the weak grain interface plays a critical role in the overall strength and the crack
path of the model. The average thickness of the IMC layer has the greatest impact on the overall strength
and the failure mode of the solder joint. The wavelength and the roughness of the solder/IMC interface
have little impact on the overall strength but do have an impact on the failure mode of the solder joint.
The predicted failure mode agrees well with the experimental observation in solder joints. The presented
approach is feasible for simulating microcracking and the failure behavior of the IMC layer in solder joints
and other quasi-brittle polycrystalline materials.
We extend the weighted Nitsche’s method proposed in the first part of this study to include multiple intersecting embedded interfaces. These intersections arise either inside a computational domain – where two internal interfaces intersect; or on the boundary of the computational domain – where an internal interface intersects with the external boundary. We propose a variational treatment of both the interfacial kinematics and the external Dirichlet constraints within Nitsche’s framework. We modify the numerical analysis to account for these intersections and provide an explicit expression for the weights and the method parameters that arise in the Nitsche variational form in the presence of junctions. Finally, we demonstrate the performance of the method for both perfectly-tied interfaces and perfectly-plastic sliding interfaces through several benchmark examples.
We investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method. In particular, we focus on explicit time integration of the semi‐discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime. We reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the Courant–Friedrichs–Levy condition on primal formulations. We propose an alternate, consistent formulation for the primal method and derive an estimate for the stabilization parameter, which is more amenable in an explicit dynamics framework. Importantly, the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary. We also show that with interfacial constraints, existing mass lumping schemes can lead to prohibitively small critical time steps. Accordingly, we propose a mass lumping procedure, which provides a more favorable estimate. These techniques are then demonstrated on several benchmark numerical examples, where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints. Copyright © 2012 John Wiley & Sons, Ltd.
We present a derivation of a new interface formulation via a merger of continuous Galerkin and discontinuous Galerkin concepts, enhanced by the variational multiscale method. Developments herein provide treatment for the pure‐displacement form and mixed form of small deformation elasticity as applied to the solution of two problem classes: domain decomposition and contact mechanics with friction. The proposed framework seamlessly accommodates merger of different element types within subregions of the computational domain and nonmatching element faces along embedded interfaces. These features are retained in the treatment of multibody small deformation contact problems as well, where an unbiased treatment of the contact interface stands in contrast to classical master/slave constructs. Numerical results for problems in two and three spatial dimensions illustrate the robustness and versatility of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.
SUMMARYA stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C 1‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C 2 B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.
Failure analyses of rock masses are frequently encountered in major civil engineering construction and mining, and in particular for projects related to slope excavation. The proposed procedure determines the seismic factor of safety against sliding along a joint plane.
In this paper, an extended finite element method is employed to simulate the presence of discontinuities caused by the contact surface. In X-FEM, the need for mesh adaption to interface is neglected and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the elements. The modified level set technique and the Heaviside enrichment function are employed to approximate the discontinuous displacement field of elements located on the contact surface. The penalty method is used to impose the contact constraints and establish the non-penetration condition. An efficient numerical algorithm is employed to model the large deformation contact behavior based on the node-to-segment technique. Finally, numerical examples are presented to demonstrate the accuracy and capability of proposed X-FEM technique in modeling of large deformation – large sliding contact problems.
In this paper, the shear behaviour of rock joints are numerically simulated using the discrete element code PFC2D. In PFC, the intact rock is represented by an assembly of separate particles bonded together where the damage process is represented by the breakage of these bonds. Traditionally, joints have been modelled in PFC by removing the bonds between particles. This approach however is not able to reproduce the sliding behaviour of joints and also results in an unrealistic increase of shear strength and dilation angle due to the inherent micro-scale roughness of the joint surface. Modelling of joints in PFC was improved by the emergence of the smooth joint model. In this model, slip surfaces are applied to contacts between particles lying on the opposite sides of a joint plane. Results from the current study show that this method suffers from particle interlocking which takes place at shear displacements greater than the minimum diameter of the particles. To overcome this problem, a new shear box genesis approach is proposed. The ability of the new method in reproducing the shear behaviour of rock joints is investigated by undertaking direct shear tests on saw-tooth triangular joints with base angles of 15°, 25° and 35° and the standard joint roughness coefficient profiles. A good agreement is found between the results of the numerical models and the Patton, Ladanyi and Archambault and Barton and Choubey models. The proposed model also has the ability to track the damage evolution during the shearing process in the form of tensile and shear fracturing of rock asperities.
In this paper, some new generalized Newton’s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones. A numerical experimentation is performed to compare the different methods, especially with respect to the sensitivity to the method parameter. Two different strategies to approximate the contact and friction condition are considered: a nodal and an integral one. Existence and uniqueness results of the solution to the discretized problem are also discussed.
This paper presents the application of a new method for interfacial modeling utilizing a merger of continuous Galerkin and discontinuous Galerkin concepts to simulate the behavior of mechanical joints. The interfacial flux terms arising naturally from the discontinuous Galerkin treatment provide a mechanism to embed friction models in a variationally consistent fashion. Due to the unbiased implementation of the interface, facilitated by avoiding the master–slave concept, the deformation of the two interacting surfaces conforms to the local material and geometric attributes of the surfaces. This results in a better preservation of physics in interface mechanics. Additionally, the method is incorporated into a Variational Multiscale framework that comes equipped with a built-in error estimation module, providing numerical estimation of convergence and distinguishing discretization errors from modeling errors. A series of quasi-static numerical simulations of a lap joint under fretting conditions are conducted to compare the performance of two friction models: (i) classical Coulomb friction model and (ii) physics-based multiscale model. Hysteresis study of a three-dimensional double-bolted lap joint for the two friction models is also presented and the computed results are shown to be consistent between conforming and nonconforming meshes.
In this paper, a new computational technique is presented based on the eXtended Finite Element Method (X-FEM) in pressure-sensitive plasticity of powder compaction considering frictional contact. In X-FEM, the need for mesh adaption to discontinuity interface is neglected and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the domain of the elements. The technique is applied by employing additional functions, which are added to approximate the displacement field of the elements located on the interface. The double-surface cap plasticity model is employed within the X-FEM framework in numerical simulation of powder material. The plasticity model includes a failure surface and an elliptical cap, which closes the open space between the failure surface and hydrostatic axis. The moving cap expands in the stress space according to a specified hardening rule. The frictional behavior of contact between two bodies is modeled by using the X-FEM technique and applying the Heaviside enrichment function. The application of X-FEM technique in simulation of pressure-sensitive material is presented in an incremental manner and the role of sub-elements in simulation of contact treatment is demonstrated. Finally, several numerical examples are analyzed with special reference to plasticity forming of powder compaction.
This paper represents a continuation of our earlier studies into the numerical analysis of contact problems with non-classical friction laws
In the first part of this work, the theoretical basis of a frictional contact domain method for two-dimensional large deformation problems is presented. Most of the existing contact formulations impose the contact constraints on the boundary of one of the contacting bodies, which necessitates the projection of certain quantities from one contacting surface onto the other. In this work, the contact constraints are formulated on a so-called contact domain, which has the same dimension as the contacting bodies. This contact domain can be interpreted as a fictive intermediate region connecting the potential contact surfaces of the deformable bodies. The introduced contact domain is subdivided into a non-overlapping set of patches and is endowed with a displacement field, interpolated from the displacements at the contact surfaces. This leads to a contact formulation that is based on dimensionless, strain-like measures for the normal and tangential gaps and that exactly passes the contact patch test. In addition, the contact constraints are enforced using a stabilized Lagrange multiplier formulation based on an interior penalty method (Nitsche method). This allows the condensation of the introduced Lagrange multipliers and leads to a purely displacement driven problem. An active set strategy, based on the concept of effective gaps as entities suitable for smooth extrapolation, is used for determining the active normal stick and slip patches of the contact domain.
The author gives necessary and sufficient conditions for existence and uniqueness of a class of problems of ″saddle point″ type which are often encountered in applying the method of Lagrangian multipliers. A study of the approximation of such problems by means of ″discrete problems″ (with or without numerical integration) is also given, and sufficient conditions for the convergence and error bounds are obtained.
In this study, we propose a segment-to-segment contact formulation (mortar-based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X-FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point-to-point approach. This is due to the non-linear character of the X-FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment-to-segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.
A key challenge while employing non-interpolatory basis functions in finite-element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B-spline basis functions, with application to both second- and fourth-order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.
A stabilized global–local quasi-static contact algorithm for 3D non-planar frictional crack is presented in the X-FEM/level set framework. A three-field weak formulation is considered and allows an independent discretization of the bulk and the crack interface. Then, a fine discretization of the interface can be defined according to the possible complex contact state along the crack faces independently from the mesh in the bulk. Furthermore, an efficient stabilized non-linear LATIN solver dedicated to contact and friction is proposed. It allows solving in a unified framework frictionless and frictional contact at the crack interface with a symmetric formulation, no iterations on the local stage (unilateral contact law with/without friction), no calculation of any global tangent operator, and improved convergence rate. 2D and 3D patch tests are presented to illustrate the relevance of the proposed model and an actual 3D frictional crack problem under cyclic fretting loading is modeled. Copyright © 2011 John Wiley & Sons, Ltd.
A model based on the penalty method for 3-D contact problems with friction is proposed. The friction forces are assumed to follow the Coulomb law, with a slip criterion treated in the context of a standard return mapping algorithm. Consistent linearization of the field equations is performed which leads to a fully implicit scheme with non-symmetric tangent stiffness which preserves asymptotic quadratic convergence of the Newton-Raphson method. Numerical results are obtained for some representative examples and compared with existing solutions.
The extended finite element method (X-FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X-FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X-FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability. Copyright © 2008 John Wiley & Sons, Ltd.
Many methods have been proposed to model joints in rocks or the interface between soil and a structure. Many analysts have reported numerical problems when using zero thickness interface elements while others have presented satisfactory results without comment of such difficulties. The numerical behaviour of zero thickness interface elements is further investigated in this paper. Some simple examples illustrate the application of interface elements to practical situations and highlight the numerical difficulties that may be encountered. Both ill-conditioning of the stiffness matrix and high stress gradients were found to cause numerical instability. Ill-conditioning can be reduced by careful selection of the size of the 2D elements adjacent to the interface. The problem of steep stress gradients is entirely one of inadequate mesh design. Contrary to other reports, this paper shows that the Newton–Cotes integration scheme has no benefit over Gaussian integration.
Analyses of a retaining wall using interface elements confirm the analytical values of active and passive earth pressure coefficients which are commonly used in analysis and design of retaining walls.
The performance of partition of unity-based discontinuous elements was studied by means of a numerical study. In particular, it was shown that conventional interface elements and partition of unity-based discontinuous elements share the same structure of the stiffness matrix governing interface behaviour and, under specific circumstances, the same shortcomings (i.e. oscillations in the traction profile). The effect of various integration schemes on the interface contribution was studied through the analysis of a linear-elastic notched beam. Further, an eigenvalue analysis of the partition of unity-based discontinuous element was conducted to gain more insight into its mechanical behaviour. Copyright © 2004 John Wiley & Sons, Ltd.
We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S-shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.
We present an incremental quasi-static contact algorithm for path-dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non-linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.
We develop both stable and stabilized methods for imposing Dirichlet constraints on embedded, three-dimensional surfaces in finite elements. The stable method makes use of the vital vertex algorithm to develop a stable space for the Lagrange multipliers together with a novel discontinuous set of basis functions for the multiplier field. The stabilized method, on the other hand, follows a Nitsche type variational approach for three-dimensional surfaces. Algorithmic and implementational details of both methods are provided. Several three-dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches. The results indicate that both methods yield optimal rates of convergence in various quantities of interest, with the primary differences being in the surface flux. The utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques. Copyright © 2011 John Wiley & Sons, Ltd.
Earthquakes have been recognized as resulting from a stick–slip frictional instability along the faults between deformable rocks. An arbitrarily-shaped contact element strategy, named the node-to-point contact element strategy, is proposed, applied with the static-explicit characters to handle the friction contact between deformable bodies with stick and finite frictional slip and extended here to simulate the active faults in the crust with a more general nonlinear friction law. An efficient contact search algorithm for contact problems among multiple small and finite deformation bodies is also introduced. Moreover, the efficiency of the parallel sparse solver for the nonlinear friction contact problem is investigated. Finally, a model for the plate movement in the north-east zone of Japan under gravitation is taken as an example to be analyzed with different friction behaviors. Copyright © 2002 John Wiley & Sons, Ltd.
We present an extended finite element (FE) approach for the simulation of slow-rate frictional faulting in geologic media incorporating bulk plasticity and variable friction. The method allows the fault to pass through the interior of FEs without remeshing. The extended FE algorithm for frictional faulting, advocated in two recent articles, emanates from a variational equation formulated in terms of the relative displacement on the fault. In the present paper we consider the combined effects of bulk plasticity and variable friction in a two-dimensional plane strain setting. Bulk plasticity is localized to the fault tip and could potentially be used as a predictor for the initiation and propagation of new faults. We utilize a variable velocity- and state-dependent friction, known as the Dieterich–Ruina or ‘slowness’ law, formulated in a slip-weakening format. The slip-weakening/variable friction model is then time-integrated according to the generalized trapezoidal rule. We present numerical examples demonstrating the convergence properties of a global Newton-based iterative scheme, as well as illustrate some interesting properties of the variable friction model. Copyright © 2009 John Wiley & Sons, Ltd.
This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method. In contrast to earlier approaches, we do not work with an interior penalty formulation as, e.g. for Nitsche techniques, but impose the constraints weakly in terms of Lagrange multipliers. Roughly speaking a stable and optimal discrete Lagrange multiplier space has to satisfy two criteria: a best approximation property and a uniform inf–sup condition. Owing to the fact that the interface does not match the edges of the mesh, the choice of a good discrete Lagrange multiplier space is not trivial. Here we propose a new algorithm for the local construction of the Lagrange multiplier space and show that a uniform inf–sup condition is satisfied. A counterexample is also presented, i.e. the inf–sup constant depends on the mesh-size and degenerates as it tends to zero. Numerical results in two-dimensional confirm the theoretical ones. Copyright © 2008 John Wiley & Sons, Ltd.
This work concerns finite-element algorithms for imposing frictional contact constraints on intra-element, or embedded surfaces. Existing techniques typically rely on the underlying bulk mesh to implicitly partition the surface, a strategy that can give rise to overconstraint. In the present work, we first apply a mortaring algorithm to the modeling of frictional contact conditions on arbitrary interfaces. The algorithm is based upon a projection of the bulk and surface fields onto independent mortar fields at the interface. We examine the advantages of this approach when combined with extended finite-element approximations to the bulk fields. In particular, the method allows for bulk and surface domains to be partitioned separately, as well as enforce nonlinear contact constraints on surfaces that are not explicitly “fitted” to the bulk mesh. Results from several benchmark problems in frictional contact are provided to demonstrate the accuracy and efficacy of the method, as well as the improvement in robustness compared to existing techniques. We also provide an example that illustrates the effectiveness of the approach in high-speed machining simulation.
A new technique for treating the mechanical interactions of overlapping finite element meshes is proposed. Numerous names
have been applied to related approaches, here we refer to such techniques as embedded mesh methods. Such methods are useful
for numerous applications e.g., fluid-solid interaction with a superposed meshed solid on an Eulerian background fluid grid
or solid-solid interaction with a superposed meshed particle on a matrix background mesh etc. In this work we consider the
interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background
mesh and is often a grid. We first employ a classical mortar type approach [see Baaijens (Int J Numer Methods Eng 35:743–761,
2001)] to impose constraints on the interface. It turns out that this approach will work well except in special cases. In fact,
many related approaches can exhibit mesh locking under certain conditions. This motivates the proposed version of Nitsche’s
method which is shown to eliminate the locking phenomenon in example problems.
KeywordsEmbedded mesh–Nitsche’s method–Interfaces
This paper presents a new time-stepping algorithm for frictional contact problems that exhibits unconditional positive energy dissipation. More specifically, the proposed scheme preserves a priori stability estimates of the continuum problem for both frictionless and frictional contact, leading to improved numerical stability properties in particular. For the normal contact component, the algorithm exhibits full energy conservation between released states, while the energy does not increase over its initial value due to the enforcement of the normal contact constraint during persistent contact. A penalty regularization is considered to this purpose. A new regularization of the stick conditions is considered for the frictional part. The new scheme is shown rigorously to exhibit positive energy dissipation like the continuum physical problem in this frictional case. Coulomb friction is assumed. Complete analyses of these considerations, as well as a detailed description of their finite element implementation, are included in the general finite deformation range. Representative numerical simulations are presented to assess the performance of the newly proposed methods.