Weak Discontinuities Research Articles

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations, including crack propagation and strain localization is developed. By this approach, a peridynamic model capable of effectively treating strong and weak discontinuities is superimposed in the critical regions over an underlying finite element mesh placed over the entire problem domain. A rigorous variational framework of coupling local finite element and nonlocal peridynamics approximations that is free of blending parameters is developed. Several numerical examples involving mixed-model fracture, three-dimensional adaptive crack propagation and strain localization induced ductile failure demonstrate the rational and efficiency of the proposed superposition-based coupling approach.

P.406 Intrinsically disordered regions of synapsin hold together synaptic vesicles of the reserve pool in a living synapse

A novel computational methodology is proposed in this work to simulate the nonlinear hydro-mechanical process in saturated porous media containing crossing fractures. Specifically, the weak form of mechanical coupling and mass transfer equations is derived based on Biot’s theory and lubrication theory, respectively. The weak discontinuity for pressure field around T-shaped crack is described by a new junction enrichment function. The nonlinear hydro-mechanical coupled equations are obtained by Extended Finite Element Method (XFEM) discretization and solved by Newton-Raphson method. This proposed computational model is verified by comparing numerical results with analytical solution. The robustness of this numerical method is demonstrated by several examples. The effects of crack orientation, matrix stiffness, and confining stress on the equivalent permeability of fractured porous medium are investigated. Numerical results indicate that the interaction of cracks plays a critical role on the hydro-mechanical behavior of the saturated porous media containing crossing fractures.

Weak Shock Waves in Charged Gas

The problem of plane discontinuities of relatively small amplitude propagating in a layer of nonuniformly charged gas in the variable electric and constant gravity fields is considered within the framework of ideal electrohydrodynamics model. The transport equations of variational nature are derived for their amplitudes and an exact solution of the problem of weak discontinuity is given. Then, the problem of weak shock wave is solved in the approximation to the solution obtained. Examples of propagation of a wave through an arbitrary equilibrium initial state and along a gas layer moving as a solid body in a variable electric field are considered.

Adaptive isogeometric discretizations for diffuse modeling of discontinuities

AbstractIn this contribution, we discuss the modeling of discontinuities using non‐conforming meshes. For weak discontinuities we propose to regularize the sharp material interface over a characteristic length scale. To define the material properties in the transition region, homogenization assumptions are used that fulfill the kinematic compatibility across the interface and the static equilibrium at the interface. For strong discontinuities we use a phase‐field model that is able to account for failure of adhesive interfaces and extend it to general heterogeneous materials. To improve the efficiency of the approach, adaptive isogeometric analysis is used to discretize the computational domain.

Modelling strong and weak discontinuities with the scaled boundary finite element method through enrichment

In this paper, a technique to model strong and weak discontinuities with the scaled boundary finite element method through enrichment is proposed. The main advantage of the method is that the enriched elements, in the spirit of the extended finite element method (XFEM), do not need to physically conform to the geometry of features, e.g. internal interfaces and cracks, and remeshing is unnecessary as the interfaces evolve. All the advantages of the SBFEM and the XFEM are retained. The stress singularity at the crack tip can be captured accurately and the stress intensity factors (SIFs) can be directly computed based on the singular displacement or stress at the crack tip within the framework of the SBFEM. The numerical properties and performance for the proposed method are assessed using several numerical examples. In particular, problems with discontinuities, e.g. voids, inclusions, and cracks are analysed. The results show that the accuracy and convergence rate of the new approach for solving void or inclusion problems are identical to those of the XFEM, but requires less number of degrees-of-freedom than the XFEM. For crack problems, compared with the XFEM with topological enrichment, the developed method is superior.

Improving the modified XFEM for optimal high‐order approximation

SummaryThis paper investigates the accuracy of high‐order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method, is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher‐order contributions to the approximation space. An improved modified XFEM is presented, with basis functions “corrected” by projecting out higher‐order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre‐computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order p ≤ 4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity and are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order p ≤ 4.

Solution of Navier–Stokes equations on a fixed mesh using conforming enrichment of velocity and pressure

Simulation of fluid flows of multi-materials is an intriguing topic in computational mechanics. Capturing the physics of the interface between different materials poses a challenge because of the discontinuities that may occur on the interface. Several methods have been proposed in the literature to deal with this issue. In this paper, a technique based on Nitsche’s method has been employed on a fixed mesh combined with the PFEM-2 strategy for the solution of Navier–Stokes equations on multi-fluid flows. The novelty of this technique is its capability of capturing the strong and weak discontinuities and its compatibility for the application of various types of boundary conditions on the interface.

10 CASE-CASE GWAS OF TWO DIFFERENT DISORDERS USING CASE-CONTROL SUMMARY STATISTICS

Chapter 6 presents formulations based on extended Isogeometric Analysis (XIGA). Isogeometric analysis (IGA) is an emerging field in structural engineering introduced in [10], [20]. The concept of IGA is to employ the same functions, i.e. B-splines and non-uniform rational B-splines (NURBS) to describe the geometry of the computational domain. In this chapter, a brief review of B-splines, NURBS and the analysis using splines will be presented. We also give an overview of PHT-splines, which are useful for h-adaptive refinement procedures within IGA. Implementation details of IGA for problems in linear elastostatics are provided before different XIGA approaches for weak discontinuities are discussed in detail.

A single crystal plasticity finite element formulation with embedded deformation twins

Deformation twinning is an important plastic deformation mechanism in some polycrystalline metals such as titanium and magnesium. In this paper, we present a novel crystal plasticity finite element framework that accounts for deformation twinning explicitly, in addition to crystallographic slip. Within this computational framework, deformation twins are treated as weak discontinuities embedded within individual finite elements, such that a jump in the velocity gradient field is introduced (via the discretized gradient operator) between the twinned and untwinned crystalline regions, taking into account compatibility and traction continuity conditions at the interface between these two regions. The deformation gradient is multiplicatively split into elastic and plastic parts in the untwinned region, as is customary in finite-deformation crystal plasticity formulations. A different multiplicative decomposition of the deformation gradient into elastic, plastic (slip), and twinning parts is adopted in the twinned region, allowing deformation twinning to be accounted for as an additional mode of plastic deformation. A stochastic model is used to predict twin nucleation at grain boundaries, and the evolution of the length and thickness of the twinned region under subsequent deformation is taken into account. The linearization of the single-crystal plasticity model, required in order to enforce traction continuity at the interface between the twinned and untwinned regions using a Newton iterative scheme, is described in detail. Simulations of the initiation and propagation of {101¯2} tensile twins in hexagonal close packed (HCP) titanium are presented to demonstrate the capabilities of the proposed computational framework.

A stable interface‐enriched formulation for immersed domains with strong enforcement of essential boundary conditions

SummaryGenerating matching meshes for finite element analysis is not always a convenient choice, for instance, in cases where the location of the boundary is not known a priori or when the boundary has a complex shape. In such cases, enriched finite element methods can be used to describe the geometric features independently from the mesh. The Discontinuity‐Enriched Finite Element Method (DE‐FEM) was recently proposed for solving problems with both weak and strong discontinuities within the computational domain. In this paper, we extend DE‐FEM to treat fictitious domain problems, where the mesh‐independent boundaries might either describe a discontinuity within the object, or the boundary of the object itself. These boundaries might be given by an explicit expression or an implicit level set. We demonstrate the main assets of DE‐FEM as an immersed method by means of a number of numerical examples; we show that the method is not only stable and optimally convergent but, most importantly, that essential boundary conditions can be prescribed strongly.

Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity

In this paper, we propose a smoothed stable extended finite element method (S2XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions and/or voids in hyperelastic matrix materials. The interface geometries are implicitly represented through level sets and a geometry based error indicator is used to resolve the geometry. For the unknown fields, the mesh is refined based on a recovery based error indicator combined with a quadtree decomposition guarantee the method's accuracy with respect to the computational costs. Elements with hanging nodes (due to the quadtree meshes) are treated as polygonal elements with mean value coordinates as the basis functions. The accuracy and the convergence properties are compared to similar approaches for several numerical examples. The examples indicate that S2XFEM is computationally the most efficient without compromising the accuracy.

Modeling curved interfaces without element‐partitioning in the extended finite element method

SummaryIn this paper, we model holes and material interfaces (weak discontinuities) in two‐dimensional linear elastic continua using the extended finite element method on higher‐order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one‐dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used. For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h‐refinement (p=1,2,…,5) and exponential rates of convergence with p‐refinement (p=1,2,…,7) for elastostatic problems with holes and material inclusions on Cartesian pth‐order spectral finite element meshes.

Optimum Compression in a Wave Compressor with Hybrid Gas Bearings

The concept of a wave compressor using a centered isentropic wave for air compression is considered, and shock-wave structures are studied, which are arisen when a centered wave is focused. It is concluded that the characteristic centered compression wave with reflected weak discontinuity is optimal.

Modeling branched and intersecting faults in reservoir‐geomechanics models with the extended finite element method

SummaryFaults are geological entities with thickness several orders of magnitude smaller than the grid blocks typically used to discretize geological formations. On using the extended finite element method (X‐FEM), a structured mesh suffices and the faults can arbitrarily cut the elements in the mesh. Modeling branched and intersecting faults is a challenge, in particular when the faults work as internal fluid flow conduits that allow fluid flow in the faults as well as to enter/leave the faults. By appropriately selecting the enrichment function and the nodes to be enriched, we are able to capture the special characteristics of the solution in the vicinity of the fault. We compare different enrichment schemes for strong discontinuities and develop new continuous enrichment functions with discontinuous derivatives to model branched and intersecting weak discontinuities. Symmetric fluid flows within the regions embedded by branched, coplanar intersecting, and noncoplanar intersecting faults are considered to verify the effectiveness of the proposed enrichment scheme. For a complex fault consisting of branched and intersecting faults, the accuracy and efficiency of the X‐FEM is compared with the FEM. We also demonstrate different slipping scenarios for branched and intersecting faults with the same friction coefficient. In addition, fault slipping triggered by an injection pressure and three‐dimensional fluid flows are modeled to show the versatility of the proposed enrichment scheme for branched and intersecting weak discontinuities.

A new methodological approach to assess the stability of discontinuous rocky cliffs using in-situ surveys supported by UAV-based techniques and 3-D finite element model: a case study

This paper aims to present a new methodological approach for the stability assessment of coastal cliffs constituted of discontinuous rock masses. The method entails in situ specific geostructural and geomechanical surveys, three-dimensional UAV-based Photogrammetric structural models, laboratory geotechnical tests and, two- and three-dimensional finite element analysis (FEM). The application of the method to a case study is then presented and discussed; it regards a 600 m long sea rocky cliff located at Sant'Andrea (Melendugno, South of Apulia, SE Italy) and faced to the southern Adriatic Sea. Here the cliff is made up of an about 15 m-thick sequence of laminate calcisiltites alternate with bioturbated calcarenites belonging to the Uggiano la Chiesa Fm. (Middle-Upper Pliocene).The structural discontinuities detected with photogrammetry techniques were compared and validated with those derived from conventional in situ survey methods. Later on, the paper discusses assumptions and results of two- and three-dimensional finite element models developed to investigate the potential failure mechanisms of the sea cliff accounting for pre-existing weak planes or discontinuities with unfavourable orientation.The failure mechanisms obtained by both FEM analysis agree well with those typically observed in the study area.

Experimental investigation on the failure mechanism of a rock landslide controlled by a steep-gentle discontinuity pair

A type of rock landslide is very common in practical engineering, whose stability is mainly controlled by the rock bridge between the steep tensile crack at the crest and the low-inclination weak discontinuities at the toe (namely, ligament is the term for the locking section in the slope). To obtain a deeper understanding into the failure process of this kind of landslide, twenty-four physical slope models containing a steep-gentle discontinuity pair (a steep crack in the upper part and a low-inclination discontinuity in the lower part) were tested by applying vertical loads at the crests. The results indicate that the inclination angle of the ligament (θ) has great influence on the failure and stability of this type of rock slope. With the change of θ, three failure patterns (five subtypes) concerning the tested slopes can be observed, i.e., tensile failure of the ligament (Type 1), tension-shear failure of the ligament (Type 2) and two-stage failure of the main body (Type 3). The failure process of each failure mode presents five stages in terms of crack development, vertical load, horizontal/vertical displacements and strains in the ligaments. The specific range of the ligament angle between different failure patterns is summarized. The discussion on the failure resistances and ductility of different failure patterns, and the guiding significances of the experimental findings to the stability evaluation and the reinforcement were conducted.

On the unsteady, stagnation point flow of a Maxwell fluid in 2D

A two-dimensional unsteady stagnation-point flow of an incompressible viscoelastic fluid is studied theoretically assuming that the fluid obeys the upper convected Maxwell model. To achieve better understanding of the main properties of the governing equations, the system of non-linear equations is transformed to Lagrangian variables. As a result, a closed system of equations of the mixed elliptic–hyperbolic type is obtained. These equations are decomposed into a hyperbolic submodel and a quadrature. The hyperbolic part is responsible for the transport of nonlinear transverse waves in an incompressible Maxwell medium. The system of equations guarantees the existence of the energy integral, which allows one to analyze discontinuous solutions to these equations. It is demonstrated that solutions with strong discontinuities are impossible, though a solution with weak discontinuities can exist. Several numerical examples of the problems of practical interest show that perturbations induced by weak discontinuities in the initial data propagate with a finite speed, which confirms the hyperbolic character of the system.

A stable discontinuity-enriched finite element method for 3-D problems containing weak and strong discontinuities

A new enriched finite element technique, named the Discontinuity-Enriched Finite Element Method (DE-FEM), was introduced recently for solving problems with both weak and strong discontinuities in 2-D. In this mesh-independent procedure, enriched degrees of freedom are added to new nodes collocated at the intersections between discontinuities and the sides of finite elements of the background mesh. In this work we extend DE-FEM to 3-D and describe in detail the implementation of a geometric engine capable of handling interactions between discontinuities and the background mesh. Several numerical examples in linear elastic fracture mechanics demonstrate the capability and performance of DE-FEM in handling discontinuities in a fully mesh-independent manner. We compare convergence properties and the ability to extract stress intensity factors with standard FEM. Most importantly, we show DE-FEM provides a stable formulation with regard to the condition number of the resulting system stiffness matrix.

Three-dimensional explicit finite element formulation for shear localization with global tracking of embedded weak discontinuities

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.

Interaction of elementary waves with a weak discontinuity in an isothermal drift-flux model of compressible two-phase flows

In this paper, we study the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows. We construct the solution of the Riemann problem in terms of a one parameter family of curves. Using the properties of elementary waves, we prove a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of a left shock, contact discontinuity, and a right shock. Moreover, we derive the amplitudes of weak discontinuity and discuss the interactions of weak discontinuity with shocks and contact discontinuity. Finally, we carry out some tests to investigate the effect of shock strength and initial data on the jump in shock acceleration and the amplitudes of reflected and transmitted waves.