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This paper aims at investigating fracture behavior of single and multiple cracks in two-dimensional solids by an adaptive extended isogeometric analysis (XIGA) based on locally refined (LR) B-splines. The adaptive XIGA is capable of modeling cracks without considering the location of crack faces due to the local enrichment technique based on partition-of-unity concept. The XIGA approximation is locally enriched by Heaviside function and crack tip enrichment functions to capture the discontinuity across crack faces and singularity in the vicinity of crack tips. The LR B-splines, which are generalized by B-splines and NURBS, not only inherent desirable properties of the B-splines and NURBS but also can be locally refined, ideally suitable for adaptive analysis. Structured mesh refinement strategy is applied to perform local refinement for LR B-splines based on a posteriori error estimator. According to the recovery technique proposed by Zienkiewicz and Zhu, the smoothed strain field is obtained to construct the posteriori error estimation based local refinement. The stress intensity factors (SIFs) are evaluated using the contour interaction integral technique. Several benchmark numerical examples are illustrated in comparison to analytical or reference solutions to verify the accuracy and efficiency of the developed approach. The proposed adaptive XIGA method is also applied to a curved crack, multiple cracks and complicated structure with a crack, which sufficiently presents the applicability of the proposed method in crack modeling. In addition, the convergence rate of the adaptive local refinement strategy is numerically studied and compared with that of the global refinement approach. The convergence rate of the adaptive local refinement is shown to be faster than that of the global refinement.

The variation of physical properties across the interface and the inaccuracy of spatial derivative on discontinuity contribute to the numerical instability in multiphase flow with large density or viscosity ratio. With integral basic equation in Peridynamic theory under updated Lagrangian formulation and stress tensor calculation by Moving Particle Semi-implicit method(MPS), a new numerical approach on dealing with the abrupt physical quantities is introduced. Besides, a modified matrix inspired by the shape function in Peridynamics is added in the gradient and divergence operators to eliminate the error caused by irregular particle distribution. In addition, to decrease the calculation error caused by large properties difference, a decomposition process on the stress tensor and its divergence are proposed, the large acceleration difference by the discontinuity of density on the interface is smoothed after a mean density and the accuracy is validated by a simple example. Four benchmark cases, oscillating two-layer elliptical region, Rayleigh–Taylor instability, two-phase dam-break and bubble rising examples, are simulated to demonstrate the feasibility and accuracy of the proposed method in multiphase flow with both small and large physical quantity ratio.

A quadrilateral bi-cubic G 1 -conforming finite element for the analysis of Kirchhoff plates is presented. The rational version of the Gregory patch proposed by Loop et al. (2009) is the starting point of our formulation. This version of the Gregory patch consists in rational enhancement of the bi-cubic Bezier interpolation representing a suitable tool for designing G 1 -conforming quadrilateral element on C 0 -conforming un-structured meshes. The element includes as additional degrees of freedom the edge rotations like in the Loof-formulations but is only displacement based. Because of the presence of the rational functions, the second derivatives of the interpolation present a finite discontinuity at the corners of the element, that prevent the element from passing the bending patch test. The element so formulated does not present optimal rate of convergence under h -refinement operation. The formulation is enhanced enforcing these discontinuities to be zero by means of Lagrange multipliers. It is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh. In this way the rational conforming approximation collapses into a conforming re-arrangement of the complete bi-cubic Bezier interpolation. Some examples and benchmarks are presented in order to test the performance of the element for the Kirchhoff plate model.

In this work, a local moving extended phase field method (LMXPFM) is proposed for the failure analysis of brittle materials. In this approach, nonlinear phase field crack evolution equations are solved in an adaptively refined small region (which is prone to crack nucleation or propagation). In the proposed LMXPFM, phase field method (PFM), extended finite element method (XFEM) and multiscale finite element method (MsFEM) are combined together such that the MsFEM acts as a medium to couple coarse mesh with fine mesh, and the XFEM is used to represent a discontinuity (such as crack) outside the phase field region. Thus, the LMXPFM not only provides an adaptively refined mesh in a small region near the smeared discontinuity but also fulfills the prerequisite of refined mesh for solving phase field crack evolution equations. This local approach provides a huge saving in the computational efforts as compared to PFM applied in the entire domain. The efficiency and accuracy of the LMXPFM are verified by solving and comparing the results with standard PFM and literature solutions.

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface . We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG’s intrinsic (upwind) dissipation being biased towards high frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations , similar properties are achievable through the addition of artificial diffusion such as spectral vanishing viscosity (SVV). However although SVV is recognised as very useful in CG-based high-fidelity turbulence simulations, this approach has been observed to be sub-optimal when compared to DG at intermediate polynomials orders ( P ≈ 3 ). In this paper we explore an alternative stabilisation approach through the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP’s P -dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier–Stokes equation, as we achieve higher quality flow solutions at P = 3 using GJP, whereas SVV performs marginally worse at P = 5 with twice as many degrees of freedom in total.

It is often highly desirable to simultaneously optimize the layout of embedded functional components and the topology of the host structure supporting these components to achieve the best overall performance while still ensuring the structural integrity. We propose a topology optimization framework to account for connecting interface behaviors between the components and the host structure. Here we treat the connecting interfaces with the cohesive zone model to reflect the adhesively bonded interface behaviors. A conforming mesh in conjunction with interface elements is employed to discretize the evolving structure while accounting for the strong discontinuity of displacement field across the material interfaces . To give a clear representation of structural boundaries and the connecting interfaces, we also suggest a multi-material interpolation model in the level set framework, which can conveniently define the connecting interface locations and describe multi-material distribution without redundant phase in the design domain. The objective function is defined as the sum of the strain energy and the work done by the traction on the connecting interface, and the evolution velocities of the level set and the embedded components are treated as design variables. These design variables are updated with the MMA optimizer on the basis of adjoint-variable sensitivity analysis. This optimization formulation allows multiple constraints and mixed design variables ( i.e level set design variables and components’ design variables) to be easily handled in level set based optimization. The design is advanced by the Hamilton–Jacobi equation with the velocity design variables as input. Numerical examples demonstrate the validity and applicability of the proposed method.

This paper presents a new stabilized method that is endowed with variationally derived Discontinuity Capturing (DC) features to model steep advection fronts and discontinuities that arise in multi-phase flows as well as in mixing flows of immiscible incompressible fluids. Steep fronts and discontinuities also arise in hypersonic compressible flows. The new method finds roots in the Variational Multiscale (VMS) framework that yields a coupled system of coarse and fine-scale variational problems. Augmenting the space of functions for the fine-scale fields with weak and/or strong discontinuities results in fine-scale models that naturally accommodate jumps in the fine fields. Variationally embedding the discontinuity enriched fine-scale models in the coarse-scale formulation leads to the Variational Multiscale Discontinuity Capturing (VMDC) method where stabilization tensors are naturally endowed with discontinuity capturing structure. In the regions with sharp gradients, these variationally projected fine-scale models augment the stability of the coarse-scale formulation to accurately capture sharply varying coarse solutions. Since the proposed method relies on local enrichment, it does not require either the complete or the dynamic enrichment algorithms that are invariably employed in methods that use global enrichment ideas. The scalar advection equation serves as a model problem to investigate the variational structure of the DC terms. The VMDC method is then applied to the Navier–Stokes equations and tested on problems involving two-phase flows with and without surface tension. These test problems highlight that fine-scale models not only stabilize the weak form, variationally derived fine models that are endowed with sharp discontinuities also augment the coarse scale solutions with features that are otherwise not adequately resolved by variational formulations that act only at the coarse-scale levels.

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points . Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO.

Many state-of-the-art approaches for predicting discrete fracture within the finite element framework were designed by enhancing the continuous displacement field with a discontinuous part. Broadly categorised into elemental and nodal enrichment approaches, they provided unprecedented versatility in discretising domains with prescribed and evolving boundaries. However, there can be an additional computational burden when global tracking schemes are used to enforce the continuity of tractions/jumps along crack paths. Many current approaches also entail involved integration procedures over cut subdomains, which pose challenges of efficiency and accuracy. In addition, although cracks do not need to conform to the geometry of the mesh, the discretisation of cracks is still linked to that of the underlying elements, which can lead to the inaccurate representation of traction profiles along stiff interfaces. Aiming at overcoming these issues, the present work introduces a new formulation with minimal enhanced degrees of freedom. This is achieved by decoupling the discretisation of bulk and cracks, a feature that is enabled by a novel clustered rigid body definition of enhancement field in the set of the variational equations . A new method, termed Cracked Zone Clustering Method (CZCM), is then proposed for discrete fracture propagation by deploying only one rotational degree of freedom per set of cracked elements within the same zone. The stability of the method is first studied using element examples. The method is also shown to resolve the issue of traction oscillations over stiff interfaces. The accuracy and efficiency of the independent discretisation strategy are studied using several known crack propagation benchmarks. Results are compared with the Extended/Generalised Finite Element Method (XFEM/GFEM) and the Discrete Strong Discontinuity Approach (DSDA). In summary, it is shown that the cracked zones can encompass a multitude of elements, thus significantly decreasing the required number of enhancement degrees of freedom by orders of magnitude and without compromising accuracy.

The material point method (MPM) combines the advantages of both mesh-free and grid-based methods for simulating complex cracking problems without the limitations of the grid. However, the conventional MPM cannot deal with cracks directly because of the continuous nodal velocity field . Therefore, in this study, the conventional MPM is improved using three new algorithms to simulate the propagation of multiple branched cracks. The first proposed algorithm is the phantom node method using a particle cutting technique, which allows discontinuity of the velocity field around arbitrary cracks, and the second one is the multi-crack contact algorithm to deal with the contact problem between multiple objects cut by multiple cracks, avoiding crack penetration. The last one is particle interaction integral method based on classical fracture mechanics to get the dynamic stress intensity factors (DSIF) and realize the crack propagation . And all algorithms are validated by comparing with the extended finite element method . Subsequently, uniaxial compression tests of three rock samples with different pre-existing cracks are simulated using the extended MPM. Three types of crack coalescence are found, and the simulated results agree with the previous experimental results, which demonstrate the feasibility and flexibility of the proposed extended MPM for simulating the propagation of multiple branched cracks.

This paper presents a dimensionally reduced thermal model and gradient-based shape optimization scheme for the 3D computational design of actively cooled panels. A correction method previously used in wire-based electromagnetics is applied to address convergence issues associated with the singularity of the thermal solution along the microchannels. The numerical solution is obtained with the interface-enriched generalized finite element method (IGFEM), which greatly simplifies mesh generation by allowing for the use of a non-conforming mesh to capture the temperature gradient discontinuity across the microchannels. The temperature distribution calculated with the IGFEM on coarse meshes agrees with that obtained using significantly more complex and costly ANSYS FLUENT simulations. We then combine the IGFEM with a sensitivity analysis and the sequential quadratic programming algorithm in MATLAB to solve two sets of shape optimization problems related to actively cooled microvascular composite panels. These problems demonstrate a key advantage of the IGFEM in avoiding severe mesh distortion during shape optimization. Lastly, we present a semi-analytical model based on the concept of the zone of influence of a channel to estimate the maximum temperature of an actively cooled plate with straight embedded microchannels.

This paper presents the development of new three-dimensional finite elements in the finite deformation range for the numerical modeling of material failure characterized by propagating crack-type surfaces in solids, commonly referred to as strong discontinuities. The singular fields associated with such discontinuous solutions are captured by an element-based enhancement, locally within a multi-scale framework. The complex kinematics arising from the general propagating discontinuity surfaces, especially in the considered finite deformation setting, is captured by the direct identification of the element strain fields associated to the separation modes of the discontinuity, rather than attempting to interpolate directly the corresponding discontinuous deformation fields. To avoid the over-stiff response known as stress-locking, by which the discontinuities cannot open as needed by the material response, the new brick elements consider full linear interpolations of the displacement jumps. This requires a total of nine local parameters for each localized finite element, parameters that are condensed out at the element level. The resulting formulation does not change then neither the number nor the connectivity of the global degrees of freedom employed in the finite element resolution of the underlying global mechanical problem, thus leading to a computationally efficient numerical technique that captures sharply these discontinuous solutions. In particular, the localized dissipative mechanism associated to the material failure is captured objectively through the consideration of the proper cohesive law along the discontinuity surface. Special care is taken to assure the material frame indifference of the final formulation. Complete details of the implementation of the new elements, including the consistent linearization of the governing equations, are also presented. The results obtained in numerical simulations of representative benchmark problems illustrate the performance of the new finite elements with embedded 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.

In this work, a particle regeneration technique is developed for Smoothed Particle Hydrodynamics (SPH). In traditional SPH, the particle disorder phenomenon will occur when dealing with the strongly compressible flow problem. To solve this, in the present work, uniformly distributed background particles filled in the computational domain are adopted. The particle regeneration technique is that the fluid particles replaced by the background particles when the fluid density changes to a specific limitation. The fluid variables of the background particles are approximated by the fluid variables of the initial particles in their support domain. For the multiphase flow, the multiphase interface is calculated by an interface reproducing algorithm, in which, we defined an indicator variable, and set the indicator value discontinuity between different materials. By setting the threshold value, the multiphase interface is reconstructed. Meanwhile, the momentum equation for the Riemann SPH is modified to eliminate the instability in the light phase. Several numerical examples are studied to verify the present algorithm.

Thermally induced cracking occurs in many engineering problems such as drying shrinkage cracking of concrete, thermal shock induced fracture, micro cracking of two-phase composite materials etc. The computational simulation of such a fracture is complicated, but the use of phase-field models (PFMs) is promising as they can seamlessly model complex crack patterns like branching, merging, and fragmentation by treating the crack discontinuity as thin band of diffuse damage. Despite the success of phase-field models there are two major issues in previous PFMs of thermally induced fracture. Firstly, these models, which are mostly based on a PFM using a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale. Secondly, they are limited to brittle fracture only. As a solution, we extend the phase-field regularized cohesive zone model ( PF-CZM ) of Wu [JMPS, 103 (2017)], with a rational degradation function dependent on elasticity and fracture related material parameters, to thermoelastic solids. Furthermore, we present a monolithic BFGS algorithm to solve the three-field (displacements, phase-field and temperature) coupling equations altogether. Multiple thermally induced fracture problems, simulated within the framework of the finite element method, are presented with predictions in good agreement with previous findings and experiments. The proposed model is shown to give better performance in capturing the physics of thermally induced fracture: it provides length scale insensitive responses. And the monolithic solver is 4 ∼ 5 times faster than the conventional alternating minimization solver.

The concept of the partition of unity (PU) enabled the development of nodal enrichment strategies, such as the Extended Finite Element Method (XFEM) and Generalised Finite Element Method (GFEM), for realistic simulations of structural behaviour with applications to both static and dynamic problems. Nonetheless, the majority of existing methodologies still inherit instability issues for arbitrary discontinuity geometries when using an explicit time integration scheme. To address this, the discrete strong discontinuity approach is herein developed for the simulation of dynamic crack propagation. The formulation is fully variationally consistent and a new mapping approach is introduced to embed the rigid body movements associated with discontinuities while keeping the critical time step bounded. Multiple discontinuities within a single element are also considered for the accurate modelling of crack branching. The stability of the new technique is first verified in one- and two-dimensional elements. Next, the accuracy and efficiency are validated with structural examples including tensile and mixed-mode loadings. A concrete L-specimen, where different loading rates produce significant variations in failure patterns and strength, is also considered. Results show the good overall agreement with experimental data and other numerical studies available in the literature. The new formulation, however, is able to capture complex crack propagation phenomena, such as crack branching, without any specific additional criterion (e.g. based on crack tip velocity). The formulation presented in this paper is, to the best of the authors’ knowledge, the first PU-based consistent finite element with intrinsically bounded critical time step for explicit time integration.

In this paper, the Interface Immersed Particle Difference Method (IIPDM) for weak discontinuity in elliptic problems is presented. Heat conduction and potential flow problems with smooth and non-smooth interfaces are considered. The previously developed Particle Difference Method (PDM) accurately solves this class of interfacial singularity problems, however, it requires additional difference equations on the interfacial points. In contrast, the IIPDM no longer requires the additional interfacial equations since the interface condition is already immersed in the particle derivative approximation through the moving least squares procedure. The method successfully captures both singularities and discontinuities in the derivative field due to the interface geometry regardless of its smoothness. In fact, enforcement of the interface condition is conducted both in an implicit manner and in an explicit manner enriching the polynomial basis as well as the local approximation. Under the constrained derivative approximation, discretization of the elliptic PDE in its strong form leads to a difference scheme or a point collocation scheme involving only the particles. Consequently, the strong formulation can avoid the increase in total system size and improves the computational efficiency. Numerical experiments show that the IIPDM can sharply capture discontinuities and singularities within a solution field. Furthermore, convergence studies of various elliptic interface problems demonstrate efficiencies captured from the IIPDM when comparing convergence rates against the PDM.

In this paper, an adaptive refinement technique using the scaled boundary finite element method (SBFEM) is proposed. The salient feature of this technique is that it is not required to regenerate the mesh for the whole model during the iterations. To this end, a local mesh refinement strategy is implemented based on a polytree algorithm in three dimensions, which can be applied to polyhedral elements with arbitrary number of nodes, edges and faces. These elements constructed by the SBFEM can be used in analysis with their boundaries discretized only, which reduce the difficulty to connect elements with different sizes. An explicit residual based error indicator is developed using the discontinuity of the stress field to guide the adaptive mesh refinement. The accuracy and efficiency of the proposed method are demonstrated using five numerical examples, including complex geometry and stress singularity.

International audience; In this paper we present a new stabilized finite element method to solve the two phase compressible–incompressible fluid flow problems using the level set method. An anisotropic mesh adaptation with a regularized interface is adopted to deal with the high discontinuity in the material properties. The coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum equation. Therefore, the same set of primitive unknowns and equations is described for both phases. The unified system is solved using a new derived Variational MultiScale stabilized finite element method. It is tested on three time-dependent liquid–gas interface problems. The numerical results show good stability and accuracy properties, permitting also to deal with important compressibility effect, high density ratio, and extremely stretched anisotropic elements.

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

We propose a change of variable approach and discontinuity capturing methods to ensure physical constraints for advection-reaction equations discretized by the finite element method. This change of variable confines the concentration below an upper bound in a very natural way. For the non-negativity constraint, we propose to use a discontinuity capturing method defined on the reference element that is combined with an anisotropic crosswind-dissipation operator. This discontinuity capturing cannot completely eliminate negative values but effectively minimizes their occurrence. The proposed methods are applied to different biophysical models and show a good agreement with experimental results for the FDA benchmark blood pump for a physiological red blood cell pore formation model., 19 pages, 15 figures

Recently developed phase field models of fracture require a diffusive crack representation based on an introduction of a crack phase field. We outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles. In this work, the numerical implementation of the phase field model for brittle fracture using natural neighbor Galerkin method is presented. Phase field method diffuses the sharp discontinuity by introducing a regularized crack functional. The notion of regularized crack functional provides a basis for defining suitable convex dissipation functions which govern the evolution of the crack phase field. The use of natural neighbors to dynamically decide the compact support at a nodal point makes it truly mesh free and nonlocal. The method uses smooth non-polynomial type Sibson interpolants, which are C 0 at a given node and C ∞ everywhere else. This allows to capture the interface region in a phase field model more accurately with only few nodes. For a given accuracy, there is an improvement in the computational time with the use of natural neighbor interpolants. Several benchmark problems were solved to investigate the efficiency of the numerical implementation of phase field model using natural neighbor Galerkin method.

Reliable prediction and thorough understanding of crack growth in engineering materials and structures are of challenging problems, but scientific and technical community has been continuously pursuing an efficient numerical approach to crack propagation simulation. We present in this paper a significant extension and development of the adaptive extended isogeometric analysis (XIGA) based on LR B-splines (locally refined B-splines), which facilitates the local refinement, for accurately modeling crack growth in isotropic and orthotropic media. The discontinuity and singularity of displacements and stresses induced by cracks are locally captured by special enrichment functions, for both isotropic and orthotropic media. This allows the representation of crack to be independent of the computational mesh and avoids the requirement for re-meshing in crack growth simulation. The numerical integration accuracy is improved with the use of the almost-polar integration and sub-triangle technique for the crack-tip and cut elements, respectively. The posteriori error estimator based on Zienkiewicz–Zhu error estimation is employed to define refinement domains, and thereby, drives the adaptivity. On the other hand, the maximum circumferential stress criterion is adopted to determine the direction of crack growth. Several crack growth numerical examples for both isotropic and orthotropic media are studied in different scenarios to show the performance and accuracy of the developed approach. Numerical results reveal that the present approach integrated with the adaptive local refinement scheme offers higher accuracy and convergence rate than the standard XIGA. Furthermore, the adaptive XIGA in a tandem with Nitche’s method is applied to simulate the crack growth of complicated structures. We additionally explore the effects of some numerical aspects or factors on crack path. Smaller crack growth increment could be used to improve the accuracy of crack path within a finer mesh in the adaptive XIGA.

Geological solid modeling has been widely studied and applied in geological analysis relevant to rock engineering, mining, geophysics, geo-hazard assessment, and geographic information systems (GIS). The discontinuities and other faces that may be planes or surfaces need to be fully considered in an attempt to yield solids formed by the mutual intersection and cutting between faces in practical geological modeling. As one kind of geological solid modeling approach, 3D block cutting analysis can simultaneously identify tens of thousands of solids (blocks) from a complicated 3D discontinuity network without user intervention. However, this analysis can only address planes and build flat blocks. In this paper, we propose a method of 3D curved block cutting analysis. The corresponding algorithm is presented completely, with some key processes of analysis thoroughly accounted for, such as solving the surface-to-surface intersection, analyzing the composition of vertices, and analyzing relevant loops. Our proposed method fully exploits the advantages of topology and computational geometry. The 3D curved blocks can be accurately constructed and represented with minor computation and memory. Analysis examples showed that several hundreds of curved blocks can be automatically constructed within one minute using a microcomputer. This method is of significance for the development of geological solid modeling and can provide a reference for related studies.

In this work we investigate the possibility of using physics-informed neural networks (PINNs) to approximate the Euler equations that model high-speed aerodynamic flows. In particular, we solve both the forward and inverse problems in one-dimensional and two-dimensional domains. For the forward problem, we utilize the Euler equations and the initial/boundary conditions to formulate the loss function, and solve the one-dimensional Euler equations with smooth solutions and with solutions that have a contact discontinuity as well as a two-dimensional oblique shock wave problem. We demonstrate that we can capture the solutions with only a few scattered points clustered randomly around the discontinuities. For the inverse problem, motivated by mimicking the Schlieren photography experimental technique used traditionally in high-speed aerodynamics, we use the data on density gradient ∇ ρ ( x , t ) , the pressure p ( x ∗ , t ) at a specified point x = x ∗ as well as the conservation laws to infer all states of interest (density, velocity and pressure fields). We present illustrative benchmark examples for both the problem with smooth solutions and Riemann problems (Sod and Lax problems) with PINNs, demonstrating that all inferred states are in good agreement with the reference solutions. Moreover, we show that the choice of the position of the point x ∗ plays an important role in the learning process. In particular, for the problem with smooth solutions we can randomly choose the position of the point x ∗ from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point x ∗ from the domain between the initial discontinuous point and the shock position of the final time. We also solve the inverse problem by combining the aforementioned data and the Euler equations in characteristic form, showing that the results obtained by using the Euler equations in characteristic form are better than that obtained by using the Euler equations in conservative form. Furthermore, we consider another type of inverse problem, specifically, we employ PINNs to learn the value of the parameter γ in the equation of state for the parameterized two-dimensional oblique wave problem by using the given data of the density, velocity and the pressure, and we identify the parameter γ accurately. Taken together, our results demonstrate that in the current form, where the conservation laws are imposed at random points, PINNs are not as accurate as traditional numerical methods for forward problems but they are superior for inverse problems that cannot even be solved with standard techniques.

Owing to the solution space enriched with a priori known knowledge through the partition of unity, the extended or generalized finite element method (XFEM/GFEM) has been widely applied to the modeling of localized failure in solids. As far as cohesive cracks are concerned in this work, on the one hand, the XFEM enriched with Heaviside function (or the shifted one with respect to nodal values) is of optimal accuracy, but is prone to the issue of ill-conditioned or even singular system matrix for arbitrary crack propagation. On the other hand, the stable XFEM (s-XFEM) circumvents the ill-conditioning issues by introducing the residual Heaviside function with respect to its linear interpolant, at the cost of losing accuracy (e.g., spurious stress locking) for discontinuities with non-uniform displacement jumps. Aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix, this work addresses a simple yet effective improved stable XFEM (Is-XFEM) with a novel enrichment function. Both the XFEM and s-XFEM are recovered as its particular cases, with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale. In the proposed Is-XFEM, as the enrichment basis functions are not contained in the standard polynomial space, conditioning of the resulting system matrix is nearly insensitive to the mesh/discontinuity configuration even if the crack path gets close to element nodes. Furthermore, the Is-XFEM with a small stabilization parameter is almost of the same accuracy as the XFEM. Particularly, pathological numerical results polluted by spurious stress locking exhibited in the s-XFEM are not observed across a fully softened discontinuity with non-uniform displacement jumps. As only the enrichment function is modified, the proposed Is-XFEM, together with appropriate crack propagation criterion and tracking algorithm, can be implemented easily in the existing XFEM codes. Several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed Is-XFEM, regarding both accuracy of the solution and conditioning of the system matrix.

In computational simulations of hydraulic fracturing problems, consideration of interactions between the propagating fracture zone and the fluid flow through the porous material requires an appropriate up-scaling procedure from the spatial scale of the local crack, which usually is much smaller compared to the scale of typical finite elements in poromechanics problems. This scale transition refers to both the displacement field (discontinuity across cracks) as well as to the fluid flow (accelerated flow within cracks and the interaction with the flow in the bulk material). To resolve the small and the large scale portion of the solution, the Generalized Finite Element Method (GFEM) exploiting the partition of unity property of shape functions is used. Accordingly, the displacements u and the liquid pressure pl are locally enriched to better resolve their distribution in the vicinity of cracks by means of an additive decomposition into a large and a small scale part. As far as the representation of cracks is concerned, the Extended Finite Element Method (XFEM) is used by enriching the displacement field by means of a jump function as well as crack tip functions. In the framework of the GFEM physically motivated enrichment functions for the local enrichment of the C1 discontinuity of the liquid pressure field across cracks are proposed in the paper. The space and time variant analytical solutions obtained from the 1D transient response of saturated porous materials subjected to the liquid pressure within the crack are applied as enrichment functions to locally improve the approximation of the liquid pressure field at discontinuities. Applying these space and time variant functions, which are the exact solutions of the pressure field in the vicinity of cracks, as local enrichment functions lead to a significant improvement in the local approximation of the pressure field at discontinuities. The new GFEM model is formulated in a poromechanics framework for fully saturated porous materials. Representative analyses demonstrate the improvement of the solution quality compared to existing FEM and XFEM models.

This paper addresses novel extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids. On one hand, based upon the unified multiscale kinematics of strong discontinuities, standard finite elements at the coarse scale are enriched with fine scale kinematics in which non-uniform discontinuity modes are incorporated. On the other hand, the traction continuity condition across the discontinuity is enforced in the statically optimal form as the fine scale statics. The admissible discontinuity modes satisfying a priori the traction continuity condition are derived. In addition to the relative rigid body motions (translations and rotations), more general relative stretching modes that induce discontinuous strain/stress fields in the bulk, are obtained with no limit of a vanishing Poisson’s ratio. The proposed model then particularizes to 2D quadrilateral elements. The displacement jumps at the discontinuity nodes are selected as the enrichment parameters and regarded as global variables shared by neighboring elements. The resulting model not only is intrinsically stress locking free for a fully softened discontinuity, but also automatically guarantees global continuity of the displacement jumps. Another benefit is that the numerical instability suffered under the unfavorable element/discontinuity configuration can be easily circumvented. Furthermore, the vanishing displacement jumps at the discontinuity tip can be enforced trivially as conventional essential boundary conditions and those complex strategies introduced in existing models are avoided. Together with the failure criterion related to the element average stress and the propagation orientation based on a simplified nonlocal stress, a local crack tracking algorithm is adopted to guarantee global continuity of the discontinuity path. Representative numerical simulations of element and structure benchmark tests are presented to verify the performance of the proposed approach in the modeling of arbitrary discontinuity propagation in solids.

This paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints, i.e. , constraints with a linear force/displacement characteristic whenever active. The presented procedure relies on an event-driven strategy for the handling of the contact constraints, in combination with one-step schemes dedicated to the time integration of the second-order equations of motion. Efficiency of the procedure follows from the use of cubic Hermite interpolation to continuously extend the normal gap functions that reflect the openings of the contact interfaces. Robustness follows from the proper handling of complex numerical situations, e.g. , numerical grazing or discontinuity sticking, through appropriate algorithm structure and numerical implementation. And, integration stability is guaranteed by the very nature of the algorithm and that of the one-step integration scheme. Following a detailed coverage of the integration procedure and the countermeasures to the expected numerical difficulties, three application examples are treated for illustration purposes. A MATLAB implementation of the procedure is provided online; download and usage information are given in Appendix A .

This paper presents new finite elements with embedded strong discontinuities to model failure in three dimensional electromechanical coupled materials. Following the strong discontinuity approach for plane electromechanical problems, the coupled boundary value problem is decomposed into a continuous global part and into a discontinuous local part where strong discontinuities in the displacement field and electric potential are introduced. Those are incorporated into general three-dimensional brick finite elements through nine mechanical separation modes and three new electrical separation modes. All the local enhanced parameters related to those modes can be statically condensed out on the element level, yielding a computationally efficient framework to model failure in electromechanical coupled materials. Impermeable electric boundary conditions are assumed along the strong discontinuities. Their initiation and orientation is detected through a configurational force driven failure criterion. A marching cubes based crack propagation concept is used to obtain smooth failure surfaces in the three dimensional problems of interest. Several representative numerical simulations are included and compared with experimental results of failure in piezoelectric ceramics to outline the performance of the new finite elements.

Integrated design of the structure topology and involved component layout is a challenging design issue when compared with traditional topology optimization. In this paper, we propose an implicit modeling approach that works completely on an Eulerian finite element mesh throughout the whole optimization process. To this aim, implicit level-set functions and R-functions are employed to describe geometrical shapes of movable components. In particular, a modified arctan function is adopted to depict the material discontinuity along the interface between the structure domain and each component domain. They are then used for material interpolations and analytical sensitivity analysis w.r.t. both pseudo-density design variables and location design variables related to the host structure and components, respectively. Based on a variety of numerical tests, it is demonstrated that considered design problems with movable components can easily be solved by extending the SIMP material model based topology optimization approach using an Eulerian mesh and the gradient-based optimization algorithm.

An interface-enriched generalized finite element method (GFEM) is introduced for 3D problems with discontinuous gradient fields. The proposed method differs from conventional GFEM by assigning the generalized degrees of freedom to the interface nodes, i.e., nodes generated along the interface when creating integration subdomains, instead of the nodes of the original mesh. A linear combination of the Lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field. This approach provides a great flexibility for evaluating the enrichment functions, including for cases where elements are intersected with multiple interfaces. We show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional GFEM. The potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber-reinforced composites and in actively-cooled microvascular materials.

In this contribution a new geometrically nonlinear, discontinuous solid-like shell finite element is presented for the simulation of cracking phenomena in thin shell structures. The discontinuous shell element is based on the solid-like shell element, having a layout similar to brick elements but better performance in bending. The phantom node method is employed to achieve a fully discontinuous shell finite element, which incorporates a discontinuity in the shell mid-surface, director and in thickness stretching field. This allows the element to model arbitrary propagating cracks in thin shell structures in combination with geometrical non-linearities. The kinematics of the discontinuous shell element as well as the detailed finite element formulation and implementation are described. Several numerical examples are presented to demonstrate the performance of the element.

Submitted by Jairo Amaro (jairo.amaro@sibi.ufrj.br) on 2019-06-28T13:02:25Z No. of bitstreams: 1 2012_CARMO_CMAME_v201-204_p127-138-min.pdf: 572016 bytes, checksum: 729db0e4147a9d67362147e8e213b2a2 (MD5) Made available in DSpace on 2019-06-28T13:02:25Z (GMT). No. of bitstreams: 1 2012_CARMO_CMAME_v201-204_p127-138-min.pdf: 572016 bytes, checksum: 729db0e4147a9d67362147e8e213b2a2 (MD5) Previous issue date: 2011-10-08 Indisponível. This paper presents a methodology for the development of discontinuity-capturing operators for general elastodynamics. These operators are indicated for problems with sharp gradients in the space and in the time. The development here presented is based on the methodology for obtaining discontinuity-capturing operators developed by Dutra do Carmo and Galeão (1986) for diffusion–convection problems and is inspired in the works presented by Hughes and Hulbert (1988 and 1990). It is shown that their operator belongs to the families of operators developed here. The formulation is applied to one-dimensional and two-dimensional problems. The results show that the method produces better results than classic methods for the one dimensional case and presents robust performance for the two-dimensional case.

The extended finite element method (XFEM) provides a natural way to incorporate strong and weak discontinuities into discretizations. It alleviates the need to mesh discontinuities, allowing simulation meshes to be nearly independent of discontinuity geometry. Currently, both quasistatic deformation and dynamic earthquake rupture simulations under standard FEM are limited to simplified fault networks, as generating meshes that both conform with the faults and have appropriate properties for accurate simulation is a difficult problem. In addition, fault geometry is not well known; robustness of solution to fault geometry must be determined. Remeshing with varying geometry would make such tests computationally unfeasible. The XFEM makes a natural choice for discretization in these crustal deformation simulations on complex fault systems. Here, we develop a method based upon the XFEM using Nitsche’s method to apply boundary conditions, enabling the solution of static deformation and dynamic earthquake models. We compare several approaches to calculating and applying frictional tractions. Finally, we demonstrate the method with two problems: an earthquake community dynamic code verification benchmark and a quasistatic problem on a fault system model of southern California.

We study the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. We consider discretizations that use regularly-shaped meshes, so that the interface “cuts” through the cells (elements or volumes) without respecting the regular geometry of the mesh. Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. This motivates the derivation of computational error estimates that yield accurate estimates for specified quantities of interest. For this purpose, we adapt the well-known adjoint based a posteriori error analysis technique used for finite element methods. In order to employ this method, we describe a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.

This paper presents a sharp discontinuity multiscale approach to address key challenges in tracking the behavior modeling granular media: accommodation discontinuities in the kinematic fields, and the direct link to the underlying grain scale information. Concepts assumed enhanced strain (AES) are taken to improve elements of the post-localization analysis, but reformulated in a hierarchical multiscale computational framework recently proposed. Unlike traditional methods of AES, in material properties are usually constant or assumed to evolve with some arbitrary phenomenological laws, the framework provides a bridge to extract key parameters evolutions of material, such as friction and dilatancy based data of computational or experimental scale grain. More importantly, the softening module phenomenological methods typically used in AES is no longer necessary. Numerical examples of plane strain compression tests are presented to illustrate the applicability of this method and analyze numerical performance. Cuenca volumen 200; cuestiones 33-36

A time-dependent convection–diffusion-reaction problem is discretized in space by a continuous finite element method with local projection stabilization and in time by a discontinuous Galerkin method. We present error estimates for the semidiscrete problem after discretizing in space only and for the fully discrete problem. Numerical tests confirm the theoretical results.

A stabilized mixed discontinuous Galerkin (SMDG) method based on Brezzi–Hughes–Marini–Masud [F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Masud, Mixed discontnuous Galerkin methods for Darcy flow, J. Sci. Comput. 22 (2005) 119–145.] is proposed to solve a thermally coupled nonlinear elliptic system modeling a large class of engineering problems. A fixed point algorithm is adopted to solve the nonlinear systems. Convergence analysis and error estimates are presented for equal order linear or bilinear discontinuous Lagrangian finite element interpolations for all fields. Numerical results are presented confirming the predicted convergence rates and illustrating the performance of the proposed formulation solving problems with globally stable and blowing up solutions.

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.

We present a continuous-discontinuous finite element method for the Mindlin-Reissner plate model based on continuous polynomials of degree k >= 2 for the transverse displacements and discontinuous polynomials of degree k - 1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.

A scalable algorithm for modeling dynamic fracture and fragmentation of solids in three dimensions is presented. The method is based on a combination of a discontinuous Galerkin (DG) formulation of the continuum problem and cohesive zone models (CZM) of fracture. Prior to fracture, the flux and stabilization terms arising from the DG formulation at interelement boundaries are enforced via interface elements, much like in the conventional intrinsic cohesive element approach, albeit in a way that guarantees consistency and stability. Upon the onset of fracture, the traction–separation law (TSL) governing the fracture process becomes operative without the need to insert a new cohesive element. Upon crack closure, the reinstatement of the DG terms guarantee the proper description of compressive waves across closed crack surfaces. The main advantage of the method is that it avoids the need to propagate topological changes in the mesh as cracks and fragments develop, which enables the indistinctive treatment of crack propagation across processor boundaries and, thus, the scalability in parallel computations. Another advantage of the method is that it preserves consistency and stability in the uncracked interfaces, thus avoiding issues with wave propagation typical of intrinsic cohesive element approaches. A simple problem of wave propagation in a bar leading to spall at its center is used to show that the method does not affect wave characteristics and, as a consequence, properly captures the spall process. We also demonstrate the ability of the method to capture intricate patterns of radial and conical cracks arising in the impact of ceramic plates, which propagate in the mesh impassive to the presence of processor boundaries.

This paper proposes a time-discontinuous and space-continuous variational integration (TDSC-VI) method for accurate elastic stress wave propagation computations in one-dimensional bar. The algorithm employs a limiter, akin to classical artificial viscosity treatment, to mitigate the deleterious Gibbs jumps across the stress discontinuities, and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the Courant stability limit, which becomes necessary for elastic unloading and internal reflections in plastic deformation problems. Stability and accuracy analyses of the proposed TDSC-VI method are carried out and compared with several well-known traditional integration algorithms. Numerical experiments are carried out with the proposed implicit and explicit methods, which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method.

Submitted by Jairo Amaro (jairo.amaro@sibi.ufrj.br) on 2019-07-09T17:36:25Z No. of bitstreams: 1 ARRUDA_v199_p3227-3236-min.pdf: 475762 bytes, checksum: 247960e6f6647063ba218940c955d752 (MD5) Made available in DSpace on 2019-07-09T17:36:25Z (GMT). No. of bitstreams: 1 ARRUDA_v199_p3227-3236-min.pdf: 475762 bytes, checksum: 247960e6f6647063ba218940c955d752 (MD5) Previous issue date: 2010-07-01 Indisponível. In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.

In this paper a finite element framework based on the incomplete interior penalty Galerkin formulation, a non-symmetric discontinuous Galerkin method, is consistently formulated for modeling plasticity problems with small deformation. Because of its pure displacement-based framework, this proposed discontinuous Galerkin method is possibly able to completely preserve numerical integration algorithms efficiently developed in the traditional continuous Galerkin framework. Besides stresses on element interior quadrature points, stresses on element surface quadrature points are also required to return on yielding surfaces in this discontinuous Galerkin framework, which is able to provide more accurate material yielding profiles than the continuous Galerkin framework. The performance of the proposed discontinuous Galerkin framework has been evaluated in detail for J 2 and pressure-dependent plasticities using perfect plasticity, plasticity with hardening, and associative and non-associative material models. Quadratic convergent rates compatible to the tradition continuous Galerkin method for modeling plasticity problems have been achieved within a large penalty range in a nodal-based discontinuous Galerkin implementation.

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in Engineering, 2004) where the crack is modeled by a set of cracked segments. However, in contrast to the above mentioned paper, we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity. Instead, the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics. We apply this method to several two- and three-dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any “mesh” orientation bias.

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.

International audience; Using a kinetic approximation of a linear diffusion operator, we propose an algorithm that allows one to deal with the simulation of a multi-dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities. Numerical examples are given in dimensions one to three on PDEs or stochastic PDEs with or without source terms.

In this work we present a new modelling paradigm for computing the complete failure of metal frames by combining the stress-resultant beam model and the shell model. The shell model is used to compute the material parameters that are needed by an inelastic stress-resultant beam model; therefore, we consider the shell model as the meso-scale model and the beam model as the macro-scale model. The shell model takes into account elastoplasticity with strain-hardening and strain-softening, as well as geometrical nonlinearity (including local buckling of a part of a beam). By using results of the shell model, the stress-resultant inelastic beam model is derived that takes into account elastoplasticity with hardening, as well as softening effects (of material and geometric type). The beam softening effects are numerically modelled in a localized failure point by using beam finite element with embedded discontinuity. The original feature of the proposed multi-scale (i.e. shell-beam) computational model is its ability to incorporate both material and geometrical instability contributions into the stress-resultant beam model softening response. Several representative numerical simulations are presented to illustrate a very satisfying performance of the proposed approach.