Discontinuous Formulation Research Articles

An a posteriori error analysis for an augmented discontinuous Galerkin method applied to Stokes problem

AbstractThis paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.

Discontinuous bifurcation of FRCC with zero-thickness interface modeling

In this work, firstly a fracture-based interface constitutive theory, aimed at simulating the cracking mechanisms of Fiber Reinforced Cementitious Composites (FRCCs), is presented. The discontinuous formulation assumes a hyperbolic maximum strength criterion in terms of normal and shear joint stresses. The latter are evaluated on each crack front to simulate the failure behavior of plain and FRCC systems. A non-associated plastic flow rule, in conjunction with a post-cracking softening law, is defined to complete the modeling approach. On the other hand, the use of the most-classical Mixture Theory is followed for taking into account the actions of fibers in concrete matrix. The bridging mechanisms between fibers and active cracks are defined in terms of fiber-to-concrete bond–slip rule and dowel effects. Secondly, a normalized Cracking Indicator (CI) for discrete crack is proposed in the spirit of Hill’s indicator for loss of stability of inelastic continua, to effectively evaluate the most critical direction for further loading in terms of the resulting energy release and crack opening, while accounting for the fiber direction and content. After presenting the constitutive theory and, particularly, the novel concept of the CI, numerical analyses at constitutive level are performed to evaluate the evolution of the fracture energy, post-peak strength, and critical cracking directions under variable fiber contents. Different load scenarios are evaluated, and the numerical predictions are compared with experimental data.

Piecewise integrable neural network: An interpretable chaos identification framework.

Artificial neural networks (ANNs) are an effective data-driven approach to model chaotic dynamics. Although ANNs are universal approximators that easily incorporate mathematical structure, physical information, and constraints, they are scarcely interpretable. Here, we develop a neural network framework in which the chaotic dynamics is reframed into piecewise models. The discontinuous formulation defines switching laws representative of the bifurcations mechanisms, recovering the system of differential equations and its primitive (or integral), which describe the chaotic regime.

New low order Runge–Kutta schemes for asymptotically exact global error estimation of embedded methods without order reduction

Embedded Runge–Kutta (RK) methods provide an attractive methodology for solution of initial value problems (IVP) by means of adaptive step size control. According to Fehlbergs suggestion, only one extra function calculation is required to estimate the local error of the embedded method. In the present paper, this methodology is applied to three prominent low order implicit RK-schemes. To this end, new embedded Butcher-tableaus are presented for the implicit Euler-method, the trapezoidal-rule and the midpoint-rule by means of corresponding higher order RK-schemes. In order to account for the global error, an asymptotically exact global error estimator is derived for the so-called reversed embedded RK-method. Moreover, a cause of order reduction is illustrated by aid of a weak discontinuous formulation. It is prevented in a new hybrid integration algorithm by means of a Hermite interpolation for the source term of the IVP, and its implementation renders a convenient “node-to-node” stepping scheme for a laser beam simulation. Two numerical examples are presented with the following main results: The higher order convergence behaviors of the proposed new RK-schemes are verified, its successful performances for asymptotically exact global error estimation of the reversed embedded RK-method are shown, the proposed methodology is illustrated for two different global error functions, and the desired effect of the hybrid algorithm to prevent order reduction is illustrated.

A Simplified Discontinuous Galerkin Self-Dual Integral Equation Formulation for Electromagnetic Scattering From Extremely Large IBC Objects

A Simplified Discontinuous Galerkin Self-Dual Integral Equation Formulation for Electromagnetic Scattering From Extremely Large IBC Objects

Closed form solutions for the strain localization problem in a softening circular bar in pure torsion with the continuum damage and the embedded discontinuity models

The boundary value problems and closed form solutions for the strain localization in a softening bar under pure torsion with the continuum damage and the embedded discontinuity formulations are developed. In this formulation, quasi-brittle materials such as concrete or mortar are considered. The boundary value problem is developed from a variational formulation and the corresponding Euler-Lagrange differential equation. These formulations include not only the continuum damage model, in which the twist angle jump and the angular strain concentration are smeared into the volume of the bar, but also the embedded discontinuity model, in which the twist angle jump and the angular strain localization are lumped into a zero thickness localization zone. The non-linear behaviour of the materials is described by a continuous constitutive model for the continuum damage model and by a discrete constitutive model for the embedded discontinuity model; both constitutive relationships include linear or exponential softening. Solutions to overcome the problem of crack bandwidth dependency when modelling strain localization with continuous models are developed, these solutions are based on a rational analysis of the kinematics of the strain model and on the fracture energy per unit area. To validate the developed closed form solutions for the strain localization problem in a softening bar under torsion, three examples of concrete cylindrical specimens under torsion are presented. The relationship between the total area under the twisting moment-twist angle curves and the fracture energy of the material guarantees the correctness of the developed closed form solutions.

Machine learning the derivative discontinuity of density-functional theory

Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.

Discontinuity formulation of restitutions in Aceh qanun against the protection of rape victims

This study discusses the fulfilment of the rights of rape victims in Aceh in the form of restitution payments. Restitution which is the main punishment in Qanun Aceh No. 6 of 2014 concerning the Jinayat Law, has never been included in the judge's decision in the cases of rape. Basically Qanun Jinayah has shown its side with rape victims by formulating punishments for perpetrators which are certain to have a deterrent effect such as the length of prison sentences and the number of sentences handed down to the perpetrators. However, the problem of hampering the fulfilment of restitution to victims is a new problem in the realm of enforcing the Jinayah Qanun in Aceh. This raises the question of why restitution was never included in the decision of the Sharia Court for the rape case in Aceh. This research is a literature study using a normative approach which is then explained descriptively. The results of this study showed that the restitution was never included in the decision is due to the incompatibility of the restitution formulation mentioned in the Jinayat Law Qanun and later referred to differently in the Jinayat Procedural Law, namely as compensation. The fulfilment of this right to restitution is also hampered due to the absence of a governor's regulation that contains technical rules regarding the mechanism for executing restitution for victims. The impact of this discontinuity in the formulation of restitution makes judges unable to include sanctions for restitution in their decisions.

Crack propagation simulation without crack tracking algorithm: Embedded discontinuity formulation with incompatible modes

We show that for the simulation of crack propagation in quasi-brittle, two-dimensional solids, very good results can be obtained with an embedded strong discontinuity quadrilateral finite element that has incompatible modes. Even more importantly, we demonstrate that these results can be obtained without using a crack tracking algorithm. Therefore, the simulation of crack patterns with several cracks, including branching, becomes possible. The avoidance of a tracking algorithm is mainly enabled by the application of a novel, local (Gauss-point based) criterion for crack nucleation, which determines the time of embedding the localisation line as well as its position and orientation. We treat the crack evolution in terms of a thermodynamical framework, with softening variables describing internal dissipative mechanisms of material degradation. As presented by numerical examples, many elements in the mesh may develop a crack, but only some of them actually open and/or slide, dissipate fracture energy, and eventually form the crack pattern. The novel approach has been implemented for statics and dynamics, and the results of computed difficult examples (including Kalthoff’s test) illustrate its very satisfying performance. It effectively overcomes unfavourable restrictions of the standard embedded strong discontinuity formulations, namely the simulation of the propagation of a single crack only. Moreover, it is computationally fast and straightforward to implement. Our numerical solutions match the results of experimental tests and previously reported numerical results in terms of crack pattern, dissipated fracture energy, and load–displacement curve.

Closed form solutions for the strain localization problem in a softening bar under tension with the continuum damage and the embedded discontinuity kinematics

The boundary value problems and closed form solutions for the strain localization in a softening bar with the continuum damage and the embedded discontinuity formulations are developed. The boundary value problem is developed from a variational formulation and the corresponding Euler-Lagrange differential equation. These formulations include not only the classical continuum damage model, in which the displacement jump and the strain concentration are smeared into the bar length, but also the embedded discontinuity model, in which the displacement jump and the strain concentration are lumped into a zero thickness localization zone. For the classical continuum damage model, the non-linear behaviour of the materials is described by a continuum constitutive model and for the embedded discontinuity model by a discrete constitutive model; both constitutive relationships may consider linear or exponential softening. Solutions to overcome the problem of crack bandwidth dependency when modelling strain localization with continuum models are developed. These solutions are based on a rational analysis of the kinematics of the continuum damage model and on the fracture energy per unit area. To validate the developed closed form solutions for the strain localization problem in a softening bar, two examples of cylindrical specimens of concrete under tension are presented. The relationship between the total area under the load-displacement curves and the fracture energy of the material guarantees the correctness of the developed closed form solutions.

Fracture of quasi-brittle solids by continuum and discrete-crack damage models and embedded discontinuity formulation

We derive a finite element formulation for modelling fracture in 2d quasi-brittle solids. The kinematics of isoparametric quadrilateral is enriched by strong discontinuity in displacements in order to capture the discrete crack opening and sliding. To describe both the bulk and crack-surface dissipative phenomena induced by the crack propagation, a combination of continuum and discrete damage models is used. The continuum damage model describes dissipation in the bulk, which mainly occurs ahead of the crack front. The discrete damage model includes two uncoupled rigid-damage cohesive laws to deal with fracture energy release due to crack propagation in modes I and II. The numerical solution strategy is based on a local-global operator-split procedure and incorporates a crack-tracking algorithm for enforcing continuity of the crack path. Numerical examples illustrate very satisfying performance of the derived formulation in terms of mesh independency and robustness. The sensitivity of results on material parameters for mixed mode fracture is also illustrated.

Simulation of the Damage Process in Quasi‐Brittle Materials by a Modified Finite Element Method Using the Consistent Embedded Discontinuity Formulation

This paper investigates the variational finite element formulation and its numerical implementation of the damage evolution in solids, using a new discrete embedded discontinuity approach. For this purpose, the kinematically optimal symmetric (KOS) formulation, which guarantees kinematics, is consistently derived. In this formulation, rigid body motion of the parts in which the element is divided is obtained. To guarantee equilibrium at the discontinuity surfaces, the length of the discontinuity is introduced in the numerical implementation at elemental level. To illustrate and validate this approach, two examples, involving mode‐I failure, are presented. Numerical results are compared with those reported from experimental tests. The presented discontinuity formulation shows a robust finite element method to simulate the damage evolution processes in quasi‐brittle materials, without modifying the mesh topology when cohesive cracks propagate.

On weak Dirichlet boundary conditions for elliptic problems in the continuous Galerkin method

We combine continuous and discontinuous Galerkin methods in the setting of a model diffusion problem. Starting from a hybrid discontinuous formulation, we replace element interiors by more general subsets of the computational domain – groups of elements that support a piecewise-polynomial continuous expansion. This step allows us to identify a new weak formulation of Dirichlet boundary condition in the continuous framework. We show that the boundary condition leads to a stable discretization with a single parameter insensitive to mesh size and polynomial order of the expansion. The robustness of the approach is demonstrated on several numerical examples.

Discontinuous isogeometric boundary element (IGABEM) formulations in 3D automotive acoustics

The isogeometric boundary element method (IGABEM) is a technique that employs non-uniform rational B-splines (NURBS) as basis functions to discretise the solution variables as well as the problem geometry in a boundary element formulation. IGABEM has shown improved convergence properties over the conventional boundary element method (BEM) algorithms. However, in acoustics, IGABEM has only been applied to problems with simple smooth boundary conditions. In most real-world engineering design and analysis acoustic problems, geometric corners and discontinuities in boundary conditions can give rise to more complexity in the solution field that may be more efficiently modelled using a discontinuous approach.In the current work, we develop a discontinuous IGABEM formulation based on discontinuous elements and a suitable collocation scheme. Continuous and discontinuous formulations are compared. In this paper, a three dimensional model with different sets of boundary conditions is presented to explore the conditions under which a discontinuous formulation outperforms the continuous IGABEM. A simple car passenger compartment model characterised by panels with piecewise continuous impedance boundaries is presented to illustrate the potential of the proposed method for integrated engineering design and analysis.

Nonlinear Diffusion Acceleration of the Least-Squares Transport Equation in Geometries with Voids

In this paper we show the extension of nonlinear diffusion acceleration (NDA) to geometries containing small voids using a weighted-least-squares (WLS) high-order equation. Even though the WLS equation is well defined in voids, the low-order drift-diffusion equation was not defined in materials with a zero cross section. This paper derives the necessary modifications to the NDA algorithm. We show that a small change to the NDA closure term and a nonlocal definition of the diffusion coefficient solve the problems for void regions. These changes do not affect the algorithm for optically thick material regions while making the algorithm well defined in optically thin ones. We use a Fourier analysis to perform an iterative analysis to confirm that the modifications result in a stable and efficient algorithm. Later in the paper, numerical results of our method are presented. We test this formulation with a small, one-dimensional test problem. Additionally, we present results for a modified version of the C5G7 benchmark containing voids as a more complex, reactor-like problem. We compared our results to Texas A&M’s transport code PDT, utilizing a first-order discontinuous formulation as reference and the self-adjoint angular flux equation with void treatment (SAAF), a different second-order form. The results indicate that the NDA WLS performed comparably or slightly worse then the asymmetric SAAF while maintaining a symmetric discretization matrix.

A three dimensional adaptive multiscale method for crack growth in Silicon

A three dimensional concurrently coupled adaptive multiscale method is introduced here to simulate complex crack growth patterns in Silicon, by combining several numerical techniques across the length scales. The coarse scale material is modeled using the virtual atom cluster approach. The strong kinematic discontinuities in the bulk are simulated based on a three dimensional version of the phantom node method. A molecular statics model placed around the crack tip is concurrently coupled with the phantom-based discontinuous formulation, where the coupling between the fine and coarse scales is realized through the use of ghost atoms, whose positions are interpolated based on the coarse scale solution. The boundary conditions to the fine scale model, at the coupling region, are assigned by enforcing the interpolated displacements of ghost atoms. In order to optimize the computation costs, adaptivity schemes for adjustment of the fine scale region as the crack propagates, and coarse graining of the region behind the crack tip, are proposed. The crack tip location is detected based on an energy criterion. All the molecular simulations in the pure atomistic as well as the multiscale model are carried out using the LAMMPS software, triggered through the system command in MATLAB. The performance of the developed framework in terms of computation cost, robustness and versatility, is assessed through several numerical examples concerning crack growth in Silicon. Therefore, the diamond cubic lattice structure of Silicon is used at the fine scale, where the atom-atom interactions are modeled based on the Tersoff potential function. According to the numerical examples presented in this study, savings in computational time using the present multiscale method are observed to be up to 87%, as compared to the pure atomistic model.

Análise estratégica do processo de formulação da PM&A-SUS: lições aprendidas e desafios

RESUMO O artigo apresenta a Análise Estratégica (AE) do processo de formulação da Política Nacional de Monitoramento e Avaliação do Sistema Único de Saúde (PNM&A-SUS). O estudo qualitativo utilizou-se da análise documental e revisão de literatura considerando as três dimensões da AE: pertinência do problema, ou seja, necessidade institucional da política; dos objetivos propostos e das parcerias mobilizadas. A análise mostrou que a formulação vem acontecendo de forma descontínua e não há definição de conceitos, proposições e usos unanimemente aceita entre os stakeholders. O processo interrompido necessita ser retomado para a elaboração consensuada da PNM&A-SUS, considerando as lições aprendidas e a apropriação realista do contexto.

Material softening and strain localization in spatial geometrically exact beam finite element method with embedded discontinuity

When some critical condition is reached at a material point of a solid body, a localized strain starts developing which makes the strain field discontinuous and highly accelerates local damaging of material. The present paper addresses this kind of strain localization in spatial geometrically exact beams. Here we propose a new beam finite element formulation which accounts for softening of material by applying the embedded strong strain discontinuity technology. The formulation is essentially an extension of the original strain-based formulation, upgraded such to allow for detecting the onset of strain localization and to introduce additional equations for evaluating singular strain peaks and jumps of displacements and rotations at the localized section in further deformation. The consistency condition that the equilibrium and the constitutive stress-resultants are equal is shown to be naturally suited for the implementation into the discontinuous formulation. The condition for the onset of strain localization at a beam cross-section is here related to the loss of uniqueness of the beam cross-sectional constitutive equations. If the condition for a unique inverse is violated, two solutions are possible for cross-sectional strains. In a subsequent deformation, one of the two solutions follows the softening regime of material. The discontinuous increments in strains, displacements and rotations at the softening cross-section are obtained from the equations of the structure supplemented by the consistency conditions of the softening cross-section.

A discontinuous Petrov–Galerkin (DPG) formulation for the FEM applied to the Helmholtz equation for high wavenumbers

Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is widely studied in a large number of papers, e.g., (Ihlenburg and Babuska, Comput Math Appl 30(9):9–37, 1995; Ihlenburg, Finite element analysis of acoustic scattering applied mathematical sciences, vol 132. Springer, New York, 1998). Robust methodologies for structured square meshes have been developed in recent years. This work seeks to develop a methodology based on Petrov–Galerkin discontinuous formulation, to minimize phase error for Helmholtz equation for both structured and unstructured meshes. A Petrov–Galerkin finite element formulation is introduced for the Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the triangular mesh, a global basis function for the weighting space is obtained, adding bilinear $$C^0$$ Lagrangian weighting function linear combinations. The optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimal criteria. This is done numerically through a preprocessing technique that is naturally applied to non-uniform and unstructured meshes. In particular, for uniform meshes a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by (Babuska et al., Comput Meth Appl Mech Eng 128:325–359, 1995) is obtained. The numerical results indicate a better performance in relation to the classic discontinuous Galerkin method.

RETRACTED: Effect of joint spacing and joint dip on the stress distribution around tunnels using different numerical methods

Different conditions may affect the stability of tunnels by the geometry (spacing and orientation) of joints in the surrounded rock mass. In this study, by comparing the results obtained by the three novel numerical methods i.e. finite element method (Phase2), discrete element method (UDEC) and indirect boundary element method (TFSDDM), the effects of joint spacing and joint dips on the stress distribution around rock tunnels are numerically studied. These comparisons indicate the validity of the stress analyses around circular rock tunnels. These analyses also reveal that for a semi-continuous environment, boundary element method gives more accurate results compared to the results of finite element and distinct element methods. In the indirect boundary element method, the displacements due to joints of different spacing and dips are estimated by using displacement discontinuity (DD) formulations and the total stress distribution around the tunnel are obtained by using fictitious stress (FS) formulations.