Discontinuous Element Research Articles

Stability Issues of Entropy-Stable and/or Split-form High-order Schemes

The focus of the present research is on the analysis of local energy stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we further demonstrate numerically that such schemes cause exponential growth of errors during the simulation. Furthermore, we encounter a similar abnormal behaviour for the compressible Euler equations, for a smooth exact solution of a density wave. Finally, for the same case, we demonstrate numerically that other commonly known split-forms, such as the Kennedy and Gruber splitting, are also locally energy unstable.

Dual interpolation boundary face method for 3-D potential problem based on binary tree grids

This paper presents the dual interpolation boundary face method (DiBFM) based on a binary tree grid for solving the 3-D potential problems. The subdivision method based on binary tree algorithm is capable to generate both continuous and discontinuous grids, and to achieve the grids​ generation for arbitrary complex model much easier and automatically. By adding virtual nodes on the vertices and edges of traditional discontinuous element, the dual interpolation elements are introduced while the order is increased by two. The values of physical variables are approximated by Lagrange interpolation polynomial in the first-layer interpolation, and meshless interpolation is used to condense the degree of freedom of virtual nodes in the second-layer interpolation. Since no requirement is needed for the continuity of grids in the DiBFM, the dilemma of discontinuous grids are avoidable. In this paper, the DiBFM is implemented based on the binary tree grids to solve general problems with thin-wall structures and “geometrical construction noise”.

Fully coupled thermo-hydro-mechanical approach to model fracture response to injection process in enhanced geothermal systems using displacement discontinuity and finite element method

Assessing, predicting, and controlling injection-induced seismicity is a major challenge for developing enhanced geothermal systems (EGS) due to the complexity of coupled thermo-hydro-mechanical (THM) processes. This study aimed to develop a fully coupled numerical model to assess the complex behavior of a low permeable matrix-fracture during non-isothermal single-phase fluid injection process. A thermo-poroelastic displacement discontinuity (DD) method combining with different forms of finite element method are implemented to encapsulate the fractured medium response and transport processes, respectively. The nonlinear characteristics of normal (changing the joint to hydraulic fracture status) and shear (changing the stick to slip fracture status) fracture deformation are taken into account through fracture constitutive relations. Developed numerical approach was applied to simulate cool water injection into fracture/matrix systems, analyzing the role of coupled processes on spatiotemporal variation of matrix-fracture stresses, temperature and pore pressure and assessing induced seismicity (slippage) and permeability alteration. Numerical simulations demonstrate a fully coupled relation between matrix dilation, shrinkage, and non-linear fracture deformation. The redistribution of dynamic and kinematic parameters along with non-linear fracture deformation showed that although the poroelastic effects are dominant in the early stages, thermoelastic effects dominate in the long-term injection stages. • A fully-coupled Thermo-Hydro-Mechanical approach was developed to model injection process in EGS. • Displacement discontinuity and finite element methods were combined to encapsulate fracture deformation and transport processes, respectively. • Fracture constitutive laws were implemented to model non-linear crack deformation. • Poroelastic effects are dominant in the early stages, and thermoelastic effects dominate in the long-term injection stages.

Fully-Discrete Spatial Eigenanalysis of Discontinuous Spectral Element Methods: Insights into Well-Resolved and Under-Resolved Vortical Flows

This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalizing previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge-Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang and Jameson [1] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. It is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems. The effect of mesh regularity is also considered, where time integration errors are found to be, in the case of irregular grids, less pronounced than those of the spatial discretisation. In fact, for the under-resolved vortical flows considered, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. Nevertheless, for well-resolved nonlinear simulations, the effect of time integration errors could still be recognized. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behaviour of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.

The Threat of Religious Fundamentalism and the European Immigration Crisis

A significant part of the crisis of our contemporary European societies can be attributed to misplaced and abused religious zeal in various forms of religious fundamentalism, both domestically grown as well as imported and shared by the immigrants to Europe from third-world countries. To deal with this complex phenomenon in the European environment, it is necessary to conceive the analysis of the presented issue into a sociological scheme based on three premises: (1) diagnosis of migration processes in the context of growing population movements in Europe, (2) identification of determinants and factors that cause these movements, as well as (3) a proposal to solve the current situation in the spirit of social teaching of the Roman Catholic Church. Our paper is an attempt to interpret and compare the opinions of selected experts on this sensitive issue and, with the help of their opinions, to present some guiding ideas on the path to possible solutions to the current situation. We begin by describing the ideological deconstruction of the moral and cultural world as evidenced in the postmodern society, accompanied by processes of subjectivization and individualization, which acquired a societal context in Europe and North America in the 1960s. We then turn to exploring the context of religious change (from a theological-sociological perspective). The religiosity of postmodern man becomes a mixture of various correct, albeit often contradictory, discontinuous elements , involving a small dose of love for one's neighbor, often taking the form of friendly affection and showing emotions towards animals and the external environment, ideologically correct psychology as well as parapsychology, supplemented by esoteric, occult and astrological notions, while staying open to the possibility for Eastern philosophies and the sects. Islamic fundamentalism is seen as a reaction to this religious-cultural context that is perceived (by conservative Muslims and Christians alike) as hostile to traditional values, ideas about the world, and ideals. The context of contemporary Islam’s influence on the European religious landscape and culture is scrutinized in the next section of our paper. In Europe, the number of Christians will fall from 74.5% to 65.2% between 2010 and 2050, while the number of non-believers (nones) will increase from 18.8% to 23.3%, and the number of Muslims will also almost double from 5.9% to 10.2%. The growth of Muslims in Europe will be affected by both birth rates and migration. A part of our critical analysis points to the self-destructive tendencies of some European elites and cultural influencers/policy makers. After outlining some forecasts and developments, and offering initial critical views on the transpiring phenomena, we move on to delineating possible solutions to this situation. Due to the complexity of the problem, there is no ready-made, simple way to handle this situation. While immigrants have always played in important role in the European history, a growing number of political scientists talk in particular about the internal protection of Europe, that is, the inevitability of protecting its constitutive, fundamental values and rights. If Europe is not to lose its face and cultural/moral fiber, it is important to uphold its constitutive values. This will not be possible without an intentional struggle to reinvent its moral and spiritual heritage with every new generation without forfeiting the fundamentals upon which our culture and civilization has been built. The concluding section of our paper focuses on the Catholic Church's position on this issue and its recent proposals for resolving the migration crisis. The Church’s teaching that state officials and others who profess Christianity but reject refugees are hypocrites because Jesus would accept these people should be balanced by a critical call to be aware that our obligation to love and care for our neighbor extends not only to the immigrants and their families but also to the families and individuals of the European host countries. Our fear of Islamization of Christian Europe may be an indication that we Europeans have very little confidence in our own faith. Accordingly, we will not be able to preserve the Christian faith by living it secluded in our churches, but by presenting our Christian spirit - by accepting these refugees and by helping them in their concrete circumstances, and by engaging them (as well as our secular counterparts) publicly with due respect in an open-ended discourse of metanarratives.

Zonal direct-hybrid aeroacoustic simulation of trailing edge noise using a high-order discontinuous Galerkin spectral element method

We present the extension of a discontinuous Galerkin framework to zonal direct-hybrid aeroacoustic simulations. This extension provides the ability to simultaneously perform a zonal large eddy simulation (LES), solving the compressible Navier–Stokes equations, and an acoustic propagation simulation, solving the acoustic perturbation equations. In doing so, the acoustic sources are exchanged without using the file system, and the bottleneck of I/O operations is avoided. This approach is well suited for large-scale simulations done in high-performance computing. The zonal LES uses the recently introduced recycling rescaling anisotropic linear forcing as a turbulent inflow method. We present a methodology to model the required Reynolds stresses based on the distribution of the turbulent kinetic energy obtained from solving the Reynolds-averaged Navier–Stokes equations. We show at the example of a turbulent flow over a flat plate and a NACA 64418 trailing edge simulation that the chosen model of the Reynolds stresses is valid. Direct-hybrid simulation results of a NACA 0012 airfoil, including tonal self-noise and a NACA 64418 trailing edge, demonstrate the presented approach’s applicability. This zonal direct-hybrid simulation approach shows great potential for efficient hybrid computational aeroacoustic simulations in high-performance computing.

Oscillation-free nodal discontinuous spectral element method for the simulation of compressible multicomponent flows

The nodal discontinuous spectral element method (DSEM) is improved to avoid pressure and velocity oscillations from material interfaces in the solution of multi-component compressible flows. A novel double flux method compatible with the DSEM scheme is developed to account for variations in the specific heats ratio at material interfaces. The double flux method consists of a modified Harten-Lax-van Leer-Contact (HLLC) Riemann solver and an averaging technique to freeze the temperature-dependent specific heats ratio during the solution update to remove oscillations. The double flux model is combined with a conservative method of calculating the convective flux function to avoid unnecessary energy conservation error at shock fronts. Numerical diffusion terms are also introduced for capturing a shock wave, contact discontinuity, and material interface, without inducing oscillations at the material interface. A series of one- and two-dimensional tests are conducted to evaluate the efficacy of the numerical scheme in capturing sharp discontinuities, as well as vortices, without introducing significant conservation error. According to the benchmark results, the new approach maintains the pressure, velocity, and temperature equilibriums across material interfaces. Artificial diffusion is added to the equation in a controlled manner that enables a higher Courant-Friedrichs-Lewy (CFL) number for high polynomial orders. The present scheme's performance is validated by comparing the spatiotemporal evolution of characteristic points in the shock-bubble interaction and Richtmyer-Meshokov's instability problems.

Algunas observaciones sobre el funcionamiento del adverbio dizque, un elemento de discontinuidad diacrónica y diatópica en español

Some Observations Regarding the Functioning of the Adverb dizque, an Element of Diachronic and Diatopic Discontinuity in Spanish. The adverb dizque lies at the core of a double discontinuity in the landscape of Spanish language. On the one hand, it marks a diachronic discontinuity, as its modal-epistemic and pragmatic values have developed significantly since the 13th century, when it was first registered in Old Spanish, with the evidential-reportative meaning that stems from its etymology. On the other hand, dizque is affected by a diatopic discontinuity, as it is becoming obsolete in the Iberian Peninsula, but is extremely productive in Hispanic American varieties. This paper traces the evolution of the modal and discursive values of dizque and observes its syntactic behaviour by means of a corpus analysis, concluding that, in today’s Hispanic American Spanish, this adverb has reinforced this position in the left periphery, either as a sentence modifier or as a constituent modifier, with clearly established syntactic peculiarities. Keywords: evidentiality, epistemic, syntax, pragmatics

A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s model

We consider the system of partial differential equations stemming from the time discretization of the two-field formulation of the Biot’s model with the backward Euler scheme. A typical difficulty encountered in the space discretization of this problem is the robustness with respect to various material parameters. We deal with this issue by observing that the problem is uniformly stable, irrespective of all parameters, in a suitable nonsymmetric variational setting. Guided by this result, we design a novel nonconforming discretization, which employs Crouzeix-Raviart and discontinuous elements. We prove that the proposed discretization is quasi-optimal and robust in a parameter-dependent norm and discuss the consequences of this result.

Assessment of turbulence models using DNS data of compressible plane free shear layer flow

The present paper uses the detailed flow data produced by direct numerical simulation (DNS) of a three-dimensional, spatially developing plane free shear layer to assess several commonly used turbulence models in compressible flows. The free shear layer is generated by two parallel streams separated by a splitter plate, with a naturally developing inflow condition. The DNS is conducted using a high-order discontinuous spectral element method (DSEM) for various convective Mach numbers. The DNS results are employed to provide insights into turbulence modelling. The analyses show that with the knowledge of the Reynolds velocity fluctuations and averages, the considered strong Reynolds analogy models can accurately predict temperature fluctuations and Favre velocity averages, while the extended strong Reynolds analogy models can correctly estimate the Favre velocity fluctuations and the Favre shear stress. The pressure–dilatation correlation and dilatational dissipation models overestimate the corresponding DNS results, especially with high compressibility. The pressure–strain correlation models perform excellently for most pressure–strain correlation components, while the compressibility modification model gives poor predictions. The results of an a priori test for subgrid-scale (SGS) models are also reported. The scale similarity and gradient models, which are non-eddy viscosity models, can accurately reproduce SGS stresses in terms of structure and magnitude. The dynamic Smagorinsky model, an eddy viscosity model but based on the scale similarity concept, shows acceptable correlation coefficients between the DNS and modelled SGS stresses. Finally, the Smagorinsky model, a purely dissipative model, yields low correlation coefficients and unacceptable accumulated errors.

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

Discontinuous Galerkin methods through the lens of variational multiscale analysis

In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.

The effect of entrance flow development on vortex formation and wall shear stress in a curved artery model

We numerically investigate the effect of entrance condition on the spatial and temporal evolution of multiple three-dimensional vortex pairs and the wall shear stress distribution in a curved artery model. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e., uniform). We discuss the connection along the axial direction between regions of organized vorticity observed at various cross sections of the model and compare results between the different entrance conditions. We model a human artery with a simple, rigid 180° curved pipe with a circular cross section and constant curvature, neglecting the effects of taper, torsion, and elasticity. Numerical results are computed from a discontinuous high-order spectral element flow solver. The flow rate used in this study is physiological. We observe differences in secondary flow patterns, especially during the deceleration phase of the physiological waveform where multiple vortical structures of both Dean-type and Lyne-type coexist. The results indicate that decreased axial velocities under an undeveloped condition produce smaller secondary flows that ultimately inhibit growth of any interior flow vortices. We highlight the effect of the entrance condition on the formation of these structures and subsequent appearance of abnormal inner wall shear stresses, which suggest there may be a lower prevalence of cardiovascular disease in curved arteries where the flow is rather undeveloped—a potentially physiologically significant result to help understand the influence of blood flow development on disease.

Numerical-simulation-based Landslide Warning System and Its Application

Landslide early warning is a systematic project involving multidisciplinary integration, i.e. geology, mechanics, engineering, monitoring technology, and information technology. When conducting landslide studies, due to the limitation of failure theory of geological body, landslide warning system is purely based on monitoring data nowadays, so it is difficult to integrate interdisciplinary techniques. A trigger condition based prediction theory and disaster stage judgment approach are introduced, with which the landslide time prediction is converted to the disaster stage judgment. The definition and significance of fracture degree is presented, by which the disaster stage judgment is convert to inner fracture state analysis. At last, the numerical-simulation-based landslide warning system is discussed in detail. This system contains four parts, namely parameter acquisition part, disaster kernel analysis system, current state back analysis part, and landslide reliability evaluation part, respectively. In the first part, the essential parameters and its acquisition method is presented, and then the necessity and advantages of dimensional analysis is discussed. In the second part, the new numerical method named continuous discontinuous element method (CDEM) is introduced, and main features, i.e. Lagrange equation, strain strength distribution criteria, element crack strategy, and indented point & indented edge contact model is introduced. In third part, the inverse analysis method of current parameters of geological body based on monitoring data and numerical simulation is discussed. In the fourth part, the reliability evaluation steps are presented in detail. Finally, the sliding occurrence probability of Liangshuijing landslide in Chongqing, China is discussed, which demonstrated to show the precision and rationality of the proposed landslide warning system.

An Energy Conservative hp-method for Liouville\u2019s Equation of Geometrical Optics

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of 10^{-6}.

Boundary-Layer Dynamics in Wall-Resolved LES Across Multiple Turbine Stages

Modern turbomachinery relies on accurate prediction of the flow and especially the state of turbulence to achieve the required level of performance. Transition, relaminarization, wake interactions, and interrow influence form complex, highly unsteady flow patterns. Large eddy simulation (LES) emerges as a promising method to deliver improved accuracy over Reynolds-averaged Navier–Stokes (RANS) approaches as the major energy-carrying scales are fully resolved. However, for wall-bounded flows, modeling of (parts of) the boundary layer might still be inevitable to keep the computational costs manageable. In this paper, we aim to characterize and analyze the boundary-layer state in such a scenario. We employ the high-order discontinuous Galerkin spectral element method to perform a wall-resolved LES of a stator-rotor-stator cascade ( up to 0.65, up to ). The interfaces between the blade rows are treated with a high-order accurate sliding mesh approach. Time-averaged performance characteristics are compared against experimental and numerical data. The temporal evolution of the solution is first assessed through the phase-averaged flowfield at different stator-rotor positions. Subsequently, special emphasis is placed on the spatiotemporal evolution of turbulence near the blade surface. Boundary-layer profile and energy spectra analysis are used to give insight into the turbulence development. This investigation not only reveals the complexities of the boundary-layer dynamics but can also serve as benchmark and reference for evaluation and development of wall-modeled LES approaches for turbomachinery applications.

A Split-form, Stable CG/DG-SEM for Wave Propagation Modeled by Linear Hyperbolic Systems

We present a hybrid continuous and discontinuous Galerkin spectral element approximation that leverages the advantages of each approach. The continuous Galerkin approximation is used on interior element faces where the equation properties are continuous. A discontinuous Galerkin approximation is used at physical boundaries and if there is a jump in properties at a face. The approximation uses a split form of the equations and two-point fluxes to ensure stability for unstructured quadrilateral/hexahedral meshes with curved elements. The approximation is also conservative and constant state preserving on such meshes. Spectral accuracy is obtained for all examples, which include wave scattering at a discontinuous medium boundary.

Behaviour of prefabricated steel-concrete composite slabs with a novel interlocking system–Numerical analysis

Steel-concrete composite structures have been widely used in the on-site constructions due to the benefit of composite actions provided by the steel and concrete. However, prefabricated steel–concrete composite constructions for building structures are not very popular in practice due to insufficient research studies and the lack of information on the connection systems, especially for the connection of prefabricated composite slabs. Prefabricated slabs are usually designed and constructed as simply supported and discontinuous elements. However, on-site composite slabs are continuous and connected to steel beams using shear connectors. Hence, the capacity of the composite slab constructed on-site is higher than that of conventional prefabricated slabs. Therefore, there is a significant need to develop a prefabricated continuous composite slab system to utilise the full benefit of the composite action. This study proposes interlocking connection systems for prefabricated steel–concrete composite slabs. The proposed prefabricated composite slabs with an interlocking system are analysed using finite element (FE) modelling and compared the results with the conventional slabs such as simply supported non-composite slabs, non-composite continuous slab and composite continuous slab. The ultimate load capacity of the proposed prefabricated composite slabs with the interlocking connection system has been found to be 40% higher than that of simply supported non-composite slabs and 10% lower than that of the continuous composite slab.

A Weak Form Electromagnetic Transient Simulation Approach for Cole-Cole Medium

Although the finite-difference time-domain (FDTD) method is usually used to simulate the Cole-Cole model, its relatively high requirements on the grid size and time step interval limit its application in the calculation of electrically large target. Therefore, a weak form electromagnetic transient simulation method for Cole-Cole model is proposed, which is based on the discontinuous Galerkin spectral element time-domain (DG-SETD) method. However, the implementation of the Cole-Cole model using the time-domain method is difficult because of the fractional order differentiators. The z-transform (ZT) method and second order Taylor series are used to solve this difficulty in this paper. The effectiveness of this method is shown by a numerical example.

A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Considering droplet phenomena at low Mach numbers, large differences in the magnitude of the occurring characteristic waves are presented. As acoustic phenomena often play a minor role in such applications, classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction. In this work, a novel scheme based on a specific level set ghost fluid method and an implicit-explicit (IMEX) flux splitting is proposed to overcome this timestep restriction. A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface. In this part of the domain, the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases. It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method. Applications to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.