Nodal Method Research Articles

Elastodynamic analysis of rotating solids by novel nodal position finite element method

This study develops a novel 8-node isoparametric hexahedral element using the Nodal Position Finite Element Method (NPFEM) for elastodynamic analysis of rotating solids. The element also incorporates the flexural modes directly into its element shape function to alleviate the shear locking when modeling the bending deformation of solids. Unlike conventional displacement-based finite element methods, which require the decoupling of elastic deformation from rigid-body motions, the NPFEM eliminates this process by directly representing strain and kinetic energies through nodal position coordinates, which avoids potential approximation errors in the decoupling process. To validate the accuracy and efficacy of this new NPFEM solid element, numerical simulations of a beam under static and dynamic loads are conducted and benchmarked against the theoretical solutions. Then, dynamic analysis of a rotating blade demonstrates that the NPFEM element can directly account for the centrifugal stiffening effect and superharmonic resonance of rotating blades without resorting to conventional methods. The successful implementation of this NPFEM element in complex simulations highlights its potential to provide significant advancements in computational mechanics.

3D Controlled‐source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

AbstractQuantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

Insights into calculating Reference Discontinuity Factors with Serpent Monte Carlo code

This study explores the calculation of Reference Discontinuity Factors (RDFs) using the Serpent Monte Carlo code, focusing on the methodology and potential pitfalls. In two-step reactor analyses, consistently generated RDFs are crucial for aligning homogeneous nodal diffusion results with the reference heterogeneous transport solution. However, the Serpent internal diffusion solver, based on the Analytic Function Expansion Nodal (AFEN) method, may not be compatible with other nodal methods such as the Nodal Expansion Method (NEM). Additionally, the solver can suffer from instabilities, particularly in multi-group calculations, leading to erroneous RDFs. Despite these challenges, Serpent can generate the necessary raw data for RDF calculation, which can be accurately processed using external diffusion solvers. Two numerical examples − a 1D fuel-reflector model and a 2D SMR core model − illustrate the effects of consistent and inconsistent RDFs on simulation accuracy. The study emphasizes the importance of using compatible diffusion solvers and thoroughly assessing RDFs to avoid errors in reactor simulations.

Barrel rifling node offset detection and subsequent optimization based on thin film in-mold decoration characteristics of the Johnson–Cook model

In this paper, a nodal detection method for the detection and optimization of barrel helix offsets is proposed. The barrel used in this experiment is a 6-helix barrel. Moreover, the special properties of the film of Polyetheretherketone (PEEK) material are used to cover the surface of the barrel helix with a virtual in-mold decoration (IMD) film, and the unique nature of the film die offset in the IMD is utilized to detect the position of the barrel helix. The area with a large die index is the area with a large helix offset in the barrel, and the IMD die index is introduced to quantify the data. The IMD die index is used to determine the helix offset of the damaged barrel. The novelty of this work is that each node can use the die index to efficiently detect the position of the barrel helix deviation, carry out subsequent optimization and repair work through the optimization of the injection molding parameters and the design optimization of the barrel and verify the experiment by comparing the results. Through the steady-state simulation research mode, different permutations and combinations of the two process parameters are simulated to study their effects. Quantitative reference indicators include but are not limited to dependent variables such as the fluid flow velocity, shear rate, temperature distribution and phase transition, and the shear heating process of the injection screw is taken into account in the mold flow analysis to ensure that the die index value is more accurate. It can be seen from the analysis results that the temperature of the barrel changes after the groove depth and groove width are changed, and the effect ratio of the groove depth is lower than that of the groove width in the same degree of size change.

Способи зниження складності обчислень при розрахунках потокорозподілу в мережах нестисливої рідини при фіксованих коефіцієнтах гідравлічного опору

Based on the analysis of the process of solving the problem of flow distribution calcula-tion in compressible fluid systems, which boils down to finding the root of a nonlinear system of equations using the Newton — Raphson method, methods of reducing the complexity of calculations at the stages of the iterative process are proposed. The studies belong to the class of hydraulic systems for which the total pressure change on the branch elements is a monotonic smooth function of the flow rate G. For the iterations of the Newton-Raphson method, the stages are analyzed: linearization of the nonlinear system of equations, determination of the discrepancy and error estimation, determination of cost growth and obtaining a new iterative cost value. A method and algorithm for determining the discrepancy and estimating the error with linear complexity relative to the number of branches of graph E is proposed. A method of reducing the number of unknowns in the system of flow distribution equations at the stage of forming a system of linear equations based on the use of the nodal convolution method is described. A number of variants of heuristic algorithms for determining the parameter that takes into account the rate of in-crease in costs when forming a new iterative value of costs and their analysis are present-ed.

Three-Dimensional MT Conductive Anisotropic and Magnetic Modeling Using A − ϕ Potentials Employing a Mixed Nodal and Edge-Based Element Method

Magnetotelluric (MT) sounding is a geophysical technique widely utilized in mineral resource surveys, where conductivity and magnetic permeability serve as essential physical parameters for forward modeling and inversion. However, the effects of conductive anisotropy and non-zero magnetic susceptibility are usually ignored. In this study, we present a three-dimensional (3D) MT modeling algorithm using Coulomb-gauged electromagnetic potentials, incorporating a mixed nodal and edge-based finite element method capable of simulating MT responses for conductive anisotropic and magnetic anomalies. Subsequently, the algorithm’s accuracy was validated in two steps: first, it was compared with analytical solutions for a 1D magnetic model; then, a comparison was made with previously published numerical results for a 3D generalized conductive anisotropic model. The results of two tests show that the maximum relative error is below 0.5% for both models. Furthermore, representative models were computed to comprehensively analyze the responses of MT. The findings illustrate the relationship between anisotropic parameters and electric fields and emphasize the significance of considering the impact of magnetic susceptibility in magnetite-rich regions.

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

Modelling of courtyard acoustics using the nodal discontinuous Galerkin finite element method to validate a newly developed PML

Evaluating the acoustics of courtyards has gained increasing attention from a sustainable urban design perspective, which requires numerical predictions tools for estimation of noise pollution and assessment of the influence of acoustic conditions. Accurate modeling of the acoustics of a courtyard is scarce because of the variety of factors to include: meteorological effects, building material properties, and geometry of the inner façades. We consider the nodal Discontinuous Galerkin finite element method framework in a 3D static and uniform medium and use it to model the sound propagation in a courtyard. A new and efficient Perfectly Matched Layer formulation is implemented at the open boundary of the courtyard model to absorb outgoing acoustic waves and this way limiting the size of the computational domain. The numerical results are validated against experimental measurements carried out in a scaled model of a courtyard with an elementary design to implement an accurate corresponding numerical model. The experiments were conducted in an anechoic chamber where the meteorological effects are controlled, and the courtyard facades are modelled as flat surfaces with known material properties.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

A micro-channel cooling system with two-phase looped thermosyphon for a supercritical CO2 Brayton cycle

The supercritical Carbon Dioxide (sCO2) closed Brayton cycle is a promising power generation technology, while the efficient cooling of CO2 and precise control of Compressor Inlet Temperature (CIT) is crucial for the high cycle efficiency and stable compressor operation due to the acute variation of thermophysical properties in the near-critical region. In conventional indirect cooling scheme, an intermediate single-phase water circuit is used to transfer heat from CO2 to water at the precooler, and then from water to the environment at the cooling tower, having the disadvantage of large pumping work consumption and high thermal resistance. In this work, a self-driven two-phase looped thermosyphon that significantly enhances the heat transfer by internal evaporation and condensation, is proposed to replace the water circuit. An experimental ultra-compact cooling system, consisting of a looped thermosyphon combined with micro-channel evaporator and condenser, filled with R134a coolant, is designed, fabricated, and tested. Visualized observation of the two-phase flow pattern and simultaneous measurement of the temperature, pressure and mass flow rates are conducted. A nodal analysis method is adopted, and a MATLAB code is developed for analyzing the internal fluid flow and the coupled sCO2-R134a-Air heat transfer, which is validated by experiment data. The results show that, the CO2 temperature could be accurately maintained at a specified near-critical point with a fluctuation of less than 1 K, and the average heat-releasing temperature can be reduced, while considerable pumping work, usually accounting for 2–5 % of the rated power output can be saved, thus contributing to increased cycle efficiency and system compactness.

Systematic derivation of [formula omitted] equations, its discretization using GTIN method and development of a switchable SP3 to [formula omitted] neutron transport solver

The Generalized Simplified Spherical Harmonics (GSPN) method has drawn recent interest owing to the fact that theoretically, it is equivalent to the PN and does not rely on any assumption other than piecewise homogeneity. Its level wise approach to PN offers an agreeable balance between the accuracy and computational efficiency making it a valuable mathematical tool for analyzing neutron transport problems pertaining to nuclear engineering and reactor physics. The lowest level approximation for N = 3, expressed as GSP3(0), offers reducing the complexity while still capturing the essential physics. In the current work, GSP3(0) equations are derived along with the interface and boundary conditions for linear anisotropic scattering. These equations are then discretized using the generalized transverse integration nodal method for a two-dimensional system of piecewise homogeneous rectangular regions. The discretization is done with an option kept for reducing the GSP3(0) formalism to the conventional SP3. Subsequently, a computer code has been developed to solve these discretized equations in the multigroup structure for vacuum, reflective or albedo boundaries. At present, this switchable SP3/GSP3(0) code has the capability of estimating k-eigenvalue and position dependent scalar neutron flux within a given two-dimensional rectangular geometry. The code has been verified by solving SP3, GSP3(0) and other benchmark problems from available literature. The results obtained from our code demonstrate that GSP3(0) is superior to the conventional SP3 and comparable to the recently proposed SDPN (N = 1,2,3) [M. Nazari, A. Zolfaghari, M. Abbasi, Prog. Nucl. Energy, 2023; 166, 104,933] in terms of accuracy as well as computation time.

Coupled nodal integral-immersed boundary method (NI-IBM) for simulating convection-diffusion physics

The Nodal Integral Method (NIM), which is a coarse grid numerical method, works quite efficiently on orthogonal grids. However, there are many challenges to the implementation of this method in the complex shaped domains due to its intricate discretization process. In contrast, the Immersed Boundary Method (IBM) offers the advantage of implementing boundary conditions in an arbitrarily shaped domain using rectilinear, non-conformal body grids. This work presents the NIM, coupled with the IBM methodology, to solve convection-diffusion physics in complex shaped domains. IBM is utilized to enforce the boundary effect on the non-body-conformal Cartesian grid, while the NIM serves as a numerical technique to discretize the governing equations. The study describes the detailed implementation of coupled NI-IBM (Nodal Integral-Immersed Boundary Method) for convection-diffusion physics. The validation and verification of the NI-IBM methodology is done through the comparison of the numerical results with analytical solutions of the diffusion and convection-diffusion test cases. Results demonstrate that the scheme maintains its accuracy in the coarse grid structure as well as second-order accurate. Furthermore, the numerical results produced by NI-IBM are at par or more accurate with respect to the NIM based schemes for arbitrarily shaped domains and the FV-IBM (Finite Volume-Immersed Boundary Method) method for the convection-diffusion physics. The results also indicate that even with coarse grids and higher convection, the NI-IBM method preserves its nature to produce reasonably accurate numerical results.

Advancing Fixed-Source Reactor Analysis Through OpenNode Implementation

OpenNode, a new open-source Fortran code, enables the simulation of fixed-source reactors. Powered by the nodal expansion method (NEM) and seamless Python integration, OpenNode competes with existing nuclear reactor simulation tools. The fixed-source problem plays a crucial role in radioprotection, providing flux calculations and detailed insights into heating effects and dose rates. Users can define and configure reactor models via a JSON input file, specifying critical parameters like geometry, materials, cross-section data, boundary conditions, and fixed sources. OpenNode further empowers users with a Python interface for preprocessing and postprocessing, streamlining result analysis. Distinguishing itself from conventional NEM codes, OpenNode supports three-dimensional (3D) Cartesian geometries, facilitating intricate reactor design simulations. Customization options, including mesh sizes, polynomial orders, and calculation modes, enhance precision and efficiency. In our comprehensive study, we verified OpenNode’s fixed-source mode using the 3D-IAEA reactor, comparing it with the SANM code KOMODO. The results underscore OpenNode’s exceptional accuracy in computing critical parameters, like the effective multiplication factor, power distribution, and nodal flux, within fixed-source reactors. OpenNode stands as a reliable, user-friendly tool poised to advance nuclear reactor simulations.

Research on Intelligent Ventilation System of Metal Mine Based on Real-Time Sensing Airflow Parameters with a Global Scheme

In ventilation systems of metal mines, the real-time measurement of the airflow field and a reduction in pollutants are necessary for clean environmental management and human health. However, the limited quantitative data and expensive detection technology hinder the accurate assessment of mine ventilation effectiveness and safety status. Therefore, we propose a new method for constructing a mine intelligent ventilation system with a global scheme, which can realize the intelligent prediction of unknown points in the mine ventilation system by measuring the airflow parameters of multiple known points. Firstly, the nodal wind pressure method combined with the Hardy–Cross iterative algorithm is used to solve the mine ventilation network, and the airflow parameters under normal operation and extreme working conditions are simulated, based on which an intelligent ventilation training database is established. Secondly, we compared the airflow parameter prediction ability of three different machine learning models with different neural network models based on the collected small-sample airflow field dataset of a mine roadway. Finally, the depth learning method is optimized to build the intelligent algorithm model of the mine ventilation system, and a large number of three-dimensional simulation data and field measurement data of the mine ventilation system are used to train the model repeatedly to realize the intelligent perception of air flow parameters of a metal mine ventilation network and the construction of an intelligent ventilation system. The results show that the maximum error of a single airflow measurement point is 1.24%, the maximum overall error is 3.25%, and the overall average error is 0.51%. The intelligent algorithm has a good model training effect and high precision and can meet the requirements of the research and application of this project. Through case analysis, this method can predict the airflow parameters of any position underground and realize the real-time control of mine safety.

Survival Impact of Sentinel Lymph Node Biopsy in Patients with Early-Stage Cervical Cancer.

To assess whether there were statistically significant differences in terms of overall survival (OS) and progression-free survival (PFS) between pelvic lymphadenectomy (PL) and sentinel lymph node biopsy (SLNB) alone as a nodal assessment method in patients with early-stage cervical cancer (IA1 with ILV to IB2 or IIA1 of the FIGO 2018 classification). A retrospective study was conducted among patients with early-stage cervical cancer who underwent radical surgery with pelvic lymph node assessment at La Paz University Hospital between 2005 and 2022. For nodal staging, either PL, SLNB + PL, or exclusive SLNB were performed, depending on the time period. Kaplan-Meier survival curves were compared between the PL and SLNB groups. Predictors of bilateral sentinel lymph node (SLN) detection were identified with Cox proportional hazard models. Among the 128 patients included, PL ± SLNB was performed in 79 (61.7%) patients and exclusive SLNB in 49 (38.3%) patients. There was no difference between PL and SLNB in OS (log-rank 0.0730) or PFS (log-rank 0.0189). Lower limb lymphedema (LLL) was significantly lower in the SLNB group (p = 0.001). Pelvic nodal assessment with SLNB alone did not worsen survival rates compared with the standard PL in patients with early-stage cervical cancer, and it is associated with a lower rate of LLL.

The Extended Diamond Difference - Constant Nodal Method with Decoupled Cell Iteration Scheme in Two-Dimensional Discrete Ordinate Transport Problems

Analytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems.

Development and Verification of Neutron Kinetics Parameter Calculation Program Applicable to Subcritical Systems of Long-Life Reactors

The fundamental prompt neutron delayed constant α, the effective delayed neutron fraction β eff , and the effective neutron generation time Λ eff are important for accurate measurements of keff in subcritical systems. In this paper, based on the open-source program OpenMC, a module with the function of calculating the α-eigenvalue is developed based on the k-α iterative method. Based on the nodal expansion method (NEM), an adjoint neutron flux calculation program was developed, which in turn used the idea of coupling the k-α iterative method with the NEM to obtain the neutron kinetics parameters (α, β eff , and Λ eff ) of the subcritical system and allowed for a more accurate keff under the subcritical system as compared to the traditional k-mode program. In this paper, the results of the α-eigenvalues calculated by the program are validated using the Godiva benchmark and the MUSE-4 device as computational objects. The accuracy of the results for β eff and Λ eff was verified with the KUCA (Kyoto University Critical Assembly) benchmark. The results of the keff calculations using k-mode and α-mode are compared with the ADS benchmark. The results show that the calculation program (α-mode) can obtain keff under the subcritical system more accurately, which makes up for the shortcomings of the traditional program, which cannot be calculated under the deep subcritical system. The relative error between the calculated and reference values was less than 750 pcm at deep subcriticality (keff = 0.899). The error was less than 400 pcm at the deeper subcritical system (keff = 0.952), which can satisfy the demand for neutron dynamics parameters and keff calculation in the subcritical system. The program can also continue to be upgraded in the later stage in the calculation method and other aspects.

Numerical analysis of downward progressive landslides in long natural slopes with sensitive clay

Landslides occurring in sensitive clay often result in widespread destruction, posing a significant risk to human lives and property due to the substantial decrease in undrained shear strength during deformation. Assessing the consequences of these landslides is challenging and necessitates robust numerical methods to comprehensively investigate their failure mechanisms. While studies have extensively explored upward progressive landslides in sensitive clays, understanding downward progressive cases remains limited. In this study, we utilised the nodal integration-based particle finite element method (N-PFEM) with a nonlinear strain-softening model to analyse downward progressive landslides in sensitive clay on elongated slopes, induced by surcharge loads near the crest. We focused on elucidating the underlying failure mechanisms and evaluating the effects of different soil parameters and strain-softening characteristics. The simulation results revealed the typical pattern for downward landslides, which typically start with a localised failure in proximity to the surcharge loads, followed by a combination of different types of failure mechanisms, including single flow slides, translational progressive landslides, progressive flow slides, and spread failures. Additionally, inclined shear bands occur within spread failures, often adopting distinctive ploughing patterns characterised by triangular shapes. The sensitive clay thickness at the base, the clay strength gradient, the sensitivity, and the softening rate significantly influence the failure mechanisms and the extent of diffused displacement. Remarkably, some of these effects mirror those observed in upward progressive landslides, underscoring the interconnectedness of these phenomena. This study contributes valuable insights into the complex dynamics of sensitive clay landslides, shedding light on the intricate interplay of factors governing their behaviour and progression.

Equilibrium state of axially symmetric electric solar wind sail at arbitrary sail angles

This paper studies the equilibrium state and trajectory dynamics of an axially symmetric Electric solar wind sail (E-sail) at arbitrary sail angles. The E-sail is assumed operating in a heliocentric-ecliptic orbit at approximately one astronomic unit (au) from the Sun, and experiencing various dynamic disturbances like solar wind pressure, tether tension oscillations, and centrifugal forces. The study derives analytical expressions for the E-sail’s equilibrium state and its maximal coning angle under small coning angle assumption. Subsequently, an improved propulsion model is developed for the E-sail in this equilibrium state. To assess the precision of these formulations, a high-fidelity E-sail dynamic model is constructed using the nodal position finite element method, where the tethers are modeled as two-noded tensile elements and the central spacecraft and remote units are simplified as lumped masses. Through thorough parametric analyses, this paper conclusively demonstrates that the operation of the E-sail at the equilibrium state can be achieved in accordance with the derived analytical prediction of the equilibrium state. Furthermore, the improved propulsion model is employed in trajectory analyses for a mission to reach the solar system’s boundary. The study provides valuable insights and findings and foundation for the practical application and further advancement of the E-sail technology.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.