Discontinuous Galerkin Methods

Understanding how fractures and fluids can influence elastic-wave propagation remains a complex puzzle, driving the exploration of the relationship between fluid properties and P-wave propagation through fractured media. Unraveling fluid viscosity and density from Pwave recordings still poses challenges, and the literature does not provide univocal answers. Therefore, we conduct both laboratory and numerical experiments to examine the effects of fluid viscosity and density on P-wave propagation when fractures are interpreted in terms of the linear-slip model. The medium consists of stacked aluminum discs with parallel horizontal fractures. We consider 1, 5 and 10 fractures and use water, silicone oil and honey as infill materials. In the laboratory, we obtain the static and dynamic, normal and tangential compliances of parallel fluid-filled fractures. We performed numerical simulations using the discontinuous Galerkin method, incorporating the dynamic compliances obtained from the experiments. Our laboratory findings indicate that fluid density correlates positively with Pwave velocity, transmission coefficient, and quality factor. Furthermore, there is an inverse correlation with the number of fractures. In addition, the normal and tangential fracture compliances and their ratio vary between dry and saturated conditions and decrease when the number of fractures increases. The static compliance is, in general, higher than the dynamic. The numerical results showed good agreement in discriminating between different fluids, although numerical attenuation was slightly underestimated compared to experimental observations. The results highlight the impact of fluid properties on wave behavior in fractured media and provide insights into wave sensitivity to fracture characteristics.

Optimal order a aposteriori error bounds for semilinear parabolic equation are derived by using discontinuous Galerkin method in time and continuous (conforming) finite element in space. Our main tools in this analysis the reconstruction in time and elliptic reconstruction in space with Gronwall's lemma and continuation argument.

We study space–time finite element methods for semilinear parabolic problems in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the discontinuous Galerkin timestepping method with implicit treatment of the linear terms and either implicit or explicit multistep discretisation of the zeroth order nonlinear reaction terms. Conforming finite element methods are used for the space discretisation. For this implicit-explicit IMEX–dG family of methods, we derive a posteriori and a priori energy-type error bounds and we perform extended numerical experiments. We derive a novel hp–version a posteriori error bounds in the L∞(L2) and L2(H1) norms assuming an only locally Lipschitz growth condition for the nonlinear reactions and no monotonicity of the nonlinear terms. The analysis builds upon the recent work in [60], for the respective linear problem, which is in turn based on combining the elliptic and dG reconstructions in [83, 84] and continuation argument. The a post...

The main goal of this study is to obtain the dominant frequencies and wavenumbers related to the main flow structures of a zero-net-mass-flux (ZNMF) jet. For this purpose, numerical simulations have been carried out at Reynolds Number 1000. The influence of including a cavity for the ZNMF piston is also studied. With this aim, the data obtained from the numerical simulations are analysed using higher order dynamic mode decomposition. The method has been applied in both directions, time and space. The results show that the effect of including a cavity increases the flow complexity.

The flux vector splitting (FVS) method has firstly been incorporated into the Runge-Kutta Discontinuous Galerkin (RKDG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional RKDG approaches that typically utilize the Lax-Friedrichs flux scheme or classical Riemann solvers such as HLLC. The control equations are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. Then, FVS-RKDG is implemented in two-dimensional case by splitting the normal flux on cell interfaces instead of splitting dimension by dimension in the x and y directions. Finally, the concept of “flux vector splitting based on Jacobian eigenvalue decomposition” has been applied to the conservative linear scalar transport equations and the nonlinear Burgers' equation. This approach has led to the rederivation of the classical Lax-Friedrichs flux scheme and the provision of a Steger-Warming flux scheme for scalar cases.

We use a multiwavelet basis with the Discontinuous Galerkin (DG) method to produce a multi-scale DG method. We apply this Multiwavelet DG method to convection and convection-diffusion problems in multiple dimensions. Merging the DG method with multiwavelets allows the adaptivity in the DG method to be resolved through manipulation of multiwavelet coefficients rather than grid manipulation. Additionally, the Multiwavelet DG method is tested on non-linear equations in one dimension and on the cubed sphere.

Accuracy of velocity-vorticity ( V, ω)-formulations over other formulations in solving Navier-Stokes equation has been established in recent times. However, the issue of non-satisfaction of solenoidality conditions on vorticity is not addressed in the literature which can possibly lead to non-physical solution. In this respect, here, we have developed and reported conservative rotational form of the ( V, ω)-formulation which preserves the solenoidality condition on vorticity in a much simpler way compared to other formulations. Superiority of rotational form over the conventional Laplacian form of ( V, ω)-formulation is also shown [by comparing the results for flows inside cubical lid driven cavity (LDC)]. For solving the 3D Navier-Stokes equation using a staggered grid, we use optimized compact schemes for (a) interpolation and (b) evaluation of first and second derivatives. As illustrations, we have solved problems of (i) flow inside a 3D lid driven cavity (LDC), whose solutions are compared with experimental results reported by Koseff and Street [J. Fluids Engg. 106, 390-398 (1984)] and (ii) 3D

Accuracy of velocity-vorticity ( V, ω)-formulations over other formulations in solving Navier-Stokes equation has been established in recent times. However, the issue of non-satisfaction of solenoidality conditions on vorticity is not addressed in the literature which can possibly lead to non-physical solution. In this respect, here, we have developed and reported conservative rotational form of the ( V, ω)-formulation which preserves the solenoidality condition on vorticity in a much simpler way compared to other formulations. Superiority of rotational form over the conventional Laplacian form of ( V, ω)-formulation is also shown [by comparing the results for flows inside cubical lid driven cavity (LDC)]. For solving the 3D Navier-Stokes equation using a staggered grid, we use optimized compact schemes for (a) interpolation and (b) evaluation of first and second derivatives. As illustrations, we have solved problems of (i) flow inside a 3D lid driven cavity (LDC), whose solutions are compared with experimental results reported by Koseff and Street [J. Fluids Engg. 106, 390-398 (1984)] and (ii) 3D

The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids. Global linear instability • Matrix-forming • Collocated grids • Pressure boundary conditions Communicated by Ati Sharma.

We apply the second-order Israel-Stewart theory of relativistic fluid- and thermodynamics to a physically realistic model of a radiative fluid in a simple anisotropic cosmological background. We investigate the asymptotic future of the resulting cosmological model and review the role of the dissipative phenomena in the early Universe. We demonstrate that the transport properties of the fluid alone, if described appropriately, do not explain the presently observed accelerated expansion of the Universe. Also, we show that, in constrast to the mathematical fluid models widely used before, the radiative fluid does approach local thermal equilibrium at late times, although very slowly, due to the cosmological expansion.

Ultrasonic imaging and reconstruction tools are commonly used to detect, identify and measure defects in different mechanical parts. Due to the complexity of the underlying physics, and due to the evergrowing quantity of acquired data, computation time is becoming a limitation to the optimal inspection of a mechanical part. This article presents the performances of several implementations of a computational heavy algorithm, named Total Focusing Method, on both graphics processing units (GPU) and general purpose processors (GPP). The scope of this study is narrowed to planar parts tested for defects in immersion. Using algorithmic simplifications and architectural optimizations, the algorithm has been drastically accelerated resulting in memory-bound implementations. On GPU, high performances can be achieved by profiting from GPU long cache-lines and from hand managed memory. Whereas on GPP, computations cost are overrun by memory access resulting in less efficient performances compared to the computing capabilities available. The following study constitutes the first step toward analyzing the target algorithm for diverse hardware in the nondestructive testing environment.

Periodicity can change materials properties in a very unintuitive way. Many wave propagation phenomena, such as waveguides, light bending structures or frequency filters can be modeled through finite periodic structures designed using optimization techniques. Two different kind of problems can be found: those involving linear waves and those involving nonlinear waves. The former have been widely studied and analyzed within the last few years and many interesting results have been found: cloaking devices, superlensing, fiber optics The latter is a topic of high interest nowadays and a lot of work still needs to be done, since it is far more complicated and very little is known. Nonlinear wave phenomena include acoustic amplitude filters, sound bullets or elastic shock mitigation structures, among others. The wave equation can be solved accurately using the Hybridizable Discontinuous Galerkin Method both in time and in frequency domain. Furthermore, convex optimization techniques can be used to obtain the desired material properties. Thus, the path to follow is to implement a wave phenomena simulator in 1 and 2 dimensions and then formulate specific optimization problems that will lead to materials with some particular and special properties. Within the optimization problems that can be found, there are eigenvalue optimization problems as well as more general optimal control topology optimization problems. This thesis is focused on linear phenomena. An HDG simulation code has been developed and optimization problems for the design of some model devices have also been formulated. A series of numerical results are also included showing how effective and unintuitive such designs are.

A new class of methods for propagation of uncertainty through complex models nonlinear-in-parameters is proposed. It is derived from a recent idea of propagating covariance within the unscented Kalman filter. The nonlinearity could be due to a pole-zero parametrization of a dynamic model in the Laplace domain, finite element model (FEM) or other large computer models, models of mechanical fatigue etc. Two approximate methods of this class are evaluated against Monte Carlo simulations and compared to the application of the Gauss approximation formula. Three elementary static models illustrate pros and cons of the methods, while one dynamic model provides a realistic simple example of its use.

For the model Poisson problem we propose a method combining the discontinuous Galerkin method with a mixed formulation. In the method independent and fully discontinuous basis functions are used both for the scalar unknown and its flux. The continuity requirement is imposed by Nitsche's technique . In the implementation the flux is eliminated by local condensing. We show that the method is stable and optimally convergent for all positive values of the stability parameter. We also perform an a posteriori error analysis. The theoretical results are verified by numerical computations.

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the topof-the-line one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, convolution with PSIAC filters reduces to a sequences of small inner products with local DG output and hence provides a more stable and efficient implementation. The paper focuses on the remarkable fact that, for the canonical hyperbolic test equation, new piecewise constant PSIAC filters of small support outperform the top-of-the-line boundary filters in the literature. Numerical experiments show that these least-degree filters reduce the error at the boundaries to less than the error even of the symmetric filters of the same support that are applied in the interior. Due to their simplicity, and since this least degree filter has an exact symbolic form, convolution is stable as well as efficient; and derivatives of the convolved output are easy to compute.

Convolving the output of Discontinuous Galerkin (DG) computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. We interpret these filters as instances of a general class of position-dependent spline filters that we abbreviate as PSIAC filters. These filters may have a non-uniform knot sequence and may leave out some B-splines of the sequence. For general position-dependent filters, we prove that rational knot sequences result in rational filter coefficients. We derive symbolic expressions for prototype knot sequences, typically integer sequences that may include repeated entries and corresponding B-splines, some of which may be skipped. Filters for shifted or scaled knot sequences are easily derived from these prototype filters so that a single filter can be re-used in different locations and at different scales. Moreover, the convolution itself reduces to executing a single dot product making it more stable and efficient than the existing approaches based on numerical integration. The construction is demonstrated for several established and one new boundary filter.

The paper presents a short survey of some topics related to speed control of electrical drives based on fuzzy PI controllers. In the beginning the conventional control systems of the main three motors mostly used in practice: DC motors, induction motors ...

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.

The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the Gradient Scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full H 2 -regularity or for non-linear models.

In this paper, we introduce a hybridizable discontinuous Galerkin method for Stokes flow. The method is devised by using the discontinuous Galerkin methodology to discretize a velocity-pressure-gradient formulation of the Stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization. One of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix. Moreover, by using an augmented lagrangian method, the globally coupled unknowns are further reduced to the numerical trace of the velocity only. Another important feature is that the approximations of the velocity, pressure, and gradient converge with the optimal order of k þ 1 in the L 2 -norm, when polynomials of degree k P 0 are used to represent the approximate variables. Based on the optimal convergence of the HDG method, we apply an element-by-element postprocessing scheme to obtain a new approximate velocity, which converges with order k þ 2 in the L 2 -norm for k P 1. The postprocessing performed at the element level is less expensive than the solution procedure. Numerical results are provided to assess the performance of the method.

This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approxima-tions, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L 1 norm and a very fast convergence speed, but only first order accuracy in the L ∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L ∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.

Reverse Time Migration (RTM) is one of the most widely used techniques for Seismic Imaging, but it induces very high computational cost since it is based on many successive solutions to the full wave equation. High-Order Discontinuous Galerkin Methods (DGM), coupled with High Performance Computing techniques, can be used to solve accurately this equation in complex geophysic media without increasing the computational burden. However, to fully exploit the high-order space discretization, it is necessary to use a high-order time discretization. In this work, we propose a new high order time scheme, the so-called Nabla-p scheme. This scheme does not increase the storage costs since it is a single step method and does not require the storage of auxiliary unknowns. Numerical results show that it requires less storage than the ADER scheme for a given accuracy and that it can be efficiently implemented in an RTM algorithm.

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We present a new method for simulating incompressible immiscible two-phase flow in porous media. The semi-implicit method decouples the wetting phase pressure and saturation equations. The equations are discretized using a hybridizable discontinuous Galerkin (HDG) method. The proposed method is of high order, conserves global/local mass balance, and the number of globally coupled degrees of freedom is significantly reduced compared to standard interior penalty discontinuous Galerkin methods. Several numerical examples illustrate the accuracy and robustness of the method. These examples include verification of convergence rates by manufactured solutions, common 1D benchmarks and realistic discontinuous permeability fields.

This work is devoted to the numerical solution of the three dimensional (3D) compressible Navier-Stokes system on unstructured mesh. The numerical simulation are performed using a fractional step method treating separately the convection and the diffusion parts. The discretization is done by the finite volume method for the two parts. The convective flux is computed by the Godunov method and a new scheme is presented for the treatment of the diffusive flux. The technique is illustrated by solving various numerical problems.

This work applies a finite element method (FEM) approach with periodic boundary conditions to the Navier-Stokes equations for an incompressible fluid moving inside a planar hexagonally-arranged domain. We highlight the usability of a model which decomposes the total pressure into a linear part and a periodic component to ensure the maintenance of mass flow across the domain. Besides, we apply a strategy to eliminate degrees of freedom directly inside the global matrix, so enforcing the periodicity for Dirichlet and Neumann conditions accordingly. Numerical simulations under low Reynolds number are presented for staggered and nonstaggered array configurations. Additionally, the periodic implementation is verified in contrast to the classical analytical solution of a Taylor vortex, thus providing good representation of the flow desired.

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising highaccuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising highaccuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today's CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

Financé par le Centre d'Excellence Africain en Sciences Mathématiques et Applications (CEA-SMA) Remerciements Je tiens à exprimer ma profonde reconnaissance aux Professeurs GOUDJO Aurélien et AHOUNOU Bernadin pour l'attention qu'ils ont portée à mon égard. Par leur disponibilité permanente, leurs conseils et leurs encouragements, ils m'ont apporté l'encadrement nécessaire à la conduite de mes travaux de thèse. Nos remerciements vont également à l'endroit de l'Institut de Mathématiques et de Sciences Physiques (IMSP) pour nous avoir soutenu financièrement à travers le projet CEA-SMA de la Banque Mondiale. Nous remercions le président du jury, les rapporteurs, l'examinateur et les superviseurs d'avoir acceptés de lire cette thèse. Nous remercions également le professeur Regis Duvigneau pour les remarques et suggestions faites afin de me permettre d'améliorer la qualité scientifique de cette thèse. Nous tenons à remercier aussi le Docteur HOUEDANOU Koffi Wilfrid et le Docteur ADETOLA Jamal pour leur contribution scientifique à la rédaction de mon article : An isogeometric error estimate for transport equation in 2D. Nous adressons également mes sincères remerciements aux amis du laboratoire d'analyse fonctionnel, d'analyse numérique et applications(Labo 3A) de la FAST, à ma famille et à ma belle famille, pour le soutien moral et la solidarité à mon égard. Nous terminerons en ayant une pensée tournée vers toutes ces personnes qui ont contribué de près ou de loin à la réalisation de ce projet de thèse.

We discuss a new, conservative, fully implicit 2D-3V particle-in-cell algorithm for non-radiative, electromagnetic kinetic plasma simulations, based on the Vlasov-Darwin model. 2 Unlike earlier linearly implicit PIC schemes and standard explicit PIC schemes, fully implicit PIC algorithms are unconditionally stable and allow exact discrete energy and charge conservation. This has been demonstrated in 1D electrostatic 3 and electromagnetic 4 contexts. In this study, we build on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the Darwin field and particle orbit equations for multiple species in multiple dimensions. 5 The Vlasov-Darwin model is very attractive for PIC simulations because it avoids radiative noise issues in non-radiative electromagnetic regimes. The algorithm conserves global energy, local charge, and particle canonical-momentum exactly, even with grid packing. The nonlinear iteration is effectively accelerated with a fluid preconditioner, which allows efficient use of large timesteps, O(√ m i me c v eT) larger than the explicit CFL. In this presentation, we will introduce the main algorithmic components of the approach, and demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 1D and 2D.

We analyse numerical errors (dissipation and dispersion) introduced by the discretisation of inviscid and viscous terms in energy stable discontinuous Galerkin methods. First, we analyse these methods using a linear von Neumann analysis (for a linear advection-diffusion equation) to characterise their properties in wave-number space. Second, we validate these observations using the 3D Taylor-Green Vortex Navier-Stokes problem to assess transitional/turbulent flows. We show that the dissipation introduced by upwind Riemann solvers affects primarily high wave-numbers. This dissipation may be increased, through a penalty parameter, until a critical value. However, further augmentation of this parameter leads to a decrease of dissipation, reaching zero for very large values. Regarding the dissipation introduced by second order derivatives, we show that this dissipa- tion acts at low and medium wave-numbers (lower wave-numbers compared to upwind Riemann solvers). In addition, we analyse ...

In this paper a comparison of the performance of two ways of discretizing the nonlinear convection-diffusion equation in a one-dimensional (1D) domain is performed. The two approaches can be considered within the class of high-order methods. The first one is the discontinuous Galerkin method, which has been profusely used to solve general transport equations, either coupled as the Navier-Stokes equations, or on their own. On the other hand, the ENATE procedure (Enhanced Numerical Approximation of a Transport Equation), uses the exact solution to obtain an exact algebraic equation with integral coefficients that link nodal values with a three-point stencil. This paper is the first of thorough assessments of ENATE by comparing it with well-established high-order methods. Several test cases of the steady Burgers' equation with and without source have been chosen for comparison.

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional L 2 norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term (SBP-SAT) properties (e.g. DG with Gauss-Lobatto points). Numerical experiments are included to illustrate the theoretical findings.

This paper presents comparisons of high order Discontinuous Galerkin (DG) solvers using both compressible and incompressible formulations for the solution of the Navier-Stokes equations. The main purpose of this paper is to provide qualitative and quantitative comparisons to assess the limits of accuracy of compressible solvers when computing low Reynolds and low Mach number flows. Comparisons encompass flow simulations for two test-cases included in the 2nd International Workshop on High Order Methods May 27-28, 2013 to be held in Cologne, Germany: "C1.3 Flow over the NACA0012 airfoil" and "C1.4 Flat plate Boundary Layer". Both high order codes provide h/p refinement capabilities and enable grid independent flow comparisons using either h or p refinement. Namely, drag coefficients and boundary layer profiles (e.g. displacement thickness) are compared together with theoretical estimates, when available (e.g. Blasius solution). It is shown that both compressible and incompressible formulations provide similar results even at low Reynolds numbers. However, this paper shows that the accuracy of the compressible computations is inversely proportional to both Reynolds and Mach numbers and quantifies these

Pre-placed aggregate concrete (PAC) is widely used in civil engineering construction. PAC concrete contains a higher percentage
of coarse aggregate because the coarse aggregate is deposited directly into the mold. PAC concrete has a higher compressive strength, elastic
modulus, and lower shrinkage than conventional concrete. However, PAC has low tensile strength, so it is necessary to look for materials that
can increase the tensile strength of the PAC, one of which is rebar wire fiber. The type of fiber used in this research work is straight rebar wire
fiber with a diameter (d) of 1 mm and a length (l) of 60 mm. The mortar with a cement-to-sand ratio of 1.0 and a water-to-cement ratio of 0.45
is used as grouting material for filling the voids between coarse aggregate. The compressive and tensile strength tests showed that the addition
of 1% rebar wire fiber contributed to the compressive strength and tensile strength of the pre-placed aggregate concrete (PAC). The
compressive strength and tensile strength increased by 11.23% and 38.84%, respectively.

We study the propagation of high-frequency electromagnetic waves in randomly heterogeneous bianisotropic media with dissipative properties. For that purpose we consider randomly fluctuating optical responses of such media with correlation lengths comparable to the typical wavelength of the waves. Although the fluctuations are weak, they induce multiple scattering over long propagation times and/or distances such that the waves end up traveling in many different directions with mixed polarizations. We derive the dispersion and evolution properties of the Wigner measure of the electromagnetic fields, which describes their angularly-resolved energy density in this propagation regime. The analysis starts from Maxwell's equations with general constitutive equations. We first ignore the random fluctuations of the optical response and obtain uncoupled transport equations for the components of the Wigner measure on the different propagation modes (polarizations). Then we use a multi-scale expansion of the Wigner measure to obtain the radiative transfer equations satisfied by these components when the fluctuations are no longer ignored. The radiative transfer equations are coupled through their collisional parts, which account for the scattering of waves by the random fluctuations and their possible changes in polarization. The collisional kernels describing these processes depend on the power and cross-power spectral densities of the fluctuations at the wavelength scale. The overall derivation is based on the interpretation of Wigner transforms and Wigner measures in terms of semiclassical pseudo-differential operators in their standard quantization.

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.

Explicit local time-stepping methods are derived for the time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fullly explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leapfrog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretization in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1 + t 2), where p denotes the polynomial degree, h the mesh size, and t the time step.

Schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized. For hybridizable discontinuous Galerkin (HDG) methods, this is a relatively new field of research, because HDG methods impose continuity across elements using a Robin condition, while classical Schwarz solvers use Dirichlet transmission conditions. Robin conditions are used in optimized Schwarz methods to get faster convergence compared to classical Schwarz methods, and this even without overlap, when the Robin parameter is well chosen. We present in this paper a rigorous convergence analysis of Schwarz methods for the concrete case of hybridizable interior penalty (IPH) method. We show that the penalization parameter needed for convergence of IPH leads to slow convergence of the classical additive Schwarz method, and propose a modified solver which leads to much faster convergence. Our analysis is entirely at the discrete level, and thus holds for arbitrary interfaces between two subdomains. We then generalize the method to the case of many subdomains, including cross points, and obtain a new class of preconditioners for Krylov subspace methods which exhibit better convergence properties than the classical additive Schwarz preconditioner. We illustrate our results with numerical experiments.

Computational fluid dynamics (CFD) simulations, particularly those employing high-order methods like Discontinuous Galerkin (DG) or Hybridized Discontinuous Galerkin (HDG), offer superior accuracy over lower-order methods like the standard second-order Finite Volume Method at similar degrees of freedom. These methods enable arbitrary order of accuracy on unstructured meshes and are suitable for h- and hp- mesh adaptation. Metric-field based mesh adaptation is a popular framework for unstructured mesh adaptation, providing solid mathematical background and robustness for both 2D and 3D simulations. Metric-field based mesh adaptation is implemented either through solution-based mesh adaptation or adjoint-based mesh adaptation. Solution-based adaptation focuses on minimizing the interpolation error of a solution variable. Higher-order solution derivatives are obtained in order to find the anisotropy and density of desired mesh elements. While adjoint-based adaptation focuses on minimising error in some output functional such as drag or lift. In addition to higher-order solution derivatives, adjoint-based adaptation requires an adjoint solution, in order to predict error in output functional. The necessity to get higher-order derivatives and adjoint solutions adds extra computational overhead, thereby increasing mesh adaptation costs.
This work focuses on using machine learning models to predict the adjoint-based local error estimate (elemental contribution to the error in the output functional), thus reducing the computational cost involved in solving an extra adjoint system of equations for doing the adjoint-based mesh adaptation. This study aims to streamline the approach for the advection-diffusion equation to maintain a focused and manageable scope. To achieve this goal, we propose employing ensemble models and Graph Convolutional Network (GCN) architectures. The ensemble models are chosen for their robust generalization capabilities, while GCN architectures are proposed because of their ability to incorporate spatial information. Furthermore, transfer learning enables models trained on isotropic meshes to predict errors effectively on anisotropic meshes. The required datasets for training these models are generated using an in-house HDG solver, with ensemble models trained sequentially and GCN models in parallel. The evaluation metrics include the R^2 score for ensemble models and training loss for GCN models.
Results reveal that GCN models outperform ensemble models for isotropic mesh adaptation, while ensemble models demonstrate slightly superior performance for anisotropic meshes. Despite the superior performance with isotropic mesh, GCN models face challenges with mesh coarsening compared to ensemble models. However, when paired with transfer learning, GCN models demonstrate effectiveness, leading to reduced errors compared to isotropic meshes generated using exact error. Similarly, ensemble models also benefit from transfer learning, showcasing superior performance when complemented by this technique compared to GCN models.
In conclusion, ensemble models exemplify robust machine learning approaches with minimal need for hyperparameter tuning, whereas GCN models, despite their superior performance in some cases, require extensive hyperparameter tuning. Additionally, transfer learning holds promise for more resource-intensive simulations, reducing costs associated with solving adjoint equations and facilitating the use of isotropic models for anisotropic scenarios.

(In Persian).
This work focuses on the dynamical behavior of CARBON NANOTUBES, including vibration, wave propagation, and fluid-structure interaction. The present research investigates the TRANSVERSE VIBRATION of nanofluid conveying CARBON NANOTUBES. To this end, based on the nonlocal and strain-inertia gradient continuum elasticity theories and by using rod and Euler-Bernoulli beam models, the system’s dynamical behavior is modeled and then, the governing equation of motion is solved and discretized by applying the weighted-residual GALERKIN APPROXIMATE METHOD. Moreover, the effect of considering nano-scale fluid flowing through the nanotube, the boundary conditions, the different elastic mediums, and the van der Walls interaction between the layers of multi-walled CARBON NANOTUBES on the natural frequencies, critical velocities, and stability of the system are considered. The results show that the passing fluid flow and the axially moving of nanotubes decrease the system’s natural frequencies, especially for nanotubes with large internal radii and in high fluid flow and axially moving speeds of nanotube. In addition, it is observed that the natural frequencies and stability of the system strongly depend on the small-scale parameter (nano-scale), mainly in the longitudinal vibration.

This paper investigates the capabilities of two subgrid-scale (SGS) models suitable for unstructured grids for predicting the complex flow in transitional separated bubbles. The flow over a NACA 0012 airfoil at Reynolds number Re = 5 Â 10 4 and angles of attack (AOA) AOA = 5°and 8°is here considered. The SGS models investigated are: the wall-adapting eddy viscosity model within a variational multiscale method (VMS-WALE) and the QR eddy-viscosity model. Both are well suited for large-eddy simulations (LES) in complex geometries with unstructured grids. The models are assessed and compared to the results of direct numerical simulations (DNS) on the basis of first and second order statistics. Based on the good results obtained, specially with the VMS-WALE model, challenging simulations at high Reynolds numbers and various AOA are also performed. It has been found that predictions of the lift and drag coefficients agree reasonably well with experimental data.

We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection-diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is ðp þ 1Þth order accurate and when polynomials of degree p P 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p þ 1 in the L 2-norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p þ 1 in the L 2-norm. The new approximate scalar variable is shown to converge with order p þ 2 in the L 2-norm. For the time-dependent case, the postprocessing does not need to be applied at each time-step but only at the times for which an enhanced solution is required. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method. Published by Elsevier Inc.