Galerkin Method Research Articles

A Weak Galerkin Method for the Coupled Darcy-Stokes Problem with a Nonstandard Transmission Condition on the Stokes Boundary

A Weak Galerkin Method for the Coupled Darcy-Stokes Problem with a Nonstandard Transmission Condition on the Stokes Boundary

Cartesian mesh adaptation: Immersed boundary method based on high-order discontinuous Galerkin method

Immersed boundary method (IBM) can easily distinguish fluid and solid regions in the computational region, thereby the workload of complex grid generation can be reduced. To accurately characterize the solid geometry, a large number of cells are required near the solid surface. The h-adaptive algorithm is adopted to reduce the requirement for the number of cells. In addition, considering the inherent adaptability to the h-adaptive Cartesian grids of the discontinuous Galerkin method, a high-order discontinuous Galerkin solver with an IBM is developed. To validate the h-adaptive algorithm and the solver, three cases are tested, including the steady flow past the National Advisory Committee for Aeronautics 0012 airfoil, the steady flow past a cylinder, and the unsteady flow past a cylinder. Compared with the non-adaptive cases, the h-adaptive cases need smaller total number of cells, and the numerical accuracy is significantly improved with an increasing degree of mesh refinement.

Nonlinear free vibration analysis of the rectangular conductive elastic plate in magnetic field based on homotopy perturbation method

AbstractThis article tries to investigate the nonlinear free vibrations of the rectangular conductive elastic plate in uniform magnetic fields under the classic plate theory considering nonlinear strain‐displacement. The formulation of the governing equations integrates the electromagnetic forces arising from the motion of the plate. The nonlinear motion equations are dimensionless, which takes the effect of in‐plane inertia into account. The equations are solved by the Galerkin method and homotopy perturbation method (HPM). The effectiveness of the solution is verified. According to the solutions by HPM, the effects of the initial conditions, length‐to‐thickness ratio, and magnetic induction intensity on the nonlinear free vibrations behavior of conductive elastic plates are discussed.

General non-linear fragmentation with discontinuous Galerkin methods

ABSTRACT Dust grains play a significant role in several astrophysical processes, including gas/dust dynamics, chemical reactions, and radiative transfer. Replenishment of small-grain populations is mainly governed by fragmentation during pair-wise collisions between grains. The wide spectrum of fragmentation outcomes, from complete disruption to erosion and/or mass transfer, can be modelled by the general non-linear fragmentation equation. Efficiently solving this equation is crucial for an accurate treatment of the dust fragmentation in numerical modelling. However, similar to dust coagulation, numerical errors in current fragmentation algorithms employed in astrophysics are dominated by the numerical overdiffusion problem – particularly in three-dimensional hydrodynamic simulations where the discrete resolution of the mass-density distribution tends to be highly limited. With this in mind, we have derived the first conservative form of the general non-linear fragmentation with a mass flux highlighting the mass transfer phenomenon. Then, to address cases of limited mass density resolution, we applied a high-order discontinuous Galerkin scheme to efficiently solve the conservative fragmentation equation with a reduced number of dust bins. An accuracy of $0.1{\!-\!}1~{{\ \rm per\ cent}}$ is reached with 20 dust bins spanning a mass range of 9 orders of magnitude.

Dynamic response of a pipe conveying pulsating fluid under random vibration

This paper studies the dynamic response of a pipe conveying pulsating fluid under random vibration. Its nonlinear equation of motion is first developed using the method of Newton. The method of Galerkin is then used to discretize the equation of motion into a system of ordinary differential equations. Afterwards, the statistical moments and mean peak values of the displacement and velocity responses of the pipe are obtained using an efficient Monte Carlo simulation (MCS) method based on the explicit time-domain method (ETDM). Furthermore, the displacement-sensitive and velocity-sensitive locations of the pipe are obtained for different boundary conditions. The impacts of the fluid, random vibration, and structural parameters on the mean peak value of the dynamic response of the pipe are finally analyzed. The obtained results provide a reference for better understanding the dynamic characteristics of pulsating fluid transport pipes under random vibration and for accurately designing pipe structures.

Superconvergent method for weakly singular Fredholm-Hammerstein integral equations with non-smooth solutions and its application

In this article, we propose shifted Jacobi spectral Galerkin method (SJSGM) and iterated SJSGM to solve nonlinear Fredholm integral equations of Hammerstein type with weakly singular kernel. We have rigorously studied convergence analysis of the proposed methods. Even though the exact solution exhibits non-smooth behaviour, we manage to achieve superconvergence order for the iterated SJSGM. Further, using smoothing transformation, we improve the regularity of the exact solution, which enhances the convergence order of the SJSGM and iterated SJSGM. We have also shown the applicability of our proposed methods to high-order nonlinear weakly singular integro-differential equations and achieved superconvergence. Several numerical examples have been implemented to demonstrate the theoretical results.

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

Effects of sinusoidal wall temperature on thermal dynamics and irreversibility around an inclined plate embedded in a square cavity

Abstract Effective thermal and flow control within complex geometries is essential for engineering applications. In this study, an in-depth examination of flow dynamics, entropy, and thermal regulation is undertaken within a square cavity featuring sinusoidal wall temperature. To introduce complexity, an inclined plate obstacle is strategically positioned within the cavity with an inclination angle of 45o, and the investigation spans three distinct scenarios: adiabatic, cold, and hot conditions. The initial physical model is developed by formulating a system of partial differential equations, which are then transformed into a dimensionless representation using relevant variables. Subsequently, the Galerkin method is employed for approximated analysis of the simplified fluid flow model, and the computational code is verified in tabular format. The embedded physical parameters are constrained to specific numerical values to ensure the convergence of the physical model in each scenario. The physical characteristics of isotherms, streamlines, Nusselt numbers, entropy, and Bejan numbers are investigated. Notably, the results demonstrate that the introduction of a cold inclined plate leads to peak values in generating the entropy and average heat transfer rates. When comparing the cold inclined plate to the heated inclined plate, an increase of approximately 20% in the average heat transfer rate and a 15% rise in the entropy generation rate was found for the cold inclined plate. Furthermore, the Bejan number showed a 10% decrease for the cold inclined plate compared to the heated inclined plate. Additionally, increasing the amplitude and wavenumber led to a rise in average heat transfer and entropy generation rates, with 25% and 30% increases, respectively.

Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes

Abstract It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

Variational consistent one-point integration with Taylor's expansion-based stabilization in the second-order meshfree Galerkin method for strain gradient elasticity

A generalized variational principle with five independent variables is proposed for strain gradient elasticity, including displacement, strain, strain gradient, stress, and double stress. Based on the principle, a one-point integration scheme is designed for the second order meshfree Galerkin method through nodal smoothed derivatives and their high order derivatives by Taylor's expansion. Since the proposed integration scheme meets the orthogonality conditions, it is variational consistent. The weak form expanded with Taylor's polynomials can be well evaluated by nodal smoothed derivatives and their high order derivatives on one quadrature point. Numerical one- and two-dimensional case studies show that the proposed integration scheme performs better than the standard Gaussian integration method in terms of accuracy, convergence, efficiency, and stability.

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

In Dzanic (2024) [12], a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

Nonlinear Vibrations of the Piezoelectric-Laminated Composite Cantilever Rectangular Plate Subjected to the Transverse and Parametric Excitation

The intelligent structure will be the key to the industrial manufacturing field in the future, piezoelectric materials are the most commonly used active materials in many mechatronic and vibration control systems. In this paper, we illustrated the nonlinear vibration behaviors of the piezoelectric-laminated composite cantilever rectangular plate subjected to transverse and parametric excitation. Two piezoelectric layers are attached to the upper and lower surfaces of graphite/epoxy composite laminated plate. The nonlinear dynamic equations are derived via the classical laminated plate theory, von Karman large deformation theory, the constitutive equation of the piezoelectric materials, and the Hamilton principle. Considering the Galerkin method, the nonlinear ordinary differential equations for the two-order vibration mode of the piezoelectric laminated composite cantilever rectangular plate are achieved. The multiple scale method is applied to obtain the average equations to study the nonlinear vibration characteristic of the stable motion. The primary resonance and the 1:1 internal resonance of the piezoelectric laminated composite cantilever rectangular plate are investigated. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase curves, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristic of the piezoelectric laminated composite cantilever rectangular plate. The detailed parametric analyses show that bubble response curves are separated from the amplitude-frequency response curves with the continuous increase of the external excitation due to the internal resonance. In addition, the results reveal the energy transform between the various order vibration modes of the system. This work is expected to provide theoretical guidance for the nonlinear vibration of the smart piezoelectric laminated composite structure.

A novel coupled p(x) and fractional PDE denoising model with theoretical results

In this paper, we formulate a new coupled PDE-based configuration for image denoising. We elaborate a new class of coupled PDE that involves a p ( x ) -Laplace and a controlled fractional-type operator, which takes into account the texture and smooth components during the recovering process. We give some essential theoretical results and we establish the well-posedness of the suggested coupled equation based on Galerkin approximation. Finally, we perform a fully discretization part of the system and illustrate some various numerical realizations to ensure the efficiency of our coupled PDE by conducting some comparison experiments against state of art PDE denoising models.

Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume corrections

We present a novel quasi-conservative arbitrary high order accurate ADER (Arbitrary-Derivative) discontinuous Galerkin method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax–Wendroff theorem. Within this formalism, we also reformulate classical results concerning smooth solutions, contact discontinuities and moving interfaces. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax–Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics.To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude the presentation by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble for which we are able to avoid the development of any spurious oscillations.

Large eddy simulation based on an improved high-precision interior penalty discontinuous Galerkin method: flow past cylinders and airfoils

Large eddy simulation based on an improved high-precision interior penalty discontinuous Galerkin method: flow past cylinders and airfoils

A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

Efficient energy stable schemes for incompressible flows with variable density

We present novel numerical schemes for the incompressible Navier-Stokes equations with variable density and address two critical concerns, i.e., the preservation of lower density bounds and the preservation of energy inequality under the gravitational force. Spatial discretization is performed using upwind discontinuous Galerkin methods for density, and continuous finite element methods are used for velocity and pressure. Additionally, velocity fields are projected onto divergence-free Raviart-Thomas finite element space in the transport equation. This relaxes using divergence-free finite elements and therefore simplifies the implementation. The proposed schemes are tested on benchmark examples to demonstrate the excellent performance in accurately capturing the complex dynamics of these problems. Overall, the proposed numerical schemes exhibit enhanced stability, efficiency, and accuracy, making them suitable for modeling and simulating incompressible flows with variable density in practical applications.

Thermal analysis of convective-radiative porous fin heat sinks made from functionally graded materials using the Galerkin method

Thermal analysis of convective-radiative porous fin heat sinks made from functionally graded materials using the Galerkin method

An efficient hybrid approach for numerical study of two-dimensional time-fractional Cattaneo model with Riesz distributed-order space-fractional operator along with stability analysis

In this article, we study and analyze the two-dimensional time-fractional Cattaneo model with Riesz space distributed-order. To obtain approximate solutions of this type of fractional model the combined and effective numerical approach based on the ADI Galerkin method and the Legendre spectral method used the ADI Galerkin numerical method uses the finite difference approach. The ADI Galerkin numerical method is used to approximate the proposed model in terms of the time variable, and the Legendre spectral method is applied to discretize the fractional model with respect to the space variable. Also, the convergence analysis and stability of the proposed method are discussed and reviewed in this manuscript. In the end, some numerical examples are tested for the effectiveness and accuracy of the proposed method. As well as, in the numerical examples section, the presented numerical approach is compared with two numerical methods and the results are reported in a table.

Oscillation-free implicit pressure explicit concentration discontinuous Galerkin methods for compressible miscible displacements with applications in viscous fingering

The system of compressible miscible displacements is widely adopted to model surfactant flooding in enhanced oil recovery techniques, where a low-viscosity fluid is injected underground to replace the high-viscosity oil. When the mobility ratio of the injected fluid to oil is high, the waterflood front tends to be unstable and exhibits a finger-like growth pattern, known as viscous fingering. Due to its unstable nature, the viscous fingering phenomenon is sensitive to mesh orientation and numerical discretization. Therefore, high-order numerical methods are preferable to reduce numerical artifacts and mesh dependence. In this paper, we propose a high-order discontinuous Galerkin method for the coupled nonlinear system of compressible miscible displacements to simulate the viscous fingering fluid instability in porous media. We adopt the IMplicit Pressure Explicit Concentration time marching approach based on implicit-explicit Runge-Kutta methods to achieve high-order temporal accuracy. Additionally, we introduce an oscillation-free damping term to control the spurious oscillations encountered in the waterflood front due to the large gradient of saturation. We have conducted ample numerical tests in two space dimensions to demonstrate the effectiveness and robustness of the proposed schemes in recovering viscous fingering.