Galerkin Formulation Research Articles

Element Free Galerkin (EFG) Formulation in Solving Kinematic Wave Equation for Hydrological Modelling

Meshfree methods such as Element Free Galerkin Method (EFG) are the cutting edge of numerical methods as it does not require user to perform meshing which often is time consuming and tedious. The method being "meshfree" for the user can theoretically generate more accurate models which follow more complex boundary conditions such as natural rivers etc. The time saving and potential of higher accuracy in modelling makes EFG formulation an interesting research topic. EFG formulation is used to solve kinematic wave equations which is detailed in this paper. Galerkin weighted residual method using EFG’s shape function parameters is used to discretize the partial differential equation (PDE) of kinematic wave equations. To discretize the equations, a forward difference scheme is used for temporal discretization and Picard direct substitution method is employed to solve the nonlinear system at each time step. The formulations are validated by making a comparison between well-established numerical techniques against actual recorded datasets. After which it is found that EFG method agrees with conventional established techniques, thus validating the formulations as useful. To further investigate the usefulness of this formulation, an optimal performance study was also conducted to determine the most optimal shape function parameters. The optimal shape function values improved the performance to where it was found that established methods such as FDM were surpassed and on par with FEM. This shows that EFG is a good alternative for hydrological modelling.

Mode-matching technique for analyzing scattering in elastic shell chambers: Applications in trifurcated waveguide systems

This study presents a comprehensive investigation into the behavior of elastic shell chambers used for noise control and vibration mitigation in engineering applications. The study focuses on the analysis of fluid–structure coupled waves in a cylindrical waveguide system with trifurcated structures and membrane discs. The mode-matching technique is employed to analyze the scattering characteristics of the elastic chamber shell. Numerical simulations and analytical techniques are used to explore the interplay between material properties, geometric configurations, and loading conditions. The Galerkin formulation is utilized to model the response of membrane bridging discs and elastic shell regions. Linear algebraic systems are solved numerically to obtain the scattering amplitudes, and the convergence of the obtained solution is assessed. The energy conservation law and transmission loss are analyzed to evaluate the accuracy and convergence behavior of the system. The numerical results provide insights into the scattering behavior of the structure, showing the influence of frequency, chamber radius, and edge conditions. The findings highlight the importance of parameter selection and boundary conditions in guiding structures and provide a basis for design optimization.

Numerical homogenization of spatial network models

We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.

A hybridizable discontinuous Galerkin formulation for the Euler–Maxwell plasma model

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler–Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations.

Enhanced domain decomposition Schwarz solution schemes for isogeometric collocation methods

Isogeometric collocation methods have been introduced as an alternative to isogeometric Galerkin formulations, aiming at improving the computational cost of simulation by reducing the cost of assembly of the corresponding matrices. However, in contrast to their Galerkin counterparts, collocation formulations result in non-symmetric matrices of much higher dimensions, for a specified level of accuracy, which require special attention when addressing large-scale simulations. Furthermore, the presence of mixed boundary conditions may lead to indefinite collocation matrices, which hinder the convergence properties of domain decomposition-based iterative solution methods of the corresponding algebraic equations. To address these shortcomings, two-level hybrid, domain decomposition-based preconditioners are proposed, related to both the basic and the enhanced collocation formulations, in conjunction with the generalized minimal residual method and Schwarz-based preconditioners. The proposed solution schemes improve the computational efficiency of the isogeometric collocation simulations and exhibit numerical scalability for an increased number of subdomains. This is attributed to the fact that the iterations are performed on the Schur complement level of the corresponding matrices, leading to stable, computationally efficient, and robust solutions of the corresponding non-symmetric algebraic equations.

The Source-Stabilized Galerkin Formulation for Linear Moving Conductor Problems With Edge Elements

The Source-Stabilized Galerkin Formulation for Linear Moving Conductor Problems With Edge Elements

Fourier analysis of membrane locking and unlocking

Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler–Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.

A face-centred finite volume approach for coupled transport phenomena and fluid flow

We present a particular derivation of the face-centred finite volume (FCFV) method and study its performance in non-linear, coupled transport problems commonly encountered in geoscientific and geotechnical applications. The FCFV method is derived from the hybridisable discontinuous Galerkin formulation, using a constant degree of approximation for the discretization of the unknowns defined on the mesh faces (edges in two dimensions). The piecewise constant degrees of freedom are determined in a global problem over the mesh skeleton. Then, the solution and its gradient are recovered at the cells centroid in a set of element-by-element independent postprocesses, both exhibiting linear convergence. The formulation of the transient advection-diffusion-reaction equation is presented in detail and the numerical analysis under challenging advective/diffusive regimes is studied. Finally, we use several numerical examples to illustrate the advantages and limitations of the FCFV method to solve problems of geoscientific and geotechnical relevance governed by the non-linear coupling between advection-diffusion-reactive transport and Stokes flow. Our results show that the FCFV method is an attractive and highly competitive alternative to other commonly used methods.

A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density

Abstract This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

AbstractIn this article, a novel weak‐form zonal Petrov–Galerkin free element method is proposed for two‐ and three‐dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong‐form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two‐ and three‐dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.

Convex optimization-based structure-preserving filter for multidimensional finite element simulations

In simulation sciences, capturing the real-world problem features as accurately as possible is desirable. Methods popular for scientific simulations such as the finite element method (FEM) and finite volume method (FVM) use piecewise polynomials to approximate various characteristics of a problem, such as the concentration profile and the temperature distribution across the domain. Polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies to obtain accurate simulations. This often entails dealing with negative values for the concentration of chemicals, exceeding a percentage value over 100, and other such problems. We consider these inconsistencies in the context of partial differential equations (PDEs). We propose an innovative filter based on convex optimization to deal with the inconsistencies observed in polynomial-based simulations. In two or three spatial dimensions, additional complexities are involved in solving the problems related to structure preservation. We present the construction and application of a structure-preserving filter with a focus on multidimensional PDEs. Methods used such as the Barycentric interpolation for polynomial evaluation at arbitrary points in the domain and an optimized root-finder to identify points of interest, improve the filter efficiency, usability, and robustness. Lastly, we present numerical experiments in 2D and 3D using discontinuous Galerkin formulation and demonstrate the filter's efficacy to preserve the desired structure. As a real-world application, implementation of the mathematical biology model involving platelet aggregation and blood coagulation has been reviewed and the issues around FEM implementation of the model are resolved by applying the proposed structure-preserving filter.

Space–time computational flow analysis: Unconventional methods and first-ever solutions

The Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method was introduced in 1990 as a moving-mesh method for computational analysis of flows with moving boundaries and interfaces (MBI), which is a wide class of problems that includes fluid–particle and fluid–structure interactions and free-surface and multi-fluid flows. The method was inspired by Thomas Hughes’s 1987 work on space–time finite element methods for elastodynamics. The original DSD/SST method is now called “ST-SUPS”, reflecting its stabilization components, which are the Streamline-Upwind/Petrov–Galerkin (SUPG) method, pioneered by Hughes, and the Pressure-Stabilizing/Petrov–Galerkin (PSPG) method, inspired by Hughes’s work on a Stokes-flow Petrov–Galerkin formulation allowing equal-order interpolations for velocity and pressure. Hughes’s work on the residual-based variational multiscale (RBVMS) method inspired the ST-VMS method, which is the VMS version of the DSD/SST. A number of special methods were introduced in connection with the core methods ST-SUPS and ST-VMS. Hughes’s work on isogeometric analysis (IGA) inspired one of those special methods, the “ST-IGA”, with IGA basis functions not only in space but also in time. The core and special ST methods enabled first-ever solutions in some of the most challenging classes of MBI problems, including particle-laden flows, spacecraft parachute fluid–structure interactions, and car and tire aerodynamics. We provide an overview of the ST methods inspired by Hughes’s work and highlight some of the first-ever solutions in these three classes of MBI problems.

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. A concise proposal on how to overcome these shortcomings has been raised during the last decade by the Discontinuous Petrov–Galerkin (DPG) methodology. However, DPG has also some limitations and difficulties: the method requires ultraweak variational formulations, obtained through a hybridization process, which is not trivial to implement at the discrete level.Our motivation is to offer a simpler alternative for the case of parametric PDEs, which can be used with any variational formulation. Indeed, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently in an online stage, for a given range of parameters. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE parameter, to the matrix of coefficients of optimal test functions (in some basis expansion) associated with that PDE parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE parameters). When solving online the resulting (compressed) Petrov–Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix–vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure is as fast as an (unstable) Galerkin approach. We illustrate our findings by means of 2D–3D Eriksson–Johnson problems, together with 2D Helmholtz equation.

Discontinuous Galerkin Integral Equation Approach for Electromagnetic Modeling of Realistic and Complex Radiating Systems

An electric field integral equation (EFIE) discontinuous Galerkin (DG) formulation for the electromagnetic radiation analysis of realistic antennas is presented. Radiation problems are known to be computationally challenging, involving complex multiscale geometries often made up of a collection of closed and open surfaces. This complexity points to the convenience of separately building and meshing parts with different characteristics, either physical or geometric. This piecewise modelling approach is also desirable in the context of design optimization processes. The proposed formulation fits perfectly into this context by enabling the assembly of independently-meshed surfaces, yet without the need to enforce mesh conformity along the junction contours. The continuity of the normal components of the equivalent current at junctions is enforced using the interior penalty (IP) concept, which is extended to multiple conducting surfaces meeting at a junction.

A modified symplectic discontinuous Galerkin method for acoustic and elastic wave simulations

Numerically solving seismic wave equations is vital to large-scale forward modeling and full waveform inversion. In this paper, a new modified symplectic discontinuous Galerkin (MSDG) method is proposed to solve the acoustic and elastic equations. The MSDG method employs a symmetric interior penalty Galerkin formulation as the space discretization. The time discretization is based on a modified symplectic partitioned Runge–Kutta scheme with minimized phase error. Thus, the MSDG method has the advantages of high accuracy, being flexible to deal with complex geometric boundaries and internal structures, and stable for long time simulations. The numerical stability conditions, dispersion and dissipation are investigated in detail for the MSDG method. To validate the method, we carry out several numerical examples for solving the acoustic and elastic wave equations in various media. The numerical results show that the MSDG method can effectively suppress the numerical dispersion and is suitable for wavefield simulations.

A novel artificial neural network-based interface coupling approach for partitioned fluid–structure interaction problems

This paper presents a novel interface technique employing Artificial Neural Networks (ANN) for efficient data transfer between incompressible fluid and deformable solid domains in a partitioned fluid–structure interaction (FSI) framework. Governing differential equations (GDE) with potential flow assumptions are solved to obtain pressure values at computational nodes in the fluid domain. Pressure values are then used to train an ANN-based model to interpolate pressure loads for application at non-collocated nodes on the solid boundary. Both finite element method (FEM) and meshfree element free Galerkin (EFG) formulations are used to calculate the solid deformation. Proposed methodology is applied to solve a two-dimensional (2D) unidirectional flow problem, consisting of a flexible cantilever beam immersed in a laminar, steady, and inviscid fluid. Overwhelming mathematical intricacies of computational solvers are avoided for the sake of simplistic interface treatment, flux transfer and associated phenomena. Dedicated partitioned solvers are developed for both solid and fluid domains, which are individually benchmarked for established problems from the literature. The presented ANN-based interpolation scheme provides an alternative to higher-order polynomial algorithms at the interface boundary. The proposed scheme is simple to employ, computationally efficient, and offers competitive accuracy as well as stability.

A simple staggered approach for comprehensive analysis of forced convection heat transfer using the improved element-free Galerkin–reduced integration penalty method to solve the fluid dynamics problem

Forced convection heat transfer problems are successfully solved in this work via a simple staggered approach based on the improved element-free Galerkin (IEFG) formulation, which involves a fluid-dynamic analysis conducted in the framework of Navier–Stokes equations written in terms of velocity via a reduced integration penalty method (RIPM). Subsequently, the velocity field achieved from the solution of the Navier–Stokes equations is used in the IEFG-based solution of the internal energy balance formulated in terms of temperature. The proposed approach is tested in the solution and analysis of forced convection heat transfer problems concerning (i) the simultaneously (thermally and fluid-dynamically) developing flow between parallel plates and, (ii) the confined recirculating flow in non-isothermal lid-driven square cavities (also including viscous dissipation effects). A comprehensive parametric analysis developed in terms of characteristic dimensionless numbers is conducted for both problems, providing reliable benchmark results that will be useful to assess the performance of numerical techniques to model forced convection heat transfer phenomena. The versatility and reliability of the proposed approach are also demonstrated in the solution of a more complex problem with curved geometry, such as the non-isothermal lid-driven semicircular cavity with a circular obstacle. The accuracy and feasibility of solving forced convection heat transfer problems via the mesh-less procedure proposed in this communication are proven by comparison with results achieved via mesh-based techniques, and also with analytical solutions available in the literature. The outcomes demonstrate that the appropriate implementation of such numerical technique allows the achievement of accurate and stable results in a straightforward and remarkably simple manner, even under markedly advection-dominated flow conditions in both thermal and fluid dynamics problems.

H-adaptation for high-order discontinuous Galerkin schemes built on local multiwavelet analysis

We develop and analyze error estimators and mesh adaptation strategies within a discontinuous Galerkin formulation. The basic idea of the study is to reduce the computational cost of the simulation by employing mesh adaptation as a better alternative to the use of uniform grids. The novelty of the study resides in the use of multiwavelets and how their remarkable properties may shed new light on driving the adaptation process. This is motivated by the fact that multiwavelets break any input apart into a hierarchy of low resolution data and subsequently finer details. Our error estimator makes use of multiwavelets’ properties while being local to the element, thereby maintaining the parallel efficiency of the solver. Early tests on the one-dimensional viscous Burgers equation have shown convincing results (García Bautista et al. [1]). This work is focused on the laminar backward-facing step configuration to assess the performance of the method in higher dimensions.

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.