Petrov-Galerkin Finite Element Research Articles

Reconstruction of electromagnetic parameters of the scattering from overfilled cavities using a complex-valued convolutional neural network

We reconstruct the electromagnetic parameters of the scattering from overfilled cavities by developing a complex-valued convolutional neural network (CV-CNN) method. Since the scattering field and electromagnetic parameters are usually complex values, the input, output and weight parameters of this method are set to complex values in order to capture the characteristics of the electromagnetic field and parameters more accurately. The complex-valued matrix of the input layer is composed of the scattering field intensity received by the receivers outside the overfilled cavity, which is solved by the Petrov–Galerkin finite element interface method based on uniform non-body-fitted meshes. By means of the received scattering field intensity outside the overfilled cavities, the proposed CV-CNN can accurately reconstruct the electromagnetic parameters of arbitrarily shaped overfilled cavities with homogeneous, inhomogeneous and anisotropic media. The experimental results show that the reconstruction accuracy of CV-CNN is significantly higher than real-valued convolutional neural network (RV-CNN) with the same architecture. Moreover, the appropriate optimizer and filter size have a positive influence on the reconstruction effects. Our study can provide a new development in the electromagnetic parameter reconstruction of the cavity scattering.

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.

Performance Analysis of Thermal and Surface Roughness Effect of Slider Bearings with Unsteady Fluid Film Lubricant Using Finite Element Method

The streamline upwind Petrov‐Galerkin (SUPG) finite element method was used in this study to investigate the thermal and surface roughness effects on an inclined slider bearing with an unsteady fluid film. One‐dimensional transverse and longitudinal surface roughness models were considered with the supposition that roughness is stochastic and has a Gaussian random distribution. For simplicity of numerical computation, the irregularity caused by the texture of the surface is transformed into a regular domain. The bearing performance of the combined effect is lower than the thermal and surface roughness effects of the one‐dimensional longitudinal surface roughness for all modified Reynolds numbers of nonparallel slider bearings; this means that for nonparallel (w = 0.4) between the surface roughness effect and the combined effect condition, there is a decrease of 13% in load‐carrying capacity performance and a minimal change in friction force, respectively. However, in the case of nonparallel one‐dimensional transverse type slider bearings, the bearing performance of the thermal effect is lower than the combined and surface roughness effects for all modified Reynolds numbers, where between the combined effect and the thermal effect condition, there is a reduction of 19% in load‐carrying capacity performance and 2% in friction force practically for all changed Reynolds values, respectively. Furthermore, the combined effects at various temperatures have been investigated. As a result, in both longitudinal and transverse models, in the case of the pad temperature being lower than the slider, the load‐carrying capacity performance is higher than in other cases for nonparallel slider bearings, whereas when the slider temperature is lower than the pad temperature, the drag frictional force is the leading one in both models. In general, considering surface texture and inertial effects will increase the performance of a slider. The results obtained are displayed using figures and tables.

An implicit-in-time DPG formulation of the 1D1V Vlasov-Poisson equations

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. We believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations.

A three-dimensional Petrov-Galerkin finite element interface method for solving inhomogeneous anisotropic Maxwell's equations in irregular regions

In this paper, a three-dimensional Petrov-Galerkin finite element interface (3D PGFEI) method with non-body-fitted grids is proposed for solving the inhomogeneous anisotropic time-harmonic Maxwell's equations in irregular regions. We derive the weak variational form of the anisotropic Maxwell interface problem. The irregular region is discretized by uniform tetrahedral grids while the media interface is not aligned with the grids. The level-set functions are applied to capture the discontinuity on the anisotropic media interface. Meanwhile, we construct the special basis functions according to the jump conditions across the inhomogeneous interface, where the permittivity and permeability are discontinuous three-dimensional tensors. The resulting linear system is a banded sparse matrix and demands fewer memory resources than dense matrix. Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the relative L∞(Ω) and relative L2(Ω) error norms.

A continuous hp-mesh model for discontinuous Petrov-Galerkin finite element schemes with optimal test functions

A continuous hp-mesh model for discontinuous Petrov-Galerkin finite element schemes with optimal test functions

Application of the finite element method to the multicomponent general dynamic equation of aerosols

This paper focuses on the numerical approximation of the multicomponent dynamic equation of aerosols (MCGDE), an integro-partial differential equation that describes the temporal evolution of the aerosol number distribution containing several compounds. More specifically, we compare the performance of the finite element method (FEM) and a commonly used sectional method. In addition to the standard Galerkin FEM, we introduce the Petrov–Galerkin Finite Element Method (PGFEM) to the MCGDE. These methods are compared to analytical solutions of the MCGDE and to accurate solutions given by the discrete multicomponent GDE which models the condensation process numerically at the monomer level. Results show that in cases where condensation is the dominating process, the accuracy of the FEM and PGFEM can be significantly better than the accuracy of the sectional method. They also generally achieve a given accuracy of the solution with a significantly lower number of discretization points than the sectional method. Moreover, the sparser the discretization is, the more beneficial is the use of the PGFEM, because it stabilizes the solution of condensation dominated MCGDEs. In the case of pure coagulation, all three numerical methods yield a similar accuracy.

A point-to-point convolutional neural network for reconstructing electromagnetic parameters of multiple cavities scattering with inhomogeneous anisotropic media

We propose a point-to-point convolutional neural network (CNN) model to reconstruct the electromagnetic parameters of multiple cavities filled with inhomogeneous anisotropic media. The input of the model is a grid matrix of electromagnetic field magnitude solved by Petrov–Galerkin finite element interface method, and the output is a distribution matrix of the electromagnetic parameters of the cavities. The uniform non-body-fitted mesh is set in the computational domain, which does not need to be updated repeatedly at the complex interfaces of different media. Compared with the standard finite element, this mesh reduction method effectively saves the cost of mesh generation. Level set functions are applied to capture these arbitrarily shaped interfaces. The matrix coefficients caused by the dielectric constant and permeability tensors of anisotropic media can also be handled with this method. The point-to-point CNN model effectively reconstructs the value and the distribution of electromagnetic parameters in the cavities, also predicting the shape and the position of the interfaces. Three optimized schemes are then proposed to improve reconstruction accuracy. Numerical results show that the size of the convolution kernel and the number of fully connected layers are the crucial parameters in determining interfaces and reducing the number of artefacts.

The electromagnetic scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media

We study the electromagnetic scattering problem from multiple arbitrarily shaped cavities filled with inhomogeneous anisotropic media. A local-refined Petrov-Galerkin finite element interface (PGFEI) method is developed for solving the scattering problem. By introducing the transparent boundary condition, the unbounded scattering problem can be reduced into bounded domain problem with coupled boundary conditions on the apertures of multiple cavities. The multiple cavity scattering in transverse magnetic (TM) and transverse electric (TE) polarizations are both investigated based on uniform non-body-fitted meshes. We adopt the level-set functions to capture the irregular cavity wall and the interfaces of inhomogeneous media. The special basis is constructed to satisfy the jump conditions across the interfaces of different materials further. The local-refined algorithm is proposed to enhance the accuracy of the solutions around the discontinuous interfaces. The generation of the grid points facilitates a special numbering scheme which enables a stepwise Schur complement decomposition of the discrete linear system derived by the PGFEI method. It could be an alternative method for solving the scattering problems which have similar structures. The proposed method can handle matrix coefficients caused by the permittivity and permeability tensor of anisotropic media. Numerical experiments illustrate the validity and efficiency of the method for solving the scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media in TM and TE polarization.

Thermal and Surface Roughness Effects on Performance Analysis of Slider Bearing via Petrov–Galerkin Finite Element Method

In this study, the performance of slider bearing with impacts of temperature and roughness of surface on 1D longitudinal and transverse roughness type is investigated using the Streamline Upwind Petrov–Galerkin (SUPG)-finite element method (FEM). It is considered that the roughness is stochastic and Gaussian in random distribution. It is also considered that viscosity and density depend upon temperature. The domain’s irregularity caused by surface roughness is changed into a regular domain for numerical computation purposes. To determine the capacity of load-carrying and pressure distribution, the continuity, momentum, modified Reynolds, and energy equations are decoupled and solved by the SUPG-FEM. It can be demonstrated that in the case of the longitudinal model of nonparallel slider bearing, the load-bearing bearing capacity and drag force of friction due to the case i of the combined effects are lower than those attributable to the influence of surface roughness. Nevertheless, the thermal case’s impact on the 1D transverse type capacity of load-carrying capability and drag friction force is below that of the combined and surface roughness impact instances. The reverse is true for parallel slider bearings, however, for a 1D transverse type, the load-carrying ability is not particularly impressive.

Investigation of the shear band evolution in soil-rock mixture using the assumed enhanced strain method with the meshes of improved numerical manifold method

Falling into the framework of the Petrov-Galerkin finite element formulation with embedded discontinuity, the assumed enhanced strain (AES) method could simulate the structural localization phenomenon and the associated loss of bearing capacity in a way insensitive to mesh alignment and refinement. For another hand, the improved numerical manifold method (INMM) naturally excels at dealing with heterogeneous problems such as the soil-rock mixture (SRM) slope with embedded soft soil region. Therefore, to simulate the shear band evolution in heterogeneous slope more efficiently, precisely and also mesh independently, this paper proposes to invoke the AES method with the meshes of the INMM. Beside, this paper proposes a simple and efficient method to address the constitutive integration for multi-surface plasticity induced by cutting the tensile part of the Drucker Prager model. Finally, the efficiency and accuracy of the proposed approaches are tested on some typical examples including heterogeneous block and slope.

A high-fidelity solver based on hybrid numerical methods on unstructured grids for incompressible multiphase flows

A high-fidelity solver is developed on the unstructured grids to simulate incompressible multiphase flows with free surface based on volume of fluid (VOF) method. In this framework, the velocity is defined by both volume integrated average (VIA) and point value (PV) as computational variables which are updated simultaneously by volume-average/point-value multi-moment (VPM) method at each time step. Compared with our previous work, a node-based finite element (FE) method is used instead of cell-centered finite volume (FV) method for the discretization of pressure Poisson equation, which provides a new pattern that can naturally treat with the velocity and pressure coupling. It does not require Rhie–Chow interpolation or the use of a Petrov–Galerkin finite element formulation with tunable parameters to circumvent the well-known instability conditions. Moreover, we propose a novel balanced-force formulation with FE discretization to accommodate the surface tension force, which effectively eliminates the large numerical errors associated with the non-orthogonality of unstructured grids. In VOF method, the volume fraction is defined as VIA to identify the moving interface which is updated by solving advection equation through the FV formulation. The resulting model that combines hybrid discretizations, rigorously preserves the mass and momentum conservation, substantially improves the numerical accuracy and robustness of pressure field, and significantly suppresses the spurious velocity to nearly machine error even for multiphase flows with large density ratios. Various numerical examples have been presented to demonstrate the excellent capabilities of present solver which can provide high-fidelity solutions for practical applications in presence of largely deformed interface and highly complex geometry.

A Petrov–Galerkin finite element approach for the unsteady boundary layer upper-convected rotating Maxwell fluid flow and heat transfer analysis

A numerical investigation is carried out in the present study to view heat transfer framing indifferent Cattaneo–Christov hyperbolic heat flux equation. A numerical scheme built on the Petrov–Galerkin finite element method (PG-FEM) with shooting technique is incorporated successfully to examine the flow regime. In addition, mathematical explanation of energy equation in detail is studied which is in a transient state and unmatched in the accessible literature. Pertinent unsteady parameter for boundary layer flow is analyzed for physical implication. Moreover, momentum and energy profiles are inspected with various physical effects adjacent to thermal relaxation time, rotational parameters, Prandtl and Deborah numbers. A refined numerical strategy based on the PG-FEM scheme is implemented and validated for the results. These results found strong agreement in limiting cases to Mustafa [Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. AIP AdvAIP Advances. 2015;5:047109] and with other numerical results available in existing literature as well.

Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation

Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation

Influence of manufacturing errors and misalignment on the performances of air journal bearings considering inertia effects based on SUPG finite element method

The combined influence of manufacturing errors, misalignment and inertia effects on bearing performances is studied. The extended Reynolds equation considering inertia force of compressible air film is derived and solved by streamline upwind Petrov-Galerkin finite element method (SUPG FEM). The theoretical models of manufacturing errors and misalignment are established and verified. The results show that manufacturing errors have a significant impact on bearing performances. The influence of misalignment on moment is great, while the effects on load capacity, radial stiffness and angular stiffness are relatively small. Inertia effects can reduce bearing performances except for the journal with taper error, and the influence of inertia effects on bearing performances could be ignored when the film thickness is less than 20 μm.

Finite Element Simulation of a Taylor Bubble in Two-Phase Gas-Liquid Slug Flows using Petrov-Galerkin Formulation

Petrov-Galerkin finite element scheme for systematic analysis of the dynamics of a rising Taylor bubble and general free surface flow problems is derived and implemented. The validity of the scheme is confirmed by simulating the buoyancy-driven motion of a Taylor bubble through a stagnant Newtonian liquid in a vertical pipe characterised by dimensionless inverse viscosity number and Eötvös number of magnitude 111 and 189, respectively. Comparison of the numerical results for the steady state features defining the nose, film, and bottom regions around the bubble with the experiment shows a good agreement between the numerical simulation and the experiment. The percentage deviation of the numerical computed rise velocity, equilibrium film thickness, and stabilisation length ahead of the bubble from the experimental determined values are 8.4%, 2.3%, and 9.5%, respectively.

A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations

In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks.

A space-time certified reduced basis method for quasilinear parabolic partial differential equations

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

A finite element method for the band structure computation of photonic crystals with complex scatterer geometry

In this paper, a Petrov–Galerkin finite element interface method (PGFEIM) is proposed to compute the band structures of 2D photonic crystals (PtCs) with complex scatterer geometry, which is formulated as a generalized eigenvalue problem (GEP) for given wave vectors. The key idea of this method is to choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis, and choose a special finite element basis independent of the interface to be the test function basis to remove the boundary term upon the assertion of Bloch boundary conditions. Non-body-fitting projected grid is employed to implement this approach. Both isotropic and anisotropic materials are considered and discussed for two- and three-component PtCs with square or triangular lattice. Taking advantage of the PGFEIM in dealing with sharp-edged interfaces, PtCs with various peculiar scatterer geometries are studied. Particularly, some distinctive three-component structures with triple-junction points between different materials are fabricated, and the middle frequencies of its absolute photonic band gaps are higher than conventional three-component structures. Numerical examples demonstrate that the bands move to higher or lower frequency regions, which is determined by the component materials, with the increase of number of sharp corners on the surface of entire scatterer.