Multi-hp adaptive discontinuous Galerkin methods for simplified PN approximations of 3D radiative transfer in non-gray media

In this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified approximations of radiative transfer in non-gray media

A finite volume method for scalar conservation laws with stochastic time-space dependent flux function

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions

A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem

This work presents a boundary element method formulation for the solution of the anomalous diffusion problem. By keeping the fractional time derivative as it appears in the governing differential equation of the problem, and by employing a Weighted Residuals Method approach with the steady state fundamental solution for anisotropic media playing the role of the weighting function, one obtains the boundary integral equation of the proposed formulation. The presence of a domain integral with the fractional time derivative as part of its integrand, and the evaluation of this fractional time derivative as a Caputo derivative, constitute the main feature of the formulation. The analyses of some examples, in which the numerical results are always compared with the corresponding analytical solutions, show the robustness of the formulation, as accurate results are obtained even for small values of the order of the time derivative

Enriched finite elements for initial-value problem of transverse electromagnetic waves in time domain

This paper proposes a partition of unity enrichment scheme for the solution of the electromagnetic wave equation in the time domain. A discretization scheme in time is implemented to render implicit solutions of systems of equations possible. The scheme allows for calculation of the field values at different time steps in an iterative fashion. The spatial grid is partitioned into a finite number of elements with intrinsic shape functions to form the bases of solution. Furthermore, each finite element degree of freedom is expanded into a sum of a slowly varying term and a combination of highly oscillatory functions. The combination consists of plane waves propagating in multiple directions, with a fixed frequency. This significantly reduces the number of degrees of freedom required to discretize the unknown field, without compromising on the accuracy or allowed tolerance in the errors, as compared to that of other enriched FEM approaches. Also, this considerably reduces the computational costs in terms of memory and processing time. Parametric studies, presented herein, confirm the robustness and efficiency of the proposed method and the advantages compared to another enrichment method

An enriched finite element model with q-refinement for radiative boundary layers in glass cooling

Radiative cooling in glass manufacturing is simulated using the partition of unity finite element method. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary simplified P1 approximation for the radiation in non-grey semitransparent media. To integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework. A class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls. Using an industrial electromagnetic spectrum, the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h -version finite element method. Furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries. The time integration choice is essential to implement a q -refinement procedure introduced in the current study. The enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment. The performance of the proposed method is analysed for glass annealing in two enclosures where the simplified P1 approximation solution with the partition of unity method, the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical Rosseland model

Directional enrichment functions for finite element solutions of transient anisotropic diffusion

The present study proposes a novel approach for efficiently solving an anisotropic transient diffusion problem using an enriched finite element method. We develop directional enrichment for the finite elements in the spatial discretization and a fully implicit scheme for the temporal discretization of the governing equations. Within this comprehensive framework, the proposed class of exponential functions as enrichment enhance the approximation of the finite element method by capturing the directional based behaviour of the solution. The incorporation of these enrichment functions leverages a priori knowledge about the anisotropic problem using the partition of unity technique, resulting in significantly improved approximation efficiency while retaining all the advantages of the standard finite element method. Consequently, the proposed approach yields accurate numerical solutions even on coarse meshes and with significantly fewer degrees of freedom compared to the standard finite element methods. Moreover, the choice of mesh coarseness remains independent of the anisotropy in the problem, enabling the use of the same mesh regardless of changes in the anisotropy. Using extensive numerical experiments, we consistently demonstrate the efficiency of the proposed method in attaining the desired levels of accuracy. Our approach not only provides reliable and precise solutions but also extends the capabilities of the finite element method to effectively address aspects of the heterogeneous anisotropic transient diffusion problems that were previously considered ineffective when using this method

Development and verification of a finite volume model for hydraulics over multi-layered erodible beds

The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in multi-layered beds. The governing equations consist of two components, namely an hydraulic part described by the shallow water equations and a sediment part described by the transport and suspended sediments equations. To allow exchange between layers in the bed we propose an equation of balance laws for the bed elevation. As a numerical solver we consider a modified Roe method in the finite volume framework. A well balanced discretization is also used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both near- and far-bed simulations in sediment transport. The proposed model is verified for a dam-break problem over erodible multi-layered bed and the obtained results demonstrate it abilities to resolve the vertical features in multi-layered sediment transport problems

Mixed enrichment for the finite element method in heterogeneous media

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand, different enriched finite element methods such as the partition of unity, which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work, we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context, the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials

Identifying the wavenumber for the inverse Helmholtz problem using an enriched finite element formulation

We investigate the inverse problem of identifying the wavenumber for the Helmholtz equation. The problem solution is based on measurements taken at few points from inside the computational domain or on its boundary. A novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method. Starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method. Then the secant/descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points. In the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions. The iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met. To evaluate the estimate accuracy we propose to utilize extra data points. To validate the approach and test its efficiency two wave applications with known analytical solutions are studied. The results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise. Despite the clear advantages that were previously shown in the literature for solving the forward Helmholtz problem, this work presents a first attempt to solve the inverse Helmholtz problem with an enriched finite element approach