Discontinuous Finite Element Research Articles

Approximations of Energy Minimization in Cell-Induced Phase Transitions of Fibrous Biomaterials: $\Gamma$-Convergence Analysis

We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on the discontinuous Galerkin treatment of higher gradients and used succesfully in numerical simulations of experiments. We show that the discrete minimizers converge in the limit to minimizers of the continuous problem. This is achieved by employing the theory of $\Gamma$-convergence of the approximate energy functionals to the continuous model when the discretization parameter tends to zero. The analysis is involved due to the structure of numerical approximations which are defined in spaces with lower regularity than the space where the minimizers of the continuous variational problem are sought. This fact leads to the development of a new approach to $\Gamma$-convergence, appropriate for discontinuous finite element discretizations, which can be applied to quite general energy minimization problems. Furthermore, the adoption of exponential terms penalizing the interpenetration of matter requires a new framework based on Orlicz spaces for discontinuous Galerkin methods which is developed in this paper as well.

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

Adaptive configurational force‐based propagation for brittle and fatigue crack analysis

AbstractThis article presents a ‐adaptive crack propagation method for highly accurate 2D crack propagation paths which requires no a priori knowledge of the tip solution. The propagation method is designed to be simple to implement, only ‐adaptivity is required, with the propagation step size independent of the initial mesh allowing users to obtain high fidelity crack path predictions for domains containing multiple cracks propagating at different rates. The proposed method also includes a crack path derefinement scheme, where elements away from the crack tip are derefined whilst elements close to the crack tip are small so capture the fidelity of the crack path. The result is that the propagation of cracks over an increasingly larger distances has negligible increased computational effort and effect on the propagation path prediction. The linear elastic problem is solved using the hp discontinuous Galerkin symmetric interior penalty finite element method, which is post‐processed to obtain the configurational force at each tip to a user defined accuracy. Several numerical examples are used to demonstrate the accuracy, efficiency, and capability of the method. Due to the method's high accuracy crack path solutions of benchmark problems that are prolifically used in the literature are challenged.

A Full-Wave Discontinuous Galerkin Time-Domain Finite Element Method for Electromagnetic Field Mode Analysis

A Full-Wave Discontinuous Galerkin Time-Domain Finite Element Method for Electromagnetic Field Mode Analysis

A LIMITER OF DISCONTINUOUS FINITE ELEMENT METHOD FOR THERMAL RADIATION HEAT TRANSFER

Due to the irregular shape of the high-temperature equipment, the thermal radiation processes are susceptible to discontinuous and shielding effects. Achieving the high accuracy of numerical solution for such processes is still a significant challenge. In this paper, the discontinuous finite element method is developed to discretize the angle and space for the numerical solution of the radiative transfer equation. A limiter, which is based on the Barth-Jespersen limiter borrowing the hierarchical limiting strategy, is imposed for the angular and spatial domains of radiative intensity to suppress the nonphysical oscillations of numerical solutions. Several benchmark problems involving discontinuous boundary conditions or irregular geometrical systems with the participating medium are solved. Besides, by comparing numerical results with exact solutions of radiative intensity in azimuthal angle, it is verified that the limiter can effectively suppress the numerical oscillation caused by the shielding effect. Finally, the three-dimensional angular distributions of radiative intensity on space nodes are depicted. In a word, the limiter, which is specially designed for the discontinuous finite element method, is a useful technical means to solve radiative heat transfer with discontinuous boundary conditions or irregular geometrical systems.

ORDER TWO SUPERCONVERGENCE OF THE CDG METHOD FOR THE STOKES EQUATIONS ON TRIANGLE/TETRAHEDRON

<abstract> A new conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the Stokes equations. The CDG method gets its name by combining good features of both conforming finite element method and discontinuous finite element method. It has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. This new CDG method is not only stabilizer free but also has two order higher convergence rate than the optimal order. This CDG method uses discontinuous $ P_k $ element for velocity and continuous $ P_{k+1} $ element for pressure. Order two superconvergence is derived for velocity in an energy norm and the $ L^2 $ norm. The superconvergent $ P_k $ solution is lifted elementwise to a $ P_{k+2} $ velocity which converges at the optimal order. The numerical experiments confirm the theories. </abstract>

Space-Time Finite Element Method for Transient and Unconfined Seepage Flow Analysis

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium. The proposed method is based on the time discontinuous Galerkin Space-Time Finite Element Method (ST/FEM). It solves the seepage problem in the saturated region. The primary unknown in ST/FEM is piezometric pressure. Fluid velocities are derived from the pressure using Darcy's law. Further, an iterative algorithm has been proposed in this paper to implement the proposed method. In each iteration step, the computation domain is updated according to the flow velocity on the phreatic boundary. Subsequently, internal nodes are moved using the mesh moving technique to accommodate the newly updated computation domain. The mesh moving technique, which is discussed in this paper, is based on an elasticity problem. ST/FEM is employed to analyze several unconfined seepage flow problems, and results of steady state solutions are compared with those available in the literature to demonstrate the efficacy of the proposed scheme. • Moving mesh type finite element method is developed for transient and unconfined seepage flow analysis. • The proposed scheme requires 2 to 3 iterations in a time step to achieve convergence. • Elasticity equation based automatic mesh moving technique is employed to move the internal nodes. • Initial computation domain significantly affects the deformation characteristics of mesh. • Proposed iterative scheme can be employed with any type of mesh moving technique.

An asymptotic preserving method for the linear transport equation on general meshes

While many numerical methods for the linear transport equation are available in the literature in 1D or on Cartesian meshes, fewer works are dedicated to the resolution of this model on unstructured meshes. In the context of radiative hydrodynamics, we need a method capable to handle a wide range of radiation regimes going from free-streaming to diffusion and to be coupled with a Lagrangian hydrodynamics solver. In this paper we design a method based on the micro-macro paradigm and to the Discrete Ordinates (SN) angular discretization, which fulfills these requirements. It allows to choose the limit transport scheme and the limit diffusion scheme. It is compared on challenging test problems to a Discontinuous Finite Element (DFE) method.

Stability analysis of high order methods for the wave equation

In this paper, we investigate the stability of a numerical method for solving the wave equation. The method uses explicit leap-frog in time and high order continuous and discontinuous (DG) finite elements using the standard Lagrange and Hermite basis functions in space. Matrix eigenvalue analysis is used to calculate time-step restrictions. We show that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, such as equidistributed Lagrange nodes and Gauss–Lobatto nodes. We show that the time-step restriction for the symmetric interior penalty DG schemes with the usual penalty terms is tighter than for continuous Lagrange finite elements. Finally, we conclude that the best time-step restriction is obtained for continuous Hermite finite elements up to polynomial degrees p=13.

A strongly conservative finite element method for the coupled Stokes and dual-porosity model

In this work, we propose a numerical finite element discretization with strong mass conservation for the coupled Stokes and dual-porosity model. Based on divergence-conforming finite element spaces and piecewise discontinuous finite element spaces, this strongly conservative discretization is constructed by utilizing the symmetric interior penalty Galerkin method and mixed finite element method to discrete the governing equations of Stokes region and dual-porosity domain, respectively. In light of a discrete inf–sup condition, we present the well-posedness of discrete scheme and prove priori error estimates. After using uniformly matching meshes and the lower order finite element spaces of velocity and pressure, some numerical examples are given to validate the analysis of convergence and strong mass conservation. Further, these numerical results support our findings and illustrate the applicability of the coupled Stokes–dual-porosity model.

Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness

Spatial convergence study of the Lax-Friedrichs fast sweeping method based on the classical third order WENO scheme (WENO3) without a priori smoothness indicators is performed on the variants of Larsen’s benchmark problem with nonsmooth solutions. For comparison, third order upwind (UPWD3), weighted diamond difference, step method, bilinear discontinuous finite element method are also included. In optically thick regimes of the cases with continuous angular fluxes, WENO3 detects steep gradients of angular fluxes and handles them as discontinuities to suppress the unphysical oscillation, which improves the accuracy significantly compared with UPWD3. In the cases with discontinuous angular fluxes, WENO3 achieves non-oscillatory solutions with limited numerical diffusion. Nonlinear Gauss-Seidel iteration is necessary to obtain convergent solution for the nonlinearity of WENO3, which leads to low computational efficiency. The convergence behavior of nonlinear Gauss-Seidel iteration is affected by nonsmoothness of the exact solution, mesh number and the relaxation parameter.

Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

English

English

Higher Order Composite DG approximations of Gross–Pitaevskii ground state: Benchmark results and experiments

Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of the approximation algorithm for the ground state computation of a Gross–Pitaevskii equation, which is an eigenvalue problem with eigenvector nonlinearity. We will adapt the convergence results of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penalty hp-adaptive algorithm against the performance of the hp-DGCFEM.

A discontinuous finite element method based on B-spline wavelet on the interval for solving first-order neutron transport equation with discrete ordinate (SN) angular discretisation

In this paper, a discontinuous Galerkin finite element method (DGFEM) based on B-spline wavelet on the interval (BSWI) is developed and applied to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Then, using the scaling function of BSWI function as the basis function instead of the usual Lagrange basis function, the neutron transport equation based on BSWI discontinuous finite element is derived. Owing to the high precision characteristic of BSWI function, the method proposed in this paper can obtain fast convergence and statistical numerical accuracy under the condition that few spatial elements are used, and comparing with the results of the traditional discontinuous finite element method. Finally, some numerical examples are utilized to verify the accuracy and efficiency of the proposed method, and the results show that the method has high efficiency, reliable and stability, and the calculation accuracy is higher than that of the conventional discontinuous finite element method.

Coupled Electromagnetic and Hydrodynamic Modeling for Semiconductors Using DGTD

A hydrodynamic model (HDM) solver based on discontinuous Galerkin time domain finite element method (DGTD-FEM) has been developed in order to simulate the transient charge transport in semiconductors. The bipolar transport equations have been numerically solved together with the Poisson equation to realize ballistic charge transport in semiconductor devices. Furthermore, the developed solver is coupled with an FEM-based full-wave Maxwell solver in order to model the behavior of semiconductors under external electromagnetic illumination. This multiphysics coupled solver is capable of simulating transient behavior of photoactive semiconductor devices that are illuminated by externally modulated light at high frequencies. With the time domain HDM solver, the inertia effects and ballistic transport are also accounted in the accurate transient simulations of high-frequency photoactive devices.

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part II

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (Ye and Zhang (2020)). Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem.

Consistency and local conservation in finite-element simulation of flow in porous media

Computational models of subsurface flow involve the coupling between fluid velocity and transport equations. While solving these problems independently can be challenging, we expose the additional complication associated with coupling them. We focus on numerical methods that can readily accommodate unstructured meshes, with emphasis on the mixed finite-element method for the velocity-pressure formulation. The transport equation is discretized using continuous and discontinuous finite elements, and also finite volumes. We theoretically identify the discrete requirements, in space and time, to be satisfied for the constant-preserving test to pass when coupling these two discrete entities: a unit tracer concentration should be discretely preserved to machine precision under the right conditions. In our work, the constant-preserving test is referred to as the consistency test. In particular, we show that a discrete formulation that relies entirely on continuous finite elements is consistent. The overarching conclusion is that consistency is achieved when the discrete tracer equation reduces to the discrete pressure equation upon setting the tracer concentration to one. Equivalently, we conclude that consistency is achieved when coupling discrete formulations that rely on identical local-conservation properties. In explaining this result, we also make yet another attempt at debunking the myth that continuous finite elements are not locally conservative and cannot be used for flow simulation. We emphasize that continuous finite elements are locally conservative in a way that is derived from the discrete form, but not in a finite-volume sense, that is, not by computing the integral of the normal velocity on any element boundary.

Unified analysis of discontinuous Galerkin andC0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

We provide a unified analysis ofa posteriorianda priorierror bounds for a broad class of discontinuous Galerkin andC0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions inH2of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for thea priorierror analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields.

A discontinuous finite element approximation to singular Lane-Emden type equations

In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the L∞ norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.