New Crouzeix-Raviart elements of even degree: theoretical aspects, numerical performance and applications to the Stokes’ equations

We develop a new finite element method for solving planar elasticity problems involving heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the ‘broken’ Crouzeix–Raviart P 1 -nonconforming finite element method for elliptic interface problems [D.Y. Kwak, K.T. Wee and K.S. Chang, SIAM J. Numer. Anal. 48 (2010) 2117–2134]. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method [D.N. Arnold, SIAM J. Numer. Anal. 19 (1982) 742–760; D.N. Arnold and F. Brezzi, in Discontinuous Galerkin Methods. Theory, Computation and Applications , edited by B. Cockburn, G.E. Karniadakis, and C.-W. Shu. Vol. 11 of Lecture Notes in Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101; M.F. Wheeler, SIAM J. Numer. Anal. 15 (1978) 152–161.]. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace–Young condition along the interface of each element. We prove optimal H 1 and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lame parameters μ and λ and locking free as λ → ∞.

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

A novelp‐adaptive discontinuous Galerkin (DG) method has been developed to solve the Euler equations on three‐dimensional tetrahedral grids. Hierarchical orthogonal basis functions are adopted for the DG spatial discretization while a third order TVD Runge‐Kutta method is used for the time integration. A vertex‐based limiter is applied to the numerical solution in order to eliminate oscillations in the high order method. An error indicator constructed from the solution of order and is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. The developed method is implemented with under the Charm++ parallel computing framework. Charm++ is a parallel computing framework that includes various load‐balancing strategies. Implementing the numerical solver under Charm++ system provides us with access to a suite of dynamic load balancing strategies. This can be efficiently used to alleviate the load imbalances created byp‐adaptation. A number of numerical experiments are performed to demonstrate both the numerical accuracy and parallel performance of the developedp‐adaptive DG method. It is observed that the unbalanced load distribution caused by the parallelp‐adaptive DG method can be alleviated by the dynamic load balancing from Charm++ system. Due to this, high performance gain can be achieved. For the testcases studied in the current work, the parallel performance gain ranged from 1.5× to 3.7×. Therefore, the developedp‐adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method withoutp‐adaptation.

In this article, we present a new high‐order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty‐free DG. In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an augmented admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral, and weakly convex polygons) meshes as well as a three‐dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the norm and the energy seminorm.

Purpose – The purpose of this paper is to develop an efficient non-iterative model combining advanced numerical methods for solving buoyancy-driven flow problems. Design/methodology/approach – The solution strategy is based on two independent numerical procedures. The Navier-Stokes equation is solved using the non-conforming Crouzeix-Raviart (CR) finite element method with an upstream approach for the non-linear convective term. The advection-diffusion heat equation is solved using a combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods. To reduce the computational time due to the coupling, the authors use a non-iterative time stepping scheme where the time step length is controlled by the temporal truncation error. Findings – Advanced numerical methods have been successfully combined to solve buoyancy-driven flow problems on unstructured triangular meshes. The accuracy of the results has been verified using three test problems: first, a synthetic problem for which the authors developed a semi-analytical solution; second, natural convection of air in a square cavity with different Rayleigh numbers (103-108); and third, a transient natural convection problem of low Prandtl fluid with horizontal temperature gradient in a rectangular cavity. Originality/value – The proposed model is the first to combine advanced numerical methods (CR, DG, MPFA) for buoyancy-driven flow problems. It is also the first to use a non-iterative time stepping scheme based on local truncation error control for such coupled problems. The developed semi analytical solution based on Fourier series is also novel.

A dissipation-rate reserving DG method for wave catching-up phenomena in a nonlinearly elastic composite bar

Cavities are strong scattering parts for an aircraft. A discontinuous Galerkin (DG) method is presented to efficiently optimize cavity scattering. The DG method is shown to have ability of only remeshing the optimized parts and not remeshing the whole cavity in the computing process of scattering from the whole cavity. An effective and simple preconditioner, called distance sparse preconditioner (DSP), is applied to the DG method, different from the conventional block-diagonal preconditioner (BDP) and sparse approximate inverse preconditioner (SAI). The former one is usually recommended in the DG methods and the latter one is generally used in method of moment (MoM). Numerical results show that the DSP is more efficient than the BDP and SAI in the DG method for the cavity analysis. The effects of cavity opening shape, area, and transition shape are studied for scattering from cavities. Some interesting and useful numerical results are presented for cavity design.

In this paper, the characteristic basis function method is adapted to analyze the problem of scattering from multiple and multi-scale impenetrable targets in conjunction with the discontinuous Galerkin method, and the monopolar RWG functions are chosen as the basis functions. This method enables us to analyze multiple and multi-scale targets using nonconforming discretizations. The use of the CBFs helps reduce the size of the impedance matrix of the associated method of moment significantly, enabling us to employ direct solvers as opposed to iterative solvers. The process of generating the reduced matrix is naturally parallel, and the reduced matrix is well conditioned, which obviates the need for pre-conditioning. In addition, the adaptive cross-approximation algorithm is implemented to reduce the complexity of the computation. Numerical results are included to demonstrate the accuracy and efficiency of the present approach when analyzing scattering from multiple and multi-scale targets using nonconforming discretizations.

A Sequentially- Hybridized Locally Conservative Non-conforming Finite Element Scheme for Two-phase Flow Simulation through Heterogeneous Porous Media

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

ABSTRACTThe discontinuous Galerkin (DG) method is a numerical algorithm that is widely used in various fields. It has high accuracy and low numerical dispersion with advantages of easy handling boundary conditions and good parallel performance. In this study, we develop an efficient parallel weighted Runge–Kutta discontinuous Galerkin (WRKDG) method on unstructured meshes for solving 3D seismic wave equations. The DG method we use is based on the first-order formulation of a hyperbolic system with an explicit weighted Runge–Kutta time discretization. We describe the numerical scheme and parallel implementation in detail, and analyze the stability condition and numerical dispersion and dissipation. We carry out a convergence test on unstructured meshes and investigate the parallel efficiency of the implementation of the WRKDG method. The speedup curve shows that the method has good parallel performance. Finally, we present several numerical simulation examples, including acoustic and elastic wave propagations in isotropic and anisotropic media. Numerical results further verify the effectiveness of the WRKDG method in solving 3D wave propagation problems.

ABSTRACTIn the recent years, there has been an increasing interest in the analysis of finite element methods for vector‐valued flow problems on curved geometries. In this contribution, we derive an error analysis for a vector‐valued Crouzeix–Raviart element. The derivation is performed on the vector‐valued Laplace problem, which includes the symmetrical strain rate tensor, an important operator for modeling flow problems. The approximation of the strain rate tensor with the Crouzeix–Raviart element leads to oscillations in the velocity field due to a nontrivial kernel. We derive a stabilized form of the equation and present optimal error bounds in the ‐norm for the Crouzeix–Raviart finite element. The theoretical findings are supported by numerical results.

Histotripsy with ultrasound is an emerging noninvasive therapeutic modality that uses cavitation to precisely destroy diseased soft tissue. Accurate simulations of histotripsy are needed for treatment planning and device design. These simulations must model transient pressure fields, span hundreds of wavelengths, and must handle strong shocks and discontinuities between materials, such as the brain and the skull. The discontinuous Galerkin (DG) method is an outstanding candidate for such simulations. DG methods possess the following qualities: 1) high order accuracy, 2) geometric flexibility, 3) excellent dissipation properties, and 4) excellent scalability on massively parallel machines. The objective of this work is to introduce our efforts to develop a model nonlinear ultrasound simulations that are ultimately intended for histotripsy simulations in the brain. We have developed a 3D nonlinear wave equation solver using a time-explicit DG method [1]. The governing equations are expressed in first-order flux form, which models the effects of diffraction, attenuation, and nonlinearity. A Rusanov numerical flux is formulated and the eigenvalues of the flux Jacobian are calculated. A third-order, strong stability preserving Runge-Kutta (RK) time-integrator is used for time-discretization. Frequencysquared attenuation is modeled via a second-order diffusion term, which is evaluated using the local DG method. To stabilize the method and guarantee a non-oscillatory solution near shocks, a parameter-free stabilization scheme is implemented [2]. Full-wave 3D simulations are simulated for both linear and nonlinear problems. Numerical results for a planar waveguide and a pulsed rectangular piston are presented and compared to existing analytical solutions and to the FOCUS package [3]-[7]. Our nonlinear DG captures strong shocks and resolves diffraction, absorption, and nonlinear for all problems considered. An approach for coupling DG with FOCUS is proposed. These results suggest DG is a competitive method for transient biomedical acoustics simulations.

A coupled continuous-discontinuous FEM approach for convection diffusion equations