A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.     

A high‐order hybridizable discontinuous Galerkin method for elliptic interface problems

We present a high‐order hybridizable discontinuous Galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface. The hybridizable discontinuous Galerkin method is endowed with several distinct characteristics. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the global degrees of freedom. Second, they provide, for elliptic problems with polygonal interfaces, approximations of all the variables that converge with the optimal order of k + 1 in the L²(Ω)‐norm where k denotes the polynomial order of the approximation spaces. Third, they possess some superconvergence properties that allow the use of an inexpensive element‐by‐element postprocessing to compute a new approximate solution that converges with order k + 2. However, for elliptic problems with finite jumps in the solution across the curvilinear interface, the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric. The discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence. To recover the optimal convergence for the approximate solution and gradient, we propose to use superparametric elements at the interface. Copyright © 2012 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method with plane waves for sound‐absorbing materials

Poro‐elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper, we present a discontinuous Galerkin method (DGM) with plane waves for poro‐elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave‐based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave‐based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material. Copyright © 2015 John Wiley & Sons, Ltd.     

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

A high‐order time‐domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued‐fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued‐fraction approach for bounded domains is proposed, which yields numerically more robust time‐domain formulations. The coefficient matrices of the corresponding continued‐fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high‐order time‐domain formulation for bounded domains with a high‐order transmitting boundary suggested previously is also proposed. In the time‐domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency‐independent coefficient matrices, which can be solved efficiently using standard time‐integration schemes. Numerical examples for modal and time‐domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.     

GPU‐accelerated boundary element method for Helmholtz' equation in three dimensions

Recently, the application of graphics processing units (GPUs) to scientific computations is attracting a great deal of attention, because GPUs are getting faster and more programmable. In particular, NVIDIA's GPUs called compute unified device architecture enable highly mutlithreaded parallel computing for non‐graphic applications. This paper proposes a novel way to accelerate the boundary element method (BEM) for three‐dimensional Helmholtz' equation using CUDA. Adopting the techniques for the data caching and the double–single precision floating‐point arithmetic, we implemented a GPU‐accelerated BEM program for GeForce 8‐series GPUs. The program performed 6–23 times faster than a normal BEM program, which was optimized for an Intel's quad‐core CPU, for a series of boundary value problems with 8000–128000 unknowns, and it sustained a performance of 167 Gflop/s for the largest problem (1 058 000 unknowns). The accuracy of our BEM program was almost the same as that of the regular BEM program using the double precision floating‐point arithmetic. In addition, our BEM was applicable to solve realistic problems. In conclusion, the present GPU‐accelerated BEM works rapidly and precisely for solving large‐scale boundary value problems for Helmholtz' equation. Copyright © 2009 John Wiley & Sons, Ltd.     

A conservative high‐order discontinuous Galerkin method for the shallow water equations with arbitrary topography

A conservative high‐order Godunov‐type scheme is presented for solving the balance laws of the 1D shallow water equations (SWE). The scheme adopts a finite element Runge–Kutta (RK) discontinuous Galerkin (DG) framework. Based on an overall third‐order accurate formulation, the model is referred to as RKDG3. Treatment of topographic source term is built in the DG approximation. Simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations. Owing to the adverse effects caused by using an uncontrolled global limiting process in an RKDG3 scheme (RKDG3‐GL), a new conservative RKDG3 scheme with user‐parameter‐free local limiting method (RKDG3‐LL) is designed to gain better accuracy and conservativeness. The advantages of the new RKDG3‐LL model are demonstrated by applying to several steady and transient benchmark flow tests with irregular (either differentiable or non‐differentiable) topography. Copyright © 2010 John Wiley & Sons, Ltd.     

Time‐dependent absorbing boundary conditions for elastic wave propagation

This paper develops a finite element scheme to generate the spatial‐ and time‐dependent absorbing boundary conditions for unbounded elastic‐wave problems. This scheme first calculates the spatial‐ and time‐dependent wave speed over the cosine of the direction angle using the Higdon's one‐way first‐order boundary operator, and then this operator is used again along the absorbing boundary in order to simulate the behaviour of unbounded problems. Different from other methods, the estimation of the wave speed and directions is not necessary in this method, since the wave speed over the cosine of the direction angle is calculated automatically. Two‐ and three‐dimensional numerical simulations indicate that the accuracy of this scheme is acceptable if the finite element analysis is appropriately arranged. Moreover, only the displacements along absorbing boundary nodes need to be set in this method, so the standard finite element method can still be used. Copyright © 2001 John Wiley & Sons, Ltd.     

A quasi‐static discontinuous Galerkin configurational force crack propagation method for brittle materials

This paper presents a framework for r‐adaptive quasi‐static configurational force (CF) brittle crack propagation, cast within a discontinuous Galerkin (DG) symmetric interior penalty (SIPG) finite element scheme. Cracks are propagated in discrete steps, with a staggered algorithm, along element interfaces, which align themselves with the predicted crack propagation direction. The key novelty of the work is the exploitation of the DG face stiffness terms existing along element interfaces to propagate a crack in a mesh‐independent r‐adaptive quasi‐static fashion, driven by the CF at the crack tip. This adds no new degrees of freedom to the data structure. Additionally, as DG methods have element‐specific degrees of freedom, a geometry‐driven p‐adaptive algorithm is also easily included allowing for more accurate solutions of the CF on a moving crack front. Further, for nondeterminant systems, we introduce an average boundary condition that restrains rigid body motion leading to a determinant system. To the authors' knowledge, this is the first time that such a boundary condition has been described. The proposed formulation is validated against single and multiple crack problems with single‐ and mixed‐mode cracks, demonstrating the predictive capabilities of the method.     

Higher‐order extensions of a discontinuous Galerkin method for mid‐frequency Helmholtz problems

Recently, a discontinuous Galerkin method with plane wave basis functions and Lagrange multiplier degrees of freedom was proposed for the efficient solution of Helmholtz problems in the mid‐frequency regime. In this paper, this method is extended to higher‐order elements. Performance results obtained for various two‐dimensional problems highlight the advantages of these elements over classical higher‐order Galerkin elements such as Q₂ and Q₄ for the discretization of interior and exterior Helmholtz problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Continued fraction absorbing boundary conditions for convex polygonal domains

Continued fraction absorbing boundary conditions (CFABCs) are highly effective boundary conditions for modelling wave absorption into unbounded domains. They are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries. In this paper, CFABCs are extended to the more general case of polygonal computational domains. The key to the current development is the surprising link found between the CFABCs and the complex co‐ordinate stretching of perfectly matched layers (PMLs). This link facilitates the extension of CFABCs to oblique corners and, thus, to polygonal domains. It is shown that the proposed CFABCs are easy to implement, expected to perform better than PMLs, and are effective for general polygonal computational domains. In addition to the derivation of CFABCs, a novel explicit time‐stepping scheme is developed for efficient numerical implementation. Numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain. Although this paper focuses on wave propagation, its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients. Copyright © 2005 John Wiley & Sons, Ltd.     

A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications

A discontinuous Galerkin formulation of the boundary value problem of finite‐deformation elasticity is presented. The primary purpose is to establish a discontinuous Galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the DG concept can show its superiority compared to the continuous formulation. The method is based on a general Hu–Washizu–de Veubeke functional allowing for displacement and stress discontinuities in the domain interior. It is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation. The consistency and linearized stability of the method in the non‐linear range as well as its convergence rate are proven. An implementation in three dimensions is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner. In order to demonstrate the versatility, accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

A discontinuous‐Galerkin‐based immersed boundary method

A numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced. The proposed method is conceptually simple and free of user‐defined parameters. Starting with a conforming finite element mesh, the key ingredient is to switch those elements intersected by the Dirichlet boundary to a discontinuous‐Galerkin approximation and impose the Dirichlet boundary conditions strongly. By virtue of relaxing the continuity constraint at those elements, boundary locking is avoided and optimal‐order convergence is achieved. This is shown through numerical experiments in reaction–diffusion problems. Copyright © 2008 John Wiley & Sons, Ltd.     

SPEED: SPectral Elements in Elastodynamics with Discontinuous Galerkin: a non‐conforming approach for 3D multi‐scale problems

This work presents a new high performance open‐source numerical code, namely SPectral Elements in Elastodynamics with Discontinuous Galerkin, to approach seismic wave propagation analysis in visco‐elastic heterogeneous three‐dimensional media on both local and regional scale. Based on non‐conforming high‐order techniques, such as the discontinuous Galerkin spectral approximation, along with efficient and scalable algorithms, the code allows one to deal with a non‐uniform polynomial degree distribution as well as a locally varying mesh size. Validation benchmarks are illustrated to check the accuracy, stability, and performance features of the parallel kernel, whereas illustrative examples are discussed to highlight the engineering applications of the method. The proposed method turns out to be particularly useful for a variety of earthquake engineering problems, such as modeling of dynamic soil structure and site‐city interaction effects, where accounting for multiscale wave propagation phenomena as well as sharp discontinuities in mechanical properties of the media is crucial. Copyright © 2013 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

A hybridizable discontinuous Galerkin method for two‐phase flow in heterogeneous porous media

We present a new method for simulating incompressible immiscible two‐phase flow in porous media. The semi‐implicit method decouples the wetting phase pressure and saturation equations. The equations are discretized using a hybridizable discontinuous Galerkin method. The proposed method is of high order, conserves global/local mass balance, and the number of globally coupled degrees of freedom is significantly reduced compared to standard interior penalty discontinuous Galerkin methods. Several numerical examples illustrate the accuracy and robustness of the method. These examples include verification of convergence rates by manufactured solutions, common one‐dimensional benchmarks, and realistic discontinuous permeability fields.     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical study of a semi‐algebraic multilevel preconditioner for the local discontinuous Galerkin method

In this work, we consider the local discontinuous Galerkin (LDG) method applied to second‐order elliptic problems arising in the modeling of single‐phase flows in porous media. It has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of ??(h^?2) on structured and unstructured meshes, where h is the mesh size. Thus, efficient preconditioners are mandatory. We present a semi‐algebraic multilevel preconditioner for the LDG method using local Lagrange‐type interpolatory basis functions. We show, numerically, that its performance does not degrade, or at least the number of iterations increases very slowly, as the number of unknowns augments. The preconditioner is tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media. Copyright © 2007 John Wiley & Sons, Ltd.     

A p‐adaptive method for electromagnetic wave propagation

The discontinuous Galerkin FEM is used for the numerical solution of the three‐dimensional Maxwell equations. Control of errors in the numerical level for the divergence‐free constraint of the magnetic field can be obtained through the use of divergence‐free vector bases. In this work, the so‐called perfectly hyperbolic formulation of the Maxwell equations is used to retain both divergence‐free magnetic field and in the presence of charges to satisfy the Gauss constraint for the electric field at the numerical level. For both approaches, it is found that higher‐order approximations have favorable effect on the preservation of the divergence constraints and that the perfectly hyperbolic formulations retains these errors to a lower level. It is shown that high‐order accuracy in space and time is achieved in unstructured meshes using implicit time marching. For nonuniform meshes, local resolution refinement is used using p‐type adaptivity to ensure accurate electromagnetic wave propagation. Thus, the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated.