A Discontinuous Galerkin Method for Simulations in Complex Domains

The aim of this work is to provide a solver for viscoelastic multi-phase flows within the Bounded Support Spectral Solver (BoSSS) currently under development at the Chair of Fluid Dynamics at the Technical University of Darmstadt. The discretisation in BoSSS consists of a high-order Discontinuous Galerkin (DG) method for single-phase flow and a high-order eXtended Discontinuous Galerkin (XDG) method for the multi-phase purpose. The solver shall be used to investigate numerically the behaviour of viscoelastic droplets. The macroscopic Oldroyd B model which is used in a wide range of applications is chosen as the constitutive model. A detailed derivation of the system of equations including the modeling principles for the Oldroyd B model is presented. A DG discretisation of the system of equations including the Local Discontinuous Galerkin (LDG) method is presented after introducing the field of the DG method. The derivation of appropriate flux functions for the constitutive equations and the extra stress tensor are one of the key derivations of this scientific work. Difficulties arising in the numerical solution of viscoelastic flow problems for higher Weissenberg numbers for different discretisation methods are due to the convection dominated, mixed hyperbolic-elliptic-parabolic nature of the system of equations. Several strategies are presented which overcome these problems and are known from the literature. A key achievement of this scientific work is the application of the LDG method, originally developed for a hyperbolic system of equations for a Newtonian fluid, on the viscoelastic system of equations which renders methods for preserving ellipticity unnecessary. Furthermore, various strategiesn to enhance and to support convergence of the solution of the DG discretised system are presented. These are the Newton method with different approaches determining the Jacobian of the system, a homotopy continuation method based on the Weissenberg number for a better initial guess for the Newton method, and a troubled cell indicator combined with an artificial diffusion approach or an adaptive mesh refinement strategy, respectively. For the completeness of this work the XDG method is presented using a sharp interface approach with a signed distance level-set function as it is implemented in BoSSS. The single-phase solver is combined with these methods and appropriate flux functions for the interface are implemented to enable multi-phase applications for viscoelastic fluid. Several numerical experiments are conducted to verify and to validate the viscoelastic singlephase solver and to show the capability of the viscoelastic multi-phase solver to simulate viscoelastic droplets. Advantages and disadvantages of the implementation and an outlook for future research can be found in the conclusion.

This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties. The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics. Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases. Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.

The purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretization of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DGdiscretization is adapted in cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection-dominated boundary-value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of cubic polynomials, the boundary condition treatment appears quite effective in handling such complex situations.

Coupling local discontinuous and continuous galerkin methods for flow problems

A reconstructed discontinuous Galerkin (RDG) method based on a Hierarchical WENO reconstruction, termed HWENO(P1P2) in this paper, is developed for computing shock waves on 3D hybrid grids. The HWENO(P1P2) method is designed not only to enhance the accuracy of the discontinuous Galerkin (DG) method but also to ensure the nonlinear stability of the RDG method. In the HWENO(P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a WENO reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities and thus ensure the nonlinear stability of the RDG method. The developed HWENO(P1P2) method is used to compute a variety of shock wave problems on 3D hybrid meshes to demonstrate its accuracy, robustness, versatility, and essentially non-oscillatory property. The numerical experiments indicate that the HWENO(P1P2) method is able to provide sharp resolution of discontinuities essentially without any spurious oscillations, and achieve the designed third-order of accuracy for smooth flows.

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for simulations of flows over variable geometries

The hydraulic conductivity of a confined aquifer is estimated using the quasi-linear geostatistical approach (QLGA), based on measurements of dependent quantities such as the hydraulic head or the arrival time of a tracer. This requires the solution of the steady-state groundwater flow and solute transport equations, which are coupled by Darcy's law. The standard Galerkin finite element method (FEM) for the flow equation combined with the streamline diffusion method (SDFEM) for the transport equation is widely used in the hydrogeologists' community. This work suggests to replace the first by the two-point flux cell-centered finite volume scheme (CCFV) and the latter by the Discontinuous Galerkin (DG) method. The convection-dominant case of solute (contaminant) transport in groundwater has always posed a special challenge to numerical schemes due to non-physical oscillations at steep fronts. The performance of the DG method is experimentally compared to the SDFEM approach with respect to numerical stability, accuracy and efficient solvability of the occurring linear systems. A novel method for the reduction of numerical under- and overshoots is presented as a very efficient alternative to local mesh refinement. The applicability and software-technical integration of the CCFV/DG combination into the large-scale inversion scheme mentioned above is realized. The high-resolution estimation of a synthetic hydraulic conductivity field in a 3-D real-world setting is simulated as a showcase on Linux high performance computing clusters. The setup in this showcase provides examples of realistic flow fields for which the solution of the convection-dominant transport problem by the streamline diffusion method fails.

The discontinuous Galerkin (dG) method was introduced by Reed and Hill [73] in 1973 for steady-state neutron transport as an hyperbolic problem. This was followed by other studies; by Bassi and Rebay [12] for the compressible Navier-Stokes equations, Cockburn and Shu [31] developed the local discontinuous Galerkin (ldG) method for advection-diffusion equations, and Peraire and Persson [69] introduced the compact discontinuous Galerkin (cdG) method. Independent of the dG methods, interior penalty (IP) methods have been developed for elliptic and parabolic problems by Douglas and Dupont [40] and Wheeler [95]. Then, in the 1980’s, Arnold et al. [6] proposed a unified classification and analysis of various kinds of dG methods. Later on, the dG methods were developed for elliptic problems [8, 24, 77] and for problems with advection [7, 14, 51, 56].

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

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.

Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows

Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method

We are interested here in the numerical modeling of time-harmonic electromagnetic wave propagation problems in irregularly shaped domains and heterogeneous media. In this context, we are naturally led to consider volume discretization methods (i.e. finite element method) as opposed to surface discretization methods (i.e. boundary element method). Most of the related existing work deals with the second order form of the time-harmonic Maxwell equations discretized by a conforming finite element method [14]. More recently, discontinuous Galerkin (DG) methods have also been considered for this purpose. While the DG method keeps almost all the advantages of a conforming finite element method (large spectrum of applications, complex geometries, etc.), the DG method has other nice properties which explain the renewed interest it gains in various domains in scientific computing: easy extension to higher order interpolation (one may increase the degree of the polynomials in the whole mesh as easily as for spectral methods and this can also be done locally), no global mass matrix to invert when solving time-domain systems of partial differential equations using an explicit time discretization scheme, easy handling of complex meshes (the mesh may be a classical conforming finite element mesh, a non-conforming one or even a mesh made of various types of elements), natural treatment of discontinuous solutions and coefficient heterogeneities and nice parallelization properties.

En esta tesis se presentan varias técnicas de solución para métodos de Galerkin discontinuo (DG, p.s.s.e.i.1) de alto orden. Si bien el objetivo principal es el desarrollo de técnicas precisas y computacionalmente eficientes para resolver las ecuaciones de Navier-Stokes de flujo compresible en 3D por medio del método Discontinuous Galerkin Spectral Element Method (DGSEM), todos los métodos desarrollados en esta tesis se pueden aplicar a otras ecuaciones de conservación no lineales y a otros esquemas DG. Las técnicas de solución acá presentadas pueden ser clasificadas en dos grupos: (i) las que están orientadas a acelarar la integración temporal, y (ii) las que mejoran la discretización espacial con adaptación-p local. El primer grupo de técnicas disminuye el tiempo de cálculo dada una discretización espacial con un número fijo de grados de libertad (NDOF, p.s.s.e.i.) mediante la mejora de los algoritmos de solución. Por ese motivo, se presta especial atención al desarrollo de algoritmos eficientes para integración temporal implícita y de métodos multimalla. También se detalla una metodología para obtener el Jacobiano analítico DG para ecuaciónes de advección-difusión. Además, se demuestra que la discretización implícita del DGSEM en nodos de Gauss-Lobatto puede ser resuelta eficientemente mediante el método static condensation. Lo anterior permite resolver el problema con sistemas lineales más pequeños y mejor condicionados, lo cual disminuye los tiempos de cálculo para órdenes moderados. El segundo grupo de técnicas consiste en modificar localmente la discretización espacial para reducir NDOF sin deteriorar la precisión del método. Para lograr este obetivo, en esta tesis se desarrolla un nuevo estimador de error para el método DGSEM p-anisitrópico basado en el método _-estimation, el cual es más preciso y requiere menos recursos computacionales que enfoques previos. Posteriormente, este estimador se usa con éxito para realizar adaptación-p anisotrópica en casos estacionaros y no estacionarios, y se demuestra que éste se puede acoplar con métodos multimalla para acelerar más las simulaciones. Adicionalmente, se desarrollan dos técnicas de adaptación-p para casos no estacionarios: un algoritmo dinámico y un algoritmo estático, que es computacionalmente más eficiente en algunos casos estadísticamente estacionarios. ----------ABSTRACT---------- In this thesis, we develop space and time solution techniques for high-order Discontinuous Galerkin (DG) methods. Even though the main focus is the accurate and computationally efficient solution of the 3D compressible Navier-Stokes equations using the Discontinuous Galerkin Spectral Element Method (DGSEM), the methods developed here can be applied to other systems of nonlinear conservation laws and other DG methods. The solution techniques can be classified into two groups: (i) improved solution techniques for temporal discretizations, and (ii) local p-adaptation techniques that enhance the spatial discretization. The first group of techniques decreases the computational time for a given spatial discretization with a fixed number of degrees of freedom (NDOF) by improving the linear and nonlinear solver algorithms. In that regard, special attention is paid to the development of efficient implicit time-integration solvers and to multigrid methods (both explicit and implicit in time). We derive a methodology to obtain the analytical Jacobian of DG discretizations of nonlinear advectiondiffusion equations. Furthermore, we show that time-implicit DGSEM discretizations on Gauss-Lobatto points can be formulated as Schur complement problems and solved using the static-condensation method. This enables one to solve smaller, better-conditioned problems, which yields shorter computation times for moderate polynomial orders. The second group of techniques modifies the spatial discretization by reducing the NDOF while maintaining similar accuracy. To do that, we develop a novel truncation error estimator that uses the _-estimation method for the p-anisotropic DGSEM. The new method is more accurate and computationally cheaper than previous approaches. The error estimator is then successfully applied to perform local anisotropic p-adaptation for steady and unsteady flows. We show that the truncation error estimator can be readily coupled with multigrid techniques to obtain further speed-ups. Moreover, we develop two p-adaptation strategies for unsteady flows. One of them is a dynamic _-based anisotropic p-adaptation method. The other one is a static _-based anisotropic p-adaptation method, which may be computationally cheaper in statistically steady flows.