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We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modeling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require fewer integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element., Applied Geophysics and Petrophysics

We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modeling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the convergence rate. The quadrature rules are designed for the specific function spaces of recently developed mass-lumped tetrahedra, which consist of standard polynomial function spaces enriched with higher-degree bubble functions. For the degree-2 mass-lumped tetrahedron, the most efficient quadrature rule seems to be an existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we construct new quadrature rules that require fewer integration points than those currently available in the literature. Several numerical examples confirm that this approach is more efficient than computing the stiffness matrix exactly and that an optimal order of convergence is maintained, even when material properties vary within the element., Applied Geophysics and Petrophysics

Spectral elements with mass lumping allow for explicit time stepping and are therefore attractive for modelling seismic wave propagation. Their formulation on rectangular elements is straighforward, but for tetrahedra only elements up to degree 3 are known. To preserve accuracy after mass lumping, these elements require additional nodes that make them computationally more expensive. Here, we propose a new, less restrictive accuracy condition for the construction of these continuous mass-lumped elements. This enables us to construct several new tetrahedral elements. The new degree-2 and degree-3 elements require 15 and 32 nodes, while the existing ones have 23 and 50 nodes per element, respectively. We also developed degree-4 tetrahedral elements with 60, 61, or 65 nodes per element. Numerical examples confirm that the various mass-lumped elements maintain the optimal order of accuracy and show that the new elements are significantly more efficient in terms of accuracy versus compute time than the existing ones., Applied Geophysics and Petrophysics

Spectral elements with mass lumping allow for explicit time stepping and are therefore attractive for modelling seismic wave propagation. Their formulation on rectangular elements is straighforward, but for tetrahedra only elements up to degree 3 are known. To preserve accuracy after mass lumping, these elements require additional nodes that make them computationally more expensive. Here, we propose a new, less restrictive accuracy condition for the construction of these continuous mass-lumped elements. This enables us to construct several new tetrahedral elements. The new degree-2 and degree-3 elements require 15 and 32 nodes, while the existing ones have 23 and 50 nodes per element, respectively. We also developed degree-4 tetrahedral elements with 60, 61, or 65 nodes per element. Numerical examples confirm that the various mass-lumped elements maintain the optimal order of accuracy and show that the new elements are significantly more efficient in terms of accuracy versus compute time than the existing ones., Applied Geophysics and Petrophysics

An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady flow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion.

A mixed continuous and discontinuous Galerkin finite element discretization has been constructed for a generalized vorticity-streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The advantages of this finite-element model are the allowance of complex shaped domains and (fixed) mesh refinement, and a (spatial) discretization preserving energy and vorticity, while the discrete enstrophy is L2-stable. Verification examples support our error estimates. The method is fully described in Bernsen et al. (2005, 2006). To illustrate our method, we therefore focus here on finite-element simulations of curved critical layers in two- dimensional vortical flows using our (dis)continuous Galerkin finite element method.

A method to numerically solve the Euler equations for fluids with general equations of state is presented. It is based on a formulation solving the conservation equations for either pressure primitive variables or entropy variables, instead of the commonly used conservation variables. We use a space-time discontinuous Galerkin finite-element discretization, which yields a highly local, potentially higher-order scheme. The algorithm is applied to test cases for compressible fluids to demonstrate its capabilities and the performance of the different variable sets.

A mixed continuous and discontinuous Galerkin finite element discretization has been constructed for a generalized vorticity-streamfunction formulation in two spatial dimensions. This formulation consists of a hyperbolic (potential) vorticity equation and a linear elliptic equation for a (transport) streamfunction. The advantages of this finite-element model are the allowance of complex shaped domains and (fixed) mesh refinement, and a (spatial) discretization preserving energy and vorticity, while the discrete enstrophy is L2-stable. Verification examples support our error estimates. The method is fully described in Bernsen et al. (2005, 2006). To illustrate our method, we therefore focus here on finite-element simulations of curved critical layers in two- dimensional vortical flows using our (dis)continuous Galerkin finite element method.

A method to numerically solve the Euler equations for fluids with general equations of state is presented. It is based on a formulation solving the conservation equations for either pressure primitive variables or entropy variables, instead of the commonly used conservation variables. We use a space-time discontinuous Galerkin finite-element discretization, which yields a highly local, potentially higher-order scheme. The algorithm is applied to test cases for compressible fluids to demonstrate its capabilities and the performance of the different variable sets.

An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady flow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion.

Electrical Engineering, Mathematics and Computer Science

Electrical Engineering, Mathematics and Computer Science