Efficient Quadrature Rules for Computing the Stiffness Matrices of Mass-Lumped Tetrahedral Elements for Linear Wave Problems

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 Petrophysic

Speeding up a mass-lumped tetrahedral finite-element method for wave propagation

Mass-lumped finite elements on tetrahedra offer more flexibility than their counterpart on hexahedra for the simulation of seismic wave propagation, but there is no general recipe for their construction, unlike as with hexahedra. Earlier, we found new elements up to degree 4 that have significantly less nodes than previously known elements by sharpening the accuracy criterion. A similar approach applied to numerical quadrature of the stiffness matrix provides a speed improvement in the acoustic case and an additional factor 1.5 in the isotropic elastic case. We present numerical results for a homogeneous and heterogeneous isotropic elastic test problem on a sequence of successively finer meshes and for elements of degrees 1 to 4. A comparison of their accuracy and computational efficiency shows that a scheme of degree 4 has the best performance when high accuracy is desired, but the one of degree 3 is more efficient at intermediate accuracy.Accepted Author ManuscriptApplied Geophysics and Petrophysic

New Continuous Mass-lumped Finite Elements for 3D Wave Propagation

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 Petrophysic

New mass-lumped tetrahedral elements for 3D wave propagation modelling

We present a new accuracy condition for constructing mass-lumped elements. This condition is less restrictive than the one previously used and enabled us to construct new mass-lumped tetrahedral elements for 3D wave propagation modelling. The new degree-2 and degree-3 elements require significantly fewer nodes than previous versions and mass-lumped tetrahedral elements of higher degree had not been found before. We also present a new accuracy condition for evaluating the stiffness matrix-vector product. This enabled us to obtain tailored quadrature rules for the new elements that further reduce the computational cost.Applied Geophysics and Petrophysic

New higher-order mass-lumped tetrahedral elements for wave propagation modelling

We present a new accuracy condition for the construction of continuous mass-lumped elements. This condition is less restrictive than the one currently used and enabled us to construct new mass-lumped tetrahedral elements of degrees 2 to 4. The new degree-2 and degree-3 tetrahedral elements require 15 and 32 nodes per element, respectively, while currently, these elements require 23 and 50 nodes, respectively. The new degree-4 elements require 60, 61, or 65 nodes per element. Tetrahedral elements of this degree had not been found until now. We prove that our accuracy condition results in a mass-lumped finite element method that converges with optimal order in the $L^2$-norm and energy-norm. A dispersion analysis and several numerical tests confirm that our elements maintain the optimal order of accuracy and show that the new mass-lumped tetrahedral elements are more efficient than the current ones.Applied Geophysics and Petrophysic

Preparation and X-Ray Structures of Complexes of 18-Membered Crown Ethers with Polyfunctional Guests - Urea and (O-Alkyliso) Uronium Salts

The preparation and X-ray structure determinations of six complexes of urea and (O-n-butyliso)uronium salts with crown ethers are presented. Urea forms isostructural 5:1 adducts with 18-crown-6 (1) and aza-18-crown-6 (2), in which two urea molecules are each hydrogen bonded to two neighbouring hetero atoms of the macroring. The remaining urea molecules form two-dimensional layers alternating with crown ether layers. In both complexes the macroring has theg+g+a ag−a ag−a g−g−a ag+a ag+a conformation withCi symmetry. In the solid 1:1 complex of O-n-butylisouronium picrate with 18-crown-6 (3) two types of conformations of the macroring were observed: theg+g+a ag−a ag+a ag−g−ag−a ag+a conformation with approximateCm symmetry and to a lesser extent theg+g+a ag−a ag+a g+g+a ag−a ag+a conformation with approximateC2 symmetry. Both conformations allow the guest to form three hydrogen bonds to the macrocyclic host. Three complexes of 18-crown-6 and uronium salts have been prepared and characterized by X-ray crystallography. The 1:1 complexes with uronium nitrate (4) and uronium picrate (5) both exhibit the sameC2 conformation and the same hydrogen bonding scheme as in the least occupied form of the previous complex. A 1:2 complex with uroniump-toluenesulphonate (6) has a different hydrogen bonding scheme (two hydrogen bonds per cation to neighbouring oxygen atoms of the macroring) and a different conformation of the host molecule (theag+a ag−a ag+a ag−a ag+a ag−a conformation with almostD3d symmetry). An attempt to prepare a solid uronium nitrate complex with diaza-18-crown-6 in the same way as the 18-crown-6·uronium nitrate (1:1) complex did not yield the expected result. Instead X-ray analysis revealed that the uronium ion is dissociated, resulting in the nitrate salt of the diprotonated diaza crown ether (7)

Dispersion Properties of Explicit Finite Element Methods for Wave Propagation Modelling on Tetrahedral Meshes

We analyse the dispersion properties of two types of explicit finite element methods for modelling acoustic and elastic wave propagation on tetrahedral meshes, namely mass-lumped finite element methods and symmetric interior penalty discontinuous Galerkin methods, both combined with a suitable Lax–Wendroff time integration scheme. The dispersion properties are obtained semi-analytically using standard Fourier analysis. Based on the dispersion analysis, we give an indication of which method is the most efficient for a given accuracy, how many elements per wavelength are required for a given accuracy, and how sensitive the accuracy of the method is to poorly shaped elements.Applied Geophysics and Petrophysic