High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems

New higher-order basis functions for Curvilinear finite elements

The first contribution is a fast calculation method for tetrahedral finite element matrices which is applicable to curvilinear geometries and inhomogeneous material properties. The element matrices are obtained at a low computational cost via scaled additions of universal matrices. The proposed technique is more efficient than competing approaches and provides well-defined lower and upper bounds for the required number of matrices. In the case of tetrahedral H(div) elements, a new set of basis functions is proposed for the mixed-order Nédélec space. The specialty of the functions is a high level of orthogonality which applies to arbitrary straight-sided tetrahedra. The resulting condition numbers, compared to competing bases, are significantly lower. The remaining contributions concern hexahedral elements, where a new, mixed-order serendipity element is proposed for H(curl)-conforming functions. It allows the construction of a single set of hierarchical basis functions that can also be used to span various other finite element spaces. Therefore, it is possible to use different finite element spaces within the same mesh while maintaining conformity. In the curvilinear case, a special yet versatile way of mesh refinement is proposed along with serendipity basis functions for the interpolation of the geometry. The main advantage of the proposed methods is the resulting algebraic rate of convergence in H(curl)-norm with the least possible number of unknowns.Der erste Beitrag ist eine schnelle Berechnungsmethode von Finite-Elemente-Matrizen für Tetraeder, die auf krummlinige Geometrien und inhomogene Materialeigenschaften anwendbar ist. Die Elementmatrizen werden mit geringem Rechenaufwand durch skalierte Addition vorgefertigter Matrizen erstellt. Die vorgeschlagene Methode ist effizienter als vergleichbare Ansätze und liefert wohldefinierte obere und untere Schranken für die Anzahl der benötigten Matrizen. Für H(div)-konforme Elemente auf Tetraedern werden neue Ansatzfunktionen für den N´ed´elec-Raum gemischter Ordnung vorgestellt. Die Besonderheit dieser Funktionen ist ein hohes Maß an Orthogonalität für beliebige geradlinige Tetraeder. Im Vergleich zu anderen Ansatzfunktionen sind die resultierenden Konditionszahlen deutlich kleiner. Die übrigen Beiträge betreffen Hexaeder, für die ein neues Serentipity-Element gemischter Ordnung vorgestellt wird. Es ermöglicht die Konstruktion hierarchischer Ansatzfunktionen, die auch zum Aufspannen anderer Finite-Elemente-Räume angewandt werden kann. Daher ist es möglich, verschiedene Finite-Elemente-Räume auf dem gleichen Netz zu verwenden und dabei Konformität zu bewahren. Für den krummlinigen Fall wird eine spezielle aber vielseitige Methode zur Netzverfeinerung mit Serentipity-Ansatzfunktionen zur Interpolation der Geometrie vorgestellt. Der Hauptvorteil der vorgestellten Methoden ist die algebraische Konvergenz in der Norm des H(rot) mit der kleinstmöglichen Anzahl an Unbekannten

Applications of Special Functions in High Order Finite Element Methods

In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

Numerical analysis of a fluid droplet subject to acoustic waves

Efficient and rigorous acoustic solvers that enable high frequency sweep application over a wide range of frequencies are of great interest due to their practical importance in many engineering, physical problems or life science research that involve acoustic radiation, such as engine noise analysis, acoustic simulation in micro-fluidics and the design of lab device, etc. There is room for reduction of cost on experimental systems that can be investigated and optimised through numerical modelling of physical processes on the micro-scale level. The major difficulty that arises is the inconsistency of materials, time scales and fast oscillation nature of the solution that leads to unstable results for conventional numerical methods. However, analytical solutions are infeasible for large problems with complex geometries and sophisticated boundary conditions. Hence, the vital need for efficient solvers. In this research the development of computational methods for acoustic application is presented. The proposed method is applied to the study of propagating waves in particular to simulate acoustic phenomena in micro-droplet actuated by leaky Surface Acoustic Waves on a lithium niobate (LiNbO3) substrate. Explicitly, we introduce a new computational method for the analysis of fluids subjected to high frequency mechanical forcing. Here we solve the Helmholtz equation in the frequency domain, applying higher order Lobatto hierarchical finite element approximation in H1 space, where both pressure field and geometry are independently approximated with arbitrary and heterogeneous polynomial order. Meanwhile, a time dependent acoustic solver with arbitrary input signals is also proposed and implemented. The development of extended computational methods for the solution of the Helmholtz equation with polychromatic waves is presented, where Fourier transformation is applied to switch the incident wave and solution space from the frequency domain to the temporal domain. Consequently, the implementation and convergence rate of the numerical methods are demonstrated with benchmark problems. The numerical method is an extension of the conventional higher order finite element method and as such it relies on the definition of basis functions. In this work we implement a set of basis functions using integrated Legendre polynomials (Lobatto polynomial). Two type of basis functions are presented and compared. Therefore, the significant improvements in efficiency is demonstrated using a Lobatto hierarchical basis compared with a Legendre type basis. Moreover, a novel error estimation and automatic adaptivity scheme is outlined based on an existing a priori error estimator. The accuracy and efficiency of the proposed object oriented (predefined error level) a priori error estimator is validated through numerical assessments on a three-dimensional spherical problem and compared with uniformly h and p adaptivities. The simple and generic features of the proposed scheme allow fast frequency sweeps with low computational cost for multiple frequencies acoustic application. The current finite element approach is executed in parallel with pre-partitioned domain, which guarantees the optimal computational speed with minimal computational effort for large problems. Overall, the benefits of using the proposed acoustic solver is explained in detail. Finally, we illustrate the model's performance using an example of a micro-droplet actuated by a surface acoustic wave (SAW), which has vast applications in micro-fluidics and micro-rheology at high frequency. Conclusions are drawn, and future directions are pointed out. The proposed finite element technology is implemented in the University of Glasgow in-house open-source finite element parallel computational code, MoFEM (Mesh Oriented Finite Element Method). All algorithms and examples are publicly available for download and testing

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal