Ultraconvergence of ZZ Patch Recovery at Mesh Symmetry Points

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated

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

Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n+1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called `serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform `a priori' error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.Comment: 24 page

Adaptive grid-design methods for finite element analysis

This paper is concerned with an introduction of a concept of adaptive grid design for finite element analysis by combining numerical grid-generation methods and adaptive finite element methods. Development of a finite model is considered as a design problem similar to structural optimization problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26208/1/0000288.pd

Introduction to the finite element method

2019/202

On 3-D inelastic analysis methods for hot section components (base program)

A 3-D Inelastic Analysis Method program is described. This program consists of a series of new computer codes embodying a progression of mathematical models (mechanics of materials, special finite element, boundary element) for streamlined analysis of: (1) combustor liners, (2) turbine blades, and (3) turbine vanes. These models address the effects of high temperatures and thermal/mechanical loadings on the local (stress/strain)and global (dynamics, buckling) structural behavior of the three selected components. Three computer codes, referred to as MOMM (Mechanics of Materials Model), MHOST (Marc-Hot Section Technology), and BEST (Boundary Element Stress Technology), have been developed and are briefly described in this report

Higher-Order DGFEM Transport Calculations on Polytope Meshes for Massively-Parallel Architectures

In this dissertation, we develop improvements to the discrete ordinates (S_N) neutron transport equation using a Discontinuous Galerkin Finite Element Method (DGFEM) spatial discretization on arbitrary polytope (polygonal and polyhedral) grids compatible for massively-parallel computer architectures. Polytope meshes are attractive for multiple reasons, including their use in other physics communities and their ease in handling local mesh refinement strategies. In this work, we focus on two topical areas of research. First, we discuss higher-order basis functions compatible to solve the DGFEM S_N transport equation on arbitrary polygonal meshes. Second, we assess Diffusion Synthetic Acceleration (DSA) schemes compatible with polytope grids for massively-parallel transport problems. We first utilize basis functions compatible with arbitrary polygonal grids for the DGFEM transport equation. We analyze four different basis functions that have linear completeness on polygons: the Wachspress rational functions, the PWL functions, the mean value coordinates, and the maximum entropy coordinates. We then describe the procedure to extend these polygonal linear basis functions into the quadratic serendipity space of functions. These quadratic basis functions can exactly interpolate monomial functions up to order 2. Both the linear and quadratic sets of basis functions preserve transport solutions in the thick diffusion limit. Maximum convergence rates of 2 and 3 are observed for regular transport solutions for the linear and quadratic basis functions, respectively. For problems that are limited by the regularity of the transport solution, convergence rates of 3/2 (when the solution is continuous) and 1/2 (when the solution is discontinuous) are observed. Spatial Adaptive Mesh Refinement (AMR) achieved superior convergence rates than uniform refinement, even for problems bounded by the solution regularity. We demonstrated accuracy in the AMR solutions by allowing them to reach a level where the ray effects of the angular discretization are realized. Next, we analyzed DSA schemes to accelerate both the within-group iterations as well as the thermal upscattering iterations for multigroup transport problems. Accelerating the thermal upscattering iterations is important for materials (e.g., graphite) with significant thermal energy scattering and minimal absorption. All of the acceleration schemes analyzed use a DGFEM discretization of the diffusion equation that is compatible with arbitrary polytope meshes: the Modified Interior Penalty Method (MIP). MIP uses the same DGFEM discretization as the transport equation. The MIP form is Symmetric Positive De_nite (SPD) and e_ciently solved with Preconditioned Conjugate Gradient (PCG) with Algebraic MultiGrid (AMG) preconditioning. The analysis from previous work was extended to show MIP's stability and robustness for accelerating 3D transport problems. MIP DSA preconditioning was implemented in the Parallel Deterministic Transport (PDT) code at Texas A&M University and linked with the HYPRE suite of linear solvers. Good scalability was numerically verified out to around 131K processors. The fraction of time spent performing DSA operations was small for problems with sufficient work performed in the transport sweep (O(10^3) angular directions). Finally, we have developed a novel methodology to accelerate transport problems dominated by thermal neutron upscattering. Compared to historical upscatter acceleration methods, our method is parallelizable and amenable to massively parallel transport calculations. Speedup factors of about 3-4 were observed with our new method