skip to main content


Title: Convergence Characteristics of Geometrically Accurate Spatial Finite Elements
Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.  more » « less
Award ID(s):
1852510
NSF-PAR ID:
10217891
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Journal of Computational and Nonlinear Dynamics
Volume:
16
Issue:
1
ISSN:
1555-1415
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shownthat continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained using the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials. 
    more » « less
  2. The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques. 
    more » « less
  3. Abstract Two different cases are encountered in the thermal analysis of solids. In the first case, continua are not subject to boundary and motion constraints and all material points experience same displacement-gradient changes as the result of application of thermal loads. In this case, referred to as unconstrained thermal expansion, the thermal load produces uniform stress-free motion within the continuum. In the second case, point displacements due to boundary and motion constraints are restricted, and therefore, continuum points do not move freely when thermal loads are applied. This second case, referred to as constrained thermal expansion, leads to thermal stresses and its study requires proper identification of the independent coordinates which represent expansion degrees-of-freedom. To have objective evaluation and comparison between the two cases of constrained and unconstrained thermal expansion, the reference-configuration geometry is accurately described using the absolute nodal coordinate formulation (ANCF) finite elements. ANCF position-gradient vectors have unique geometric meanings as tangent to coordinate lines, allowing systematic description of the two different cases of unconstrained and constrained thermal expansions using multiplicative decomposition of the matrix of position-gradient vectors. Furthermore, generality of the approach for large-displacement thermal analysis requires using the Lagrange–D'Alembert principle for proper treatment of algebraic constraint equations. Numerical results are presented to compare two different expansion cases, demonstrate use of the new approach, and verify its results by comparing with conventional finite element (FE) approaches. 
    more » « less
  4. Summary

    We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.

     
    more » « less
  5. Abundances of minor and trace elements in olivine are increasingly used as petrogenetic indicators for mantle source lithologies, mantle metasomatism history, mantle potential temperatures, and magmatic differentiation. As it is common for olivine to be complexly zoned on a fine-scale, high precision analytical methods for EPMA (electron microprobe microanalyzer, or Electron Microprobe) trace element analysis under high spatial resolution have been developed. However, previous studies have focused more on analytical precision with fewer efforts in examining the accuracy of the data. In this study, we used the Cameca SXFive field emission (FE) EPMA to fully evaluate the effects of beam settings, background offsets and background regression models, and primary calibration standards on the data accuracy of 10 key petrogenetic elements (Na, Al, P, Ca, Ti, Cr, Mn, Co, Ni, and Zn) using MongOlSh11–2 olivine as a reference material. Our results indicate that high voltage, high beam current and long counting time not only improve data precision, but also improve data accuracy, especially on elements with low P/B (peak/background) ratios such as Zn and Cr. Importantly, careful background offsets and background regression models need to be obtained via high resolution WDS relative scans or step scans on each target element. Special care needs to be paid to Co element analysis to avoid or correct for peak interference of Fe Kβ. Among 10 minor and trace elements, exponential background regression models need to be applied to Al, Mn, and Ti elements, whereas other elements require linear background regression. Furthermore, to avoid Al and Zn surface contamination due to alumina polishing or brass presence, ultrasonic cleaning between each intermediate polishing steps and plasma cleaning immediately prior to EPMA experiments is highly recommended. Micro-inclusions such as chromite and spinel in olivine or adjacent Ca-rich phases need to be avoided to minimize primary or secondary fluorescence-related contamination on Al, Cr, or Ca. As a volatile element, Na element needs to be analyzed first with appropriate counting time to minimize the Na loss under high beam conditions. It needs mentioning that major elements (Mg, Fe, and Si) are best analyzed using MongOlSh11–2 or San Carlos olivine as primary standards for calibrations, which can yield more accurate data for both major elements and trace elements because of the improved matrix- corrections. Using our recommended analytical protocols, we have successfully discriminated “depleted” mantle olivine cores from an EMORB in northern East Pacific Rise (EPR) via Ca, Ti, Ni, Co, and Mn abundances. Our olivine data from Siqueiros Transform and the nearby 8◦20′ N seamounts also help reveal a metasomatized peridotite mantle beneath the northern EPR. Overall, the protocols proposed in this study can serve as a guide for accurate EPMA olivine trace element analyses, which potentially contributes to the efforts of fostering a comparable olivine database worldwide. 
    more » « less