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Title: Final Report of the Project "From the finite element method to the virtual element method"

Abstract

The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.

Authors:
 [1];  [1]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1415356
Report Number(s):
LA-UR-17-30453
DOE Contract Number:
AC52-06NA25396
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics; virtual element method

Citation Formats

Manzini, Gianmarco, and Gyrya, Vitaliy. Final Report of the Project "From the finite element method to the virtual element method". United States: N. p., 2017. Web. doi:10.2172/1415356.
Manzini, Gianmarco, & Gyrya, Vitaliy. Final Report of the Project "From the finite element method to the virtual element method". United States. doi:10.2172/1415356.
Manzini, Gianmarco, and Gyrya, Vitaliy. 2017. "Final Report of the Project "From the finite element method to the virtual element method"". United States. doi:10.2172/1415356. https://www.osti.gov/servlets/purl/1415356.
@article{osti_1415356,
title = {Final Report of the Project "From the finite element method to the virtual element method"},
author = {Manzini, Gianmarco and Gyrya, Vitaliy},
abstractNote = {The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.},
doi = {10.2172/1415356},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2017,
month =
}

Technical Report:

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  • We develop and analyze a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, that use arbitrarily regular discrete spaces V{sub h} {contained_in} C{sup {alpha}} {element_of} N. The degrees of freedom are (a) solution and derivative values of various degree at suitable nodes and (b) solution moments inside polygons. The convergence of the method is proven theoretically and an optimal error estimate is derived. The connection with the Mimetic Finite Difference method is also discussed. Numerical experiments confirm the convergence rate that is expected from the theory.
  • Tunnels buried deep within the earth constitute an important class geomechanics problems. Two numerical techniques used for the analysis of geomechanics problems, the finite element method and the boundary element method, have complementary characteristics for applications to problems of this type. The usefulness of combining these two methods for use as a geomechanics analysis tool has been recognized for some time, and a number of coupling techniques have been proposed. However, not all of them lend themselves to efficient computational implementations for large-scale problems. This report examines a coupling technique that can form the basis for an efficient analysis toolmore » for large scale geomechanics problems through the use of an iterative equation solver.« less
  • This report compares strain and stress tensors and virtual work principles of the Lagrangian and Eulerian formulations. A generalized formulation is presented that includes the Lagrangian and Eulerian formulations as special cases. This formulation uses a reference configuration that is a linear combination of the deformed and undeformed configurations. Generalized strain and stress tensors are presented that include, as special cases, Green strain and Piola--Kirchhoff stress tensors of the Lagrangian formulation and Almansi strain and Cauchy stress tensors of the Eulerian formulation. Similarly, a generalized virtual work principle is presented that includes, also as special cases, the principles of virtualmore » work for the Lagrangian and Eulerian formulations. Two approaches are presented for solving nonlinear stress analysis problems with the generalized formulation. Both use the finite-element method to approximate equilibrium: one iterates the nodal displacements, and the other iterates the incremental displacements. 3 refs.« less
  • We study the costs incurred by an implementation of the hp-version of the finite element for solving two-dimensional elliptic partial differential equations on a shared-memory parallel computer. For a collection of benchmark problems, we systematically examine the costs in central processing unit time of various individual subtasks performed by the finite element solver, including construction of local stiffness matrices, elimination of unknowns associated with element interiors, and global solution on element interfaces by a preconditioned conjugate gradient method. Our general observations are that the costs of the naturally parallel computations associated with local elements are significantly higher than any globalmore » computations, so that the latter do not represent a significant bottleneck to parallel efficiency. However, memory conflicts place some limitations on the sizes or number of local problems that can be handled efficiently in parallel.« less