Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

2016 ◽  
Vol 26 (09) ◽  
pp. 1649-1669 ◽  
Author(s):  
Jun Hu ◽  
Shangyou Zhang
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Mixed Finite Elements ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Three Dimensions ◽  
Symmetric Tensors ◽  
Element Space ◽  
Finite Element Approximations ◽  
Order Case

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

Analysis of Dynamic Response for Piezoelectric Vibration Converter by Finite Element Simulation

2007 ◽  
Vol 336-338 ◽  
pp. 335-337
Author(s):  
Xiang Cheng Chu ◽  
Ren Bo Yan ◽  
Wen Gong ◽  
Long Tu Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Dynamic Response ◽  
Finite Element Modeling ◽  
Three Dimensions ◽  
Dynamic Motion ◽  
Element Simulation ◽  
Piezoelectric Coupling ◽  
Piezoelectric Vibration

The dynamic behavior of a vibration converter of an ultrasonic motor is described using finite element method. Tetrahedral finite elements with three dimensions are formulated with the effects of piezoelectric coupling. And the solution of the coupled electroelastic equations of dynamic motion is presented. The simulated response of the vibration converter is calculated, and shows excellent consistency with experimental results, which proved that finite element modeling is a good approach to optimize piezoelectric apparatus design. A gradual optimized method is employed to ascertain the most compatible structure.

scholarly journals Finite elements for symmetric tensors in three dimensions

2008 ◽  
Vol 77 (263) ◽  
pp. 1229-1251 ◽  
Author(s):  
Douglas N. Arnold ◽  
Gerard Awanou ◽  
Ragnar Winther
Keyword(s):  
Finite Elements ◽  
Three Dimensions ◽  
Symmetric Tensors

scholarly journals Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

2019 ◽  
Vol 29 (06) ◽  
pp. 1037-1077 ◽  
Author(s):  
Ilona Ambartsumyan ◽  
Eldar Khattatov ◽  
Jeonghun J. Lee ◽  
Ivan Yotov
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Higher Order ◽  
Local Velocity ◽  
New Family ◽  
Gauss Points ◽  
Theoretical Results ◽  
Optimal Formula

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

2019 ◽  
Vol 19 (3) ◽  
pp. 415-430 ◽  
Author(s):  
Fleurianne Bertrand ◽  
Zhiqiang Cai ◽  
Eun Young Park
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Error Estimates ◽  
Stokes Equations ◽  
Displacement Velocity ◽  
Three Dimensions ◽  
Optimal Error Estimates ◽  
Finite Element Approximations ◽  
Solution Methods ◽  
Membrane Problem

AbstractThis paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the{L^{2}}norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of{H(\mathrm{div};\Omega)}for the stress, of{H^{1}(\Omega)}for the displacement/velocity, and of{L^{2}(\Omega)}for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal{L^{2}}norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.

On Higher Order Pyramidal Finite Elements

2011 ◽  
Vol 3 (2) ◽  
pp. 131-140 ◽  
Author(s):  
Liping Liu ◽  
Kevin B. Davies ◽  
Michal Křížek ◽  
Li Guan
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Basis Functions ◽  
Quadratic Functions ◽  
Polynomial Basis

AbstractIn this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions. Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one. It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials. The space of ansatz functions contains all quadratic functions on each of four subtetrahedra that form a given pyramidal element.

Efficient Residual and Matrix-Free Jacobian Evaluation for Three-Dimensional Tri-Quadratic Hexahedral Finite Elements With Nearly-Incompressible Neo-Hookean Hyperelasticity Applied to Soft Materials on Unstructured Meshes in Parallel, With PETSc and libCEED

2020 ◽  
Author(s):  
Arash Mehraban ◽  
Jed Brown ◽  
Valeria Barra ◽  
Henry Tufo ◽  
Jeremy Thompson ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Optimization Design ◽  
Solid Mechanics ◽  
Soft Materials ◽  
Biological Tissues ◽  
Higher Order ◽  
Mixed Formulation ◽  
Element Formulation ◽  
Matrix Free

Abstract Soft materials such as rubber, elastomers, and soft biological tissues mechanically deform at large strain isochorically for all time, or during their initial transient (when a pore fluid, typically incompressible such as water, does not have time to flow out of the deforming polymer or soft tissue porous skeleton). Simulating these large isochoric deformations computationally, such as with the Finite Element Method (FEM), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation typically perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEM solid mechanics solver must run efficiently and stably. Stability is ensured by the higher order finite element formulation (with possible enhancement), but efficiency for higher order FEM remains a challenge. Thus, this paper addresses efficiency from the perspective of computer science algorithms and programming. The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. For preconditioning, a scalable p-multigrid method is presented whereby a hierarchy of levels is constructed. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order, p, instead of the element size, h. A Chebyshev polynomial smoother is used on each multigrid level. Algebraic MultiGrid (AMG) is then applied to the assembled Q1 (linear) coarse mesh on the nodes of the quadratic Q2 (quadratic) mesh. This allows low storage that can be efficiently used to accelerate the convergence to solution. For a Neo-Hookean hyperelastic problem, we examine a residual and matrix-free Jacobian formulation of a tri-quadratic hexahedral finite element with enhancement. Efficiency estimates on AVX-2 architecture based on CPU time are provided as a comparison to similar simulation (and mesh) of isochoric large deformation hyperelasticity as applied to soft materials conducted with the commercially-available FEM software program ABAQUS. The particular problem in consideration is the simulation of an assistive device in the form of finger-bending in 3D.

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