Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA discrete elasticity complex on three-dimensional Alfeld splits
90
0
0.0
(
0
)
Added by
Johnny Guzman
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de~Rham complexes, and smoother finite element differential forms.
rate research
Read More
A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $textrm{inc}$ operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition of polynomial tensor spaces. The trace of the $textrm{inc}$ operator is induced from a Greens identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Our construction appears to be the first $H(textrm{inc})$-conforming finite elements on tetrahedral meshes without further splits.
We construct several smooth finite element spaces defined on three--dimensional Worsey--Farin splits. In particular, we construct $C^1$, $H^1(curl)$, and $H^1$-conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at sub-simplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation.
The main difficulty in solving the discrete constrained problem is its poor and even ill condition. In this paper, we transform the discrete constrained problems on de Rham complex to Laplace-like problems. This transformation not only make the constrained problems solvable, but also make it easy to use the existing iterative methods and preconditioning techniques to solving large-scale discrete constrained problems.
This paper focuses on studying the bifurcation analysis of the eigenstructure of the $gamma$-parameterized generalized eigenvalue problem ($gamma$-GEP) arising in three-dimensional (3D) source-free Maxwells equations with Pasteur media, where $gamma$ is the magnetoelectric chirality parameter. For the weakly coupled case, namely, $gamma < gamma_{*} equiv$ critical value, the $gamma$-GEP is positive definite, which has been well-studied by Chern et. al, 2015. For the strongly coupled case, namely, $gamma > gamma_{*}$, the $gamma$-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et. al, 2018. In this paper, we build several theoretical results on the eigenstructure behavior of the $gamma$-GEPs. We prove that the $gamma$-GEP is regular for any $gamma > 0$, and the $gamma$-GEP has $2 times 2$ Jordan blocks of infinite eigenvalues at the critical value $gamma_{*}$. Then, we show that the $2 times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next, it will bifurcate into two real eigenvalues, with one moving toward the left and the other to the right along the real axis as $gamma$ increases. A newly formed state whose energy is smaller than the ground state can be created as $gamma$ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of 3D Maxwells equations with Pasteur media.
We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the velocity field. Using the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deformations are computed from a composition of short-time submaps which can be accurately evolved on coarse grids. This method is a fundamental extension to the CM method for two-dimensional incompressible Euler equations [51]. We take a geometric approach in the 3D case where the vorticity is not a scalar advected quantity, but can be computed as a differential 2-form through the pullback of the initial condition by the characteristic map. This formulation is based on the Kelvin circulation theorem and gives point-wise a Lagrangian description of the vorticity field. We demonstrate through numerical experiments the validity of the method and show that energy is not dissipated through artificial viscosity and small scales of the solution are preserved. We provide error estimates and numerical convergence tests showing that the method is globally third-order accurate.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
