595 research outputs found
The inf-sup constant for the divergence on corner domains
The inf-sup constant for the divergence, or LBB constant, is related to the
Cosserat spectrum. It has been known for a long time that on non-smooth domains
the Cosserat operator has a non-trivial essential spectrum, which can be used
to bound the LBB constant from above. We prove that the essential spectrum on a
plane polygon consists of an interval related to the corner angles and that on
three-dimensional domains with edges, the essential spectrum contains such an
interval. We obtain some numerical evidence for the extent of the essential
spectrum on domains with axisymmetric conical points by computing the roots of
explicitly given holomorphic functions related to the corner Mellin symbol.
Using finite element discretizations of the Stokes problem, we present
numerical results pertaining to the question of the existence of eigenvalues
below the essential spectrum on rectangles and cuboids
Modeling the electron with Cosserat elasticity
We suggest an alternative mathematical model for the electron in dimension
1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose
material points can experience no displacements, only rotations. This framework
is a special case of the Cosserat theory of elasticity. Rotations of material
points are described mathematically by attaching to each geometric point an
orthonormal basis which gives a field of orthonormal bases called the coframe.
As the dynamical variables (unknowns) of our theory we choose a coframe and a
density. We then add an extra (third) spatial dimension, extend our coframe and
density into dimension 1+3, choose a conformally invariant Lagrangian
proportional to axial torsion squared, roll up the extra dimension into a
circle so as to incorporate mass and return to our original (1+2)-dimensional
spacetime by separating out the extra coordinate. The main result of our paper
is the theorem stating that our model is equivalent to the Dirac equation in
dimension 1+2. In the process of analyzing our model we also establish an
abstract result, identifying a class of nonlinear second order partial
differential equations which reduce to pairs of linear first order equations
Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids
Transformation elasticity, by analogy with transformation acoustics and
optics, converts material domains without altering wave properties, thereby
enabling cloaking and related effects. By noting the similarity between
transformation elasticity and the theory of incremental motion superimposed on
finite pre-strain it is shown that the constitutive parameters of
transformation elasticity correspond to the density and moduli of
small-on-large theory. The formal equivalence indicates that transformation
elasticity can be achieved by selecting a particular finite (hyperelastic)
strain energy function, which for isotropic elasticity is semilinear strain
energy. The associated elastic transformation is restricted by the requirement
of statically equilibrated pre-stress. This constraint can be cast as \tr
{\mathbf F} = constant, where is the deformation gradient,
subject to symmetry constraints, and its consequences are explored both
analytically and through numerical examples of cloaking of anti-plane and
in-plane wave motion.Comment: 20 pages, 5 figure
Numerical Methods in Poisson Geometry and their Application to Mechanics
We recall the question of geometric integrators in the context of Poisson
geometry, and explain their construction. These Poisson integrators are tested
in some mechanical examples. Their properties are illustrated numerically and
they are compared to traditional methods
A simple unsymmetric 4‐node 12‐DOF membrane element for the modified couple stress theory
In this work, the recently proposed unsymmetric 4‐node 12‐DOF (degree‐of‐freedom) membrane element (Shang and Ouyang, Int J Numer Methods Eng 113(10): 1589‐1606, 2018), which has demonstrated excellent performance for the classical elastic problems, is further extended for the modified couple stress theory, to account for the size effect of materials. This is achieved via two formulation developments. Firstly, by using the penalty function method, the kinematic relations between the element's nodal drilling DOFs and the true physical rotations are enforced. Consequently, the continuity requirement for the modified couple stress theory is satisfied in weak sense, and the symmetric curvature test function can be easily derived from the gradients of the drilling DOFs. Secondly, the couple stress field that satisfies a priori the related equilibrium equations is adopted as the energy conjugate trial function to formulate the element for the modified couple stress theory. As demonstrated by a series of benchmark tests, the new element can efficiently capture the size‐dependent responses of materials and is robust to mesh distortions. Moreover, as the new element uses only three conventional DOFs per node, it can be readily incorporated into the standard finite element program framework and commonly available finite element programs
- …