199 research outputs found
Geometry of polycrystals and microstructure
We investigate the geometry of polycrystals, showing that for polycrystals
formed of convex grains the interior grains are polyhedral, while for
polycrystals with general grain geometry the set of triple points is small.
Then we investigate possible martensitic morphologies resulting from intergrain
contact. For cubic-to-tetragonal transformations we show that homogeneous
zero-energy microstructures matching a pure dilatation on a grain boundary
necessarily involve more than four deformation gradients. We discuss the
relevance of this result for observations of microstructures involving second
and third-order laminates in various materials. Finally we consider the more
specialized situation of bicrystals formed from materials having two
martensitic energy wells (such as for orthorhombic to monoclinic
transformations), but without any restrictions on the possible microstructure,
showing how a generalization of the Hadamard jump condition can be applied at
the intergrain boundary to show that a pure phase in either grain is impossible
at minimum energy.Comment: ESOMAT 2015 Proceedings, to appea
Comparison results for the Stokes equations
This paper enfolds a medius analysis for the Stokes equations and compares
different finite element methods (FEMs). A first result is a best approximation
result for a P1 non-conforming FEM. The main comparison result is that the
error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or
reduced P2-P0) FEM, which is a lower bound to the error of the P1
non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The
paper discusses the converse direction, as well as other methods such as the
discontinuous Galerkin and pseudostress FEMs.
Furthermore this paper provides counterexamples for equivalent convergence
when different pressure approximations are considered. The mathematical
arguments are various conforming companions as well as the discrete inf-sup
condition
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Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes
A hierarchical a posteriori error estimator for the first-order finite
element method (FEM) on a red-refined triangular mesh is presented for the 2D
Poisson model problem. Reliability and efficiency with some explicit constant
is proved for triangulations with inner angles smaller than or equal to π/2 .
The error estimator does not rely on any saturation assumption and is valid
even in the pre-asymptotic regime on arbitrarily coarse meshes. The
evaluation of the estimator is a simple post-processing of the piecewise
linear FEM without any extra solve plus a higher-order approximation term.
The results also allows the striking observation that arbitrary local
averaging of the primal variable leads to a reliable and efficient error
estimation. Several numerical experiments illustrate the performance of the
proposed a posteriori error estimator for computational benchmarks
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