3,995 research outputs found
C1-continuous space-time discretization based on Hamilton's law of varying action
We develop a class of C1-continuous time integration methods that are
applicable to conservative problems in elastodynamics. These methods are based
on Hamilton's law of varying action. From the action of the continuous system
we derive a spatially and temporally weak form of the governing equilibrium
equations. This expression is first discretized in space, considering standard
finite elements. The resulting system is then discretized in time,
approximating the displacement by piecewise cubic Hermite shape functions.
Within the time domain we thus achieve C1-continuity for the displacement field
and C0-continuity for the velocity field. From the discrete virtual action we
finally construct a class of one-step schemes. These methods are examined both
analytically and numerically. Here, we study both linear and nonlinear systems
as well as inherently continuous and discrete structures. In the numerical
examples we focus on one-dimensional applications. The provided theory,
however, is general and valid also for problems in 2D or 3D. We show that the
most favorable candidate -- denoted as p2-scheme -- converges with order four.
Thus, especially if high accuracy of the numerical solution is required, this
scheme can be more efficient than methods of lower order. It further exhibits,
for linear simple problems, properties similar to variational integrators, such
as symplecticity. While it remains to be investigated whether symplecticity
holds for arbitrary systems, all our numerical results show an excellent
long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical
results unchange
Existence of minimizers for the d stationary Griffith fracture model
We consider the variational formulation of the Griffith fracture model in two
spatial dimensions and prove existence of strong minimizers, that is
deformation fields which are continuously differentiable outside a closed jump
set and which minimize the relevant energy. To this aim, we show that
minimizers of the weak formulation of the problem, set in the function space
and for which existence is well-known, are actually strong minimizers
following the approach developed by De Giorgi, Carriero, and Leaci in the
corresponding scalar setting of the Mumford-Shah problem
Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces
We formulate and solve the analog of the universal Conformal Ward Identity
for the stress-energy tensor on a compact Riemann surface of genus , and
present a rigorous invariant formulation of the chiral sector in the induced
two-dimensional gravity on higher genus Riemann surfaces. Our construction of
the action functional uses various double complexes naturally associated with a
Riemann surface, with computations that are quite similar to descent
calculations in BRST cohomology theory. We also provide an interpretation for
the action functional in terms of the geometry of different fiber spaces over
the Teichm\"{u}ller space of compact Riemann surfaces of genus .Comment: 38 pages. Latex2e + AmsLatex2.1. One embedded figure. One section on
the relation with the geometry of fiber spaces on the Teichmueller space and
several important references adde
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