593 research outputs found
MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle
Rahman and Valdman (2013) introduced a new vectorized way to assemble finite
element matrices. We utilize underlying vectorization concepts and extend
MATLAB codes to implementation of Bogner-Fox-Schmit C1 rectangular elements in
2D. Our focus is on the detailed construction of elements and simple computer
demonstrations including energies evaluations and their visualizations.Comment: 11 pages, 7 figure
A FEM approximation of a two-phase obstacle problem and its a posteriori error estimate
This paper is concerned with the two--phase obstacle problem, a type of a
variational free boundary problem. We recall the basic estimates of Repin and
Valdman (2015) and verify them numerically on two examples in two space
dimensions. A solution algorithm is proposed for the construction of the finite
element approximation to the two--phase obstacle problem. The algorithm is not
based on the primal (convex and nondifferentiable) energy minimization problem
but on a dual maximization problem formulated for Lagrange multipliers. The
dual problem is equivalent to a quadratic programming problem with box
constraints. The quality of approximations is measured by a functional a
posteriori error estimate which provides a guaranteed upper bound of the
difference of approximated and exact energies of the primal minimization
problem. The majorant functional in the upper bound contains auxiliary
variables and it is optimized with respect to them to provide a sharp upper
bound. A space density of the nonlinear related part of the majorant functional
serves as an indicator of the free boundary.Comment: 16 pages, 7 figures, 3 table
Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation
The quasistatic, Prandtl-Reuss perfect plasticity at small strains is
combined with a gradient, reversible (i.e. admitting healing) damage which
influences both the elastic moduli and the yield stress. Existence of weak
solutions of the resulted system of variational inequalities is proved by a
suitable fractional-step discretisation in time with guaranteed
numericalstability and convergence. After finite-element approximation, this
scheme is computationally implemented and illustrative 2-dimensional
simulations are performed. The model allows e.g. for application in geophysical
modelling of re-occurring rupture of lithospheric faults. Resulted incremental
problems are solved in MATLAB by quasi-Newton method to resolve
elastoplasticity component of the solution while damage component is obtained
by solution of a quadratic programming problem
Global injectivity in second-gradient Nonlinear Elasticity and its approximation with penalty terms
We present a new penalty term approximating the Ciarlet-Ne\v{c}as condition
(global invertibility of deformations) as a soft constraint for hyperelastic
materials. For non-simple materials including a suitable higher order term in
the elastic energy, we prove that the penalized functionals converge to the
original functional subject to the Ciarlet-Ne\v{c}as condition. Moreover, the
penalization can be chosen in such a way that all low energy deformations,
self-interpenetration is completely avoided even for sufficiently small finite
values of the penalization parameter. We also present numerical experiments in
2d illustrating our theoretical results.Comment: 37 pages, 9 figure
Error identities for variational problems with obstacles
The paper is concerned with a class of nonlinear free boundary problems,
which are usually solved by variational methods based on primal (or
primal-dual) variational settings. We deduce and investigate special relations
(error identities). They show that a certain nonlinear measure of the distance
to the exact solution (specific for each problem) is equivalent to the
respective duality gap, which minimization is a keystone of all variational
numerical methods. Therefore, the identity defines the measure that contains
maximal quantitative information on the quality of a numerical solution
available through these methods. The measure has quadratic terms generated by
the linear part of the differential operator and nonlinear terms associated
with free boundaries. We obtain fully computable two sided bounds of this
measure and show that they provide efficient estimates of the distance between
the minimizer and any function from the corresponding energy space. Several
examples show that for different minimization sequence the balance between
different components of the overall error measure may be different and
domination of nonlinear terms may indicate that coincidence sets are
approximated incorrectly
Interfacial polyconvex energy-enhanced evolutionary model for shape memory alloys
A sharp-interface model describing static equilibrium configurations of shape
mory alloys by means of interfacial polyconvex energy density introduced by
\v{S}ilhav\'y in 2010 and extended to a quasistatic situation by Kn\"upfer and
Kru\v{z}\'ik in 2016 is computationally tested. Elastic properties of variants
of martensite and the austenite are described by polyconvex energy density
functions. Volume fractions of particular variants are modeled by a map of
bounded variation. Additionally, energy stored in martensite-martensite and
austenite-martensite interfaces is measured by an interface-polyconvex
function. It is assumed that transformations between material variants are
accompanied by energy dissipation which, in our case, is positively and
one-homogeneous giving rise to a rate-independent model. Various
two-dimensional computational examples are presented and the used computer code
is made available for downloads.Comment: 23 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1508.0299
Numerical approximation of von K\'{a}rm\'{a}n viscoelastic plates
We consider metric gradient flows and their discretizations in time and
space. We prove an abstract convergence result for time-space discretizations
and identify their limits as curves of maximal slope. As an application, we
consider a finite element approximation of a quasistatic evolution for
viscoelastic von K\'{a}rm\'{a}n plates. Computational experiments are provided,
too.Comment: arXiv admin note: substantial text overlap with arXiv:1902.1003
Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems
A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner
are proposed for the solution of a nonsymmetric system of algebraic equations
arising from a general finite volume element discretization of symmetric
elliptic problems with large jumps in the entries of the coefficient matrices
across subdomains. It is shown in the analysis, that the convergence of the
preconditioned GMRES iteration with the proposed preconditioners, depends
polylogarithmically on the mesh parameters, in other words, the convergence is
only weakly dependent on the mesh parameters, and it is robust with respect to
the jumps in the coefficients
A posteriori error estimates for approximate solutions of Barenblatt-Biot poroelastic model
The paper is concerned with the Barenblatt-Biott model in the theory of
poroelasticity. We derive a guaranteed estimate of the difference between exact
and approximate solutions expressed in a combined norm that encompasses errors
for the pressure fields computed from the diffusion part of the model and
errors related to stresses (strains) of the elastic part. Estimates do not
contain generic (mesh-dependent) constants and are valid for any conforming
approximation of pressure and stress fields.Comment: 12 page
Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments
We give some theoretical as well as computational results on Laplace and
Maxwell constants. Besides the classical de Rham complex we investigate the
complex of elasticity and the complex related to the biharmonic equation and
general relativity as well using the general functional analytical concept of
Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz
domains of arbitrary topology. Our numerical aspects are presented by examples
for the de Rham complex in 2D and 3D which not only confirm our theoretical
findings but also indicate some interesting conjectures.Comment: 42 pages, 10 figure
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