# An adaptive finite element method for solving a double well problem describing crystalline microstructure

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 33, Issue: 4, page 781-796
- ISSN: 0764-583X

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topProhl, Andreas. "An adaptive finite element method for solving a double well problem describing crystalline microstructure." ESAIM: Mathematical Modelling and Numerical Analysis 33.4 (2010): 781-796. <http://eudml.org/doc/197413>.

@article{Prohl2010,

abstract = {
The minimization of nonconvex functionals naturally arises in
materials sciences where deformation gradients in certain alloys exhibit
microstructures. For example, minimizing sequences of the nonconvex
Ericksen-James energy can be associated with deformations in
martensitic materials that
are observed in experiments[2,3].
— From the numerical
point of view, classical conforming and nonconforming finite element
discretizations have been observed to give minimizers
with their quality being highly
dependent on the underlying triangulation, see [8,24,26,27] for
a survey. Recently, a new
approach has been proposed and analyzed in [15,16]
that is
based on discontinuous finite elements to reduce the pollution effect
of a general triangulation on the computed minimizer.
The goal of the present paper is
to propose and analyze
an adaptive method,
giving a more accurate resolution of laminated microstructure
on arbitrary grids.
},

author = {Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Adaptive algorithm;
finite element method;
nonconvex minimization;
multi-well problem;
microstructure;
multiscale;
nonlinear elasticity;
shape-memory alloy;
materials science.; nonconvex energy functional; laminated microstructure; improved convergence; discontinuous ansatz functions; adaptivity criterion; height of interelement jumps},

language = {eng},

month = {3},

number = {4},

pages = {781-796},

publisher = {EDP Sciences},

title = {An adaptive finite element method for solving a double well problem describing crystalline microstructure},

url = {http://eudml.org/doc/197413},

volume = {33},

year = {2010},

}

TY - JOUR

AU - Prohl, Andreas

TI - An adaptive finite element method for solving a double well problem describing crystalline microstructure

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 33

IS - 4

SP - 781

EP - 796

AB -
The minimization of nonconvex functionals naturally arises in
materials sciences where deformation gradients in certain alloys exhibit
microstructures. For example, minimizing sequences of the nonconvex
Ericksen-James energy can be associated with deformations in
martensitic materials that
are observed in experiments[2,3].
— From the numerical
point of view, classical conforming and nonconforming finite element
discretizations have been observed to give minimizers
with their quality being highly
dependent on the underlying triangulation, see [8,24,26,27] for
a survey. Recently, a new
approach has been proposed and analyzed in [15,16]
that is
based on discontinuous finite elements to reduce the pollution effect
of a general triangulation on the computed minimizer.
The goal of the present paper is
to propose and analyze
an adaptive method,
giving a more accurate resolution of laminated microstructure
on arbitrary grids.

LA - eng

KW - Adaptive algorithm;
finite element method;
nonconvex minimization;
multi-well problem;
microstructure;
multiscale;
nonlinear elasticity;
shape-memory alloy;
materials science.; nonconvex energy functional; laminated microstructure; improved convergence; discontinuous ansatz functions; adaptivity criterion; height of interelement jumps

UR - http://eudml.org/doc/197413

ER -

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