| 2018D. Dũng, M. Griebel, V. N. Huy and C. Rieger
$\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs
Journal of Complexity, 46: 66--89 2018
10.1016/j.jco.2017.12.001
|
| |
| Polona Durcik and Christoph Thiele
Singular Brascamp-Lieb inequalities
arXiv e-prints: arXiv:1809.08688 2018
https://ui.adsabs.harvard.edu/abs/2018arXiv180908688D
|
| |
| Celia Reina, Landry Fokoua Djodom, Michael Ortiz and Sergio Conti
Kinematics of elasto-plasticity: Validity and limits of applicability of $F=F_eF_p$ for general three-dimensional deformations
Journal of the Mechanics and Physics of Solids, 121: 99--113 2018
10.1016/j.jmps.2018.07.006
|
| |
| 2017Sergio Conti, Johannes Diermeier and Barbara Zwicknagl
Deformation concentration for martensitic microstructures in the limit of low volume fraction
Calc. Var. PDE, 56: 16 2017
10.1007/s00526-016-1097-1
Abstract: We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $Γ$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation. |
| |
| Francesco Di Plinio, Shaoming Guo, Christoph Thiele and Pavel Zorin-Kranich
Square functions for bi-Lipschitz maps and directional operators
arXiv e-prints: arXiv:1706.07111 2017
https://ui.adsabs.harvard.edu/abs/2017arXiv170607111D
|
| |
| 2016Patrick W. Dondl, Behrend Heeren and Martin Rumpf
Optimization of the branching pattern in coherent phase transitions
C. R. Math. Acad. Sci. Paris, 354(6): 639--644 2016
https://arxiv.org/abs/1512.06620
Abstract: Branching can be observed at the austenite-martensite interface of martensitic phase transformations. For a model problem, Kohn and Müller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents but results in a significantly lowered upper energy bound. |
| |
| 2015Sebastian Andres, Jean-Dominique Deuschel and Martin Slowik
Harnack inequalities on weighted graphs and some applications to the random conductance model
Probab. Theory Relat. Fields: 1-47 2015
http://dx.doi.org/10.1007/s00440-015-0623-y
Abstract: We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk X in an environment of ergodic random conductances taking values in (0,∞) satisfying some moment conditions. |
| |
| Sebastian Andres, Jean-Dominique Deuschel and Martin Slowik
Invariance principle for the random conductance model in a degenerate ergodic environment
Ann. Probab., 43(4): 1866-1891 2015
http://dx.doi.org/10.1214/14-AOP921
|
| |
| Annegret Y. Burtscher and Roland Donninger
Hyperboloidal evolution and global dynamics for the focusing cubic wave equation
2015
http://arxiv.org/abs/1511.08600
|
| |
| Sergio Conti and Georg Dolzmann
On the theory of relaxation in nonlinear elasticity with constraints on the determinant
Arch. Rat. Mech. Anal., 217(2): 413-437 2015
http://dx.doi.org/10.1007/s00205-014-0835-9
Abstract: We consider vectorial variational problems in nonlinear elasticity of the form I[u]=∫W(Du)dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional ∫Wqc(Du)dx is an upper bound on the relaxation of I, and coincides with the relaxation if the quasiconvex envelope W qc of W is polyconvex and has p-growth from below with p≧n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results. |
| |
| Roland Donninger
Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation
2015
http://arxiv.org/abs/1509.02041
|
| |
| Roland Donninger and Birgit Schörkhuber
Stable blowup for wave equations in odd space dimensions
2015
http://arxiv.org/abs/1504.00808
|
| |
| Dinh Dũng and Michael Griebel
Hyperbolic cross approximation in infinite dimensions
Journal of Complexity 2015
http://arxiv.org/pdf/1501.01119v1
Abstract: We give tight upper and lower bounds of the cardinality of
the index sets of certain hyperbolic crosses which reflect mixed
Sobolev–Korobov-type smoothness and mixed Sobolev-analytictype
smoothness in the infinite-dimensional case where specific
summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of
functions from the associated spaces in terms of the ε-dimension
of their unit balls. Here, the approximation is based on linear
information. Such function spaces appear for example for the
solution of parametric and stochastic PDEs. The obtained upper
and lower bounds of the approximation error as well as of the
associated ε-complexities are completely independent of any parametric
or stochastic dimension. Moreover, the rates are independent
of the parameters which define the smoothness properties
of the infinite-variate parametric or stochastic part of the solution.
These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinite-dimensional case and corresponding infinite-dimensional
problems get tractable. |
| |
| 2014Sebastian Andres, Jean-Dominique Deuschel and Martin Slowik
Heat kernel estimates for random walks with degenerate weights
2014
http://arxiv.org/abs/1412.4338
|
| |
| Sergio Conti, Georg Dolzmann and Stefan Müller
Korn's second inequality and geometric rigidity with mixed growth conditions
Calc. Var., 50: 437-454 2014
http://dx.doi.org/10.1007/s00526-013-0641-5
Abstract: Geometric rigidity states that a gradient field which is \( L^p\) -close to the set of proper rotations is necessarily \( L^p\) -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in \( L^p+L^q\) and in \( L^p,q\) interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces. |
| |
| Sergio Conti and Geog Dolzmann
Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions
Math. Models. Metods App. Sci., 24(14): 2929-2942 2014
http://dx.doi.org/10.1142/S0218202514500419
Abstract: We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant. |
| |
| Patrick Diehl and Marc A. Schweitzer
Efficient Neighbor Search for Particle Methods on GPUs
In M. Griebel and M. A. Schweitzer, editor, Meshfree Methods for Partial Differential Equations VII, Volume 100 of Lecture Notes in Computational Science and Engineering
Chapter 5, page 81-95.
Publisher: Springer International
2014
http://dx.doi.org/10.1007/978-3-319-06898-5_5
|
| |
| Landry Fokoua Djodom, Sergio Conti and Michael Ortiz
Optimal Scaling in Solids undergoing Ductile Fracture by Void Sheet Formation
Arch. Ration. Mech. Anal., 212(1): 331-357 2014
http://dx.doi.org/10.1007/s00205-013-0687-8
Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material. |
| |
| Landry Fokoua Djodom, Sergio Conti and Michael Ortiz
Optimal scaling laws for ductile fracture derived from strain-gradient microplasticity
J. Mech. Phys. Solids, 62: 295-311 2014
http://dx.doi.org/10.1016/j.jmps.2013.11.002
Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material. |
| |
| Fabian Franzelin, Patrick Diehl and Dirk Pflüger
Non-intrusive Uncertainty Quantification with Sparse Grids for Multivariate Peridynamic Simulations
In M. Griebel and M. A. Schweitzer, editor, Meshfree Methods for Partial Differential Equations VII, Volume 100 of Lecture Notes in Computational Science and Engineering
Chapter 7, page 115-143.
Publisher: Springer
2014
http://dx.doi.org/10.1007/978-3-319-06898-5_7
|
| |