Computational polyconvexification of isotropic functions

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments.Comment: 17 pages, 7 figure

Computations of quasiconvex hulls of isotropic sets

We design an algorithm for computations of quasiconvex hulls of isotropic compact sets in in the space of 2x2 real matrices. Our approach uses a recent result by the first author [Adv. Calc. Var. (2014), DOI: 10.1515acv-2012-0008] on quasiconvex hulls of isotropic compact sets in the space of 2x2 real matrices. We show that our algorithm has the time complexity of O(N log N ) where N is the number of orbits of the set. We show some applications of our results to relaxation of L∞ variational problems

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation techniques circumvent the problem of non-existence of minimizers and prevent mesh dependency of the solutions of discretized boundary value problems using finite elements. By the combination, modification and parallelization of the underlying convexification algorithms the approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step size control strategies prevents stability issues related to local minima in the energy landscape and the computation of derivatives. Special techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations

Analytical and numerical relaxation results for models in soil mechanics

A variational model of pressure-dependent plasticity employing a time-incremental setting is introduced. A novel formulation of the dissipation potential allows one to construct the condensed energy in a variationally consistent manner. For a one-dimensional model problem, an explicit expression for the quasiconvex envelope can be found which turns out to be essentially independent of the original pressure-dependent yield surface. The model problem can be extended to higher dimensions in an empirical manner. Numerical simulation exhibit well-posed behavior showing mesh-independent results.Comment: Submitted to Cont. Mech. Thermody

Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity

Reliable simulation techniques for the description of elastic deformation processes in solid mechanics are nowadays of great importance. A reasonable model should take nonlinear kinematics and a nonlinear material law into account and should coincide with Hooke’s law under small loads. In addition, a numerical method should be able to simulate compressible as well as (almost) incompressible material behavior. The calculation of good stress and displacement approximations is often of particular interest. Therefore general mixed least squares finite element methods in the context of finite hyperelasticity are considered in this work. They are based on the conservation of linear momentum and inverse stress-strain relations and will be used for the simulation of homogeneous isotropic and homogeneous transverse isotropic material behavior. For the minimization of the nonlinear least squares functionals in finite dimensional spaces a Gauss-Newton framework is applied. In the case of a specific homogeneous isotropic Neo-Hooke model an analysis is provided which proves reliability and efficiency of the nonlinear least squares functional as a-posteriori error estimator. The analysis remains valid in the incompressible limit and therefore the Poisson locking effect is excluded. The analytical results for the Neo-Hooke model are used to propose an algorithm for model adaptivity which is based on the model of linear elasticity and the Neo-Hooke model. The algorithm automatically decides in which subdomain the linear model should be locally substituted by the Neo-Hooke model. Two- and three-dimensional numerical examples for compressible and fully incompressible materials are given in order to illustrate the potential of our method. Here next-to-lowest-order Raviart-Thomas elements for the stress approximations are combined with conforming piecewise quadratic elements for the displacement approximations. A significant improvement of stress approximations in comparison to conventional discretization methods is demonstrated. In examples with corner or edge singularities almost optimal convergence rates for the nonlinear least squares functional using adaptive refinement strategies are achieved.Zuverlässige Simulationstechniken zur Beschreibung von elastischen Verformungsprozessen in der Festkörpermechanik sind heutzutage von großer Bedeutung. Ein sinnvolles Modell sollte nichtlineare Kinematik und ein nichtlineares Materialgesetz berücksichtigen und mit dem Hookeschen Gesetz unter kleinen Belastungen übereinstimmen. Ferner sollte ein numerisches Verfahren sowohl kompressibles als auch (nahezu) inkompressibles Materialverhalten simulieren können. Die Berechnung von guten Spannungs- und Verschiebungsapproximationen ist oftmals von besonderem Interesse. Aus diesen Gründen werden in dieser Arbeit allgemeine gemischte Least-Squares Finite-Element-Methoden im Rahmen der finiten Hyperelastizität betrachtet. Sie basieren auf der Impulserhaltung und inversen Spannungs-Verzerrungs-Relationen und werden zur Simulation homogen isotropen und homogen transversal-isotropen Materialverhaltens benutzt. Für die Minimierung der nichtlinearen Least-Squares Funktionale in endlichdimensionalen Räumen wird ein Gauß-Newton-Verfahren verwendet. Im Falle eines speziellen homogen isotropen Neo-Hooke Modells wird eine Analysis bereitgestellt, welche die Zuverlässigkeit und Effizienz des nichtlinearen Least-Squares Funktionals als a-posteriori Fehlerschätzer beweist. Die Analysis bleibt gleichmäßig gültig im inkompressiblen Grenzfall womit der Poisson-Locking Effekt ausgeschlossen ist. Die analytischen Resultate für das Neo-Hooke Modell werden benutzt um einen Algorithmus zur Modelladaptivität vorzuschlagen, welcher auf dem linearen Elastizitätsmodell und dem Neo-Hooke Modell basiert. Der Algorithmus entscheidet automatisch in welchem Teilgebiet das lineare Modell durch das Neo-Hooke Modell lokal ausgetauscht werden soll. Zwei- und dreidimensionale numerische Beispiele für kompressible und inkompressible Materialien werden betrachtet, um das Potenzial unserer Methode zu verdeutlichen. Hierbei werden Raviart-Thomas Elemente zweitniedrigster Ordnung für die Spannungsapproximationen mit konformen, stückweise quadratischen, Elementen für die Verschiebungsapproximationen kombiniert. Eine signifikante Verbesserung von Spannungsapproximationen im Vergleich zu herkömmlichen Diskretisierungsmethoden wird nachgewiesen. In Beispielen mit Eck- oder Kantensingularitäten werden unter Verwendung adaptiver Verfeinerungsstrategien nahezu optimale Konvergenzraten für das nichtlineare Least-Squares Funktional erreicht

Matematické modelování tenkých filmů z martenzitických materiálů

The aim of the thesis is the mathematical and computer modelling of thin films of martensitic materials. We derive a thermodynamic thin-film model on the meso-scale that is capable of capturing the evolutionary process of the shape-memory effect through a two-step procedure. First, we apply dimension reduction techniques in a microscopic bulk model, then enlarge gauge by neglecting microscopic interfacial effects. Computer modelling of thin films is conducted for the static case that accounts for a modified Hadamard jump condition which allows for austenite--martensite interfaces that do not exist in the bulk. Further, we characterize $L^p$-Young measures generated by invertible matrices, that have possibly positive determinant as well. The gradient case is covered for mappings the gradients and inverted gradients of which belong to $L^\infty$, a non-trivial problem is the manipulation with boundary conditions on generating sequences, as standard cut-off methods are inapplicable due to the determinant constraint. Lastly, we present new results concerning weak lower semicontinuity of integral functionals along (asymptotically) $\mathcal{A}$-free sequences that are possibly negative and non-coercive. Powered by TCPDF (www.tcpdf.org)Cílem této práce je matematické a počítačové modelování tenkých filmů martenzitických materiálů. Dvoustupňovém postupem odvodíme mezoskopický termodynamický model pro tenké filmy, jenž umí zachytit evoluční proces efektu tvarové paměti. Nejprve provedeme redukci dimenze v mikroskopickém 3D modelu, pak zvětšíme měřítko zanedbáním mikroskopických mezifázových vlivů. Počítačové modelování tenkých filmů je provedeno v statickém případě zahrnutím modifikované Hadamardovy podmínky skoku, jež dává slabší podmínku na kompatibilitu fází ve srovnání s 3D modelem. Dále jsou popsány $L^p$-Youngovy míry generované regulárními maticemi, popř. maticemi s kladným determinantem. Gradientní případ je vyřešen pro zobrazení, kde gradient a inverze gradientu jsou v $L^\infty$, netriviálním problémem byla manipulace s okrajovými podmínkami u generující posloupnosti, neboť standardní "ořezávací metody" nelze v našem případě aplikovat kvůli podmínce na determinant. V poslední kapitole zmíníme nové výsledky týkající se slabé zdola polospojitosti integrálních funkcionálů podél tzv. (asymptoticky) $\mathcal{A}$-free posloupností, jež mohou být záporné i nekoercivní. Powered by TCPDF (www.tcpdf.org)Matematický ústav UKMathematical Institute of Charles UniversityFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Discrete Riemannian Calculus and A Posteriori Error Control on Shape Spaces

In this thesis, a novel discrete approximation of the curvature tensor on Riemannian manifolds is derived, efficient methods to interpolate and extrapolate images in the context of the time discrete metamorphosis model are analyzed, and an a posteriori error estimator for the binary Mumford–Shah model is examined. Departing from the variational time discretization on (possibly infinite-dimensional) Riemannian manifolds originally proposed by Rumpf and Wirth, in which a consistent time discrete approximation of geodesic curves, the logarithm, the exponential map and parallel transport is analyzed, we construct the discrete curvature tensor and prove its convergence under certain smoothness assumptions. To this end, several time discrete parallel transports are applied to suitably rescaled tangent vectors, where each parallel transport is computed using Schild’s ladder. The associated convergence proof essentially relies on multiple Taylor expansions incorporating symmetry and scaling relations. In several numerical examples we validate this approach for surfaces. The by now classical flow of diffeomorphism approach allows the transport of image intensities along paths in time, which are characterized by diffeomorphisms, and the brightness of each image particle is assumed to be constant along each trajectory. As an extension, the metamorphosis model proposed by Trouvé, Younes and coworkers allows for intensity variations of the image particles along the paths, which is reflected by an additional penalization term appearing in the energy functional that quantifies the squared weak material derivative. Taking into account the aforementioned time discretization, we propose a time discrete metamorphosis model in which the associated time discrete path energy consists of the sum of squared L2-mismatch functionals of successive square-integrable image intensity functions and a regularization functional for pairwise deformations. Our main contributions are the existence proof of time discrete geodesic curves in the context of this model, which are defined as minimizers of the time discrete path energy, and the proof of the Mosco-convergence of a suitable interpolation of the time discrete to the time continuous path energy with respect to the L2-topology. Using an alternating update scheme as well as a multilinear finite element respectively cubic spline discretization for the images and deformations allows to efficiently compute time discrete geodesic curves. In several numerical examples we demonstrate that time discrete geodesics can be robustly computed for gray-scale and color images. Taking into account the time discretization of the metamorphosis model we define the discrete exponential map in the space of images, which allows image extrapolation of arbitrary length for given weakly differentiable initial images and variations. To this end, starting from a suitable reformulation of the Euler–Lagrange equations characterizing the one-step extrapolation a fixed point iteration is employed to establish the existence of critical points of the Euler–Lagrange equations provided that the initial variation is small in L2. In combination with an implicit function type argument requiring H1-closeness of the initial variation one can prove the local existence as well as the local uniqueness of the discrete exponential map. The numerical algorithm for the one-step extrapolation is based on a slightly modified fixed point iteration using a spatial Galerkin scheme to obtain the optimal deformation associated with the unknown image, from which the unknown image itself can be recovered. To prove the applicability of the proposed method we compute the extrapolated image path for real image data. A common tool to segment images and shapes into multiple regions was developed by Mumford and Shah. The starting point to derive a posteriori error estimates for the binary Mumford–Shah model, which is obtained by restricting the original model to two regions, is a uniformly convex and non-constrained relaxation of the binary model following the work by Chambolle and Berkels. In particular, minimizers of the binary model can be exactly recovered from minimizers of the relaxed model via thresholding. Then, applying duality techniques proposed by Repin and Bartels allows deriving a consistent functional a posteriori error estimate for the relaxed model. Afterwards, an a posteriori error estimate for the original binary model can be computed incorporating a suitable cut-out argument in combination with the functional error estimate. To calculate minimizers of the relaxed model on an adaptive mesh described by a quadtree structure, we employ a primal-dual as well as a purely dual algorithm. The quality of the error estimator is analyzed for different gray-scale input images