scholarly journals Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies

2021 ◽  
Author(s):  
José A. Iglesias
Keyword(s):  
Thin Shell ◽  
Shape Matching ◽  
Narrow Band ◽  
Convergence Result ◽  
Energy Functional ◽  
Implicit Surfaces ◽  
Variational Model ◽  
Scaling Limits ◽  
Gamma Convergence ◽  
Realistic Shape

In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a $\Gamma$-convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.

scholarly journals Homogenization in BV of a model for layered composites in finite crystal plasticity

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Elisa Davoli ◽  
Rita Ferreira ◽  
Carolin Kreisbeck
Keyword(s):  
Crystal Plasticity ◽  
Integral Form ◽  
Convergence Result ◽  
Variational Model ◽  
Functions Of Bounded Variation ◽  
Potential Candidate ◽  
Gamma Convergence ◽  
Two Dimensional ◽  
Rigid Component

Abstract In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space BV of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

Elasticity Equations with High Contrast Parameters

2015 ◽  
Author(s):  
Habib Ammari ◽  
Elie Bretin ◽  
Josselin Garnier ◽  
Hyeonbae Kang ◽  
Hyundae Lee ◽  
...  
Keyword(s):  
Extreme Values ◽  
Convergence Result ◽  
Energy Functional ◽  
Incompressible Limit ◽  
High Contrast ◽  
Complete Asymptotic Expansion ◽  
Shear Moduli ◽  
Elasticity Equations ◽  
Compressional Modulus ◽  
Uniformly Bounded

This chapter presents some recent results on the elasticity equations with high contrast coefficients. It first sets up the problems for finite and extreme moduli before discussing the incompressible limit of elasticity equations. It then provides a complete asymptotic expansion with respect to the compressional modulus and considers the limiting cases of holes and hard inclusions. It proves that the energy functional is uniformly bounded and demonstrates that the potentials on the boundary of the inclusion are also uniformly bounded. It also shows that these potentials converge as the bulk and shear moduli tend to their extreme values and that similar boundedness and convergence result holds true for the boundary value problem.

scholarly journals SCALING LIMITS OF THE CHERN–SIMONS–HIGGS ENERGY

2008 ◽  
Vol 10 (01) ◽  
pp. 1-16 ◽  
Author(s):  
MATTHIAS KURZKE ◽  
DANIEL SPIRN
Keyword(s):  
The Self ◽  
Scaling Limits ◽  
Gamma Convergence ◽  
Two Dimensional ◽  
Energy Formula ◽  
Simply Connected ◽  
Weak Topologies ◽  
Chern Simons ◽  
Dual Case ◽  
Dimensional Domain

We continue our study in [16] of the Gamma limit of the Abelian Chern–Simons–Higgs energy [Formula: see text] on a bounded, simply connected, two-dimensional domain where ε → 0 and με → μ ∈ [0, +∞]. Under the critical scaling, Gcsh ≈ | log ε2, we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H1 = H1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μμ → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(uε) = det (∇ uε) in terms of the Chern–Simons–Higgs energy Ecsh.

scholarly journals Nonlocal Variational Model for Saliency Detection

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Meng Li ◽  
Yi Zhan ◽  
Lidan Zhang
Keyword(s):  
Visual Attention ◽  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Temporal Evolution ◽  
Lagrange Equation ◽  
Saliency Detection ◽  
Energy Functional ◽  
Experimental Results ◽  
Variational Model ◽  
Still Images

We present a nonlocal variational model for saliency detection from still images, from which various features for visual attention can be detected by minimizing the energy functional. The associated Euler-Lagrange equation is a nonlocalp-Laplacian type diffusion equation with two reaction terms, and it is a nonlinear diffusion. The main advantage of our method is that it provides flexible and intuitive control over the detecting procedure by the temporal evolution of the Euler-Lagrange equation. Experimental results on various images show that our model can better make background details diminish eventually while luxuriant subtle details in foreground are preserved very well.

Bond Behavior of FRCM Carbon Yarns Embedded in a Cementitious Matrix: Experimental and Numerical Results

2017 ◽  
Vol 747 ◽  
pp. 305-312 ◽  
Author(s):  
Jacopo Donnini ◽  
Giovanni Lancioni ◽  
Tiziano Bellezze ◽  
Valeria Corinaldesi
Keyword(s):  
Mechanical Performance ◽  
Stress Transfer ◽  
Polymer Coatings ◽  
Energy Functional ◽  
Numerical Results ◽  
Variational Model ◽  
Experimental Session ◽  
Cementitious Matrix ◽  
Pull Out ◽  
Multifilament Yarns

The use of inorganic cement based composite systems, known as Fiber Reinforced Cementitious Matrix (FRCM), is a very promising technique for retrofitting and strengthening the existing masonry or concrete structures. The effectiveness of FRCM systems is strongly related to the interface bond between inorganic matrix and fabric reinforcement, and, since the major weakness is often located on this interface, the study of stress-transfer mechanisms between fibers and matrix becomes of fundamental importance.FRCM are usually reinforced with uni-directional or bi-directional fabrics consisting of multifilament yarns made of carbon, glass, basalt or PBO fibers, disposed along two orthogonal directions. The difficulty of the mortar to penetrate within the filaments that constitute the fabric yarns and the consequent non-homogeneous stress distribution through the yarn cross section makes difficult to access the characterization of the composite material. The use of polymer coatings on the fibers surface showed to enhance the bond strength of the interface between fibers and mortar and, as a consequence, to improve the mechanical performance of the composite. The coating does not allow the mortar to penetrate within the filaments while is able to improve the bond between the two materials and to increase the shear stress transfer capacity at the interface.An experimental session of several pull out tests on carbon yarns embedded in a cementitious matrix was carried out. Different embedded lengths have been analyzed, equal to 20, 30 and 50 mm. The carbon yarns object of this study were pre-impregnated with a flexible epoxy resin enhanced with a thin layer of quartz sand applied on the surface.A variational model was proposed to evaluate the pull-out behaviour and failure mechanisms of the system and to compare numerical results to the experimental outcomes. Evolution of fracture in the yarn-matrix system is determined by solving an incremental energy minimization problem, acting on an energy functional which account for brittle failure of matrix and yarn, and for debonding at the yarn-matrix interface. The model was able to accurately describe the three phases of the pull-out mechanism, depending on the embedded length.

scholarly journals Shape-Aware Matching of Implicit Surfaces Based on Thin Shell Energies

2017 ◽  
Vol 18 (4) ◽  
pp. 891-927 ◽  
Author(s):  
José A. Iglesias ◽  
Martin Rumpf ◽  
Otmar Scherzer
Keyword(s):  
Thin Shell ◽  
Implicit Surfaces

scholarly journals Variational model of scoliosis

2018 ◽  
Vol 45 (2) ◽  
pp. 167-175
Author(s):  
Igor Popov ◽  
Nikita Lisitsa ◽  
Yuri Baloshin ◽  
Mikhail Dudin ◽  
Stepan Bober
Keyword(s):  
Mathematical Model ◽  
Variational Principles ◽  
Three Dimensional ◽  
Energy Functional ◽  
Variational Model ◽  
Spinal Diseases ◽  
Clear Understanding ◽  
History Of ◽  
Minimal Curves ◽  
Initiating Factors

Scoliosis, being one of the most widespread spinal diseases among children, has been studied extensively throughout the history of medicine, yet there is no clear understanding of its initiating factors and the mechanogenesis of the monomorphic three-dimensional deformation due to its polyetiological nature. We present a novel mathematical model of the process of emergence of the three-dimensional deformation of the human spine based on variational principles. Typical scoliosis geometry is assumed to be described as minimal curves of a particular energy functional, which are shown to closely resemble actual scoliosis. We investigate the numerical properties of the first stage of scoliosis, which is shown to have the highest influence on the development of the disease.

scholarly journals Fuzzy Bit-Plane-Dependence Region Competition

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2392
Author(s):  
Siukai Choy ◽  
Tszching Ng ◽  
Carisa Yu ◽  
Benson Lam
Keyword(s):  
Energy Functional ◽  
Superior Performance ◽  
Projection Algorithm ◽  
Variational Model ◽  
Fuzzy Membership Function ◽  
Probability Models ◽  
Closed Form Solutions ◽  
Additional Information ◽  
Minimization Procedure ◽  
Bit Plane

This paper presents a novel variational model based on fuzzy region competition and statistical image variation modeling for image segmentation. In the energy functional of the proposed model, each region is characterized by the pixel-level color feature and region-level spatial/frequency information extracted from various image domains, which are modeled by the windowed bit-plane-dependence probability models. To efficiently minimize the energy functional, we apply an alternating minimization procedure with the use of Chambolle’s fast duality projection algorithm, where the closed-form solutions of the energy functional are obtained. Our method gives soft segmentation result via the fuzzy membership function, and moreover, the use of multi-domain statistical region characterization provides additional information that can enhance the segmentation accuracy. Experimental results indicate that the proposed method has a superior performance and outperforms the current state-of-the-art superpixel-based and deep-learning-based approaches.

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