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4D Embryogenesis image analysis using PDE methods of image processing

Paul Bourgine, Róbert Čunderlík, Olga Drblíková-Stašová, Karol Mikula, Mariana Remešíková, Nadine Peyriéras, Barbara Rizzi, Alessandro Sarti (2010)

Kybernetika

In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely...

A balanced finite-element method for an axisymmetrically loaded thin shell

Norbert Heuer, Torsten Linss (2024)

Applications of Mathematics

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.

A computational approach to fractures in crystal growth

Matteo Novaga, Emanuele Paolini (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the present paper, we motivate and describe a numerical approach in order to detect the creation of fractures in a facet of a crystal evolving by anisotropic mean curvature. The result appears to be in accordance with the known examples of facet-breaking. Graphical simulations are included.

A continuous finite element method with face penalty to approximate Friedrichs' systems

Erik Burman, Alexandre Ern (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

A continuous finite element method to approximate Friedrichs' systems is proposed and analyzed. Stability is achieved by penalizing the jumps across mesh interfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence rates in the graph norm and suboptimal of order ½ convergence rates in the L2-norm. A variant of the method specialized to Friedrichs' systems associated with elliptic PDE's in mixed form and reducing the number...

A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation

El-Hassan Benkhira, Rachid Fakhar, Youssef Mandyly (2021)

Applications of Mathematics

We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...

A convergence result for finite volume schemes on Riemannian manifolds

Jan Giesselmann (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law u t + ∇ g · f ( x , u ) = 0 on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a h 1 4 convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to h 1 2 .

A discretization method for the problem of a membrane constrained by elastic obstacle

Aldo Maceri (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro sono dati alcuni modelli matematici per il problema di contatto tra una membrana ed un suolo od ostacolo elastico. Viene costruita una approssimazione lineare a tratti della soluzione e, tramite una disequazione variazionale discreta, se ne dà il corrispondente teorema di convergenza.

A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

Yongxing Shen, Adrian J. Lew (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields...

A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity

Yongxing Shen, Adrian J. Lew (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent...

A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

Yongxing Shen, Adrian J. Lew (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields...

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