A stabilized multigrid solver for hyperelastic image registration

Image registration is a central problem in a variety of areas involving imaging techniques and is known to be challenging and ill‐posed. Regularization functionals based on hyperelasticity provide a powerful mechanism for limiting the ill‐posedness. A key feature of hyperelastic image registration approaches is their ability to model large deformations while guaranteeing their invertibility, which is crucial in many applications. To ensure that numerical solutions satisfy this requirement, we discretize the variational problem using piecewise linear finite elements, and then solve the discrete optimization problem using the Gauss–Newton method. In this work, we focus on computational challenges arising in approximately solving the Hessian system. We show that the Hessian is a discretization of a strongly coupled system of partial differential equations whose coefficients can be severely inhomogeneous. Motivated by a local Fourier analysis, we stabilize the system by thresholding the coefficients. We propose a Galerkin‐multigrid scheme with a collective pointwise smoother. We demonstrate the accuracy and effectiveness of the proposed scheme, first on a two‐dimensional problem of a moderate size and then on a large‐scale real‐world application with almost 9 million degrees of freedom.     

Elastic lattices: equilibrium, invariant laws and homogenization

In the recent years, lattice modelling proved to be a topic of renewed interest. Indeed, fields as distant as chemical modelling and biological tissue modelling use network models that appeal to similar equilibrium laws. In both cases, obtaining an equivalent continuous model allows to simplify numerical procedures. We define the basic properties of lattices: elasticity, frame-indifference, hyperelasticity. We determine rigorously the form that constitutive laws undertake under frame-indifference and hyperelasticity assumptions. Finally, we describe an homogenization technique designed for discrete structures that provides a limit continuum mechanics model and, in the special case of hexagonal lattices, we investigate the symmetry properties of the limit constitutive law.      

Application of a viscoplastic material model on a combined quasi-static-electromagnetic forming process

The prediction and simulation of material behavior by finite element methods has become indispensable. Furthermore, various phenomena in forming processes lead to highly differing results. In this work, we have investigated the process chain on a cross-shaped cup in cooperation between the Institute of Applied Mechanics (IFAM) of the RWTH Aachen and the Institute of Forming Technology and Lightweight Construction (IUL) of the TU Dortmund. A viscoplastic material model based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity has been used [1,2]. The finite strain constitutive model combines nonlinear kinematic and isotropic hardening and is derived in a thermodynamically consistent setting. This anisotropic viscoplastic model is based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong-Frederick kinematic hardening. The constitutive equations of the material model are integrated in an explicit manner and implemented as a user material subroutine in the commercial finite element package LS-DYNA with the electromagnetical module. The aim of the work is to show the increasing formability of the sheet by combining quasi-static deep drawing processes with high speed electromagnetic forming. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness of Solutions to Hyperbolic Balance Laws in Several Space Dimensions

We introduce the concept of modular family of entropy vectors for general r x r systems of balance laws. We then define the notion of entropy solution to the Cauchy problem compatible with the modular family, assuming that the system admits such a family. We show that this concept reduces to the usual one, introduced by S.N. Kruzkov, in the scalar case and when we restrict ourself to Lipschitz continuous solutions. We also show how the compatibility condition appears in the cases of symmetric systems, 2 x 2 psystems and equations of hyperelasticity and electromagnetism, the last two considered earlier by C.M. Dafermos. We demonstrate that generalized Oleinik's condition implies our compatibility condition in the case of symmetric systems. We prove the uniqueness and stability relatively to the initial data of the entropy solutions compatible with the modular family. This theorem has as corollary uniqueness results due to O.A. Oleinik, S.N. Kruzkov, A.E. Hurd, R.J. DiPerna and C. Bardos. We give also two uniqueness theorems to solutions of equations of hyperelasticity and electromagnetism.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

Applying weak equivalence of categories between partial map and pointed set against changing the condition of 2‐arms bandit problem

The main interest of the organizational learning is that the agent balances between exploring and exploiting. When the agent decides the answer in finite time, it gives rise to the problem that is trade‐off between exploring and exploiting. This problem never arises in the given enough time. The agent must decide answer from his imperfect information, when the time is given finite for the task. 2‐arms bandit problem is often taken into consideration in this problem. In previous article, we proposed the model for weak identity that is dynamically changing the hierarchy of his knowledge. Recently, the method of the heuristic model is attracted to approach this problem. In this article, we proposed different approaches for 2‐arms bandit problem, and the model we proposed can adapt to the environment when the condition of the problem is changed and shows another approach to the organizational learning. © 2010 Wiley Periodicals, Inc. Complexity 16: 10‐21, 2011     

Identification of source term for the Rayleigh‐Stokes problem with Gaussian random noise

In this paper, we study an inverse source problem for the Rayleigh‐Stokes problem for a generalized second‐grade fluid with a fractional derivative model. The problem is severely ill‐posed in the sense of Hadamard. To regularize the unstable solution, we apply a general filter method for constructing regularized solution, and the convergence rate of this method also has been investigated.     

Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors

In this paper, we discussed a general multidimensional nonisentropic hydrodynamical model for semiconductors with small momentum relaxation time. The model is self‐consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. With the help of the Maxwell‐type iteration, we prove that, as the relaxation time tends to zero, periodic initial‐value problem of certain scaled multidimensional nonisentropic hydrodynamic model has a unique smooth solution existing in the time interval where the corresponding classical drift‐diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the drift‐diffusion models from the nonisentropic hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.     

An epidemiology model suggested by yellow fever

In this work, we construct and analyze a nonlinear reaction–diffusion epidemiology model consisting of two integral‐differential equations and an ordinary differential equation, which is suggested by various insect borne diseases, for example, Yellow Fever. We begin by defining a nonlinear auxiliary problem and establishing the existence and uniqueness of its solution via a priori estimates and a fixed point argument, from which we prove the existence and uniqueness of the classical solution to the analytic problem. Next, we develop a finite‐difference method to approximate our model and perform some numerical experiments. We conclude with a brief discussion of some subsequent extensions. Copyright © 2011 John Wiley & Sons, Ltd.     

THE MITRA‐WAN FORESTRY MODEL: A DISCRETE‐TIME OPTIMAL CONTROL PROBLEM

In this paper, we present some new properties of the Mitra‐Wan forestry model written as a discrete‐time optimal control problem. For this problem, the set of stationary states is characterized. For the optimal long‐run management, we consider the following optimality criteria: average optimality, good control policies, bias optimality, and overtaking optimality. We establish relationships between these criteria and show that the value of average optimal policies is constant and equals the value in the optimal stationary state.     

On Inverse Form Finding For Anisotropic Materials

The main goal of this paper is to extend the work of Govindjee and Mihalic (1996) to anisotropic hyperelasticity that is formulated in the logarithmic strain space involving a spectral decomposition of the right Cauchy-Green tensor. Beside the theoretical aspects, we present a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

A fourth‐order parabolic equation in two space dimensions

In this paper, we consider an initial‐boundary problem for a fourth‐order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd.     

On the linearization of operators related to the full waveform inversion in seismology

In this work, we analyze the parameter‐to‐solution map of the acoustic wave equation with respect to its parameters wave speed and mass density. This map is a mathematical model for the seismic inverse problem where one wants to recover the parameters from measurements of the acoustic potential. We show its complete continuity and Fréchet differentiability. To this end, we provide necessary existence, stability, and regularity results. Moreover, we discuss various implications of our findings on the inverse problem and comment on the Born series. Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional solidification with two possible crystallization states: Existence of solutions with flow in the melt

In this paper we discuss the existence of solutions of a system of nonlinear and singular partial differential equations constituting a phase field model with convection for non‐isothermal solidification/melting of certain metallic alloys in the case where two different kinds of crystallization are possible. Each one of these crystallization states is described by its own phase field, while the liquid phase is described by another one. The model also allows the occurrence of fluid flow in non‐solid regions, which are a priori unknown, and then we have a free‐boundary value problem. Thus, the model relates the evolutions of these three phase fields, the temperature of the solidification/melting process and the fluid flow in non‐solid regions. Copyright © 2009 John Wiley & Sons, Ltd.     

A study of the fractional‐order mathematical model of diabetes and its resulting complications

Diabetes is a worldwide problem that affects one of every 11 persons nowadays. The IDF Diabetes Atlas (Eighth edition, 2017) states that approximately 415 million people in the world are living with the disease and that this number will rise to 629 million by the year 2045. It is a very serious problem of the world. A major part of the world population is affected by this disease and its resulting complications. In this paper, we propose to investigate a fractional‐order model of diabetes and its resulting complications. The mathematical model's parameters define the population of diabetic patients and those who are diabetic with complications at a given time t. We have also discussed the existence, uniqueness, and stability of the fractional‐order model, which we consider here. We make use of the homotopy decomposition method (HDM) in order to solve the problem.