Solving the Homogeneous Isotropic Linear Elastodynamics Equations Using Potentials and Finite Elements. The Case of the Rigid Boundary Condition

International audienceIn this article, elastic wave propagation in a homogeneous isotropic elastic medium with rigid boundary is considered. A method based on the decoupling of pressure and shear waves via the use of scalar potentials is proposed. This method is adapted to a finite elements discretization, which is discussed. A stable, energy preserving numerical scheme is presented, as well as 2D numerical results.Dans cet article, on s'intéresse à la propagation d'ondes élastiques dans un matériau élastique homogène et isotrope avec une condition d'encastrement. On propose une méthode basée sur le découplage des ondes de pression et de cisaillement via l'utilisation de potentiels scalaires. Cette méthode est adaptée à une discrétisation éléments finis, et on présente un schéma stable préservant une énergie discrète et des résultats numériques en 2D

Introduction and study of fourth order theta schemes for linear wave equations

International audienceA new class of high order, implicit, three time step schemes for semi-discretized wave equations is introduced and studied. These schemes are constructed using the modified equation approach, generalizing the $\theta$-scheme. Their stability properties are investigated via an energy analysis, which enables us to design super convergent schemes and also optimal stable schemes in terms of consistency errors. Specific numerical algorithms for the fully discrete problem are tested and discussed, showing the efficiency of our approach compared to second order $\theta$-schemes.Nous introduisons et étudions une nouvelle classe de schémas d'ordre élevé, implicites et à trois pas de temps pour les équations d'ondes semi-discrètes. Ces schémas sont construits sur le principe de l'équation modifiée et généralisent le theta-schéma. Nous étudions leurs propriétés de stabilité via des techniques d'énergie, ce qui nous permet de concevoir des schémas super convergents ainsi que des schémas optimaux en terme d'erreur de consistance. Des algorithmes numériques de résolution pour le problème totalement discrétisé sont testés et critiqués, montrant la supériorité de notre approche comparée aux theta shémas classiques du second ordre

Introduction and study of fourth order theta schemes for linear wave equations

A new class of high order, implicit, three time step schemes for semi-discretized wave equations is introduced and studied. These schemes are constructed using the modified equation approach, generalizing the $\theta$-scheme. Their stability properties are investigated via an energy analysis, which enables us to design super convergent schemes and also optimal stable schemes in terms of consistency errors. Specific numerical algorithms for the fully discrete problem are tested and discussed, showing the efficiency of our approach compared to second order $\theta$-schemes.Nous introduisons et étudions une nouvelle classe de schémas d'ordre élevé, implicites et à trois pas de temps pour les équations d'ondes semi-discrètes. Ces schémas sont construits sur le principe de l'équation modifiée et généralisent le theta-schéma. Nous étudions leurs propriétés de stabilité via des techniques d'énergie, ce qui nous permet de concevoir des schémas super convergents ainsi que des schémas optimaux en terme d'erreur de consistance. Des algorithmes numériques de résolution pour le problème totalement discrétisé sont testés et critiqués, montrant la supériorité de notre approche comparée aux theta shémas classiques du second ordre

Mathematical and numerical modelling of piezoelectric sensors

International audienceThe present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element / finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented

Energy decay and stability of a perfectly matched layer for the wave equation

In [25,26], a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and L^2 stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations. Numerical results validate the theory

Desmoplastic small round cell tumor: impact of 18F-FDG PET induced treatment strategy in a patient with long-term outcome

The desmoplastic small round cell tumor (DSRCT) is an uncommon and highly aggressive cancer. The role of 18F-FDG PET in management of DSRCT is little reported. We report a case of metastasized abdominal DSRCT detected in a 43-year old patient whose diagnostic and therapeutic approaches were influenced by 18F-FDG PET-CT. The patient is still alive ten years after diagnosis. 18F-FDG PET-CT seems to be a useful method for assessing therapeutic efficiency and detecting early recurrences even in rare malignancies such as DSRCT

Asymptotic analysis of abstract two-scale wave propagation problems

This work addresses the mathematical analysis, by means of asymptotic analysis, of linear wave propagation problems involving two scales, represented by a single small parameter, and written in an abstract setting. This abstract setting is defined using linear operators in Hilbert spaces and also enters the framework of semi-group theory. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem