Similarity, Symmetry and Group Theoretical Methods in Mechanics - CURRENTLY NOT SCHEDULED
September 7, 2015 — September 11, 2015
- Ivailo Mladenov (Bulgarian Academy of Sciences, Sofia, Bulgaria)
- Jean-François Ganghoffer (Université de Lorraine, Nancy, France)
The aim of the course is to bring together researchers in mechanics, applied physics and applied mathematics who use similarity and symmetry analysis of engineering problems in both solid and fluid mechanics, researchers who are developing significant extensions of these methods implement, and numerical analysts who develop and use such methods in numerical schemes.
The powerfulness of the Lie group symmetry analysis has been extensively utilized, essentially to support the finding of analytic solutions to partial differential equations. For a given DE problem, one can algorithmically calculate its admitted point symmetries – transformations of dependent and independent variables that map a problem intoItself. Knowledge of admitted symmetries allows one to construct mappings relating DE systems, find out whether or not a given nonlinear DE system can be mapped into a linear system by an invertible transformation, and find exact (group-invariant or symmetrygenerated) solutions. Lie group analysis is of further interest in setting up numerical schemes preserving the group properties of an initial boundary value problem (BVP).
Symmetries have been historically relied upon to construct Lagrangian formulations in field theory. In the context of continuum solids mechanics, Lie groups have been applied to solve the Navier and the Lame equations, or, in a similar spirit and extending this view to dissipation, to partially solve the ideal plasticity equation, to formulate conservation laws and invariance relations, to analyze the kinematics of mechanisms, and more recently to formulate the constitutive laws and master response of materials with complex rheological behaviors.
Symmetry methods have a fundamental role in Lagrangian mechanics, Eshelbian mechanics, and nonlinear elasticity. The field of Eshelbian Mechanics (so called in the honor of the works of Eshelby, but also known as Configurational Mechanics), relies on translational symmetries in the material space, for writing field equations in terms of Eshelby stresses. Those symmetries extended to rotations and dilatations have been intensively used to construct the well known J-integrals. The concept of nonlocal symmetries allows to construct novel BVP in continuum mechanics (and group invariant solutions), involving potential variables, thereby extending the classical picture relying on the traditional Lagrangian and Eulerian viewpoints.
Symmetry methods are at the basis of methodologies for finding invariance relations of the BVP of continua obeying non dissipative and dissipative behaviors, including nonlinear elasticity, plasticity and creep. Lie symmetries are useful to find conservation laws in the analysis of Euler and Navier-Stokes equations for incompressible fluids: this particularly includes very recent results on new vorticity related conservation laws for Euler and Navier-Stokes equations and others which only exist in reduced dimensions such for as for plan or helically symmetric flows. The three “complete approaches” to statistical turbulence theory are an immediate consequence of Navier-Stokes equations. Beside the classical Lie symmetries stemming from Navier- Stokes equations, these sets of equations admit more Lie symmetries, named statistical symmetries. The involvement of Lie groups as a new predictive and systematic methodology to obtain invariance properties of materials is more recent. From the knowledge of the constitutive law of a given material, Lie symmetries are able to predict its response under various control conditions, and inversely to formulate a material’s constitutive law exploiting a postulated Lie group structure satisfying the symmetries involved in the experimental data.
The proposed course will reflect the organization of the Summer School on the same topics that took place in Varna, Bulgaria (June 7–12, 2013), jointly organized by I. Mladenov and J.F. Ganghoffer. Those topics were dealt with by three main speakers, leaders in the field, who made a pedagogical introduction and laid out key issues and concepts: G. Bluman (UBC, Vancouver, Canada), N. Ibragimov (Blekinge Institute of Technology, Karlskrona University, Sweden), and C-M. Marle (Univ. Pierre et Marie Curie and French Academy of Sciences, Paris). Two of these lecturers will deliver courses during the CISM session.
The course is mostly intended for Master students in mechanics or applied mathematics (or in physics, but having a sufficiently good level in mechanics, defined by the prerequisites), for PHD students, post-doctoral students, industrial researchers and engineers interested in the more practical use of symmetry methods. Permanent researchers willing to get an overview of the field are also welcome.