Two-field variational formulations for a class of nonlinear mechanical models

A nonlinear boundary value problem arising from continuum mechanics is considered. The nonlinearity of the model arises from the constitutive law which is described by means of the subdifferential of a convex constitutive map. A bipotential [Formula: see text], related to the constitutive map and its Fenchel conjugate, is considered. Exploring the possibility to rewrite the constitutive law as a law governed by the bipotential [Formula: see text], a two-field variational formulation involving a variable convex set is proposed. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed.

Nonpotentiality of a diffusion system and the construction of a semi-bounded functional

The wide prevalence and the systematic variational principles are used in mathematics and applications due to a series of remarkable consequences among which the possibility to establish the existence of the solutions of the initial equations, and the determination of stable approximations of the solutions of the considered equations by the so-called variational methods. In this connection, it is natural for a given system of equations to investigate the problem of the existence of its variational formulations. It can be considered as the inverse problem of the calculus of variations. The main goal of this work is to study this problem for a diffusion system of partial differential equations. A key object is the criterion of potentiality. On its ground, the nonpotentiality of the operator of the given boundary value problem with respect to the classical bilinear form is proved. This system does not admit a matrix variational multiplier of the given form. Thus, the diffusion system cannot be deduced from the classical Hamilton’s principle. We posed the question that whether there exists a functional semi-bounded on solutions to the boundary value problem. We have done the algorithm of the constructive determination of such a functional. The main value of constructed functional action will be in applications of direct variational methods.

Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.

fenicsR13

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.

A numerical study of variational discretizations of the Camassa–Holm equation

We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa–Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.

A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.