Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations

We prove a multiplicity result of periodic solutions for a system of second order differential equations having asymmetric nonlinearities. The proof is based on a recent generalization of the Poincar\ue9\u2013Birkhoff fixed point theorem provided by Fonda and Ure\uf1a

A numerical study of dynamic capillary pressure effect for supercritical carbon dioxide-water flow in porous domain

This is the accepted version of the following article: DAS, D.B. ... et al., 2014. A numerical study of dynamic capillary pressure effect for supercritical carbon dioxide-water flow in porous domain. AIChE Journal, 60 (12), pp. 4266-4278, which has been published in final form at http://dx.doi.org/10.1002/aic.14577Numerical simulations for core-scale capillary pressure (Pc)–saturation (S) relationships have been conducted for a supercritical carbon dioxide-water system at temperatures between 35°C and 65°C at a domain pressure of 15 MPa as typically expected during geological sequestration of CO2. As the Pc-S relationships depend on both S and time derivative of saturation (∂S / ∂t) yielding what is known as the ‘dynamic capillary pressure effect’ or simply ‘dynamic effect’, this work specifically attempts to determine the significance of these effects for supercritical carbon dioxide-water flow in terms of a coefficient, namely dynamic coefficient (τ). The coefficient establishes the speed at which capillary equilibrium for supercritical CO2-water flow is reached. The simulations in this work involved the solution of the extended version of Darcy’s law which represents the momentum balance for individual fluid phases in the system, the continuity equation for fluid mass balance, as well as additional correlations for determining the capillary pressure as a function of saturation, and the physical properties of the fluids as a function of temperature. The simulations were carried for 3D cylindrical porous domains measuring 10 cm in diameter and 12 cm in height. τ was determined by measuring the slope of a best-fit straight line plotted between (i) the differences in dynamic and equilibrium capillary pressures (Pc,dyn – Pc,equ) against (ii) the time derivative of saturation (dS/dt), both at the same saturation value. The results show rising trends for τ as the saturation values reduce, with noticeable impacts of temperature at 50% saturation of aqueous phase. This means that the time to attain capillary equilibrium for the CO2-water system increases as the saturation decreases. From a practical point view, it implies that the time to capillary equilibrium during geological sequestration of CO2 is an important factor and should be accounted for while simulating the flow processes, e.g., to determine the CO2 storage capacity of a geological aquifer. In this task, one would require both the fundamental understanding of the dynamic capillary pressure effects for supercritical CO2-water flow as well as τ values. These issues are addressed in this article

Weak periodic solutions of the boundary value problem for nonlinear heat equation

summary:The paper deals with the existence of periodic solutions of the boundary value problem for nonlinear heat equation, where various types of nonlinearities are considered. The proofs are based on the investigation of Liapunov-Schmidt bifurcation system via Leray-Schauder degree theory

A reduced basis method for parametrized variational inequalities applied to contact mechanics

International audienceWe investigate new developments of the Reduced-Basis (RB) method for parametrized optimization problems with nonlinear constraints. We propose a reduced-basis scheme in a saddle-point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced-basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone-projected greedy algorithm that conserves the non-negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using non-matching meshes. The numerical examples confirm the efficiency of the reduction strategy