Arbitrary order monotonic finite-volume schemes for 2D elliptic problems

Monotonicity is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order monotonic method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensorvalued diffusion coefficient, while keeping the order of convergence. We assess the new scheme on several test problems

Arbitrary-order monotone finite-volume schemes for 1D elliptic problems

International audienceWhen solving numerically an elliptic problem, it is important in most applications that the scheme used preserves the positivity of the solution. When using finite volume schemes on deformed mesh, the question has been solved rather recently. Such schemes are usually (at most) second order convergent, and nonlinear. On the other hand, many high-order schemes have been proposed, that do not ensure positivity of the solution. In this paper we propose a very high-order monotone (that is, positivity preserving) numerical method for elliptic problems in 1D. We prove that this method converges to an arbitrary order and is indeed monotone. We also show how to handle discontinuous sources or diffusion coefficients, while keeping the order of convergence. We assess the new scheme, on several test problems, with arbitrary (regular, distorted, random) meshes

Monotonic diamond and DDFV type finite-volume schemes for 2D elliptic problems

The DDFV (Discrete Duality Finite Volume) method is a finite volume scheme mainly dedicated to diffusion problems, with some outstanding properties. This scheme has been found to be one of the most accurate finite volume methods for diffusion problems. In the present paper, we propose a new monotonic extension of DDFV, which can handle discontinuous tensorial diffusion coefficient. Moreover, we compare its performance to a diamond type method with an original interpolation method relying on polynomial reconstructions. Monotonicity is achieved by adapting the method from [44, 19, 49, 18] to our schemes. Such a technique does not require the positiveness of the secondary unknowns. We show that the two new methods are second-order accurate and are indeed monotonic on some challenging benchmarks as a Fokker-Planck problem

Looking over Liquid Silicone Rubbers: 2. Mechanical properties vs. network topology

International audienceIn the previous paper of this series, eight formulations were analyzed under their un-cross-linked forms to relate Liquid Silicone Rubber (LSR) chemical compositions to material network topologies. Such topologies were confirmed by swelling measurements and hardness evaluation on vulcanized samples. In this article, characterizations of cross-linked materials is further done using different mechanical measurements on final materials, including dynamic mechanical analysis, compression set, stress-strain behavior and tear resistance. It was shown that the compression set value is mainly related to the chains motion: increasing the filler-polymer interactions and/or decreasing the dangling/untethered chains content positively impact the compression resistance. Elongation at break depends on the molar mass between cross-linking points, showing an optimum value set at around 20,000 g.mol-1 i.e. the critical mass between entanglements. The distribution of elastic strands into the network has strong implications on the stress-strain curves profiles. By generating bimodal networks, the ultimate properties are enhanced. The materials cured by hydride addition on vinyl groups catalyzed by peroxide exhibit poorer compression set and tensile strength values, respectively due to post-cross-linking reaction and broad polydispersity index of elastic network chains. A final discussion relates these final mechanical properties to network topologies

Maillage de Delaunay d'un polyedre convexe en dimension d. Extension a un polyedre quelconque

CNRS 14802 E / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc