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Auxiliary Problem (auxiliary + problem)
Selected AbstractsHomogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluidMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003J. L. Ferrin We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source] An exact sinusoidal beam finite elementPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008Zdzislaw Pawlak The purpose of the paper is to derive an efficient sinusoidal thick beam finite element for the static analysis of 2D structures. A two,node, 6,DOF curved, sine,shape element of a constant cross,section is considered. Effects of flexural, axial and shear deformations are taken into account. Contrary to commonly used curvilinear co,ordinates, a rectangular co,ordinates system is used in the present analysis. First, an auxiliary problem is solved: a symmetric clamped,clamped sinusoidal arch subjected to unit nodal displacements of both supports is considered using the flexibility method. The exact stiffness matrix for the shear,flexible and compressible element is derived. Introduction of two parameters "n" and "t" enables the identification of shear and membrane influences in the element stiffness matrix. Basing on the principle of virtual work a full set of 18 shape functions related to unit support displacements is derived (total rotations of cross,sections, tangential and normal displacements along the element). The functions are found analytically in the closed form. They are functions of one linear dimensionless coordinate of x,axis and depend on one geometrical parameter of sinusoidal arch, height/span ratio "c" and on physical and geometrical properties of the element cross,section. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A numerical approximation of the thermal coupling of fluids and solidsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2009Javier Principe Abstract In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context. Copyright © 2008 John Wiley & Sons, Ltd. [source] On the structural stability of thermoelastic model of porous mediaMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2008Stan Chiri Abstract In the present paper we study the structural stability of the mathematical model of the linear thermoelastic materials with voids. We prove that the solutions of problems depend continuously on the constitutive quantities, which may be subjected to error or perturbations in the mathematical modelling process. Thus, we assume to have changes in the various coupling coefficients of the model and then we establish estimates of continuous dependence of solutions. We have to outline that such estimates play a central role in obtaining approximations to these kinds of problems. To derive a priori estimates for a solution we first establish appropriate bounds for the solutions of certain auxiliary problems. These are achieved by means of so-called Rellich-like identities. We also investigate how the solution in the coupled model behaves as some coupling coefficients tend to zero. Copyright © 2007 John Wiley & Sons, Ltd. [source] | ||||||||||