A family of one-dimensional(1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding ...A family of one-dimensional(1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method(RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.展开更多
Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency ...Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency Biot’s equations(in the porous medium).Interface conditions are introduced to model various hydraulic contacts between the two media:open pores,sealed pores,and imperfect pores.Well-posedness of the initial-boundary value problem is proven.Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context:a fourth-order ADER scheme with Strang splitting for timemarching;a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory;and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution.Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions,demonstrating the accuracy of the approach.展开更多
基金Project supported by the Desenvolvimento e Aplicaoes de Mtodos Matemticos de Homogeneizaao(CAPES)(No.88881.030424/2013-01)the Homogeneizao Reiterada Aplicada a Meios Dependentes de Múltiplas Escalas con Contato Imperfeito Entre as Fases(CNPq)(Nos.450892/2016-6and 303208/2014-7)the Caracterizacin de Propiedades Efectivas de Tejidos Biolgicos Sanos y Cancerosos(CONACYT)(No.2016–01–3212)
文摘A family of one-dimensional(1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method(RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.
文摘Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media.Wave propagation is described by the usual acoustics equations(in the fluid medium)and by the low-frequency Biot’s equations(in the porous medium).Interface conditions are introduced to model various hydraulic contacts between the two media:open pores,sealed pores,and imperfect pores.Well-posedness of the initial-boundary value problem is proven.Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context:a fourth-order ADER scheme with Strang splitting for timemarching;a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory;and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution.Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions,demonstrating the accuracy of the approach.
