Zeitschrift fur Angewandte Mathematik und Physik最新文献

We provide a new derivation of the quasi-static equations of Biot's poroelasticity from the microstructure via the asymptotic (periodic) homogenisation method (AHM) by assuming intrinsic incompressibility of both an isotropic, linear elastic solid and a low Reynolds' number Newtonian fluid, in a small deformations regime. This is done by starting from a fluid-structure interaction (FSI) problem between the two phases at the pore scale, and by introducing both a solid and a fluid pressure, as both phases are equipped with an incompressibility constraint. Upscaling by the AHM then results in the expected Biot's equation at the macroscopic scale, with coefficients which are to be computed by solving non-standard periodic cell problems at the pore scale. These latter differ from the ones arising from classical derivations of poroelasticity via the AHM, which are typically obtained by assuming that the elastic phase is compressible, and are characterised by a saddle point structure which is inherited from the equations governing the original FSI problem. The proposed approach, which is new and cannot be derived as a particular case of existing formulations, means that the poroelastic governing equations for intrinsic incompressible phases are obtained without performing any "a posteriori" assumption on the macroscale coefficients, as these latter are typically employed based on physical arguments rather than following from a rigorous analysis of the properties of the pore scale cell problems. The advantages of the current formulation for incompressible solids are as follows. (a) The formulation is derived for two genuinely incompressible phases and in particular for an incompressible solid, which means that a reduced number of input parameters is required to compute the effective stiffness. In the case of pore scale isotropy, this means that only the shear modulus is to be provided. (b) The pore scale cell problems can be solved without approximating the pore scale elastic properties to that of an incompressible solid, i.e. in the case of isotropy, no additional errors are to be introduced by utilising approximate values of the Poisson's ratio (which in a compressible formulation can be close to, but not identical to, 0.5).

In this work, we investigate data-driven elasticity problems defined on a closed interval of the real line that are spatially discretized by means of the finite element method. This one-dimensional setting allows us to gain a deeper understanding of the underlying discrete-continuous quadratic optimization problems. We provide an in-depth analysis of their structural properties and prove their global solvability. Based on this analysis, we propose a new structure-specific initialization for a solution strategy relying on an alternating direction method, and we prove that it is globally optimal in certain symmetric cases. Finally and to support our formal mathematical analysis, we also provide a series of examples that show the benefits of this kind of approach and briefly illustrate the challenges when dealing with real experimental data.

This work explores diffusion with scale-dependent coefficients, starting from a general advection-diffusion framework from a theoretical standpoint, and then focusing numerically on a purely diffusive regime. Advection-diffusion processes are central to modeling transport phenomena in natural and engineered systems. However, classical models often fail to capture the complexities of systems with spatial and temporal variability. In this work, we present a multiscale advection-diffusion model that incorporates time-dependent diffusion coefficients and spatially inhomogeneous, multiscale body forces. Using the asymptotic homogenization technique, we derive a macroscopic equation that reflects the evolution of transport properties across multiple scales, accounting for both spatial and temporal variations. A key contribution of this study is the formulation of new cell problems associated with the dual time-dependence of the diffusion coefficient and the multiscale forces, which lead to the introduction of additional source terms. Furthermore, we incorporate a novel source term arising from the nonzero divergence of the advective velocity field, which modifies the effective macroscopic advection velocity to capture source and sink effects at the microscale. We apply this model to describe water molecule diffusion in packed erythrocytes, a system exhibiting dual time scales, and by showing how our approach captures the temporal evolution of transport under dynamic diffusion.

We study the essential spectrum, which corresponds to inertia-gravity modes, of the system of equations governing a rotating and self-gravitating gas planet. With certain boundary conditions, we rigorously and precisely characterize the essential spectrum and show how it splits from the portion of the spectrum corresponding to the acoustic modes. The fundamental mathematical tools in our analysis are a generalization of the Helmholtz decomposition and the Lopantinskii conditions.

We investigate the evolution of a two-phase viscoelastic material at finite strains. The phase evolution is assumed to be irreversible: One phase accretes in time in its normal direction, at the expense of the other. Mechanical response depends on the phase. At the same time, growth is influenced by the mechanical state at the boundary of the accreting phase, making the model fully coupled. This setting is inspired by the early stage development of solid tumors, as well as by the swelling of polymer gels. We formulate the evolution problem by coupling the balance of momenta in weak form and the growth dynamics in the viscosity sense. Both a diffused- and a sharp-interface variant of the model are proved to admit solutions and the sharp-interface limit is investigated.