A domain decomposition approach to finite-epsilon homogenization of scalar transport in porous media

Modeling scalar transport by advection and diffusion in multiscale porous structures is a challenging problem, particularly in the preasymptotic regimes when non-Fickian effects are prominent. Mathematically, one of the main difficulties is to obtain macroscale models from the homogenization of conservation equations at microscale when epsilon, the ratio of characteristic lengthscales between the micro- and macroscale, is not extremely small compared to unity. Here, we propose the basis of a mathematical framework to do so. The focal idea is to decompose the spatial domain at pore-scale into a set of N subdomains to capture characteristic times associated with exchanges between these subdomains. At macroscale, the corresponding representation consists of a system of N coupled partial differential equations describing the transport of the spatially averaged scalar variable within each subdomain. Besides constructing the framework, we also compare numerically the results of our models to a complete resolution of the problem at the pore-scale, which shows great promises for capturing preasymptotic regimes, non-Fickian transport, and going toward finite-epsilon homogenization

Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations

It is generally accepted that the effective velocity of a viscous flow over a porous bed satisfies the Beavers-Joseph slip law. To the contrary, interface law for the effective stress has been a subject of controversy. Recently, a pressure jump interface law has been rigorously derived by Marciniak-Czochra and Mikeli\'c. In this paper, we provide a confirmation of the analytical result using direct numerical simulation of the flow at the microscopic level.Comment: 25 pages, preprin

International Conference on Nonlinear Differential Equations and Applications

Dear Participants, Colleagues and Friends It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA). This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics. The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

MODELLING AND IN VIVO MONITORING OF THE TIME DEPENDENT MECHANICAL PROPERTIES OF TISSUE ENGINEERING SCAFFOLDS

When organs and tissue fail either due to pre-existing disease progression or by accidental damage, current state of the art treatment involves the replacement of the damaged or diseased tissue with new donor derived organs/tissue. The limitations of these current approaches include a limited supply of tissue for treatments and the immune response of the patient’s own body against the new implanted tissue/organs. To solve these issues, tissue engineering aims to develop artificial analogs derived from a patient’s own cells instead of donor tissue/organs for treatment. To this end, a promising approach, known as scaffold-based tissue engineering, is to seed engineered constructs or scaffolds with cells to form artificial analogs, which then develop with time into new tissue/organs for implantation. The mechanical properties of the scaffold play a critical role in the success of scaffold-based treatments, as the scaffold is expected to provide a temporary support for the generation of new tissue/organs without causing failure at any time during the treatment process. It is noted that due to the degradation of scaffold in the treatment process, the mechanical properties of the scaffold are not constant but change with time dynamically. This raises two scientific issues; one is the representation of the time-dependent mechanical properties and the other one is the monitoring of these properties, especially in the in vivo environments (i.e., upon the implantation of scaffolds into animal/patient bodies). To address these issues, this research is aimed at performing a novel study on the modelling and in vivo monitoring of the time dependent mechanical properties of tissue engineering scaffolds. To represent the time-dependent mechanical properties of a scaffold, a novel model based on the concept of finite element model updating is developed. The model development involves three steps: (1) development of a finite element model for the effective mechanical properties of the scaffold, (2) parametrizing the finite element model by selecting parameters associated with the scaffold microstructure and/or material properties, which vary with scaffold degradation, and (3) identifying selected parameters as functions of time based on measurements from the tests on the scaffold mechanical properties as they degrade. To validate the developed model, scaffolds were made from the biocompatible polymer polycaprolactone (PCL) mixed with hydroxyapatite (HA) nanoparticles and their mechanical properties were examined in terms of the Young modulus. Based on the bulk degradation exhibited by the PCL/HA scaffold, the molecular weight was selected for model updating. With the identified molecular weight, the finite element model v developed was effective for predicting the time-dependent mechanical properties of PCL/HA scaffolds during degradation . To monitor and characterize scaffold mechanical properties in vivo, novel methods based on synchrotron-based phase contrast imaging and finite element modeling were developed. The first method is to represent the scaffold mechanical properties from the measured deflection. In this method, the phase contrast imaging is used to characterize the scaffold deflection caused by ultrasound radiation forces; and the finite element modelling is used to represent the ultrasonic loading on the scaffold, thus predicting the mechanical properties from the measured deflection. The second method is to characterize the scaffold degradation due to surface erosion, which involves the remote sensing of the time dependent morphology of tissue scaffolds by phase contrast imaging and the estimation of time dependent mass loss of the scaffolds from the sensed morphology. The last method is to relate the elastic mechanical property and nonlinear stress-strain behavior to the scaffold geometry, both changing with time during surface erosion. To validate the above methods, scaffolds was made from varying biomaterials (PLGA and PCL) and their mechanical properties (degradation, mass loss, and elastic modulus) were examined experimentally. The results obtained illustrate the methods developed in this research are effective to monitor and characterize scaffold mechanical properties. The significance of this research is that the model developed for the scaffold mechanical properties can be used in the design of scaffolds with the desired mechanical properties, instead of the trial and error methods typical in current scaffold design; and that these novel monitoring methods based on synchrotron imaging can be used to characterize the scaffold time-dependent mechanical properties in the in vivo environments, representing an important advance in tissue engineering

Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)

The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy. The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism