Geometrical model of lobular structure and its importance for the liver perfusion analysis

A convenient geometrical description of the microvascular network is necessary for computationally efficient mathematical modelling of liver perfusion, metabolic and other physiological processes. The tissue models currently used are based on the generally accepted schematic structure of the parenchyma at the lobular level, assuming its perfect regular structure and geometrical symmetries. Hepatic lobule, portal lobule, or liver acinus are considered usually as autonomous functional units on which particular physiological problems are studied. We propose a new periodic unit—the liver representative periodic cell (LRPC) and establish its geometrical parametrization. The LRPC is constituted by two portal lobulae, such that it contains the liver acinus as a substructure. As a remarkable advantage over the classical phenomenological modelling approaches, the LRPC enables for multiscale modelling based on the periodic homogenization method. Derived macroscopic equations involve so called effective medium parameters, such as the tissue permeability, which reflect the LRPC geometry. In this way, mutual influences between the macroscopic phenomena, such as inhomogeneous perfusion, and the local processes relevant to the lobular (mesoscopic) level are respected. The LRPC based model is intended for its use within a complete hierarchical model of the whole liver. Using the Double-permeability Darcy model obtained by the homogenization, we illustrate the usefulness of the LRPC based modelling to describe the blood perfusion in the parenchyma.

A wideband and directive metasurface FPC antenna with toroidal metal structure loading

In this paper, a novel metasurface-based Fabry−Perot cavity antenna loaded with toroidal metal structures is presented. The antenna is compact, wideband, and has high directivity and a high front-to-back ratio. The idea of the antenna is based on loading a microstrip narrow band patch antenna resonating at 4.5 GHz by a single layer metasurface superstrate and with a toroidal metal structure. The metasurface superstrate comprises a periodic array of square patch cells. Compared to conventional microstrip antenna, the front to back lobe ratio is increased from 7 to 20 dB and the directivity is increased by 7 dB. Also, the antenna impedance bandwidth is 34% which is increased four times. This is the first-ever antenna with enhanced bandwidth and directivity using a single layer of metasurface and that too made up of periodic cell array and has application as an energy harvester.

Metamaterial plate with an arrangement of different resonators

Local resonant metamaterials have been widely studied for vibration suppression in the last 20 years. They produce band gaps, which are frequency regions where the wave is not allowed to propagate. They are an alternative to reduce vibration levels at lower frequencies when compared to phononic crystals, which require larger periodic cells to create band gaps at lower frequencies. The most common configuration for a local resonant metamaterial is a periodic cell of a known structure with one attached resonator. In this study, a plate with a periodic cell using two different resonators is analyzed. Some configurations of mass and stiffness for the two resonators will be discussed to pursue the best compromise between a wider band gap and a more considerable vibration attenuation. The dispersion relation for the proposed metamaterial unit cell will be calculated using the Wave Finite Element Method to evaluate these configurations. The frequency response function for a finite structure with the proposed arrangement will also be calculated using the Finite Element Method to compare the results.

Double poroelasticity derived from the microstructure

We derive the balance equations for a double poroelastic material which comprises a matrix with embedded subphases. We assume that the distance between the subphases (the local scale) is much smaller than the size of the domain (the global scale). We assume that at the local scale both the matrix and subphases can be described by Biot’s anisotropic, heterogeneous, compressible poroelasticity (i.e. the porescale is already smoothed out). We then decompose the spatial variations by means of the two-scale homogenization method to upscale the interaction between the poroelastic phases at the local scale. This way, we derive the novel global scale model which is formally of poroelastic-type. The global scale coefficients account for the complexity of the given microstructure and heterogeneities. These effective poroelastic moduli are to be computed by solving appropriate differential periodic cell problems. The model coefficients possess properties that, once proved, allow us to determine that the model is both formally and substantially of poroelastic-type. The properties we prove are a) the existence of a tensor which plays the role of the classical Biot’s tensor of coefficients via a suitable analytical identity and b) the global scale scalar coefficient $$\bar{\mathcal {M}}$$ M ¯ is positive which then qualifies as the global Biot’s modulus for the double poroelastic material.

Две модели резинокордного слоя

Breker layers in a pneumatic tire are an important part in the tire construction. These layers have a metal cord resulting in substantial bending stiffness. When homogenizing such layers, a “shave” method is applied to the breaker layer. This results in a thinner layer having adequate stiffness in both tension and bending. In this work, a phenomenological approach is used to obtain the effective properties of a homogeneous anisotropic hyper elastic material of an equivalent layer. Two models utilize transverse isotropic or orthotropic potential used to describe the homogenized properties. Comparison is made between these models for the “shaved” rubber-cord layer based on numerical experiments. In both cases, the potentials are built on the basis of the Treloar or Mooney potentials. Note that in the case of an inhomogeneous thin layer, the traditional definition of homogenization needs to be modified. In previous works of the authors, it was proposed to determine 3D averaged elastic properties of a layer by surrounding it with a homogeneous material. This makes it possible to correctly take into account the fact that the boundary effect from the upper to lower surfaces that penetrates through the whole periodicity cell. A set of local problems formulated for the periodicity cell is proposed. This set is sufficient for determining elastic potential material parameters. Nonlinear local problems on a periodic cell are solved and the material constants of the elastic potential are determined. The applicability of the orthotropic potential (second model) is determined for the “shaved” layer. It was found that orthotropic properties are manifested relative to longitudinal shears. The results show the suitability of the proposed potential and the scheme for determining the material parameters.

Semi-Analytical Method for Computing Effective Thermoelastic Properties in Fiber-Reinforced Composite Materials

Effective elastic and thermal properties for isotropic or transversely isotropic thermoelastic fibrous composite materials are obtained. Fibers are distributed with the same periodicity along the two perpendicular directions to the fiber orientation. The periodic cell of the composite has a square or hexagonal distribution. Perfect contact between the fiber and the matrix is presented. The effective properties are calculated using a semi-analytical method. The semi-analytical method consists of obtaining the differential equations that describe the local problems using the Asymptotic Homogenization Method. Then, these equations are solved using the Finite Element Method. Effective elastic coefficient (C¯), effective thermal expansion coefficient (α¯) and the effective thermal conductivity (κ¯) are obtained. The numerical results are compared with the semi-analytical solution and with results reported by other authors. Additionally, the effective properties for a fiber with an elliptical cross section are calculated. Distributions of the fiber’s cross section with different orientations are also studied. A MATLAB program for computing the effective coefficients is presented.

Computation of Effective Elastic Properties Using a Three-Dimensional Semi-Analytical Approach for Transversely Isotropic Nanocomposites

A three-dimensional semi-analytical finite element method (SAFEM-3D) is implemented in this work to calculate the effective properties of periodic elastic-reinforced nanocomposites. Different inclusions are also considered, such as discs, ellipsoidals, spheres, carbon nanotubes (CNT) and carbon nanowires (CNW). The nanocomposites are assumed to have isotropic or transversely isotropic inclusions embedded in an isotropic matrix. The SAFEM-3D approach is developed by combining the two-scale asymptotic homogenization method (AHM) and the finite element method (FEM). Statements regarding the homogenized local problems on the periodic cell and analytical expressions of the effective elastic coefficients are provided. Homogenized local problems are transformed into boundary problems over one-eighth of the cell. The FEM is implemented based on the principle of the minimum potential energy. The three-dimensional region (periodic cell) is divided into a finite number of 10-node tetrahedral elements. In addition, the effect of the inclusion’s geometrical shape, volume fraction and length on the effective elastic properties of the composite with aligned or random distributions is studied. Numerical computations are developed and comparisons with other theoretical results are reported. A comparison with experimental values for CNW nanocomposites is also provided, and good agreement is obtained.

Effect of microstructure on dynamic compressive behavior of cellular materials

It is known that the microstructure of cellular materials has a significant impact on their compressive properties. To study these phenomena, a hierarchical Poisson disk sampling algorithm and Voronoi partitioning were used to create a 3D numerical analysis model of cellular materials. In this study, we prepared random, periodic, and ellipsoidal cell models to investigate the effects of cell shape randomness and oblateness. Numerical experiments were performed using the finite element method solver RADIOSS. In the numerical analysis, an object collided with the cellular materials at a velocity of 25 m/s. The results showed that the flow stress of the random cell model was higher than that of the periodic cell model. Further, it was found that the aspect ratio of the cell shape has a significant impact on the mechanical properties of cellular materials.

High-frequency homogenization in periodic media with imperfect interfaces

In this work, the concept of high-frequency homogenization is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discontinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenization, the homogenization is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail, resulting in an approximation of the dispersion diagram (first-order) and the wave field (zeroth-order) near these points. Finally, a uniform approximation valid for both Dirac and non-Dirac points is provided. Numerical comparisons are made with the exact solutions obtained by the Bloch–Floquet approach for the particular examples of monolayered and bilayered materials. In these two cases, convergence measurements are carried out to validate the approach, and we show that the uniform approximation remains a very good approximation even far from the edges of the Brillouin zone.