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The paper presents a new type of weakly nonlinear two-scale model of controllable periodic porous piezoelectric structures saturated by Newtonian fluids. The flow is propelled by peristaltic deformation of microchannels which is induced due to piezoelectric segments embedded in the microstructure and locally actuated by voltage waves. The homogenization is employed to derive a macroscopic model of the poroelastic medium with effective parameters modified by piezoelectric properties of the skeleton. To capture the peristaltic pumping, the nonlinearity associated with deforming configuration must be respected. In the macroscopic model, this nonlinearity is introduced through homogenized coefficients depending on the deforming micro-configurations. For this, linear expansions based on the sensitivity analysis of the homogenized coefficients with respect to deformation induced by the macroscopic quantities are employed. This enables to avoid the two-scale tight coupling of the macro- and microproblems otherwise needed in nonlinear problems. The derived reduced-order model is implemented and verified using direct numerical simulations of the periodic heterogeneous medium. Numerical results demonstrate the peristaltic driven fluid propulsion in response to the electric actuation and the efficiency of the proposed treatment of the nonlinearity. The paper shows new perspectives in homogenization-based computationally efficient modelling of weakly nonlinear problems where continuum microstructures are perturbed by coupled fields.

We present a new approach and an algorithm for optimizing the material configuration and behaviour of a fluid saturated porous medium in a two-scale setting. The state problem is governed by the Biot model describing the fluid-structure interaction in homogenized poroelastic structures. However, the approach is widely applicable to multiphysics problems involving several macroscopic fields where homogenization provides the relationship between the microconfigurations and the macroscopic mathematical model. The optimization variables describe the local microstructure design by virtue of the pore shape which determines the effective medium properties - the material coefficients - computed by the homogenization method. The main idea of the numerical optimization strategy consists in a) employing a precomputed database of the material coefficients associated to the geometric parameters and b) applying the sequential global programming (SGP) method for solving the problem of macroscopically optimized distribution of material coefficients. Although there are similarities with the free material optimization (FMO) approach, only effective material coefficients are considered admissible, for which a well-defined set of corresponding configurable microstructures exist. Due to the flexibility of the SGP approach, different types of microstructures with fully independent parametrizations can easily be handled. The efficiency of the concept is demonstrated by a series of numerical experiments. We show that the SGP method can handle simultaneously multiple types of microstructures with nontrivial parametrizations using a considerably low and stable number of state problems to be solved., Comment: 26 pages, 19 figures, 2 tables

We consider acoustic waves propagating in an inviscid fluid interacting with a rigid periodically perforated plate in the presence of permanent flows. The paper presents a model of an acoustic interface obtained by the asymptotic homogenization of a thin transmission layer in which the plate is embedded. To account for the flow, a decomposition of the fluid pressure and velocity in the steady and fluctuating parts is employed. This enables for a linearization and an efficient use of the homogenization method which leads to a model order reduction effect. The dependence of an extended Helmholtz equation on the permanent flow introduces a locally periodic velocity field in the perforated plate vicinity, so that the coefficients of the homogenized interface depend on the flow. The derived model extended by natural coupling conditions provides an implicit Dirichlet-to-Neumann operator. Numerical simulations of wave propagation in a waveguide illustrate the flow speed influence on the acoustic transmission. Also some geometrical aspects are explored.

Based on our previous work, we propose a homogenized model of acoustic waves propagating through periodically perforated elastic plates with metamaterial properties due to embedded arrays of soft elastic inclusions serving for resonators. Such structures enable to suppress the acoustic transmission for selected frequency bands. Homogenization of the vibro-acoustic fluid-structure interaction problem in a 3D complex geometry of the transmission layer leads to effective transmission conditions prescribed on the acoustic meta-surface associated with the mid-plane of the Reissner-Mindlin plate. Asymptotic analysis with respect to the layer thickness, proportional to the plate thickness and to the perforation period, yields an implicit Dirichlet-to-Neumann operator defined on the homogenized metasurface. An efficient method is proposed for computing frequency-dependent effective parameters involved in the homogenized model of the layer. These can change their signs, thus modifying the acoustic impedance and the effective mass of the metasurface. The global problem of the acoustic wave propagation in a waveguide fitted with the plate is solved using the finite element method. The homogenized interface allows for a significant reduction of the computational model. Numerical illustrations are presented.

The two-scale computational homogenization method is proposed for modelling of locally periodic fluid-saturated media subjected a to large deformation induced by quasistatic loading. The periodic heterogeneities are relevant to the mesoscopic scale at which a double porous medium constituted by hyperelastic skeleton and an incompressible viscous fluid is featured by large contrasts in the permeability. Within the Eulerian framework related to the current deformed configuration, the two-scale homogenization approach is applied to a linearized model discretized in time, being associated with an incremental formulation. For this, the equilibrium equation and the mass conservation expressed in the spatial configuration are differentiated using the material derivative with respect to a convection velocity field. The homogenization procedure of the linearized equations provides effective (homogenized) material properties are computed to constitute the incremental macroscopic problem. The coupled algorithm for the multiscale problem is implemented using the finite element method. Illustrative 2D numerical simulations of a poroelastic medium are presented including a simple validation test.

The paper deals with the homogenization of deformable porous media saturated by two-component electrolytes. The model relevant to the microscopic scale describes steady states of the medium while reflecting essential physical phenomena, namely electrochemical interactions in a dilute Newtonian solvent under assumptions of a small external electrostatic field and slow flow. The homogenization is applied to a linearized micromodel, whereby the thermodynamic equilibrium represents the reference state. Due to the dimensional analysis, scaling of the viscosity and electric permitivity is introduced, so that the limit model retains the characteristic length associated with the pore size and the electric double layer thickness. The homogenized model consists of two weakly coupled parts: the flow of the electrolyte can be solved in terms of a global pressure and streaming potentials of the two ions, independently of then the solid phase deformations which is computed afterwards for the fluid stress acting on pore walls. The two-scale model has been implemented in the SfePy finite element software. The numerical results show dependence of the homogenized coefficients on the microstructure porosity. By virtue of the corrector result of the homogenization, microscopic responses in a local representative cell can be reconstructed from the macroscopic solutions., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

The paper deals with modelling of acoustic waves which propagate in inviscid fluids interacting with perforated elastic plates. The plate can be replaced by an interface on which transmission conditions are derived by homogenization of a problem describing vibroacoustic fluid-structure interactions in a transmission layer in which the plate is embedded. The Reissner-Mindlin theory of plates is adopted for periodic perforations designed by arbitrary cylindrical holes with axes orthogonal to the plate midplane. The homogenized model of the vibroacoustic transmission is obtained using the two-scale asymptotic analysis with respect to the layer thickness which is proportional to the plate thickness and to the perforation period. The nonlocal, implicit transmission conditions involve a jump in the acoustic potential and its normal one-side derivatives across the interface which represents the plate with a given thickness. The homogenized model was implemented using the finite element method and validated using direct numerical simulations of the non-homogenized problem. Numerical illustrations of the vibroacoustic transmission are presented., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

In this paper we present the two-level homogenization of the flow in a deformable double-porous structure described at two characteristic scales. The higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid. The macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two-level asymptotic analysis, whereby a scaling ansatz is adopted to respect the pore size differences. The first level upscaling of the fluid-structure interaction problem yields a Biot continuum describing the mesoscopic matrix coupled with the Stokes flow in the channels. The second step of the homogenization leads to a macroscopic model involving three equations for displacements, the mesoscopic flow velocity and the micropore pressure. Due to interactions between the two porosities, the macroscopic flow is governed by a Darcy-Brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two-phase medium. Expressions of the effective macroscopic parameters of the homogenized double-porosity continuum are derived, depending on the characteristic responses of the mesoscopic structure. Some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed. The model has been implemented in the finite element code SfePy which is well-suited for computational homogenization. A numerical example of solving a nonstationary problem using mixed finite element method is included., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

SfePy (Simple finite elements in Python) is a software for solving various kinds of problems described by partial differential equations in one, two or three spatial dimensions by the finite element method. Its source code is mostly (85\%) Python and relies on fast vectorized operations provided by the NumPy package. For a particular problem two interfaces can be used: a declarative application programming interface (API), where problem description/definition files (Python modules) are used to define a calculation, and an imperative API, that can be used for interactive commands, or in scripts and libraries. After outlining the SfePy package development, the paper introduces its implementation, structure and general features. The components for defining a partial differential equation are described using an example of a simple heat conduction problem. Specifically, the declarative API of SfePy is presented in the example. To illustrate one of SfePy's main assets, the framework for implementing complex multiscale models based on the theory of homogenization, an example of a two-scale piezoelastic model is presented, showing both the mathematical description of the problem and the corresponding code., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

The paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. At the macroscopic level, the porous material is described by the Biot model defined in terms of the effective medium coefficients, involving the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. By virtue of the homogenization, these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid. For the purpose of optimization, the sensitivity analysis on the continuous level of the problem is derived. We provide sensitivities of objective functions constituted by the Biot model coefficients with respect to the underlying pore shape described by a B-spline box which embeds the whole representative cell. We consider material design problems in the framework of which the layout of a single representative cell is optimized. Then we propose a sequential linearization approach to the two-scale problem in which local microstructures are optimized with respect to macroscopic design criteria. Numerical experiments are reported which include stiffness maximization with constraints allowing for a sufficient permeability, and vice versa. Issues of the design anisotropy, the spline box parametrization are discussed. In order to avoid remeshing a geometric regularization technique based on injectivity constraints is applied., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

The paper is devoted to the homogenization of porous piezoelectric materials saturated by electrically inert fluid. The solid part of a representative volume element consists of the piezoelectric skeleton with embedded conductors. The pore fluid in the periodic structure can constitute a single connected domain, or an array of inclusions. Also the conducting parts are represented by several mutually separated connected domains, or by inclusions. Two of four possible arrangements are considered for upscaling by the homogenization method. The macroscopic model of the first type involves coefficients responsible for interactions between the electric field and the pore pressure, or the pore volume. For the second type, the electrodes can be used for controlling the electric field at the pore level, so that the deformation and the pore volume can be influenced locally. Effective constitutive coefficients are computed using characteristic responses of the microstructure. The two-scale modelling procedure is implemented numerically using the finite element method. The macroscopic strain and electric fields are used to reconstruct the corresponding local responses at the pore level. For validation of the models, these are compared with results obtained by direct numerical simulations of the heterogeneous structure; a good agreement is demonstrated, showing relevance of the two-scale numerical modelling approach., Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license

The paper deals with modelling fluid saturated porous media subject to large deformation. An Eulerian incremental formulation is derived using the problem imposed in the spatial configuration in terms of the equilibrium equation and the mass conservation. Perturbation of the hyperelastic porous medium is described by the Biot model which involves poroelastic coefficients and the permeability governing the Darcy flow. Using the material derivative with respect to a convection velocity field we obtain the rate formulation which allows for linearization of the residuum function. For a given time discretization with backward finite difference approximation of the time derivatives, two incremental problems are obtained which constitute the predictor and corrector steps of the implicit time-integration scheme. Conforming mixed finite element approximation in space is used. Validation of the numerical model implemented in the SfePy code is reported for an isotropic medium with a hyperelastic solid phase. The proposed linearization scheme is motivated by the two-scale homogenization which will provide the local material poroelastic coefficients involved in the incremental formulation.

The paper deals with modeling the liver perfusion intended to improve quantitative analysis of the tissue scans provided by the contrast-enhanced computed tomography (CT). For this purpose, we developed a model of dynamic transport of the contrast fluid through the hierarchies of the perfusion trees. Conceptually, computed time-space distributions of the so-called tissue density can be compared with the measured data obtained from CT; such a modeling feedback can be used for model parameter identification. The blood flow is characterized at several scales for which different models are used. Flows in upper hierarchies represented by larger branching vessels are described using simple 1D models based on the Bernoulli equation extended by correction terms to respect the local pressure losses. To describe flows in smaller vessels and in the tissue parenchyma, we propose a 3D continuum model of porous medium defined in terms of hierarchically matched compartments characterized by hydraulic permeabilities. The 1D models corresponding to the portal and hepatic veins are coupled with the 3D model through point sources, or sinks. The contrast fluid saturation is governed by transport equations adapted for the 1D and 3D flow models. The complex perfusion model has been implemented using the finite element and finite volume methods. We report numerical examples computed for anatomically relevant geometries of the liver organ and of the principal vascular trees. The simulated tissue density corresponding to the CT examination output reflects a pathology modeled as a localized permeability deficiency., Comment: 28 pages, 18 figures

We use a collection of Python programs for numerical simulation of liver perfusion. We have an application for semi-automatic generation of a finite element mesh of the human liver from computed tomography scans and for reconstruction of the liver vascular structure. When the real vascular trees can not be obtained from the CT data we generate artificial trees using the constructive optimization method. The generated FE mesh and vascular trees are imported into SfePy (Simple Finite Elements in Python) and numerical simulations are performed in order to get the pressure distribution and perfusion flows in the liver tissue. In the post-processing steps we calculate transport of a contrast fluid through the liver parenchyma., Comment: Part of the Proceedings of the 7th European Conference on Python in Science (EuroSciPy 2014), Pierre de Buyl and Nelle Varoquaux editors, (2014)

The interaction of exciton and charge transfer (CT) states plays a central role in photo-induced CT processes in chemistry, biology and physics. In this work, we use a combination of two-dimensional electronic spectroscopy (2D-ES), pump-probe measurements and quantum chemistry to investigate the ultrafast CT dynamics in a lutetium bisphthalocyanine dimer in different oxidation states. It is found that in the anionic form, the combination of strong CT-exciton interaction and electronic asymmetry induced by a counter-ion enables CT between the two macrocycles of the complex on a 30 fs timescale. Following optical excitation, a chain of electron and hole transfer steps gives rise to characteristic cross-peak dynamics in the electronic 2D spectra, and we monitor how the excited state charge density ultimately localizes on the macrocycle closest to the counter-ion within 100 fs. A comparison with the dynamics in the radical species further elucidates how CT states modulate the electronic structure and tune fs-reaction dynamics. Our experiments demonstrate the unique capability of 2D-ES in combination with other methods to decipher ultrafast CT dynamics., Comment: 14 pages, 11 figures, and Supporting information

We present a theory of vibrational modulation of two-dimensional coherent Fourier transformed electronic spectra. Based on an expansion of the system's energy gap correlation function in terms of Huang-Rhys factors, we explain the time-dependent oscillatory behavior of the absorptive and dispersive parts of two-dimensional spectra of a two-level electronic system, weakly coupled to intramolecular vibrational modes. The theory predicts oscillations in the relative amplitudes of the rephasing and non-rephasing parts of the two-dimensional spectra, and enables to analyze time dependent two-dimensional spectra in terms of simple elementary components whose line-shapes are dictated by the interaction of the system with the solvent only. The theory is applicable to both low and high energy (with respect to solvent induced line broadening) vibrations. The results of this paper enable to qualitatively explain experimental observations on low energy vibrations presented in the preceding paper [A. Nemeth et al, arXiv:1003.4174v1] and to predict the time evolution of two-dimensional spectra in ultrafast ultrabroad band experiments on systems with high energy vibrations., Comment: 14 pages, 9 figures

This is the first in a series of two papers investigating the effect of electron-phonon coupling in two-dimensional Fourier transformed electronic spectroscopy. We present a series of one- and two-dimensional nonlinear spectroscopic techniques for studying a dye molecule in solution. Ultrafast laser pulse excitation of an electronic transition coupled to vibrational modes induces a propagating vibrational wave packet that manifests itself in oscillating signal intensities and line-shapes. For the two-dimensional electronic spectra we can attribute the observed modulations to periodic enhancement and decrement of the relative amplitudes of rephasing and non-rephasing contributions to the total response. Different metrics of the two-dimensional signals are shown to relate to the frequency-frequency correlation function which provides the connection between experimentally accessible observations and the underlying microscopic molecular dynamics. A detailed theory of the time-dependent two-dimensional spectral line-shapes is presented in the accompanying paper [T. Mancal et al., arXiv:1003.xxxx]., Comment: 13 pages, 8 figures