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.

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.

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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.

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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.

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