Vladimír Lukeš

Homogenization of the acoustic transmission through a perforated layer

The paper deals with homogenized transmission conditions imposed on an interface plane separating two halfspaces occupied by an acoustic medium. The conditions are obtained as the two-scale homogenization limit of a standard acoustic problem imposed on the layer perforated by a sieve-like obstacle with a periodic structure. Both the characteristic scale of the perforations and the layer thickness are parametrized by @e->0. The limit model involving some homogenized coefficients governs the interface discontinuity of the acoustic pressure associated with the two halfspaces and the magnitude of the transverse acoustic velocity. This novel approach allows for the treatment of complicated designs of perforations and presents an alternative to the usual description of acoustic impedance, which relies on a rough averaging of the quasi-experimental data.

Modeling nonlinear phenomena in deforming fluid-saturated porous media using homogenization and sensitivity analysis concepts

A new computationally efficient nonlinear model of fluid saturated porous media.Two-scale modelling of the Biot-Darcy continuum based on the homogenization.Deformation-dependent coefficients are computed using the sensitivity analysis.Nonlinear effects captured without any need to update local microstructures.Nonlinearity is pronounced especially for complex geometries of pores. Homogenization of heterogeneous media with nonlinear effects leads to computationally complex problems requiring updating local microstructures and solving local auxiliary problems to compute characteristic responses. In this paper we suggest how to circumvent such a computationally expensive updating procedure while using an efficient approximation scheme for the local homogenized coefficients, so that the complexity of the whole two-scale modeling is reduced substantially. We consider the deforming porous fluid saturated media described by the Biot model. The proposed modeling approaches are based on the homogenization of the quasistatic fluid-structure interaction whereby differentiation with respect to the microstructure deformation is used as a tool for linearization. Assuming the linear kinematics framework, the physical nonlinearity in the Biot continuum is introduced in terms of the deformation-dependent material coefficients which are approximated as linear functions of the macroscopic response. These functions are obtained by the sensitivity analysis of the homogenized coefficients computed for a given geometry of the porous structure which transforms due to the local deformation. The deformation-dependent material coefficients approximated in this way do not require any solving of local microscopic problems for updated configurations. It appears that difference between the linear and nonlinear models depends significantly on the specific microstructure of the porous medium; this observation is supported by numerical examples.

Microstructure based two-scale modelling of soft tissues

A technique suitable for the modelling of large deforming biological tissues with a nearly periodic microstructure is presented in this work. The proposed approach takes into account the heterogeneous material constitution and geometrical arrangement of the tissues at the microstructural level. The global material properties are described in terms of the homogenized (effective) parameters. Numerical simulations are focused on the mechanical behaviour of an arterial wall.

Modeling Tissue Perfusion Using a Homogenized Model with Layer-wise Decomposition

Abstract The proposed two-scale model of perfused tissues enables to reduce the computational complexity by converting a 3D problem with heterogeneous structure into several 2D problems. The model is developed using the homogenization method applied to upscale the Darcy flow problem for each layer featured by the dual porosity. The multi-layer model involves 2D pressure fields for different channel types within each layer and inter-layer fluxes for multiple compartments. The approach combining homogenization and the layer-wise domain decomposition is convenient for an efficient simulation of perfusion in complex branching structures at lower hierarchies of perfusion trees.

Modeling of the contrast-enhanced perfusion test in liver based on the multi-compartment flow in porous media

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.

Homogenization of the fluid-saturated piezoelectric porous media

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.

Optimization of the porous material described by the Biot model

Abstract 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 guaranteeing minimum required permeability, and vice versa. Issues of the design and anisotropy and the spline box parametrization are discussed. In order to avoid remeshing, a geometric regularization technique based on injectivity constraints is applied.

Shape Optimization for Microstructure Design of Porous Materials Described by the Biot model in the Homogenization Framework

The paper deals with shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. The porous material is described as the Biot continuum derived by the homogenization method. The effective medium properties are given by the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. These are computed using characteristic responses - solutions of the state problems defined in the representative unit cell constituted by an elastic skeleton and by a fluid channel generating the porosity. The design of the channel is described by a B-spline box which embeds the whole representative cell. The sensitivity analysis for all the homogenized material coefficients is derived using the domain method of the design velocity approach. The optimality criterion is aimed to maximize stiffness of the drained porous material and allow for a sufficient permeability and vice versa. Issues of the spline box parametrization, the channel shape regularity and FE mesh updates are discussed. The maximization problems are solved using the sparse nonlinear optimizer SNOPT.

Towards Image-Based Analysis of the Liver Perfusion Using a Hierarchical Flow Model

The paper summarizes our activities in modelling tissue perfusion using a multilevel approach which is based on information retrievable form CT and micro CT images. We focus on the liver tissue for which the perfusion modelling is of great interest for both medical research and clinical practice. The blood flow in liver 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 the Poiseuille correction terms to respect the viscous 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. For the lowermost level representing the quasi-periodic lobular structure we apply the homogenization method which provides the permeability features of the hepatic sinusoids considered as the double porosity. In the paper we discuss several approaches how to determine the flow model parameters which can be combined. Also the model validation using the realistic geometries reconstructed using the CT scans is discussed.

Modeling large-deforming fluid-saturated porous media using an Eulerian incremental formulation

Abstract The paper deals with modeling 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.