Multiscale modeling of fiber reinforced materials via non-matching immersed methods

We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. Intravascular pressure and velocity are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix.

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“…A similar strategy is possible in the non-linear case, provided that one resorts to Lagrange multipliers, as in. 1 …”

Section: Methodsmentioning

confidence: 99%

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

2021

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“…Based on the observation we confirm the first order rate of convergence. We also present our approximated solution U 3 and its underlying subdivision in (2).…”

Section: Performance Of Regsolvementioning

confidence: 99%

“…The relevant literature for more complex distributions of singularities is more limited [31,30]. The motivation for such methods lies on the possibly complex geometry of the immersed domain, such as thin vascular structures in tissues [23,24,13] or fibers in isotropic materials [2], for which it is difficult to obtain a bulk mesh of Ω matching the embedded domain.…”

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Adaptive finite element approximations for elliptic problems using regularized forcing data

Heltai

Lei

2021

Preprint

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“…to its arclength, so that its effect, as well as elastic effects of the vessels, may be neglected. For a possible way to include the elastic behaviour of the vessels we refer to [2]. In each cross sectional plane of the vessel, we approximate the local behaviour of the problem as in the two-dimensional axi-symmetric case.…”

Section: The Hypersingular Problemmentioning

confidence: 99%

Multiscale modeling of vascularized tissues via nonmatching immersed methods

Heltai

Caiazzo

2019

Numer Methods Biomed Eng

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We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depend on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean pressure over cross section), thus drastically reducing the complexity of the computational model. We perform several numerical tests, ranging from simple cases with known exact solutions to the modeling of materials with random distributions of vessels. In the latter case, we use our immersed method to perform an in silico characterization of the mechanical properties of the effective biphasic material tissue via statistical simulations.

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Multiscale modeling of fiber reinforced materials via non-matching immersed methods

Cited by 10 publications

References 52 publications

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

Multiscale Coupling of One-dimensional Vascular Models and Elastic Tissues

Adaptive finite element approximations for elliptic problems using regularized forcing data

Multiscale modeling of vascularized tissues via nonmatching immersed methods

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