A mortar-type finite element approach for embedding 1D beams into 3D solid volumes

Steinbrecher, Ivo; Mayr, Matthias; Grill, Maximilian J.; Kremheller, Johannes; Meier, Christoph; Popp, Alexander

doi:10.1007/s00466-020-01907-0

Cited by 41 publications

(106 citation statements)

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“…In section 3.3, we justify formulating this constraint along the entire 1D domain

Λ_{L}

considering the connectivity between larger and smaller vessels in our cases. We have previously employed a similar strategy in our hybrid treatment of the vasculature in a multiphase tumor growth model 3 and the related solid mechanics problem of beam‐to‐solid mesh tying 36 . We follow the same approach as in the two aforementioned publications and incorporate the constraint with an additional Lagrange multiplier (LM) field into the weak form of our hybrid model, which reads as

({, \begin{array}{l} {(\frac{\partial δ p^{\overset{v}{true}}}{\partial s}, \frac{π R^{4}}{8 μ^{\overset{v}{true}}} \frac{\partial p^{\overset{v}{true}}}{\partial s})}_{Λ_{L}} + {(δ p^{\overset{v}{true}}, \frac{{\overset{\overset{false^}{v} \to l}{M}}_{leak}}{ρ^{\overset{υ}{true}}})}_{Λ_{L}} + {(δ p^{\overset{v}{true}}, λ)}_{Λ_{L}} = 0 ((), 13 normala) \\ {(∇ δ p^{v}, \frac{{bold-italick}^{v}}{μ^{u}} ∇ p^{v})}_{Ω_{v}} + {(δ p^{v}, \frac{{\overset{v \to l}{M}}_{leak}}{ρ^{v}})}_{Ω_{v}} - {(δ p^{v}, λ)}_{Λ_{L}} = 0 ((), 13 normalb) \\ () \end{array})

…”

Section: Mathematical Models and Numerical Methodsmentioning

confidence: 99%

“…Approximating those contributions with a finite element interpolation yields a mortar‐type formulation where the nodal LMs are additional degrees of freedom, condensed out with a dual approach 40,41 or a penalty regularization of the mortar method is employed to remove the additional degrees of freedom and the saddle‐point structure 42 . Here, we follow the latter approach just as in our previous work on 1D‐3D type couplings 3,36 . The contributions to the weak form of the mass balance equations, that is, the two last terms in (13a) and (13b) can be written as

δ Π_{LM, h} = ∑_{j = 1}^{n_{nodes, {normalΛ}_{normalL}}} ∑_{k = 1}^{n_{nodes, {normalΛ}_{normalL}}} λ_{j} D_{jk} δ p_{k}^{\overset{v}{true}} - ∑_{j = 1}^{n_{nodes, {normalΛ}_{normalL}}} ∑_{l = 1}^{n_{nodes, {normalΩ}_{v}}} λ_{j} M_{jl} δ p_{l}^{v}

with the so‐called mortar matrices

boldD ([], j, k) = D_{jk} = \int_{{normalΛ}_{L, h}} {\overset{false^}{normalΦ}}_{j} {\overset{false^}{N}}_{k} italicds

and

boldM ([], j, l) = M_{jl} = \int_{{normalΛ}_{L, h}} {\overset{false^}{normalΦ}}_{j} N_{l} italicds .

…”

Section: Mathematical Models and Numerical Methodsmentioning

confidence: 99%

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Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model

Kremheller

Brandstaeter

Schrefler

et al. 2021

Numer Methods Biomed Eng

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The goal of this paper is to investigate the validity of a hybrid embedded/homogenized in‐silico approach for modeling perfusion through solid tumors. The rationale behind this novel idea is that only the larger blood vessels have to be explicitly resolved while the smaller scales of the vasculature are homogenized. As opposed to typical discrete or fully resolved 1D–3D models, the required data can be obtained with in‐vivo imaging techniques since the morphology of the smaller vessels is not necessary. By contrast, the larger vessels, whose topology and structure is attainable noninvasively, are resolved and embedded as one‐dimensional inclusions into the three‐dimensional tissue domain which is modeled as a porous medium. A sound mortar‐type formulation is employed to couple the two representations of the vasculature. We validate the hybrid model and optimize its parameters by comparing its results to a corresponding fully resolved model based on several well‐defined metrics. These tests are performed on a complex data set of three different tumor types with heterogeneous vascular architectures. The correspondence of the hybrid model in terms of mean representative elementary volume blood and interstitial fluid pressures is excellent with relative errors of less than 4%. Larger, but less important and explicable errors are present in terms of blood flow in the smaller, homogenized vessels. We finally discuss and demonstrate how the hybrid model can be further improved to apply it for studies on tumor perfusion and the efficacy of drug delivery.

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“…In section 3.3, we justify formulating this constraint along the entire 1D domain

Λ_{L}

({, \begin{array}{l} {(\frac{\partial δ p^{\overset{v}{true}}}{\partial s}, \frac{π R^{4}}{8 μ^{\overset{v}{true}}} \frac{\partial p^{\overset{v}{true}}}{\partial s})}_{Λ_{L}} + {(δ p^{\overset{v}{true}}, \frac{{\overset{\overset{false^}{v} \to l}{M}}_{leak}}{ρ^{\overset{υ}{true}}})}_{Λ_{L}} + {(δ p^{\overset{v}{true}}, λ)}_{Λ_{L}} = 0 ((), 13 normala) \\ {(∇ δ p^{v}, \frac{{bold-italick}^{v}}{μ^{u}} ∇ p^{v})}_{Ω_{v}} + {(δ p^{v}, \frac{{\overset{v \to l}{M}}_{leak}}{ρ^{v}})}_{Ω_{v}} - {(δ p^{v}, λ)}_{Λ_{L}} = 0 ((), 13 normalb) \\ () \end{array})

…”

Section: Mathematical Models and Numerical Methodsmentioning

confidence: 99%

δ Π_{LM, h} = ∑_{j = 1}^{n_{nodes, {normalΛ}_{normalL}}} ∑_{k = 1}^{n_{nodes, {normalΛ}_{normalL}}} λ_{j} D_{jk} δ p_{k}^{\overset{v}{true}} - ∑_{j = 1}^{n_{nodes, {normalΛ}_{normalL}}} ∑_{l = 1}^{n_{nodes, {normalΩ}_{v}}} λ_{j} M_{jl} δ p_{l}^{v}

with the so‐called mortar matrices

boldD ([], j, k) = D_{jk} = \int_{{normalΛ}_{L, h}} {\overset{false^}{normalΦ}}_{j} {\overset{false^}{N}}_{k} italicds

and

boldM ([], j, l) = M_{jl} = \int_{{normalΛ}_{L, h}} {\overset{false^}{normalΦ}}_{j} N_{l} italicds .

…”

Section: Mathematical Models and Numerical Methodsmentioning

confidence: 99%

Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model

Kremheller

Brandstaeter

Schrefler

et al. 2021

Numer Methods Biomed Eng

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“…This constraint is enforced in a weak (integral) sense by choosing a stable discrete Lagrange multiplier basis for the embedded 1D-3D coupling in the spirit of the mortar method [3]. Additionally, different numerical integration strategies for the mortar integrals, the treatment of strong discontinuities when beams reach over sharp edges of the solid bodies and the spatial convergence behavior of the method under uniform mesh refinement need special consideration [4].…”

Section: Finite Elements For Beams and Solidsmentioning

confidence: 99%

Efficient mortar‐based algorithms for embedding 1D fibers into 3D volumes

Steinbrecher

Popp

2021

Proc Appl Math and Mech

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Many composite materials are based on 1D fibers being embedded into 3D solid volumes, e.g. carbon-fiber reinforced plastics in aerospace engineering or fiber-reinforced concrete in civil engineering to name only two prominent examples. The present contribution highlights the most important numerical methods and algorithmic building blocks for an efficient analysis of such systems based on cutting-edge finite element formulations for nonlinear beams and a novel beam-to-solid volume coupling approach inspired by classical mortar methods. A particular emphasis is put on the efficient parallel implementation for large-scale simulations, which includes suitable procedures for domain partitioning and geometry-based search.

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“…Mixed subscripts FB and BF refer to the coupling interaction. In order to deal with weak discontinuities at the fluid element boundaries as they arise during element-wise numerical integration of the integrals in (3), a segment-based integration approach is adopted from [7], cf. Figure 2.…”

Section: Load and Motion Transfer Schemementioning

confidence: 99%

Fluid‐Structure Interaction of Slender Continua with 3‐Dimensional Flow: An Embedded Finite Element Approach

Hagmeyer

Mayr

Popp

2021

Proc Appl Math and Mech

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The interaction of slender bodies with fluid flow plays an important role in many industrial processes and biomedical applications. The numerical modeling of problems involving such rod-like structures with classical continuum-based finite elements poses a challenge because it promptly leads to locking effects as well as very large system sizes. An alternative approach leading to rather well-posed problems is the use of 1-dimensional beam theory. Applications of so-called geometrically exact beam theories have proven to be a computationally efficient way to model the behavior of such slender structures. This work addresses research questions arising from the application of geometrically-exact beam theory in the context of fluid-structure interaction (FSI). In particular, we describe an embedded approach coupling geometrically exact beam finite elements to a background fluid mesh. Furthermore, we elaborate on the conversion between the beam's stress resultants and the 3-dimensional formulation of the fluid field. A preliminary numerical example will demonstrate the general applicability of the proposed approach for a one-way coupled problem.

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A mortar-type finite element approach for embedding 1D beams into 3D solid volumes

Cited by 41 publications

References 50 publications

Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model

Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model

Efficient mortar‐based algorithms for embedding 1D fibers into 3D volumes

Fluid‐Structure Interaction of Slender Continua with 3‐Dimensional Flow: An Embedded Finite Element Approach

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