Reduced one‐dimensional modelling and numerical simulation for mass transport in fluids

Köppl, Tobias; Wohlmuth, Barbara; Helmig, Rainer

doi:10.1002/fld.3728

Cited by 19 publications

(23 citation statements)

References 30 publications

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“…Bifurcations and their mathematical modelling have been the subject of many publications. () Coupling conditions for systems linked at a bifurcation can be derived by the principles of mass conservation and continuity of the total pressure. The total pressure for vessel V i is defined by

p_{t, i} = \frac{ρ}{2} {(\frac{Q_{i}}{A_{i}})}^{2} + p (A_{i}) .

Indexing the vessels at a bifurcation by

V_{i}, i \in ({}, normalI, II, III) \subset ({}, 1, ⋯, 55)

, we obtain the following 3 coupling conditions:

Q_{I} = Q_{II} + Q_{III} and p_{t, normalI} = p_{t, II}, p_{t, normalI} = p_{t, III} .

…”

Section: Simulation Of Arterial Blood Flow By Dimensionally Reduced Mmentioning

confidence: 99%

“…Bifurcations and their mathematical modelling have been the subject of many publications. 48,[64][65][66][67] Coupling conditions for systems linked at a bifurcation can be derived by the principles of mass conservation and continuity of the total pressure. The total pressure for vessel V i is defined by…”

Section: Modelling Of Bifurcationsmentioning

confidence: 99%

“…For our simulations, we place a stenosis in the right tibial artery located in the lower right leg (see Figure 1) and study the pressure and flow rate curves, i.e., the evolution of the 2 quantities at different points over a complete heartbeat, in the vicinity of this stenosis.We use for the blood flow simulations a 1-D-0-D coupled model, where we assign a 1-D flow model to the 55 main arteries of the systemic circulation. 8,18,21,48 At the outlets of this network, 0-D models are attached to the 1-D models to incorporate the influence of the omitted vessels. The stenosis is modelled by a 0-D model consisting of an ODE.…”

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confidence: 99%

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Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Köppl

Santin

Haasdonk

et al. 2018

Numer Methods Biomed Eng

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In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

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p_{t, i} = \frac{ρ}{2} {(\frac{Q_{i}}{A_{i}})}^{2} + p (A_{i}) .

Indexing the vessels at a bifurcation by

V_{i}, i \in ({}, normalI, II, III) \subset ({}, 1, ⋯, 55)

, we obtain the following 3 coupling conditions:

Q_{I} = Q_{II} + Q_{III} and p_{t, normalI} = p_{t, II}, p_{t, normalI} = p_{t, III} .

…”

Section: Simulation Of Arterial Blood Flow By Dimensionally Reduced Mmentioning

confidence: 99%

Section: Modelling Of Bifurcationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Köppl

Santin

Haasdonk

et al. 2018

Numer Methods Biomed Eng

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show abstract

“…, ξ Nm as the definition of the element shape functions. In fact, it can be easily verified that, as a result of the nodal interpolation property (14) and the quadrature rule (18), any mass matrix M χ (irrespective of the coefficient function χ) is a diagonal matrix [11]. This property allows for the application of fast explicit time stepping methods to the system of ordinary differential equations (ODEs) given in (17).…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Besides the MOC, finite difference, finite volume, discontinuous Galerkin and spectral methods have been used to simulate transient hydraulic systems [14,15,16,17,18,19]. In this context, it should be noted that numerous variations of spectral methods can be found in literature [11,20].…”

Section: Introductionmentioning

confidence: 99%

The spectral element method as an efficient tool for transient simulations of hydraulic systems

Mennemann

Marko

Schmidt

et al. 2018

Applied Mathematical Modelling

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This paper presents transient numerical simulations of hydraulic systems in engineering applications using the spectral element method (SEM). Along with a detailed description of the underlying numerical method, it is shown that the SEM yields highly accurate numerical approximations at modest computational costs, which is in particular useful for optimizationbased control applications. In order to enable fast explicit time stepping methods, the boundary conditions are imposed weakly using a numerically stable upwind discretization. The benefits of the SEM in the area of hydraulic system simulations are demonstrated in various examples including several simulations of strong water hammer effects. Due to its exceptional convergence characteristics, the SEM is particularly well suited to be used in real-time capable control applications. As an example, it is shown that the time evolution of pressure waves in a large scale pumped-storage power plant can be well approximated using a low-dimensional system representation utilizing a minimum number of dynamical states.

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A1D–0D–3Dcoupled model for simulating blood flow and transport processes in breast tissue

Fritz

Köppl

Oden

et al. 2022

Numer Methods Biomed Eng

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In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system necessitating the use of different flow models for different geometries and flow regimes. In total, we consider four different model types. First, a system of 1D nonlinear hyperbolic partial differential equations (PDEs) is considered to simulate blood flow in larger arteries with highly elastic vessel walls. Second, we assign 1D linearized hyperbolic PDEs to model the smaller arteries with stiffer vessel walls. The third model type consists of ODE systems (0D models). It is used to model the arterioles and peripheral circulation. Finally, homogenized 3D porous media models are considered to simulate flow and transport in capillaries and tissue within the breast volume. Sink terms are used to account for the influence of the venous and lymphatic systems. Combining the four model types, we obtain two different 1D-0D-3D coupled models for simulating blood flow and transport processes: The first model results in a fully coupled 1D-0D-3D model covering the complete path from the aorta to the breast combining a generic arterial network with a patient specific breast network and geometry. The second model is a reduced one based on the separation of the generic and patient specific parts. The information from a calibrated fully coupled model is used as inflow condition for the patient specific sub-model allowing a significant computational cost reduction. Several numerical experiments are conducted to calibrate the generic model parameters and to demonstrate realistic flow simulations compared to existing data on blood flow in the human breast and vascular system. Moreover, we use two different breast vasculature and tissue data sets to illustrate the robustness of our reduced sub-model approach.

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Reduced one‐dimensional modelling and numerical simulation for mass transport in fluids

Cited by 19 publications

References 30 publications

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

The spectral element method as an efficient tool for transient simulations of hydraulic systems

A1D–0D–3Dcoupled model for simulating blood flow and transport processes in breast tissue

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