A diffuse interface method for the Navier–Stokes/Darcy equations: Perfusion profile for a patient-specific human liver based on MRI scans

Stoter, Stein K. F.; Müller, Peter C.; Cicalese, Luca; Tuveri, Massimiliano; Schillinger, Dominik; Hughes, Thomas J.R.

doi:10.1016/j.cma.2017.04.002

Cited by 44 publications

(39 citation statements)

References 83 publications

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“…With these considerations in mind, we derive a second diffuse variant of Nitsche's method, where we consider only phase-field approximations for the boundary surface integrals, but leave the volume integrals in sharp format. Starting from the geometrically diffuse formulation (22) and (23) and corresponding finite element discretizations illustrated in Figure 5, we convert all diffuse volume terms back to their sharp representation by replacing the phase-field function by its sharp boundary limit H in the sense of (16). The resulting sharp domain, but diffuse boundary formulation, follows as find u h such that…”

Section: Diffuse Boundary But Sharp Volumementioning

confidence: 99%

“…For Neumann‐type boundary and interface conditions, this strategy leads to a straightforward phase‐field approximation. () For Dirichlet‐type constraints, most of the attention has been focused on penalty‐type approaches. ()…”

Section: Introductionmentioning

confidence: 99%

“…For Neumann-type boundary and interface conditions, this strategy leads to a straightforward phase-field approximation. 23,24 For Dirichlet-type constraints, most of the attention has been focused on penalty-type approaches. 4,5,25 Embedded domain finite element methods, also known as fictitious domain or immersed boundary methods, represent another class of methods that are targeted at overcoming problems related to boundary-fitted meshing and parametrization of complex domains (see, for example, previous studies [26][27][28][29][30][31][32] and the references therein).…”

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confidence: 99%

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The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Nguyen

Stoter

Ruess

et al. 2017

Numerical Meth Engineering

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Summary We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

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Section: Diffuse Boundary But Sharp Volumementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Nguyen

Stoter

Ruess

et al. 2017

Numerical Meth Engineering

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“…In recent years, many scientists and engineers have investigated the fluid flow interaction between the conduit and porous media regime . The massive applications, such as karst aquifer subsurface flow system, interaction between the surface flows and subsurface flows, petroleum extraction, industrial filtration, biochemical transport, field flow fractionation for separation and characterization of proteins, blood flow in arteries and veins, etc, attract scientists and engineers to build related fluid dynamical models, including (Navier‐)Stokes‐Darcy model, Stokes‐Darcy‐transport model, Darcy‐Stokes‐Brinkman model, etc . It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier‐)Stokes‐Darcy fluid flow system, including coupled finite element methods, domain decomposition methods, Lagrange multiplier methods, mortar finite element methods, least‐square methods, partitioned time‐stepping methods, two‐grid and multigrid methods, discontinuous Galerkin finite element methods, boundary integral methods, and many others …”

Section: Introductionmentioning

confidence: 99%

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

Mahbub

Nasu

et al. 2019

Numerical Meth Engineering

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In this paper, we propose and analyze two stabilized mixed finite element methods for the dual-porosity-Stokes model, which couples the free flow region and microfracture-matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual-porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. KEYWORDS decoupled numerical methods, dual-porosity-Stokes model, horizontal wellbore, mixed finite elements, stabilization Int J Numer Methods Eng. 2019;120:803-833. wileyonlinelibrary.com/journal/nme 804 AL MAHBUB ET AL.model, It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier-)Stokes-Darcy fluid flow system, including coupled finite element methods, 17-21 domain decomposition methods, 22-31 Lagrange multiplier methods, 3,32,33 mortar finite element methods, 34,35 least-square methods, 36-38 partitioned time-stepping methods, 39,40 two-grid and multigrid methods, 41-44 discontinuous Galerkin finite element methods, 45-50 boundary integral methods, 51,52 and many others. [53][54][55][56][57][58] Although the traditional Stokes-Darcy model has been well studied, it has limitation to describe the heterogeneity of the porous medium which contains multiple porosities. It is worth to notice that the realistic naturally fractured reservoir consists of two coexisting and interacting medium, namely, tight matrix and microfractures. Furthermore, it is more important to characterize the intrinsic properties and accurately modeled the flow interactions between each medium. 59,60 The first multiporosity model was proposed by Barenblatt et al 61 for the naturally fractured reservoir in 1960 where the microfracture and matrix systems are formulated by individual but overlapping continua. Warren and Root 62 developed a homogeneous orthotropic dual-porosity model in 1963 based on the model proposed by Barenblatt. There are many applications for the dual-porosity model such as the geothermal system, hydrogeology, pe...

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“…Buoso et al introduced a parameterized reduced‐order method for the noninvasive functional evaluation of coronary artery diseases 8 . Stoter et al used the Navier‐Stokes/Darcy equations to simulate the blood flow of a patient‐specific human liver 9 . Some recent reviews of patient‐specific blood flow simulations can be found in References 10 and 11.…”

Section: Introductionmentioning

confidence: 99%

A parallel non‐nested two‐level domain decomposition method for simulating blood flows in cerebral artery of stroke patient

Chen

Cheng

et al. 2020

Numer Methods Biomed Eng

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Numerical simulation of blood flows in patient‐specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre‐ and/or post‐surgery planning. For this purpose, we introduce a highly scalable, parallel non‐nested two‐level domain decomposition method for the three‐dimensional unsteady incompressible Navier‐Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton‐Krylov method with a two‐level Schwarz method. The key component of the method is a non‐nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.

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A diffuse interface method for the Navier–Stokes/Darcy equations: Perfusion profile for a patient-specific human liver based on MRI scans

Cited by 44 publications

References 83 publications

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

A parallel non‐nested two‐level domain decomposition method for simulating blood flows in cerebral artery of stroke patient

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