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

Chen, Rongliang; Wu, Bokai; Cheng, Zaiheng; Shiu, Wen-Shin; Liu, Jia; Liu, Liping; Wang, Yongjun; Wang, Xinhong; Cai, Xiao‐Chuan

doi:10.1002/cnm.3392

Cited by 14 publications

(10 citation statements)

References 46 publications

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“…As the finite element pair

(W_{h}, Q_{h})

does not satisfy the inf‐sup condition, 45 we use the stabilized finite element method 3,46 to spatially discretize the weak formulation (2): find

({bold-italicu}_{h} (\cdot t), p_{h} (\cdot t)) \in {bold-italicV}_{h} \times Q_{h}

, such that

\{\begin{matrix} left \\ (ρ \frac{\partial {bold-italicu}_{h}}{\partial t}, v_{h}) + (μ \nabla {bold-italicu}_{h}, \nabla {bold-italicv}_{h}) - (p_{h}, \nabla \cdot {bold-italicv}_{h}) + (q_{h}, \nabla \cdot {bold-italicu}_{h}) + (ρ {bold-italicu}_{h} \cdot \nabla {bold-italicu}_{h}, v_{h}) \\ - {(⟨⟩,, μ \nabla u_{h} \cdot bold-italicn, {bold-italicv}_{h})}_{Γ_{O}} + {false\sum}_{i = 1}^{m} R_{O}^{i} \int_{Γ_{O}^{i}} {bold-italicu}_{h} \cdot n d {normalΓ}_{O}^{i} \int_{Γ_{O}^{i}} {boldv}_{h} \cdot n d {normalΓ}_{O}^{i} + \underset{K \in {scriptT}_{h}}{false\sum} {((),, \nabla \cdot u_{h}, γ_{2} \nabla \cdot v_{h})}_{K} \\ + \underset{K \in {scriptT}_{h}}{false\sum} ((),, ρ) \end{matrix}

…”

Section: Model Problem and Its Discretizationmentioning

confidence: 99%

“…Let

{({}, x_{H, a}^{i})}_{i = 1}^{N_{a}^{c}}

be the collection of mesh points of

{scriptT}_{H}^{a}

. Following, 3,51 using radial basis functions (RBF) denoted

ϕ (r, ε)

(see Table 1) with the rescaled technique, 51 the extension operator from

(u_{H}, p_{H}) \in {bold-italicV}_{H}^{a} (Ω_{a}) \times Q_{H}^{a} (Ω_{a})

(u_{h}, p_{h}) \in {bold-italicV}_{h} (Ω_{a}) \times Q_{h} (Ω_{a})

can be defined as

Z_{h} (x_{h, a}^{i}) = \underset{x_{H, a}^{j} \in D_{H}^{i}}{false\sum} w_{italicij} Z_{H} (x_{H, a}^{j}), w_{italicij} = \frac{{\overset{true˜}{w}}_{ij}}{\underset{x_{H, a}^{k} \in D_{H}^{i}}{false\sum} {\overset{w}{true}}_{italicik}}, {\overset{w}{true}}_{i} = {normalΦ}^{- 1} ϕ_{i}, (Z = u, p)

with the vectors

{\overset{w}{true}}_{i} = (w_{ij}), ϕ_{i}

…”

Section: Two‐level Additive Schwarz Preconditioner With Mixed‐dimensi...mentioning

confidence: 99%

“…For the blood flows, we set the viscosity

ν = 0.035

g/(cm·s), the density

ρ = 1

g/cm 3 . On the inlet, we prescribe a pulsatile periodic flow velocity 3 (see Figure 6) with a parabolic profile. On each outlet

{normalΓ}_{O}^{i}

, the resistance

R_{O}^{i} = R_{italictotal} \sum_{j = 1}^{m} {(||, {normalΓ}_{O}^{j})}^{1.5} / {(||, {normalΓ}_{O}^{i})}^{1.5}

with

R_{italictotal} = 1500

dyn·s/cm 5 .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Computational hemodynamics is an important tool to study the behavior of blood flows and help understand certain vascular diseases such as the effect of the flow patterns and the distribution of the wall shear stress on the growth of the cerebral aneurysm 1,2 . With the rapid development of supercomputing and parallel algorithms in recent years, computational fluid dynamics (CFD) based technology for blood flow studies is becoming a powerful tool for research and clinic studies 3–10 . Compared with the standard imaging technologies such as computed tomography, magnetic resonance imaging, and transcranial Doppler, the CFD‐based methods provide more details of blood flows.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome the difficulty, we propose a new method that combines the 1D coarse preconditioner defined on the centerline of the normal region with a 3D coarse preconditioner 3 defined on the small 3D aneurysmal region to form a mixed‐dimensional coarse preconditioner with some compatibility conditions on the interfaces of the centerline and the aneurysm. The mixed‐dimensional Navier–Stokes model is discretized in space by a stabilized finite element method on the coarse mesh and the implicit backward Euler method in time.…”

Section: Introductionmentioning

confidence: 99%

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An aneurysm‐specific preconditioning technique for the acceleration of Newton‐Krylov method with application in the simulation of blood flows

Liu,

Qi,

Cai

2023

Numer Methods Biomed Eng

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In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two‐level additive Schwarz method with a mixed‐dimensional coarse preconditioner. The key components of the preconditioner include (1) a three‐dimensional coarse preconditioner covering the aneurysm; (2) a one‐dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three‐dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier–Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second‐order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton‐Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient‐specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

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“…As the finite element pair