Convection-Dominated Problems: Finite Element Approximations to the Convection-Diffusion-Reaction Equation

Zienkiewicz, O. C.; Taylor, Robert T.; Nithiarasu, Perumal

doi:10.1016/b978-1-85617-635-4.00002-9

Cited by 4 publications

(3 citation statements)

References 61 publications

(72 reference statements)

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“…For the temporal discretization, we used the implicit Euler method. For flow problems, especially those hyperbolic by nature, it is known that discretization using the Bubnov–Galerkin finite element method leads to numerical instabilities (Zienkiewicz and Heinrich 1978 ). Therefore, we used the Box spatial discretization scheme and applied mass-lumping for the temporal evolution term.…”

Section: Methodsmentioning

confidence: 99%

A continuum mechanical porous media model for vertebroplasty: Numerical simulations and experimental validation

Trivedi

Gehweiler

Wychowaniec

et al. 2023

Biomech Model Mechanobiol

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The outcome of vertebroplasty is hard to predict due to its dependence on complex factors like bone cement and marrow rheologies. Cement leakage could occur if the procedure is done incorrectly, potentially causing adverse complications. A reliable simulation could predict the patient-specific outcome preoperatively and avoid the risk of cement leakage. Therefore, the aim of this work was to introduce a computationally feasible and experimentally validated model for simulating vertebroplasty. The developed model is a multiphase continuum-mechanical macro-scale model based on the Theory of Porous Media. The related governing equations were discretized using a combined finite element–finite volume approach by the so-called Box discretization. Three different rheological upscaling methods were used to compare and determine the most suitable approach for this application. For validation, a benchmark experiment was set up and simulated using the model. The influence of bone marrow and parameters like permeability, porosity, etc., was investigated to study the effect of varying conditions on vertebroplasty. The presented model could realistically simulate the injection of bone cement in porous materials when used with the correct rheological upscaling models, of which the semi-analytical averaging of the viscosity gave the best results. The marrow viscosity is identified as the crucial reference to categorize bone cements as ‘high- ’or ‘low-’ viscosity in the context of vertebroplasty. It is confirmed that a cement with higher viscosity than the marrow ensures stable development of the injection and a proper cement interdigitation inside the vertebra.

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Section: Methodsmentioning

confidence: 99%

A continuum mechanical porous media model for vertebroplasty: Numerical simulations and experimental validation

Trivedi

Gehweiler

Wychowaniec

et al. 2023

Biomech Model Mechanobiol

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“…The generated 3D generic model utilized a fully coupled direct solver (PARDISO) and an automated Newton technique. In order to achieve stability in the Navier-Stokes equations, a hybrid approach using both streamline and crosswind diffusion numerical approaches is employed [31]. The simulation results were recorded at 2 ms intervals.…”

Section: D Closed-loop Windkessel Circulatorymentioning

confidence: 99%

A 3D-0D Computational Model of the Left Ventricle for Investigating Blood Flow Patterns for Cases of Systolic Anterior Motion and after Anterior Mitral Leaflet Splitting

Alharbi

2024

Applied Sciences

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Valvular heart conditions significantly contribute to the occurrence of cardiovascular disease, affecting around 2–3 million people in the United States. The anatomical characteristics of cardiac muscles and valves can significantly influence blood flow patterns inside the ventricles. Understanding the interaction between the mitral valve and left ventricle structures enables using fluid–structure interaction simulations as a precise and user-friendly approach to investigating outcomes that cannot be captured using experimental approaches. This study aims to develop a 3D-0D computational model to simulate the consequences of extending the anterior mitral leaflet towards the left ventricle in the presence of the thickness of the left ventricular septum and the mitral valve device. The simulations presented in this paper successfully showcased the ability of the model to replicate occlusion occurring at the left ventricular outflow tract and illustrated the impact of this blockage on the flow pattern and pressure gradient. Furthermore, these simulations conducted following anterior mitral leaflet splitting can emphasize the significance of this technique in reducing the obstruction at the left ventricle outflow tract. The computational model presented in this study, combining 3D and 0D elements, provides significant insights into the flow patterns occurring in the left ventricle before and after anterior leaflet splitting. Thus, expanding this model can help explore other cardiac phenomena and investigate potential post-procedural complications.

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“…An implicit characteristic Galerkin FEM is used to stabilise numerical oscillation occurred when of solving equation (6). 22 The time derivative term is discretised by the implicit characteristic method as 28 where Δ t is the timestep. For the discretisation of the spatial differential terms of equation (14), the standard Galerkin FEM is applied where w is the weight function and Ω the whole space in the mould.…”

Section: Numerical Formulationsmentioning

confidence: 99%

Numerical simulation of RTM process using the extended finite element method combined with the level set method

Jung

Kim

Han

2013

Journal of Reinforced Plastics and Composites

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International audienceNumerical simulation for resin transfer moulding manufacturing process is attempted using the extended finite element method combined with the level set method. The level set method is used to transport the resin flow front at each time step during the mould filling. Extended finite element method allows to obtaining a good numerical precision of the resin pressure near the resin flow front, where the gradient of the pressure could be discontinuous. The enriched shape functions of extended finite element method are derived using the level set values so as to correctly describe these shape functions with the resin flow front. In addition, the position of the resin flow front at each time step is calculated by an implicit characteristic Galerkin finite element method. Some examples are presented to validate our study in comparison with analytical or experimental results and to illustrate its industrial applications

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Convection-Dominated Problems: Finite Element Approximations to the Convection-Diffusion-Reaction Equation

Cited by 4 publications

References 61 publications

A continuum mechanical porous media model for vertebroplasty: Numerical simulations and experimental validation

A continuum mechanical porous media model for vertebroplasty: Numerical simulations and experimental validation

A 3D-0D Computational Model of the Left Ventricle for Investigating Blood Flow Patterns for Cases of Systolic Anterior Motion and after Anterior Mitral Leaflet Splitting

Numerical simulation of RTM process using the extended finite element method combined with the level set method

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