Domain Decomposition in Computational Fluid Dynamics for Pipe-like Domains Open Access

Wenda Zheng (Spring 2018)

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Nowadays, as the result of progress of numerical methods for solving complex problems, simulations of patient-specific cardiovascular districts are possible and used in medicine. Nevertheless, when covering large portion of the circulation, simulations on regular computers can be not affordable. In this thesis, starting from the evidence that the circulatory network is made of pipes and junctions, we test the efficacy of ``domain decomposition” techniques on our constructed 2D models. Using FreeFem++ as the simulation tool, we evaluates the Domain Decomposition techniques on 2D problems and try to find the possibilities to extend our work into 3D realistic cases. 

Table of Contents

1 Introduction  1.1  Motivation 1.2  Contribution 1.3  Outline 2 The Mathematical Problems  2.1  IntrotoHilbertSpaces 2.2  A simple 2D diffusion-reaction problem 2.3  Navier-Stokes Equations for Incompressible Fluids .

3 Domain Decomposition Techniques and Finite Element Method

3.1  Domain Decomposition Techniques

3.1.1  Test Cases Using 2D Elliptic Equation 

3.1.2  TheOverlappingMethod 

3.1.3  The Non-overlapping Method 

3.2  IntroductiontoFiniteElements 

              3.2.1 Finite Elements Using Galerkin Method               3.2.2 FreeFem++ 

4 Numerical Results 

       4.1 Results of a Single Model with Bifurcation

       4.2 Results of Modified Model with Child Bifurcation 

5 Conclusion  A FreeFem++ Code  Bibliography 

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