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# Table of Contents

<table><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>1.1</td><td>Problem of interest and general difficulties</td><td>3</td></tr><tr><td>1.2</td><td>Scope of the presented work</td><td>5</td></tr><tr><td>1.3</td><td>Thesis contribution</td><td>6</td></tr><tr><td>1.4</td><td>Outline</td><td>8</td></tr><tr><td>2</td><td>Preliminaries</td><td>9</td></tr><tr><td>2.1</td><td>Considered type of problems</td><td>11</td></tr><tr><td>2.1.1</td><td>Conservation principle</td><td>12</td></tr><tr><td>2.1.2</td><td>A founding example</td><td>20</td></tr><tr><td>2.1.3</td><td>Hyperbolicity and discontinuity propagation</td><td>27</td></tr><tr><td>2.1.4</td><td>Weak solutions</td><td>28</td></tr><tr><td>2.1.5</td><td>Solution profiles</td><td>29</td></tr><tr><td>2.1.6</td><td>Entropy solutions</td><td>36</td></tr><tr><td>2.2</td><td>A brief algebraic toolbox</td><td>39</td></tr><tr><td>2.2.1</td><td>Multivariate polynomial spaces</td><td>39</td></tr><tr><td>2.2.2</td><td>Polynomial representation and degrees of freedom</td><td>41</td></tr><tr><td>2.2.3</td><td>Basis tuning on degrees of freedom</td><td>44</td></tr><tr><td>2.2.4</td><td>Basis functions adjustment to geometry distortion</td><td>46</td></tr><tr><td>2.3</td><td>A brief numerical analysis toolbox</td><td>50</td></tr><tr><td>2.3.1</td><td>Basic concepts of discretisation</td><td>50</td></tr><tr><td>2.3.2</td><td>Essential properties of a numerical scheme</td><td>55</td></tr><tr><td>2.4</td><td>Some classical numerical schemes</td><td>58</td></tr><tr><td>2.4.1</td><td>Discontinuous Galerkin methods</td><td>58</td></tr><tr><td>2.4.2</td><td>Flux reconstruction schemes</td><td>60</td></tr><tr><td>2.4.3</td><td>Residual distribution framework</td><td>74</td></tr><tr><td>2.4.4</td><td>Methods strengths and drawbacks</td><td>81</td></tr><tr><td>2.5</td><td>An entropy conservative scheme</td><td>83</td></tr><tr><td>2.5.1</td><td>Reformulation of a Flux Reconstruction scheme</td><td>83</td></tr><tr><td>2.5.2</td><td>Discrete conservation of the recast scheme</td><td>85</td></tr><tr><td>2.5.3</td><td>Entropy conservation of the recast scheme</td><td>86</td></tr><tr><td>2.5.4</td><td>Flux Reconstruction nature of the corrected recast scheme</td><td>90</td></tr><tr><td>2.5.5</td><td>Summary of the construction</td><td>93</td></tr><tr><td>2.5.6</td><td>Conclusion and perspectives</td><td>94</td></tr></table>
3 $H(\text{div})$-conforming elements on general polytopal meshes 97

3.1 Conforming discretisation strategies 99

3.1.1 Discrete features and conformity 100

3.1.2 The variational space $H(\text{div})$ and its specificities 101

3.1.3 Application of $H(\text{div})$-conforming discretisations 102

3.1.4 Classical polytopal and $H(\text{div})$-conforming elements 104

3.1.5 Scope of the presented method 105

3.2 Description of two conforming elements 107

3.2.1 Simplicial Raviart – Thomas elements 107

3.2.2 Quadrilateral Raviart – Thomas elements 115

3.2.3 The classical Virtual Elements 117

3.2.4 The $H(\text{div})$-conforming virtual element space 119

3.3 A framework for arbitrary polytopes 121

3.3.1 The general idea in a nutshell 122

3.3.2 Necessary conditions over spaces and elements 122

3.3.3 A class of admissible approximation spaces 123

3.3.4 Definition of $H(\text{div})$-conforming elements 129

3.3.5 Summary of the construction 146

3.3.6 Two examples in two dimensions 148

3.3.7 Link to other discretisation spaces 151

3.3.8 Numerical results 154

3.3.9 Critical assessment of the proposed approach 158

4 A conservative and continuity preserving interface correction 161

4.1 Problem statement, modelling, and approximation strategy 163

4.1.1 Modelling strategy 164

4.1.2 Conservation principle 168

4.1.3 Dedicated approaches and numerical schemes 171

4.1.4 Scope of the presented approach 176

4.2 Description of three fundamental methods 177

4.2.1 The Level Set approach 177

4.2.2 The Volume of Fluid method 179

4.2.3 The THINC-LS approach 181

4.3 A continuity preserving procedure 184

4.3.1 The general idea in a nutshell 184

4.3.2 An intuitive continuous interface correction 185

4.3.3 Extensions 197

4.3.4 Numerical approximations 209

4.3.5 Numerical results 216

4.3.6 Critical assessment of the proposed approach 225

Bibliography iii
List of figures

<table><tr><td>2.1</td><td>Problem definition: spatial and variational domains</td><td>11</td></tr><tr><td>2.2</td><td>Local conservation principle: contributions diagram</td><td>12</td></tr><tr><td>2.3</td><td>Local conservation principle: global dynamic contextualisation</td><td>12</td></tr><tr><td>2.4</td><td>Motion and transport: illustration of displacement and distortion</td><td>13</td></tr><tr><td>2.5</td><td>Motion and transport: illustration of the transport theorem</td><td>15</td></tr><tr><td>2.6</td><td>Solution profiles: discontinuity creation</td><td>28</td></tr><tr><td>2.7</td><td>Solution profiles: discontinuity decomposition</td><td>30</td></tr><tr><td>2.8</td><td>Solution profiles: physical and unphysical solutions</td><td>33</td></tr><tr><td>2.9</td><td>Solution profiles: non-uniqueness of entropy solutions</td><td>34</td></tr><tr><td>2.10</td><td>Solution profiles: one-dimensional Euler equations</td><td>35</td></tr><tr><td>2.11</td><td>Solution profiles: general notion of entropy</td><td>36</td></tr><tr><td>2.12</td><td>Algebra toolbox: duality relationship with a set of basis functions</td><td>44</td></tr><tr><td>2.13</td><td>Algebra toolbox: orientation-preserving mapping</td><td>48</td></tr><tr><td>2.14</td><td>Algebra toolbox: orientation-preservation and non-convexity</td><td>49</td></tr><tr><td>2.15</td><td>Numerical analysis toolbox: representation of a discrete solution</td><td>50</td></tr><tr><td>2.16</td><td>Numerical analysis toolbox: unstructured spatial discretisation</td><td>52</td></tr><tr><td>2.17</td><td>Numerical analysis toolbox: possible mesh pathologies</td><td>53</td></tr><tr><td>2.18</td><td>Numerical analysis toolbox: convergence property</td><td>55</td></tr><tr><td>2.19</td><td>Numerical analysis toolbox: illustration of the stability property</td><td>56</td></tr><tr><td>2.20</td><td>Flux Reconstruction: update procedure I/II</td><td>64</td></tr><tr><td>2.21</td><td>Flux Reconstruction: choice of lifting functions</td><td>71</td></tr><tr><td>2.22</td><td>Flux Reconstruction: towards a geometrical flexibility</td><td>72</td></tr><tr><td>2.23</td><td>Flux Reconstruction: update procedure II/II</td><td>73</td></tr><tr><td>2.24</td><td>Residual Distribution: scheme construction, from primal to dual</td><td>76</td></tr><tr><td>2.25</td><td>Residual Distribution: geometrical and variational mesh duals</td><td>76</td></tr></table>

<table><tr><td>3.1</td><td>Raviart – Thomas elements: simplicial case representation</td><td>114</td></tr><tr><td>3.2</td><td>Raviart – Thomas elements: quadrilateral case representation</td><td>116</td></tr><tr><td>3.3</td><td>Virtual elements framework: conforming layout</td><td>120</td></tr><tr><td>3.4</td><td>Unisolvence: normal determination matrix</td><td>141</td></tr><tr><td>3.5</td><td>Unisolvence: internal determination matrix</td><td>144</td></tr><tr><td>3.6</td><td>Conforming example: adaptivity towards the order and geometry</td><td>149</td></tr><tr><td>3.7</td><td>Numerical results: normal component of internal basis functions</td><td>154</td></tr><tr><td>3.8</td><td>Numerical results: normal components of normal basis functions</td><td>155</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>3.9</td>
      <td>Numerical results: behaviour of degenerating basis functions</td>
      <td>155</td>
    </tr>
    <tr>
      <td>3.10</td>
      <td>Numerical results: basis function scaling</td>
      <td>156</td>
    </tr>
    <tr>
      <td>3.11</td>
      <td>Numerical results: degeneracy of basis functions</td>
      <td>157</td>
    </tr>
    <tr>
      <td>3.12</td>
      <td>Numerical results: investigation of the reduced setting</td>
      <td>157</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Problem definition: domain, phases and interface</td>
      <td>163</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Modelling strategy: prominent flow behaviours in horizontal pipes</td>
      <td>164</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Modelling strategy: depiction of two-phase flows modellings</td>
      <td>165</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Modelling strategy: update of the time-discretised model</td>
      <td>166</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Modelling strategy: example of interface topological changes</td>
      <td>167</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>Conservation principle: two-phase mass conservation principle</td>
      <td>168</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>Conservation principle: two-phase discrete conservation principle</td>
      <td>169</td>
    </tr>
    <tr>
      <td>4.8</td>
      <td>Conservation principle: approximation within a control volume</td>
      <td>170</td>
    </tr>
    <tr>
      <td>4.9</td>
      <td>Conservation principle: impact of the quadrature points</td>
      <td>170</td>
    </tr>
    <tr>
      <td>4.10</td>
      <td>Level Set method: definition of an implicit interface</td>
      <td>177</td>
    </tr>
    <tr>
      <td>4.11</td>
      <td>Level Set method: development of singularities</td>
      <td>178</td>
    </tr>
    <tr>
      <td>4.12</td>
      <td>Level Set method: numerical issues with perturbed function</td>
      <td>179</td>
    </tr>
    <tr>
      <td>4.13</td>
      <td>Volume of Fluid method: mixture and interface definition</td>
      <td>179</td>
    </tr>
    <tr>
      <td>4.14</td>
      <td>Volume of Fluid method: interfacial cells and mixture definition</td>
      <td>180</td>
    </tr>
    <tr>
      <td>4.15</td>
      <td>Volume of Fluid method: interface evolution</td>
      <td>180</td>
    </tr>
    <tr>
      <td>4.16</td>
      <td>THINC approach: semisequential update chart</td>
      <td>181</td>
    </tr>
    <tr>
      <td>4.17</td>
      <td>Intuitive interface correction: spatial layout</td>
      <td>185</td>
    </tr>
    <tr>
      <td>4.18</td>
      <td>Intuitive interface correction: continuity-preserving correction</td>
      <td>188</td>
    </tr>
    <tr>
      <td>4.19</td>
      <td>Extensions: need of continuous mapping in non-convex cells</td>
      <td>197</td>
    </tr>
    <tr>
      <td>4.20</td>
      <td>Extensions: continuous mapping of a non-convex cell</td>
      <td>198</td>
    </tr>
    <tr>
      <td>4.21</td>
      <td>Extensions: convex polygons of reference</td>
      <td>198</td>
    </tr>
    <tr>
      <td>4.22</td>
      <td>Extensions: spatial layout of a staggered setting</td>
      <td>202</td>
    </tr>
    <tr>
      <td>4.23</td>
      <td>Extensions: correction procedure in a staggered setting</td>
      <td>204</td>
    </tr>
    <tr>
      <td>4.24</td>
      <td>Extensions: interface layouts yielding a same volume fraction</td>
      <td>208</td>
    </tr>
    <tr>
      <td>4.25</td>
      <td>Numerical approximations: composite quadrature scheme</td>
      <td>209</td>
    </tr>
    <tr>
      <td>4.26</td>
      <td>Numerical approximations: volume of fluid estimator</td>
      <td>215</td>
    </tr>
    <tr>
      <td>4.27</td>
      <td>Numerical approximations: smooth bijective map</td>
      <td>216</td>
    </tr>
    <tr>
      <td>4.28</td>
      <td>Numerical results: corrected interface layouts</td>
      <td>217</td>
    </tr>
    <tr>
      <td>4.29</td>
      <td>Numerical results: corrected multi-component interface layouts</td>
      <td>218</td>
    </tr>
    <tr>
      <td>4.30</td>
      <td>Numerical results: behaviour of the volume of fluid estimator I/II</td>
      <td>219</td>
    </tr>
    <tr>
      <td>4.31</td>
      <td>Numerical results: behaviour of the volume of fluid estimator II/II</td>
      <td>220</td>
    </tr>
    <tr>
      <td>4.32</td>
      <td>Numerical results: continuity-preserving correction</td>
      <td>221</td>
    </tr>
    <tr>
      <td>4.33</td>
      <td>Numerical results: impact of the locator function shape I/IV</td>
      <td>222</td>
    </tr>
    <tr>
      <td>4.34</td>
      <td>Numerical results: impact of the locator function shape II/IV</td>
      <td>222</td>
    </tr>
    <tr>
      <td>4.35</td>
      <td>Numerical results: impact of the locator function shape III/IV</td>
      <td>223</td>
    </tr>
    <tr>
      <td>4.36</td>
      <td>Numerical results: impact of the locator function shape IV/IV</td>
      <td>223</td>
    </tr>
    <tr>
      <td>4.37</td>
      <td>Numerical results: limitations of the proposed approach</td>
      <td>224</td>
    </tr>
  </tbody>
</table>
# List of tables

<table><tr><td>3.1</td><td><i>H</i>(div)—conforming examples: normal degrees of freedom</td><td>147</td></tr><tr><td>3.2</td><td><i>H</i>(div)—conforming examples: 2D normal degrees of freedom</td><td>148</td></tr><tr><td>3.3</td><td><i>H</i>(div)—conforming examples: reduced setting</td><td>150</td></tr><tr><td>4.1</td><td>Numerical approximations: face parametrisation’s derivatives</td><td>212</td></tr></table>
<figure><img src="image_101.png" /></figure>
<figure><img src="image_102.png" /></figure>