# A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Finite Element Discretization of the Stokes System

## 4. Particle-In-Cell Simulation Methodology

- Interpolation of physical properties from Lagrangian particles to Eulerian grid.
- Assemblage of the Stokes system using interpolated physical properties.
- Solution of the system on an Eulerian grid (see Section 3).
- Interpolation of computed velocities to Lagrangian particles positions.
- Particles advection using the interpolated velocities.

## 5. Wavelet-Based Grid Adaptation

#### 5.1. Linear Interpolating Wavelet Transform

#### 5.2. Grid Adaptation Algorithm

- Perform the forward wavelet transform of a physical field which is considered as an adaptation criterion and get all ${c}_{k}^{0}$ and ${d}_{l}^{j}$ coefficients. If a physical property field is defined on Lagrangian particles, the interpolation from particles to grid nodes is performed first.
- Analyze wavelet coefficients ${d}_{l}^{j}$ at all levels and create a mask $\mathcal{M}$ containing grid nodes associated with significant ${d}_{l}^{j}$.
- Include into the mask $\mathcal{M}$ all grid nodes from the coarsest level, i.e. associated with coefficients ${c}_{k}^{0}$.
- Extend the mask $\mathcal{M}$ with grid nodes associated with adjacent to significant ${d}_{l}^{j}$. This is to ensure that the mask $\mathcal{M}$ includes all nodes whose coefficients can potentially become significant at the next simulation time step.
- Apply recursively the reconstruction check procedure to the mask $\mathcal{M}$. This is to guarantee that all wavelet coefficients ${d}_{l}^{j}$ necessary to perform the forward transform at the next time step will be available.
- Using the adapted mask $\mathcal{M}$, construct a new multilevel finite element grid.

## 6. Dealing with Hanging Nodes

## 7. Implementation Aspects

## 8. Numerical Benchmarks

#### 8.1. Lateral Viscosity Variation Benchmark

#### 8.1.1. Setup and Parameters

#### 8.1.2. Convergence Test

#### 8.2. Sinking Block Benchmark

#### 8.2.1. Setup and Parameters

#### 8.2.2. Effect of Viscosity Contrast

#### 8.2.3. Checkerboard Pressure Problem with ${\mathrm{Q}}_{1}{\mathrm{P}}_{0}$ Element

#### 8.2.4. Performance Analysis

`C++`.

#### 8.3. Brittle Extension/Compression Benchmark

#### 8.3.1. Setup and Parameters

#### 8.3.2. Shear Bands Formation

#### 8.3.3. Long-Term Brittle Extension

#### 8.3.4. Performance Analysis

#### 8.4. Incompressibility Issue with ${Q}_{1}{Q}_{1}$ Element

#### 8.5. Rayleigh-Taylor Instability Benchmark

#### 8.5.1. Setup and Parameters

#### 8.5.2. Growth of Diapirs

#### 8.5.3. Performance Analysis

#### 8.5.4. Effect of Voronoi Tessellation

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEM | finite element method |

${\mathrm{Q}}_{1}{\mathrm{P}}_{0}$ | bilinear form of FEM |

${\mathrm{Q}}_{1}{\mathrm{Q}}_{1}$ | bilinear form of FEM |

${\mathrm{Q}}_{2}{\mathrm{P}}_{-1}$ | biquadratic form of FEM |

CPU | central processing unit |

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**Figure 2.**${\mathrm{Q}}_{1}{\mathrm{Q}}_{1}$-based Eulerian grid combined with Lagrangian particles.

**Figure 5.**Father wavelet $\varphi \left(x\right)$ and mother wavelet $\psi \left(x\right)$ for the linear interpolating transform.

**Figure 6.**Wavelet coefficients for the Gaussian function $f\left(x\right)={e}^{-\frac{{(x-0.5)}^{2}}{{(1/64)}^{2}}}$.

**Figure 7.**The linear interpolating wavelet transform of the 2-D Gaussian function $f(x,y)={e}^{-\frac{{x}^{2}+{y}^{2}}{2{\left(0.07\right)}^{2}}}$: (

**a**) the initial mask $\mathcal{M}$ after the wavelet transform and (

**b**) the extended mask $\mathcal{M}$ used for mesh adaptation.

**Figure 8.**Multilevel bilinear finite element grid constructed using the adapted mask $\mathcal{M}$ on Figure 7b.

**Figure 9.**Typical segments of a multilevel grid for (

**a**) bilinear (${Q}_{1}$) and (

**b**) biquadratic (${Q}_{2}$) basis functions.

**Figure 11.**Dependence of the normalized error ${\parallel error\parallel}_{1}$ on the number of elements.

**Figure 14.**Results of sinking block benchmark with viscosity contrast $\frac{{\eta}_{b}}{{\eta}_{m}}={10}^{2}$. Material field with imposed numerical grid is shown.

**Figure 15.**Results of sinking block benchmark with viscosity contrast $\frac{{\eta}_{b}}{{\eta}_{m}}={10}^{7}$. Material field with imposed numerical grid is shown.

**Figure 16.**Pressure field around the block obtained with ${\mathrm{Q}}_{1}{\mathrm{Q}}_{1}$ and ${\mathrm{Q}}_{1}{\mathrm{P}}_{0}$ elements.

**Figure 17.**Performance comparisons between adaptive and non-adaptive numerical schemes for sinking block model;

**top**: 1 CPU;

**bottom**: 8 CPUs with MATLAB Parallel Computing Toolbox. Dual AMD Opteron 8380 system was used.

**Figure 19.**Extension model with $\varphi ={0}^{\circ}$ after 0.07% strain;

**top**: $log({\dot{\epsilon}}_{II}$) plot;

**middle**: numerical grid;

**bottom**: zoom as marked by red rectangles.

**Figure 20.**Extension model with $\varphi ={0}^{\circ}$ after 0.53% strain;

**top**: $log({\dot{\epsilon}}_{II}$) plot;

**middle**: numerical grid;

**bottom**: zoom as marked by red rectangles.

**Figure 21.**Compression model with $\varphi ={0}^{\circ}$ after 0.13% strain;

**top**: $log({\dot{\epsilon}}_{II}$) plot;

**middle**: numerical grid;

**bottom**: zoom as marked by red rectangles.

**Figure 22.**Compression model with $\varphi ={0}^{\circ}$ after 1.33% strain;

**top**: $log({\dot{\epsilon}}_{II}$) plot;

**middle**: numerical grid;

**bottom**: zoom as marked by red rectangles.

**Figure 23.**Orientations of shear bands in extension/compression simulations with different frictions angles.

**Figure 24.**Material fields and $log({\dot{\epsilon}}_{II}$) plots obtained with non-adaptive and adaptive grids for extension model with $\varphi ={0}^{\circ}$ after 12.33% strain.

**Figure 25.**Material fields and $log({\dot{\epsilon}}_{II}$) plots obtained with non-adaptive and adaptive grids for extension model with $\varphi ={0}^{\circ}$ after 25.33% strain. See Figure 26 for zoom as marked by a red rectangle here.

**Figure 26.**Results of adaptive extension simulation with $\varphi ={0}^{\circ}$ after 25.33% strain. Top: numerical grid; bottom: zoom as marked by a red rectangle here and on Figure 25.

**Figure 27.**Performance comparison between adaptive and non-adaptive numerical schemes for brittle extension model. Dual AMD Opteron 8380 system was used.

**Figure 30.**Results of Rayleigh-Taylor numerical experiment with $\frac{{\eta}_{1}}{{\eta}_{2}}={10}^{3}$.

**top**: material plot;

**middle**: numerical grid;

**bottom**: ${v}_{y}$-velocity plot.

**Figure 31.**Results of Rayleigh-Taylor numerical experiment with $\frac{{\eta}_{1}}{{\eta}_{2}}=1$.

**top**: material plot;

**middle**: numerical grid;

**bottom**: ${v}_{y}$-velocity plot.

**Figure 33.**Time (

**top**) and memory requirements (

**bottom**) for Cholesky factorization depending on numerical resolution. Quad AMD Opteron 8220 system was used.

**Figure 34.**Effect of the Voronoi tessellation algorithm on a Lagrangian particle distribution. The tessellation is disabled (

**top**)/enabled (

**bottom**).

**Figure 35.**Number of Lagrangian particles as a function of time when Voronoi tessellation algorithm is disabled/enabled.

Section | Benchmark Problem | Main Aspects of the Algorithm Tested by the Benchmark Problem |
---|---|---|

Section 8.1 | Lateral viscosity variation | Comparison with the analytical solution |

Section 8.2 | Sinking block | Ability to handle large $O\left({10}^{7}\right)$ viscosity contrasts |

Section 8.3 | Brittle extension/compression | Ability to capture and resolve spontaneously forming shear zones |

Section 8.4 | Incompressibility test | Influence of the artificial incompressibility |

Section 8.5 | Rayleigh-Taylor instability | Comparison with the analytical solution |

Parameter | Value |
---|---|

Block viscosity ${\eta}_{\mathrm{b}}$ | ${10}^{2}\u2013{10}^{7}$ |

Medium viscosity ${\eta}_{\mathrm{m}}$ | $1.0$ |

Block density ${\rho}_{\mathrm{b}}$ | $2.0$ |

Medium density ${\rho}_{\mathrm{m}}$ | $1.0$ |

Gravitational acceleration g | $-10.0$ |

Time step $\Delta t$ | $0.1$ |

Parameter | Value | |
---|---|---|

Weak inclusion viscosity ${\eta}_{\mathrm{w}}$ | $1.0$ | |

Medium viscosity ${\eta}_{\mathrm{m}}$ | $100.0$ | |

Weak inclusion and medium density $\rho $ | $1.0$ | |

Air viscosity ${\eta}_{\mathrm{a}}$ | $0.01$ | |

Air density ${\rho}_{\mathrm{a}}$ | $0.0$ | |

Gravitational acceleration g | $-10.0$ | |

Friction angle $\varphi $ | ${0}^{\circ}\u2013{30}^{\circ}$ | |

Strain values ${\epsilon}_{1}/{\epsilon}_{2}$ | $0.0\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}0.1$ | |

Cohesion ${c}_{1}/{c}_{2}$ | Extension | $4.0\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}1.0$ |

Compression | $20.0\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}10.0$ | |

Boundary velocity ${v}_{\mathrm{bc}}$ | Extension | $0.05$ |

Compression | $-0.5$ | |

Time step $\Delta t$ | Extension | $0.02$ |

Compression | $0.004$ | |

Nonlinear tolerance ${\u03f5}_{\mathrm{nl}}$ | ${10}^{-3}$ |

Element Type | Resolution | |||
---|---|---|---|---|

$\mathbf{100}\times \mathbf{100}$ | $\mathbf{200}\times \mathbf{200}$ | $\mathbf{400}\times \mathbf{400}$ | $\mathbf{800}\times \mathbf{800}$ | |

${\mathrm{Q}}_{1}{\mathrm{Q}}_{1}$ | $8.35\times {10}^{-7}$ | $1.04\times {10}^{-7}$ | $1.30\times {10}^{-8}$ | $1.63\times {10}^{-9}$ |

${\mathrm{Q}}_{2}{\mathrm{P}}_{-1}$ | $2.26\times {10}^{-17}$ | $2.31\times {10}^{-17}$ | $2.39\times {10}^{-17}$ | $2.89\times {10}^{-17}$ |

Parameter | Value |
---|---|

Top layer viscosity ${\eta}_{1}$ | $1\u20132000$ |

Bottom layer viscosity ${\eta}_{2}$ | $1\u20131000$ |

Top layer density ${\rho}_{1}$ | $1.1$ |

Bottom layer density ${\rho}_{2}$ | $1.0$ |

Gravitational acceleration g | $-10.0$ |

Courant number | $0.8$ |

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**MDPI and ACS Style**

Mishin, Y.A.; Vasilyev, O.V.; Gerya, T.V. A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems. *Fluids* **2022**, *7*, 221.
https://doi.org/10.3390/fluids7070221

**AMA Style**

Mishin YA, Vasilyev OV, Gerya TV. A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems. *Fluids*. 2022; 7(7):221.
https://doi.org/10.3390/fluids7070221

**Chicago/Turabian Style**

Mishin, Yury A., Oleg V. Vasilyev, and Taras V. Gerya. 2022. "A Wavelet-Based Adaptive Finite Element Method for the Stokes Problems" *Fluids* 7, no. 7: 221.
https://doi.org/10.3390/fluids7070221