Author Contributions
Data curation, Y.A.M. and T.V.G.; investigation, Y.A.M., O.V.V. and T.V.G.; methodology, Y.A.M. and O.V.V.; resources, T.V.G.; supervision, O.V.V. and T.V.G.; validation, Y.A.M.; visualization, Y.A.M.; writing, original draft preparation, Y.A.M., O.V.V. and T.V.G.; writing, review and editing, Y.A.M., O.V.V. and T.V.G. All authors read and agreed to the published version of the manuscript.
Figure 1.
Mixed velocity-pressure elements.
Figure 1.
Mixed velocity-pressure elements.
Figure 2.
-based Eulerian grid combined with Lagrangian particles.
Figure 2.
-based Eulerian grid combined with Lagrangian particles.
Figure 3.
Approximate Voronoi diagrams. Note that accuracy depends on the lattice resolution.
Figure 3.
Approximate Voronoi diagrams. Note that accuracy depends on the lattice resolution.
Figure 4.
Elimination and cloning of Lagrangian particles.
Figure 4.
Elimination and cloning of Lagrangian particles.
Figure 5.
Father wavelet and mother wavelet for the linear interpolating transform.
Figure 5.
Father wavelet and mother wavelet for the linear interpolating transform.
Figure 6.
Wavelet coefficients for the Gaussian function .
Figure 6.
Wavelet coefficients for the Gaussian function .
Figure 7.
The linear interpolating wavelet transform of the 2-D Gaussian function : (a) the initial mask after the wavelet transform and (b) the extended mask used for mesh adaptation.
Figure 7.
The linear interpolating wavelet transform of the 2-D Gaussian function : (a) the initial mask after the wavelet transform and (b) the extended mask used for mesh adaptation.
Figure 8.
Multilevel bilinear finite element grid constructed using the adapted mask
on
Figure 7b.
Figure 8.
Multilevel bilinear finite element grid constructed using the adapted mask
on
Figure 7b.
Figure 9.
Typical segments of a multilevel grid for (a) bilinear () and (b) biquadratic () basis functions.
Figure 9.
Typical segments of a multilevel grid for (a) bilinear () and (b) biquadratic () basis functions.
Figure 10.
Initial density (left) and viscosity (right) profiles.
Figure 10.
Initial density (left) and viscosity (right) profiles.
Figure 11.
Dependence of the normalized error on the number of elements.
Figure 11.
Dependence of the normalized error on the number of elements.
Figure 12.
Sinking block model setup.
Figure 12.
Sinking block model setup.
Figure 13.
Preliminary grid adaptation.
Figure 13.
Preliminary grid adaptation.
Figure 14.
Results of sinking block benchmark with viscosity contrast . Material field with imposed numerical grid is shown.
Figure 14.
Results of sinking block benchmark with viscosity contrast . Material field with imposed numerical grid is shown.
Figure 15.
Results of sinking block benchmark with viscosity contrast . Material field with imposed numerical grid is shown.
Figure 15.
Results of sinking block benchmark with viscosity contrast . Material field with imposed numerical grid is shown.
Figure 16.
Pressure field around the block obtained with and elements.
Figure 16.
Pressure field around the block obtained with and 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 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 18.
Extension/compression model setup.
Figure 18.
Extension/compression model setup.
Figure 19.
Extension model with after 0.07% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 19.
Extension model with after 0.07% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 20.
Extension model with after 0.53% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 20.
Extension model with after 0.53% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 21.
Compression model with after 0.13% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 21.
Compression model with after 0.13% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 22.
Compression model with after 1.33% strain; top: ) plot; middle: numerical grid; bottom: zoom as marked by red rectangles.
Figure 22.
Compression model with after 1.33% strain; top: ) 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 23.
Orientations of shear bands in extension/compression simulations with different frictions angles.
Figure 24.
Material fields and ) plots obtained with non-adaptive and adaptive grids for extension model with after 12.33% strain.
Figure 24.
Material fields and ) plots obtained with non-adaptive and adaptive grids for extension model with after 12.33% strain.
Figure 25.
Material fields and
) plots obtained with non-adaptive and adaptive grids for extension model with
after 25.33% strain. See
Figure 26 for zoom as marked by a red rectangle here.
Figure 25.
Material fields and
) plots obtained with non-adaptive and adaptive grids for extension model with
after 25.33% strain. See
Figure 26 for zoom as marked by a red rectangle here.
Figure 26.
Results of adaptive extension simulation with
after 25.33% strain. Top: numerical grid; bottom: zoom as marked by a red rectangle here and on
Figure 25.
Figure 26.
Results of adaptive extension simulation with
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 27.
Performance comparison between adaptive and non-adaptive numerical schemes for brittle extension model. Dual AMD Opteron 8380 system was used.
Figure 28.
Rayleigh-Taylor model initial setup.
Figure 28.
Rayleigh-Taylor model initial setup.
Figure 29.
Comparison of numerical and analytical growth rates for Rayleigh-Taylor model.
Figure 29.
Comparison of numerical and analytical growth rates for Rayleigh-Taylor model.
Figure 30.
Results of Rayleigh-Taylor numerical experiment with . top: material plot; middle: numerical grid; bottom: -velocity plot.
Figure 30.
Results of Rayleigh-Taylor numerical experiment with . top: material plot; middle: numerical grid; bottom: -velocity plot.
Figure 31.
Results of Rayleigh-Taylor numerical experiment with . top: material plot; middle: numerical grid; bottom: -velocity plot.
Figure 31.
Results of Rayleigh-Taylor numerical experiment with . top: material plot; middle: numerical grid; bottom: -velocity plot.
Figure 32.
Zoom in
Figure 31 at
, as marked by red rectangles.
Figure 32.
Zoom in
Figure 31 at
, as marked by red rectangles.
Figure 33.
Time (top) and memory requirements (bottom) for Cholesky factorization depending on numerical resolution. Quad AMD Opteron 8220 system was used.
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 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.
Figure 35.
Number of Lagrangian particles as a function of time when Voronoi tessellation algorithm is disabled/enabled.
Table 1.
Benchmark problems studied.
Table 1.
Benchmark problems studied.
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 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 |
Table 2.
Model parameters for the sinking block benchmark.
Table 2.
Model parameters for the sinking block benchmark.
Parameter | Value |
---|
Block viscosity | |
Medium viscosity | |
Block density | |
Medium density | |
Gravitational acceleration g | |
Time step | |
Table 3.
Model parameters for brittle extension/compression benchmark.
Table 3.
Model parameters for brittle extension/compression benchmark.
Parameter | | Value |
---|
| Weak inclusion viscosity | |
| Medium viscosity | |
| Weak inclusion and medium density | |
| Air viscosity | |
| Air density | |
| Gravitational acceleration g | |
| Friction angle | |
| Strain values | |
Cohesion | Extension | |
Compression | |
Boundary velocity | Extension | |
Compression | |
Time step | Extension | |
Compression | |
| Nonlinear tolerance | |
Table 4.
Velocity norms normalized by the number of elements.
Table 4.
Velocity norms normalized by the number of elements.
Element Type | Resolution |
---|
| | | |
---|
| | | | |
| | | | |
Table 5.
Model parameters for the Rayleigh-Taylor benchmark.
Table 5.
Model parameters for the Rayleigh-Taylor benchmark.
Parameter | Value |
---|
Top layer viscosity | |
Bottom layer viscosity | |
Top layer density | |
Bottom layer density | |
Gravitational acceleration g | |
Courant number | |