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Fig. 2.2 Sketch of the candidate stencils for the six-point reconstruction WENO scheme.

Figure 2 2 Sketch of the candidate stencils for the six-point reconstruction WENO scheme.

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Fig. 1.2 High resolution simulation of Rayleigh-Taylor instability [12].
Fig. 1.2 High resolution simulation of Rayleigh-Taylor instability [12].
Fig. 1.3 Multi-resolution simulation of the flow past an impulsively-started cylinder [89]. 1.2 High-performance computational fluid dynamics
Fig. 1.3 Multi-resolution simulation of the flow past an impulsively-started cylinder [89]. 1.2 High-performance computational fluid dynamics
Fig. 1.4 Mapping of an adaptive mesh using a Space-Filling Curve method [48]
Fig. 1.4 Mapping of an adaptive mesh using a Space-Filling Curve method [48]
pun | where p and u denote the density and the velocity vector, E = pe+ is the total energy. To closé written in the conservative forms as the system of equations, the ideal-gas equations-of-state (EOS) p = (y— 1)pe with 7 as the ratio
pun | where p and u denote the density and the velocity vector, E = pe+ is the total energy. To closé written in the conservative forms as the system of equations, the ideal-gas equations-of-state (EOS) p = (y— 1)pe with 7 as the ratio
Fig. 2.1 Sketch of the candidate stencils for the five-point reconstruction WENO scheme.
Fig. 2.1 Sketch of the candidate stencils for the five-point reconstruction WENO scheme.
This kernel successfully avoids the tensile instability and the paring instability, while satisfying the the whole support domain as
This kernel successfully avoids the tensile instability and the paring instability, while satisfying the the whole support domain as
For stability, the timestep is constrained by a CFL condition The governing equation system can be integrated in time with a velocity-Verlet scheme [106]. For 2.2.4 Time integration method
For stability, the timestep is constrained by a CFL condition The governing equation system can be integrated in time with a velocity-Verlet scheme [106]. For 2.2.4 Time integration method
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Fig. 1. Candidate stencils with incremental width towards high-order reconstruction. Inspired by Borges et al. [20], the nonlinear weights are computed by
Fig. 1. Candidate stencils with incremental width towards high-order reconstruction. Inspired by Borges et al. [20], the nonlinear weights are computed by
Numerical flux Faia = )onCmfm evaluated from the candidate stencil S,. Table 1 one end node at x; contains at least one-point upwind, thus decreasing the strong anti-dissipation induced by a pure down- wind stencil. In Fig. 2, we compare sketches for the reconstruction of 7th-order schemes for the classical WENO scheme and the proposed TENO scheme in the case of closely located two shock waves. For the classical WENO scheme all candidate stencils are crossed by the two discontinuities. The weighting strategy of the classical WENO schemes fails to guarantee the ENO property. However, for the TENO scheme the three-point candidate stencil Sg avoids the crossing of discontinuities and thus can be singled out as smooth candidate stencil to contribute to the final reconstruction. Since practical numerical experiments reveal that fifth-order WENO scheme is robust for such configurations, the TENO schemes also can recover the robustness of fifth-order WENO scheme even for very- high-order reconstructions. ~ 1 r om a = i ee cr on © & owes oe - 1h r ee
Numerical flux Faia = )onCmfm evaluated from the candidate stencil S,. Table 1 one end node at x; contains at least one-point upwind, thus decreasing the strong anti-dissipation induced by a pure down- wind stencil. In Fig. 2, we compare sketches for the reconstruction of 7th-order schemes for the classical WENO scheme and the proposed TENO scheme in the case of closely located two shock waves. For the classical WENO scheme all candidate stencils are crossed by the two discontinuities. The weighting strategy of the classical WENO schemes fails to guarantee the ENO property. However, for the TENO scheme the three-point candidate stencil Sg avoids the crossing of discontinuities and thus can be singled out as smooth candidate stencil to contribute to the final reconstruction. Since practical numerical experiments reveal that fifth-order WENO scheme is robust for such configurations, the TENO schemes also can recover the robustness of fifth-order WENO scheme even for very- high-order reconstructions. ~ 1 r om a = i ee cr on © & owes oe - 1h r ee
Fig. 2. Sketch of reconstruction schemes with closely located shock waves: the dashed lines represent two shock discontinuities. Classical 7th-order WENO scheme (left) and 7th-order TENO scheme (right).
Fig. 2. Sketch of reconstruction schemes with closely located shock waves: the dashed lines represent two shock discontinuities. Classical 7th-order WENO scheme (left) and 7th-order TENO scheme (right).
In the context of a K-point full stencil, an order K reconstruction can be achieved by the stencil combination US Sk. A (K + 1)-order scheme is reconstructed from the stencil combination UE S, with one additional upwind or downwind- biased stencil Sx_2. Arbitrarily high-order odd (upwind) or even (central) schemes can be devised by this recursion. Optimal weights for maximum order up to K = 8 on the full-point stencil are given in Table 2. We redefine the symbol r as the width of a specific candidate stencil. Conservative numerical fluxes Feiss /2 on different candidate stencils are evaluated with Eq. (7) in the same way as for WENO. The respective coefficients are presented in Table 1 up tok=5.
In the context of a K-point full stencil, an order K reconstruction can be achieved by the stencil combination US Sk. A (K + 1)-order scheme is reconstructed from the stencil combination UE S, with one additional upwind or downwind- biased stencil Sx_2. Arbitrarily high-order odd (upwind) or even (central) schemes can be devised by this recursion. Optimal weights for maximum order up to K = 8 on the full-point stencil are given in Table 2. We redefine the symbol r as the width of a specific candidate stencil. Conservative numerical fluxes Feiss /2 on different candidate stencils are evaluated with Eq. (7) in the same way as for WENO. The respective coefficients are presented in Table 1 up tok=5.
The dx represents the optimal weight for candidate stencil S, to assemble a K-th order scheme. Optimal weights for the upwind or central scheme for maximum order K. Table 2 a physically-motivated scale-separation mechanism while setting C = 1000 for compensation. Such a choice reduces the effectiveness of discontinuity detection, which in turn compromises the robustness in simulations with strong shock waves. On the contrary, larger integer power exponent q and smaller C are preferable for an even stronger separation between re- solved and non-resolved scales in the present study. The discontinuity-detection capability is significantly enhanced rather than sacrificed. Low dissipation for smooth regions is maintained by a tailored ENO-like stencil selection in the next section. B,.. can be evaluated following Jiang and Shu /9] as
The dx represents the optimal weight for candidate stencil S, to assemble a K-th order scheme. Optimal weights for the upwind or central scheme for maximum order K. Table 2 a physically-motivated scale-separation mechanism while setting C = 1000 for compensation. Such a choice reduces the effectiveness of discontinuity detection, which in turn compromises the robustness in simulations with strong shock waves. On the contrary, larger integer power exponent q and smaller C are preferable for an even stronger separation between re- solved and non-resolved scales in the present study. The discontinuity-detection capability is significantly enhanced rather than sacrificed. Low dissipation for smooth regions is maintained by a tailored ENO-like stencil selection in the next section. B,.. can be evaluated following Jiang and Shu /9] as
Fig. 3. Approximate dispersion and dissipation properties for five-point stencil schemes: dispersion (left) and dissipation (right).
Fig. 3. Approximate dispersion and dissipation properties for five-point stencil schemes: dispersion (left) and dissipation (right).
Fig. 4. Approximate dispersion and dissipation properties for six-point stencil schemes: dispersion (left) and dissipation (right)
Fig. 4. Approximate dispersion and dissipation properties for six-point stencil schemes: dispersion (left) and dissipation (right)
Optimal weights based on the dispersion-dissipation condition for five-point-stencil schemes. Table 3 to properly balance the dissipation and dispersion by a reasonable choice of ¢, so that physically-consistent schemes could be obtained. An interesting observation is that the ¢ distributions of both WENO-SYMBO and WENO-BO apparently fail to satisfy the dispersion-dissipation relation [29].
Optimal weights based on the dispersion-dissipation condition for five-point-stencil schemes. Table 3 to properly balance the dissipation and dispersion by a reasonable choice of ¢, so that physically-consistent schemes could be obtained. An interesting observation is that the ¢ distributions of both WENO-SYMBO and WENO-BO apparently fail to satisfy the dispersion-dissipation relation [29].
Fig. 5. Approximate dispersion-dissipation relation for various nonlinear schemes: five-point stencil schemes (left) and six-point stencil schemes (right). dp -1|+10-3 de y axis is in log form and ¢ = “a0 approximately.
Fig. 5. Approximate dispersion-dissipation relation for various nonlinear schemes: five-point stencil schemes (left) and six-point stencil schemes (right). dp -1|+10-3 de y axis is in log form and ¢ = “a0 approximately.
Fig. 6. Spectral properties of the optimized linear five-point stencil schemes: dissipation (left) and dispersion-dissipation relation (right).
Fig. 6. Spectral properties of the optimized linear five-point stencil schemes: dissipation (left) and dispersion-dissipation relation (right).
Fig. 7. Spectral property of optimized nonlinear five-point stencil schemes: dissipation property (left) and dispersion-dissipation relationship (right)
Fig. 7. Spectral property of optimized nonlinear five-point stencil schemes: dissipation property (left) and dispersion-dissipation relationship (right)
Fig. 8. Spectral properties of optimized linear six-point stencil schemes: dissipation property (left) and dispersion-dissipation relation (right)
Fig. 8. Spectral properties of optimized linear six-point stencil schemes: dissipation property (left) and dispersion-dissipation relation (right)
Table 4 Fig. 9. Spectral properties of the optimized nonlinear six-point stencil schemes: dissipation (left) and dispersion-dissipation relation (right).
Table 4 Fig. 9. Spectral properties of the optimized nonlinear six-point stencil schemes: dissipation (left) and dispersion-dissipation relation (right).
Parameters of TENO-family schemes for numerical validation. Table 5
Parameters of TENO-family schemes for numerical validation. Table 5
Fig. 10. Convergence of the L.-error: five-point stencil schemes (left) and six-point stencil schemes (right). The linear scheme represents the maximum- order background scheme of TENO and the optimized linear scheme represents the spectrally optimized background scheme of optimized TENO.
Fig. 10. Convergence of the L.-error: five-point stencil schemes (left) and six-point stencil schemes (right). The linear scheme represents the maximum- order background scheme of TENO and the optimized linear scheme represents the spectrally optimized background scheme of optimized TENO.
Fig. 11. Sod shock-tube problem: density distribution (left) and a zoom of the region near contact discontinuity (right). Top: five-point stencil schemes. Bottom: six-point stencil schemes. Discretization on 100 uniformly distributed grid points. “Exact” represents the theoretical solution of the Riemann problem.
Fig. 11. Sod shock-tube problem: density distribution (left) and a zoom of the region near contact discontinuity (right). Top: five-point stencil schemes. Bottom: six-point stencil schemes. Discretization on 100 uniformly distributed grid points. “Exact” represents the theoretical solution of the Riemann problem.
Fig. 12. Lax shock-tube problem: density distribution (left) and velocity distribution (right). Top: five-point stencil schemes. Bottom: six-point stenci schemes. Discretization on 100 uniformly distributed grid points. “Exact” represents the theoretical solution of the Riemann problem.
Fig. 12. Lax shock-tube problem: density distribution (left) and velocity distribution (right). Top: five-point stencil schemes. Bottom: six-point stenci schemes. Discretization on 100 uniformly distributed grid points. “Exact” represents the theoretical solution of the Riemann problem.
Fig. 13. Interacting blast waves problem: solution from five-point stencil schemes. Top: density distribution (left) and a zoom of the density profile (right). Bottom: zoom of the pressure profile (left) and zoom of the velocity profile (right). Discretization on 400 uniform grid points.
Fig. 13. Interacting blast waves problem: solution from five-point stencil schemes. Top: density distribution (left) and a zoom of the density profile (right). Bottom: zoom of the pressure profile (left) and zoom of the velocity profile (right). Discretization on 400 uniform grid points.
Fig. 14. Interacting blast waves problem: solution from six-point stencil schemes. Top: density distribution (left) and a zoom of the density distribution (right). Bottom: zoom of the pressure profile (left) and zoom of the velocity profile (right). Discretization on 400 uniform grid points. WENO-CU6-M fails to pass this case.
Fig. 14. Interacting blast waves problem: solution from six-point stencil schemes. Top: density distribution (left) and a zoom of the density distribution (right). Bottom: zoom of the pressure profile (left) and zoom of the velocity profile (right). Discretization on 400 uniform grid points. WENO-CU6-M fails to pass this case.
Fig. 15. Shock-density wave interaction problem: solutions from five-point stencil schemes. Density distribution (left) and a zoom of the density distributio1 (right). Discretization on 200 uniformly distributed grid points.
Fig. 15. Shock-density wave interaction problem: solutions from five-point stencil schemes. Density distribution (left) and a zoom of the density distributio1 (right). Discretization on 200 uniformly distributed grid points.
Fig. 16. Shock-density wave interaction problem: solutions from six-point stencil schemes. Top: density distribution (left) and a zoom of the density distribution (right). Bottom: zoom of the density profile (left) and zoom of the velocity profile (right). Discretization on 200 uniformly distributed grid points.
Fig. 16. Shock-density wave interaction problem: solutions from six-point stencil schemes. Top: density distribution (left) and a zoom of the density distribution (right). Bottom: zoom of the density profile (left) and zoom of the velocity profile (right). Discretization on 200 uniformly distributed grid points.
Fig. 17. Double Mach reflection of a strong shock: density contours from five-point stencil schemes. Resolution 1024 x 256. This figure is drawn with 43 density contours between 1.887 and 20.9. 5.7. Two-dimensional viscous shock-tube problem
Fig. 17. Double Mach reflection of a strong shock: density contours from five-point stencil schemes. Resolution 1024 x 256. This figure is drawn with 43 density contours between 1.887 and 20.9. 5.7. Two-dimensional viscous shock-tube problem
Fig. 18. Double Mach reflection of a strong shock: density contours from six-point stencil schemes. Resolution 1024 x 256. This figure is drawn with 43 density contours between 1.887 and 20.9. WENO-CU6-M fails to pass this case.
Fig. 18. Double Mach reflection of a strong shock: density contours from six-point stencil schemes. Resolution 1024 x 256. This figure is drawn with 43 density contours between 1.887 and 20.9. WENO-CU6-M fails to pass this case.
Fig. 19. Rayleigh-Taylor instability: density contours. Resolution 128 x 512.
Fig. 19. Rayleigh-Taylor instability: density contours. Resolution 128 x 512.
Fig. 20. Rayleigh-Taylor instability: density contours. Resolution 256 x 1024.
Fig. 20. Rayleigh-Taylor instability: density contours. Resolution 256 x 1024.
Fig. 21. Two-dimensional viscous shock-tube problem: reference density result from WENO-JS scheme. Resolution 5120 x 256(
Fig. 21. Two-dimensional viscous shock-tube problem: reference density result from WENO-JS scheme. Resolution 5120 x 256(
Fig. 22. Two-dimensional viscous shock-tube problem: density solutions from five-point stencil schemes. Resolution 1280 x 640
Fig. 22. Two-dimensional viscous shock-tube problem: density solutions from five-point stencil schemes. Resolution 1280 x 640
Fig. 23. Two-dimensional viscous shock-tube problem: density solutions from six-point stencil schemes. Resolution 1280 x 640.
Fig. 23. Two-dimensional viscous shock-tube problem: density solutions from six-point stencil schemes. Resolution 1280 x 640.
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Fig. 1. Sketch of a block-structured mesh generated by multi-resolution analysis [2] with maximum tree level 6. Each mesh element represents a grid-block consisting typically of hundreds of cells in two-dimensions.
Fig. 1. Sketch of a block-structured mesh generated by multi-resolution analysis [2] with maximum tree level 6. Each mesh element represents a grid-block consisting typically of hundreds of cells in two-dimensions.
Fig. 2. Sketch for close packing methods. The circle with dotted line illustrates a projection of sphere in the third direction. Each circle is assigned with th same number of colored particles in the second step.
Fig. 2. Sketch for close packing methods. The circle with dotted line illustrates a projection of sphere in the third direction. Each circle is assigned with th same number of colored particles in the second step.
Fig. 3. Illustration of the multi-resolution cell-linked list for two dimensions. At each level, a table based data structure, which consists of an identifier exist and a cell-linked list, is initialized and dynamically updated throughout the simulation. The orange table element is the parent of the green ones while four blue table elements are the children of the green one. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Illustration of the multi-resolution cell-linked list for two dimensions. At each level, a table based data structure, which consists of an identifier exist and a cell-linked list, is initialized and dynamically updated throughout the simulation. The orange table element is the parent of the green ones while four blue table elements are the children of the green one. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Illustration of the particle data structure. Physical_variables denote the important particle properties, such as mass, density, pressure, smoothing length. The particle data memories and the related operations, i.e. allocation and release, are managed by a “memory pool” [37]. We develop a multi-resolution neighbor search algorithm in combination with the Verlet list without margin to enabl efficient neighbor access. Motivated by scatter-type sampling, the particle pairs are mutual neighbors when
Fig. 4. Illustration of the particle data structure. Physical_variables denote the important particle properties, such as mass, density, pressure, smoothing length. The particle data memories and the related operations, i.e. allocation and release, are managed by a “memory pool” [37]. We develop a multi-resolution neighbor search algorithm in combination with the Verlet list without margin to enabl efficient neighbor access. Motivated by scatter-type sampling, the particle pairs are mutual neighbors when
As the final step, the multi-level cell-linked lists are traversed recursively from the coarsest to the finest level. For each particle registered in the cell list, if the tag Tree_level equals current loop level, then the particle neighbors can be detected in a similar way as that of cell-linked list technique with constant kernel cut-off radius, and the corresponding Verlet list is constructed at the same time. Note that particle pairs with the same Tree_level will access each other twice while only once with different values of Tree_level. Details are given in Algorithm 1.
As the final step, the multi-level cell-linked lists are traversed recursively from the coarsest to the finest level. For each particle registered in the cell list, if the tag Tree_level equals current loop level, then the particle neighbors can be detected in a similar way as that of cell-linked list technique with constant kernel cut-off radius, and the corresponding Verlet list is constructed at the same time. Note that particle pairs with the same Tree_level will access each other twice while only once with different values of Tree_level. Details are given in Algorithm 1.
Fig. 6. Illustration for sampling procedure. The blue dot represents a simulation particle while green dots denote sampling particles. The blue curve indicates the kernel function shape on its support. The particle interaction cut-off radius is increased by a factor of 1.5 to take into account that some sampling particles fail to interact with any simulation particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Sketch for dynamic ghost particle method in two dimensions. The red particle denotes a specific boundary particle, and the blue ones are its neighbors inside the computational domain. The particles without colors are ghost particles constructed by current algorithm. The left panel indicates the boundary particle near a domain edge and the right panel near a domain corner. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Illustration for sampling procedure. The blue dot represents a simulation particle while green dots denote sampling particles. The blue curve indicates the kernel function shape on its support. The particle interaction cut-off radius is increased by a factor of 1.5 to take into account that some sampling particles fail to interact with any simulation particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Sketch for dynamic ghost particle method in two dimensions. The red particle denotes a specific boundary particle, and the blue ones are its neighbors inside the computational domain. The particles without colors are ghost particles constructed by current algorithm. The left panel indicates the boundary particle near a domain edge and the right panel near a domain corner. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Shock double water-columns interaction: particle representation of the target mesh topology. The blue circles represent simulated particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 7. Shock double water-columns interaction: target block-structured initial background mesh topology and schlieren-type images of density gradient Vel.
Fig. 8. Shock double water-columns interaction: particle representation of the target mesh topology. The blue circles represent simulated particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 7. Shock double water-columns interaction: target block-structured initial background mesh topology and schlieren-type images of density gradient Vel.
Shock double water-columns interaction. The wall-clock time is measured without multi-threaded parallelization. For block-structured mesh, 385 mes! elements mean typical 98560 cells with each block of resolution 16 x 16 [2]. Maximum imbalance error measurement and wall-clock time.
Shock double water-columns interaction. The wall-clock time is measured without multi-threaded parallelization. For block-structured mesh, 385 mes! elements mean typical 98560 cells with each block of resolution 16 x 16 [2]. Maximum imbalance error measurement and wall-clock time.
Fig. 9. Shock double water-columns interaction: initial particle distribution. Uniform distribution (left) and random distribution in combination with close packing method (right). The domain is partitioned into 9 sub-domains.
Fig. 9. Shock double water-columns interaction: initial particle distribution. Uniform distribution (left) and random distribution in combination with close packing method (right). The domain is partitioned into 9 sub-domains.
Wall-clock time measurement for component algorithms. Table 2
Wall-clock time measurement for component algorithms. Table 2
Fig. 10. Shock double water-columns interaction: evolution of SPH simulation results. Uniform initial distribution (left) and random initial distribution in combination with close packing method (right).
Fig. 10. Shock double water-columns interaction: evolution of SPH simulation results. Uniform initial distribution (left) and random initial distribution in combination with close packing method (right).
3D shock water-droplet interaction: the partitioning number is 8. The statistical data of MR-SIM represents the memory consumption of all fluid variables and the AMR data structure while that of partition denotes all the memory consumptions by our partitioning method. Memory consumption. Table 3
3D shock water-droplet interaction: the partitioning number is 8. The statistical data of MR-SIM represents the memory consumption of all fluid variables and the AMR data structure while that of partition denotes all the memory consumptions by our partitioning method. Memory consumption. Table 3
Fig. 11. Shock double water-columns interaction: convergent solution of SPH simulation and domain decomposition. With initial uniform distribution (left) and random initial distribution in combination with close packing method (right). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 11. Shock double water-columns interaction: convergent solution of SPH simulation and domain decomposition. With initial uniform distribution (left) and random initial distribution in combination with close packing method (right). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 13. Shock double water-columns interaction: distribution of the particle-neighbor number (left) and the target density (right) for a convergent solution. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.) Fig. 12. Shock double water-columns interaction: post-processing procedure. Colored particles with background mesh (left) and partitioning sub-domain with particles after post processing (right).
Fig. 13. Shock double water-columns interaction: distribution of the particle-neighbor number (left) and the target density (right) for a convergent solution. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.) Fig. 12. Shock double water-columns interaction: post-processing procedure. Colored particles with background mesh (left) and partitioning sub-domain with particles after post processing (right).
Fig. 14. Shock double water-columns interaction: dynamic load-balancing. SPH simulation solution (top) and domain decomposition with background mesh (bottom). From left to right: snapshots of simulation result at time 0.1, 0.2, 0.3.
Fig. 14. Shock double water-columns interaction: dynamic load-balancing. SPH simulation solution (top) and domain decomposition with background mesh (bottom). From left to right: snapshots of simulation result at time 0.1, 0.2, 0.3.
Fig. 15. Shock double water-columns interaction: history of total particle number N, (top left panel), maximum imbalance error Ermax (top right panel) communication volume for static partitioning (bottom left panel) and dynamic partitioning (bottom right panel) versus iteration number. The partitior number is 9. The communication volume is defined as the boundary particle number normalized by the total particle number while the boundary particle number is defined as the number of particles which feature neighboring particle of different color within its smoothing region.
Fig. 15. Shock double water-columns interaction: history of total particle number N, (top left panel), maximum imbalance error Ermax (top right panel) communication volume for static partitioning (bottom left panel) and dynamic partitioning (bottom right panel) versus iteration number. The partitior number is 9. The communication volume is defined as the boundary particle number normalized by the total particle number while the boundary particle number is defined as the number of particles which feature neighboring particle of different color within its smoothing region.
Fig. 16. Shock double water-columns interaction: target block-structured background mesh topology. The mesh element number is 10485, corresponding t a mesh cell size 2684160 in 2D.
Fig. 16. Shock double water-columns interaction: target block-structured background mesh topology. The mesh element number is 10485, corresponding t a mesh cell size 2684160 in 2D.
Fig. 18. Shock double water-columns interaction: history of load-imbalance error and communication volume versus iteration number (static partitioning) The red line represents the partition number 36 while the green line represents the partition number 64. (For interpretation of the references to color it this figure legend, the reader is referred to the web version of this article.) Fig. 17. Shock double water-columns interaction: the partitioning number is 64. The simulated particle number is 41728. The overall view (top left panel) and zoom-in views (others).
Fig. 18. Shock double water-columns interaction: history of load-imbalance error and communication volume versus iteration number (static partitioning) The red line represents the partition number 36 while the green line represents the partition number 64. (For interpretation of the references to color it this figure legend, the reader is referred to the web version of this article.) Fig. 17. Shock double water-columns interaction: the partitioning number is 64. The simulated particle number is 41728. The overall view (top left panel) and zoom-in views (others).
Fig. 19. Static partitioning for uniform mesh. SPH simulation solution (left panel) and domain decomposition with background mesh (right panel). The partition number is 9. The mesh element number is 900, corresponding to 230400 mesh cells.
Fig. 19. Static partitioning for uniform mesh. SPH simulation solution (left panel) and domain decomposition with background mesh (right panel). The partition number is 9. The mesh element number is 900, corresponding to 230400 mesh cells.
Fig. 21. Static partitioning for uniform mesh. History of maximum imbalance error Erimgx (left), communication volume (middle) and wall-clock time (right versus iteration number. The partition number is 9. Fig. 20. Static partitioning for uniform mesh. SPH simulation solution (left panel) and domain decomposition (right panel). The partition number is 16. The mesh element number is 57600, corresponding to 14745600 mesh cells.
Fig. 21. Static partitioning for uniform mesh. History of maximum imbalance error Erimgx (left), communication volume (middle) and wall-clock time (right versus iteration number. The partition number is 9. Fig. 20. Static partitioning for uniform mesh. SPH simulation solution (left panel) and domain decomposition (right panel). The partition number is 16. The mesh element number is 57600, corresponding to 14745600 mesh cells.
Fig. 22. Underwater explosion: target initial background mesh topology and schlieren-type images of density gradient |Vp].
Fig. 22. Underwater explosion: target initial background mesh topology and schlieren-type images of density gradient |Vp].
(a) From left to right: initial condition, solution at iteration number 3400, domain decomposition. The consumed wall-clock time is 61.0s.
(a) From left to right: initial condition, solution at iteration number 3400, domain decomposition. The consumed wall-clock time is 61.0s.
b) From left to right: initial condition, solution at iteration number 8000, domain decomposition. The consumed wall-clock time is 112.0s. Fig. 23. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. The domain partition number is 4 (top) and 9 (bottom) respectively. The total mesh element number is 448 representing 114688 mesh cells.
b) From left to right: initial condition, solution at iteration number 8000, domain decomposition. The consumed wall-clock time is 112.0s. Fig. 23. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. The domain partition number is 4 (top) and 9 (bottom) respectively. The total mesh element number is 448 representing 114688 mesh cells.
Shock air-R22 bubble interaction. The wall-clock time is measured without multi-threaded parallelization. For block-structured mesh, 548 mesh elements mean typical 140288 cells with each block of resolution 16 x 16 [2]. Maximum imbalance error measurement and wall-clock time. Table 4
Shock air-R22 bubble interaction. The wall-clock time is measured without multi-threaded parallelization. For block-structured mesh, 548 mesh elements mean typical 140288 cells with each block of resolution 16 x 16 [2]. Maximum imbalance error measurement and wall-clock time. Table 4
Fig. 24. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. From left to right: SPH solution at fluid simulatio1 time 0.06, 0.12, 0.20. Top: particle distribution, bottom: domain decomposition. The domain partition number is 4.
Fig. 24. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. From left to right: SPH solution at fluid simulatio1 time 0.06, 0.12, 0.20. Top: particle distribution, bottom: domain decomposition. The domain partition number is 4.
Fig. 26. Underwater explosion: history of maximum imbalance error Ermax (left panel) and scale ratio S (right panel) versus iteration number. The red line represents partition number 4 while the green line represents partition number 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 25. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. From left to right: SPH solution at fluid simulatior time 0.06, 0.12, 0.20. Top: particle distribution, bottom: domain decomposition. The domain partition number is 9.
Fig. 26. Underwater explosion: history of maximum imbalance error Ermax (left panel) and scale ratio S (right panel) versus iteration number. The red line represents partition number 4 while the green line represents partition number 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 25. Underwater explosion: solution of SPH simulation and corresponding domain decomposition. From left to right: SPH solution at fluid simulatior time 0.06, 0.12, 0.20. Top: particle distribution, bottom: domain decomposition. The domain partition number is 9.
Fig. 27. Underwater explosion: history of communication volume versus iteration number. Left: for static partitioning; right: for dynamic partitioning. Th red line represents partition number 4 while the green line represents partition number 9. (For interpretation of the references to color in this figur legend, the reader is referred to the web version of this article.)
Fig. 27. Underwater explosion: history of communication volume versus iteration number. Left: for static partitioning; right: for dynamic partitioning. Th red line represents partition number 4 while the green line represents partition number 9. (For interpretation of the references to color in this figur legend, the reader is referred to the web version of this article.)
Fig. 29. Shock air-R22 bubble interaction: initial condition, solution at iteration 7200, convergent partition result. The domain partition number is 6. The total mesh element number is 548, corresponding to 140288 mesh cells. Fig. 28. Shock air-R22 bubble interaction: target background mesh topology and schlieren-type images of density gradient |V:e|
Fig. 29. Shock air-R22 bubble interaction: initial condition, solution at iteration 7200, convergent partition result. The domain partition number is 6. The total mesh element number is 548, corresponding to 140288 mesh cells. Fig. 28. Shock air-R22 bubble interaction: target background mesh topology and schlieren-type images of density gradient |V:e|
Fig. 30. Shock air-R22 bubble interaction: initial condition, solution at iteration 8000, convergent partition result. The domain partition number is 9. The total mesh element number is 548, corresponding to 140288 mesh cells.
Fig. 30. Shock air-R22 bubble interaction: initial condition, solution at iteration 8000, convergent partition result. The domain partition number is 9. The total mesh element number is 548, corresponding to 140288 mesh cells.
Fig. 31. Shock air-R22 bubble interaction: solution of SPH simulation and domain decomposition at fluid simulation time 0.0892 and 0.1386. Top: particle distribution; bottom: domain decomposition. The domain partition number is 9. The final mesh topology consists of 1244 mesh elements, corresponding to 318464 mesh cells.
Fig. 31. Shock air-R22 bubble interaction: solution of SPH simulation and domain decomposition at fluid simulation time 0.0892 and 0.1386. Top: particle distribution; bottom: domain decomposition. The domain partition number is 9. The final mesh topology consists of 1244 mesh elements, corresponding to 318464 mesh cells.
Fig. 32. Shock air-R22 bubble interaction: history of communication volume versus iteration number. Left: for static partitioning; right: for dynamic parti- tioning. The red line represents partition number 6 while the green line represents partition number 9. (For interpretation of the references to color in thi: figure legend, the reader is referred to the web version of this article.)
Fig. 32. Shock air-R22 bubble interaction: history of communication volume versus iteration number. Left: for static partitioning; right: for dynamic parti- tioning. The red line represents partition number 6 while the green line represents partition number 9. (For interpretation of the references to color in thi: figure legend, the reader is referred to the web version of this article.)
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Fig. 1. Sketch of Voronoi cell and particle update. For the left panel, the dashed lines denote the Voronoi cell edges, hjj denotes the distance betweer particle i and j, pjj denotes the pressure on the Voronoi cell edge between particle i and j. The orange circles denote the nearest neighbors of particle i represented by the green circle. For the right panel, the purple circle represents the new particle position by the pure VP method while the gray circle denotes that by pure CVT requirement. The red line with arrow denotes the particle evolution path by the VP part of CVP method while the blue lin denotes that by the CVT part. The gray line with arrow denotes the real evolution path. (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.)
Fig. 1. Sketch of Voronoi cell and particle update. For the left panel, the dashed lines denote the Voronoi cell edges, hjj denotes the distance betweer particle i and j, pjj denotes the pressure on the Voronoi cell edge between particle i and j. The orange circles denote the nearest neighbors of particle i represented by the green circle. For the right panel, the purple circle represents the new particle position by the pure VP method while the gray circle denotes that by pure CVT requirement. The red line with arrow denotes the particle evolution path by the VP part of CVP method while the blue lin denotes that by the CVT part. The gray line with arrow denotes the real evolution path. (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.)
Fig. 2. Sketch of boundary condition. The general Voronoi diagram (left panel) and the Voronoi diagram with ghost particles (right panel). The blue lines represent the Voronoi cell edges while the dashed black lines represent the computational domain boundaries. The red crosses denote partitioning particles, the green circles denote boundary particles while the dashed circles denote the ghost particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) particles, are identified. As shown on the left of Fig. 2, particles in all Voronoi cells that contain Voronoi vertices outside the computational domain are marked as boundary particles. By mirroring these particles, a set of ghost particles is constructed, and all physical properties, such as mass and pressure, are copied from the corresponding particles. In the second step, a new Voronoi diagram is constructed including these ghost particles, as shown on the right of Fig. 2. The Voronoi cells cutting the boundary are now closed by the domain-boundary segments. The ghost particles are reconstructed for each iteration. Consequently, all numerical algorithms discussed in the last section now can be applied to all particles within the computational domain.
Fig. 2. Sketch of boundary condition. The general Voronoi diagram (left panel) and the Voronoi diagram with ghost particles (right panel). The blue lines represent the Voronoi cell edges while the dashed black lines represent the computational domain boundaries. The red crosses denote partitioning particles, the green circles denote boundary particles while the dashed circles denote the ghost particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) particles, are identified. As shown on the left of Fig. 2, particles in all Voronoi cells that contain Voronoi vertices outside the computational domain are marked as boundary particles. By mirroring these particles, a set of ghost particles is constructed, and all physical properties, such as mass and pressure, are copied from the corresponding particles. In the second step, a new Voronoi diagram is constructed including these ghost particles, as shown on the right of Fig. 2. The Voronoi cells cutting the boundary are now closed by the domain-boundary segments. The ghost particles are reconstructed for each iteration. Consequently, all numerical algorithms discussed in the last section now can be applied to all particles within the computational domain.
Fig. 3. The convergence histories of maximum load-imbalance error Emax, energy function and interfacial area for three different partitioning methods.
Fig. 3. The convergence histories of maximum load-imbalance error Emax, energy function and interfacial area for three different partitioning methods.
Fig. 4. Sketch of the partitionings and Voronoi diagrams. From left to right: initial partitioning, converged partitioning and Voronoi diagram. From top to bottom: results from CVP, VP and CVT methods. The cross symbols in the right panel represent the partitioning particles.
Fig. 4. Sketch of the partitionings and Voronoi diagrams. From left to right: initial partitioning, converged partitioning and Voronoi diagram. From top to bottom: results from CVP, VP and CVT methods. The cross symbols in the right panel represent the partitioning particles.
Fig. 6. Sketch of the partitioning results for uniform mesh with few generators. The partitioning number is 5, 7, and 14. Fig. 5. Sketch of the partitioning results and convergence history. Top: initial partitioning, convergent partitioning and convergent Voronoi diagram. The black cross symbols denote partitioning particles of small target mass while the red star symbols denote those of large target mass. Bottom: convergence history of maximum load-imbalance error Emax, energy function and interfacial area. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Sketch of the partitioning results for uniform mesh with few generators. The partitioning number is 5, 7, and 14. Fig. 5. Sketch of the partitioning results and convergence history. Top: initial partitioning, convergent partitioning and convergent Voronoi diagram. The black cross symbols denote partitioning particles of small target mass while the red star symbols denote those of large target mass. Bottom: convergence history of maximum load-imbalance error Emax, energy function and interfacial area. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Sketch of adaptive particle distribution.
Fig. 7. Sketch of adaptive particle distribution.
Fig. 8. Convergence histories of maximum load-imbalance error Emax and energy function for different partitioning number.
Fig. 8. Convergence histories of maximum load-imbalance error Emax and energy function for different partitioning number.
Fig. 9. Sketch of partitionings and Voronoi diagrams. From the left to right: initial partitioning, convergent partitioning and convergent Voronoi diagram. From the top to bottom: partitioning number 7, 14, 28. The square and cross symbols denote the partitioning particles.
Fig. 9. Sketch of partitionings and Voronoi diagrams. From the left to right: initial partitioning, convergent partitioning and convergent Voronoi diagram. From the top to bottom: partitioning number 7, 14, 28. The square and cross symbols denote the partitioning particles.
Fig. 10. Convergence history comparisons between CVP and VP method. Three sets of simulations are carried out with random initial conditions. Th partitioning number is 14.
Fig. 10. Convergence history comparisons between CVP and VP method. Three sets of simulations are carried out with random initial conditions. Th partitioning number is 14.
Fig. 11. Sketch of partitioning result from VP method. Left panel: partitioning subdomains; right panel: Voronoi diagram. The partitioning number is 14. The square and cross symbols denote the partitioning particles.
Fig. 11. Sketch of partitioning result from VP method. Left panel: partitioning subdomains; right panel: Voronoi diagram. The partitioning number is 14. The square and cross symbols denote the partitioning particles.
Fig. 12. Comparison of the boundary element number. The mesh element is defined as boundary element under the condition that among its nearest element neighbors there is at least one element located in different subdomains.
Fig. 12. Comparison of the boundary element number. The mesh element is defined as boundary element under the condition that among its nearest element neighbors there is at least one element located in different subdomains.
Fig. 13. Convergence history comparisons with different choice of ~ in CVP method. The partitioning number is 28.
Fig. 13. Convergence history comparisons with different choice of ~ in CVP method. The partitioning number is 28.
Fig. 14. Sketch of partitionings and convergence history from CVT method. The partitioning numbers are 7 (top left), 14 (top right) and 28 (bottom left). Bottom right: convergence history of maximum load-imbalance error. The crosses denote the partitioning particles.
Fig. 14. Sketch of partitionings and convergence history from CVT method. The partitioning numbers are 7 (top left), 14 (top right) and 28 (bottom left). Bottom right: convergence history of maximum load-imbalance error. The crosses denote the partitioning particles.
Fig. 15. Sketch of partitionings (left) and convergence history (right) from CVT method based on the asymptotic conjecture. The partitioning number 28.
Fig. 15. Sketch of partitionings (left) and convergence history (right) from CVT method based on the asymptotic conjecture. The partitioning number 28.
Fig. 16. Sketch of partitionings and convergence history from CVP method. The partitioning number is 9. Top: particle partitioning configuration (left) and the Voronoi diagram (right). Bottom: convergence history of maximum load-imbalance error (left) and energy function (right). The crosses denote the partitioning particles.
Fig. 16. Sketch of partitionings and convergence history from CVP method. The partitioning number is 9. Top: particle partitioning configuration (left) and the Voronoi diagram (right). Bottom: convergence history of maximum load-imbalance error (left) and energy function (right). The crosses denote the partitioning particles.
Fig. 17. Sketch of the block-structured mesh (left panel) and anisotropic unstructured mesh (right panel).
Fig. 17. Sketch of the block-structured mesh (left panel) and anisotropic unstructured mesh (right panel).
Fig. 18. Sketch of partitioning for block-structured mesh for partitioning number 9, 18, 27. From the top to bottom: the partitioning number is 9, 18 and 27. From the left to right: the converged interaction particle partitioning, block-structured mesh partitioning and Voronoi diagram. The square and cross symbols denote the partitioning particles.
Fig. 18. Sketch of partitioning for block-structured mesh for partitioning number 9, 18, 27. From the top to bottom: the partitioning number is 9, 18 and 27. From the left to right: the converged interaction particle partitioning, block-structured mesh partitioning and Voronoi diagram. The square and cross symbols denote the partitioning particles.
Fig. 19. Convergence history of the partitioning for block-structured mesh. Maximum load-imbalance error Emax (left), energy function (middle) and elapsec time (right).
Fig. 19. Convergence history of the partitioning for block-structured mesh. Maximum load-imbalance error Emax (left), energy function (middle) and elapsec time (right).
Fig. 20. Sketch of the partitioning for adaptive unstructured mesh with partitioning number 250. Left: interaction particle partitioning; middle: Voronoi diagram; right: convergence history of maximum load-imbalance error Emax.
Fig. 20. Sketch of the partitioning for adaptive unstructured mesh with partitioning number 250. Left: interaction particle partitioning; middle: Voronoi diagram; right: convergence history of maximum load-imbalance error Emax.
Fig. 21. 3D partitioning simulation. (a) The 3D interaction particle distribution; (b) a zoom view in the x-y plane; (c) the partitioning topology of par- titioning number 64; (d) the partitioning Voronoi diagram of partitioning number 64; (e) the convergence history of the load-imbalance error; (f) the convergence history of energy function.
Fig. 21. 3D partitioning simulation. (a) The 3D interaction particle distribution; (b) a zoom view in the x-y plane; (c) the partitioning topology of par- titioning number 64; (d) the partitioning Voronoi diagram of partitioning number 64; (e) the convergence history of the load-imbalance error; (f) the convergence history of energy function.
Related topics:
Compressible Fluid DynamicsIncompressible FlowsGas DynamicsCompressible FlowLES and DNSTENO scheme
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