Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan[2] and Lucy[3] in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method (where the co-ordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as density.
Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations. Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid (usually water) and empty space represents the lighter fluid (usually air). For these reasons, it is possible to simulate fluid motion using SPH in real time. However, both grid-based and SPH techniques still require the generation of renderable free surface geometry using a polygonization technique such as metaballs and marching cubes, point splatting, or 'carpet' visualization. For gas dynamics it is more appropriate to use the kernel function itself to produce a rendering of gas column density (e.g., as done in the SPLASH visualisation package).
One drawback over grid-based techniques is the need for large numbers of particles to produce simulations of equivalent resolution. In the typical implementation of both uniform grids and SPH particle techniques, many voxels or particles will be used to fill water volumes that are never rendered. However, accuracy can be significantly higher with sophisticated grid-based techniques, especially those coupled with particle methods (such as particle level sets), since it is easier to enforce the incompressibility condition in these systems. SPH for fluid simulation is being used increasingly in real-time animation and games where accuracy is not as critical as interactivity.
Recent work in SPH for fluid simulation has increased performance, accuracy, and areas of application:
Smoothed-particle hydrodynamics's adaptive resolution, numerical conservation of physically conserved quantities, and ability to simulate phenomena covering many orders of magnitude make it ideal for computations in theoretical astrophysics.[22]
Simulations of galaxy formation, star formation, stellar collisions,[23] supernovae[24] and meteor impacts are some of the wide variety of astrophysical and cosmological uses of this method.
SPH is used to model hydrodynamic flows, including possible effects of gravity. Incorporating other astrophysical processes which may be important, such as radiative transfer and magnetic fields is an active area of research in the astronomical community, and has had some limited success.[25][26]
Libersky and Petschek[27][28] extended SPH to Solid Mechanics. The main advantage of SPH in this application is the possibility of dealing with larger local distortion than grid-based methods. This feature has been exploited in many applications in Solid Mechanics: metal forming, impact, crack growth, fracture, fragmentation, etc.
Another important advantage of meshfree methods in general, and of SPH in particular, is that mesh dependence problems are naturally avoided given the meshfree nature of the method. In particular, mesh alignment is related to problems involving cracks and it is avoided in SPH due to the isotropic support of the kernel functions. However, classical SPH formulations suffer from tensile instabilities[29] and lack of consistency.[30] Over the past years, different corrections have been introduced to improve the accuracy of the SPH solution, leading to the RKPM by Liu et al.[31] Randles and Libersky[32] and Johnson and Beissel[33] tried to solve the consistency problem in their study of impact phenomena.
Dyka et al.[34][35] and Randles and Libersky[36] introduced the stress-point integration into SPH and Ted Belytschko et al.[37] showed that the stress-point technique removes the instability due to spurious singular modes, while tensile instabilities can be avoided by using a Lagrangian kernel. Many other recent studies can be found in the literature devoted to improve the convergence of the SPH method.
Recent improvements in understanding the convergence and stability of SPH have allowed for more widespread applications in Solid Mechanics. Other examples of applications and developments of the method include:
The Smoothed-Particle Hydrodynamics (SPH) method works by dividing the fluid into a set of discrete moving elements i , j {\displaystyle i,j} , referred to as particles. Their Lagrangian nature allows setting their position r i {\displaystyle \mathbf {r} _{i}} by integration of their velocity v i {\displaystyle \mathbf {v} _{i}} as:
These particles interact through a kernel function with characteristic radius known as the "smoothing length", typically represented in equations by h {\displaystyle h} . This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles that lie within the range of the kernel, the latter being used as a weighting function W {\displaystyle W} . This can be understood in two steps. First an arbitrary field A {\displaystyle A} is written as a convolution with W {\displaystyle W} :
The error in making the above approximation is order h 2 {\displaystyle h^{2}} . Secondly, the integral is approximated using a Riemann summation over the particles:
where the summation over j {\displaystyle j} includes all particles in the simulation. V j {\displaystyle V_{j}} is the volume of particle j {\displaystyle j} , A j {\displaystyle A_{j}} is the value of the quantity A {\displaystyle A} for particle j {\displaystyle j} and r {\displaystyle {\boldsymbol {r}}} denotes position. For example, the density ρ ρ --> i {\displaystyle \rho _{i}} of particle i {\displaystyle i} can be expressed as:
where m j = ρ ρ --> j V j {\displaystyle m_{j}=\rho _{j}V_{j}} denotes the particle mass and ρ ρ --> j {\displaystyle \rho _{j}} the particle density, while W i j = W j i {\displaystyle W_{ij}=W_{ji}} is a short notation for W ( | r i − − --> r j | , h ) {\displaystyle W(|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|,h)} . The error done in approximating the integral by a discrete sum depends on h {\displaystyle h} , on the particle size (i.e. V j 1 / d {\displaystyle V_{j}^{1/d}} , d {\displaystyle d} being the space dimension), and on the particle arrangement in space. The latter effect is still poorly known.[43]
Kernel functions commonly used include the Gaussian function, the quintic spline and the Wendland C 2 {\displaystyle C^{2}} kernel.[44] The latter two kernels are compactly supported (unlike the Gaussian, where there is a small contribution at any finite distance away), with support proportional to h {\displaystyle h} . This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles.
Although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, the resolution of a simulation can be made to automatically adapt itself depending on local conditions. For example, in a very dense region where many particles are close together, the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where individual particles are far apart and the resolution is low, the smoothing length can be increased, optimising the computation for the regions of interest.
For particles of constant mass, differentiating the interpolated density ρ ρ --> i {\displaystyle \rho _{i}} with respect to time yields
where ∇ ∇ --> W i j = − − --> ∇ ∇ --> W j i {\displaystyle \nabla W_{ij}=-\nabla W_{ji}} is the gradient of W i j {\displaystyle W_{ij}} with respect to r i {\displaystyle {\boldsymbol {r}}_{i}} . Comparing this equation with the continuity equation in the Lagrangian description (using material derivatives),
it is apparent that its right-hand side is an approximation of − − --> ρ ρ --> ∇ ∇ --> ⋅ ⋅ --> v {\displaystyle -\rho \nabla \cdot \mathbf {v} } ; hence one defines a discrete divergence operator as follows:
This operator gives an SPH approximation of ∇ ∇ --> ⋅ ⋅ --> v {\displaystyle \nabla \cdot \mathbf {v} } at the particle i {\displaystyle i} for a given set of particles with given masses m j {\displaystyle m_{j}} , positions { r j } {\displaystyle \left\{\mathbf {r} _{j}\right\}} and velocities { v j } {\displaystyle \left\{\mathbf {v} _{j}\right\}} .
The other important equation for a compressible inviscid fluid is the Euler equation for momentum balance:
Similarly to continuity, the task is to define a discrete gradient operator in order to write
One choice is
which has the property of being skew-adjoint with the divergence operator above, in the sense that
this being a discrete version of the continuum identity
This property leads to nice conservation properties.[45]
Notice also that this choice leads to a symmetric divergence operator and antisymmetric gradient. Although there are several ways of discretizing the pressure gradient in the Euler equations, the above antisymmetric form is the most acknowledged one. It supports strict conservation of linear and angular momentum. This means that a force that is exerted on particle i {\displaystyle i} by particle j {\displaystyle j} equals the one that is exerted on particle j {\displaystyle j} by particle i {\displaystyle i} including the sign change of the effective direction, thanks to the antisymmetry property ∇ ∇ --> W i j = − − --> ∇ ∇ --> W j i {\displaystyle \nabla W_{ij}=-\nabla W_{ji}} .
Nevertheless, other operators have been proposed, which may perform better numerically or physically. For instance, one drawback of these operators is that while the divergence D {\displaystyle \operatorname {D} } is zero-order consistent (i.e. yields zero when applied to a constant vector field), it can be seen that the gradient G {\displaystyle \operatorname {\mathbf {G} } } is not. Several techniques have been proposed to circumvent this issue, leading to renormalized operators (see e.g.[46]).
The above SPH governing equations can be derived from a least action principle, starting from the Lagrangian of a particle system:
where e j {\displaystyle e_{j}} is the particle specific internal energy. The Euler–Lagrange equation of variational mechanics reads, for each particle:
When applied to the above Lagrangian, it gives the following momentum equation:
where the chain rule has been used, since e j {\displaystyle e_{j}} depends on ρ ρ --> j {\displaystyle \rho _{j}} , and the latter, on the position of the particles. Using the thermodynamic property d e = ( p / ρ ρ --> 2 ) d ρ ρ --> {\displaystyle \mathrm {d} e=\left(p/\rho ^{2}\right)\mathrm {d} \rho } we may write
Plugging the SPH density interpolation and differentiating explicitly ∂ ∂ --> ρ ρ --> j ∂ ∂ --> r i {\displaystyle {\tfrac {\partial \rho _{j}}{\partial {\boldsymbol {r}}_{i}}}} leads to
which is the SPH momentum equation already mentioned, where we recognize the G {\displaystyle \operatorname {\mathbf {G} } } operator. This explains why linear momentum is conserved, and allows conservation of angular momentum and energy to be conserved as well.[47]
From the work done in the 80's and 90's on numerical integration of point-like particles in large accelerators, appropriate time integrators have been developed with accurate conservation properties on the long term; they are called symplectic integrators. The most popular in the SPH literature is the leapfrog scheme, which reads for each particle i {\displaystyle i} :
where Δ Δ --> t {\displaystyle \Delta t} is the time step, superscripts stand for time iterations while a i {\displaystyle {\boldsymbol {a}}_{i}} is the particle acceleration, given by the right-hand side of the momentum equation.
Other symplectic integrators exist (see the reference textbook[48]). It is recommended to use a symplectic (even low-order) scheme instead of a high order non-symplectic scheme, to avoid error accumulation after many iterations.
Integration of density has not been studied extensively (see below for more details).
Symplectic schemes are conservative but explicit, thus their numerical stability requires stability conditions, analogous to the Courant-Friedrichs-Lewy condition (see below).
In case the SPH convolution shall be practiced close to a boundary, i.e. closer than s · h, then the integral support is truncated. Indeed, when the convolution is affected by a boundary, the convolution shall be split in 2 integrals,
where B(r) is the compact support ball centered at r, with radius s · h, and Ω(r) denotes the part of the compact support inside the computational domain, Ω ∩ B(r). Hence, imposing boundary conditions in SPH is completely based on approximating the second integral on the right hand side. The same can be of course applied to the differential operators computation,
Several techniques has been introduced in the past to model boundaries in SPH.
The most straightforward boundary model is neglecting the integral,
such that just the bulk interactions are taken into account,
This is a popular approach when free-surface is considered in monophase simulations.[49]
The main benefit of this boundary condition is its obvious simplicity. However, several consistency issues shall be considered when this boundary technique is applied.[49] That's in fact a heavy limitation on its potential applications.
Probably the most popular methodology, or at least the most traditional one, to impose boundary conditions in SPH, is Fluid Extension technique. Such technique is based on populating the compact support across the boundary with so-called ghost particles, conveniently imposing their field values.[50]
Along this line, the integral neglect methodology can be considered as a particular case of fluid extensions, where the field, A, vanish outside the computational domain.
The main benefit of this methodology is the simplicity, provided that the boundary contribution is computed as part of the bulk interactions. Also, this methodology has been deeply analyzed in the literature.[51][50][52]
On the other hand, deploying ghost particles in the truncated domain is not a trivial task, such that modelling complex boundary shapes becomes cumbersome. The 2 most popular approaches to populate the empty domain with ghost particles are Mirrored-Particles[53] and Fixed-Particles.[50]
The newest Boundary technique is the Boundary Integral methodology.[54] In this methodology, the empty volume integral is replaced by a surface integral, and a renormalization:
with nj the normal of the generic j-th boundary element. The surface term can be also solved considering a semi-analytic expression.[54]
Another way to determine the density is based on the SPH smoothing operator itself. Therefore, the density is estimated from the particle distribution utilizing the SPH interpolation. To overcome undesired errors at the free surface through kernel truncation, the density formulation can again be integrated in time. [54]
The weakly compressible SPH in fluid dynamics is based on the discretization of the Navier–Stokes equations or Euler equations for compressible fluids. To close the system, an appropriate equation of state is utilized to link pressure p {\displaystyle p} and density ρ ρ --> {\displaystyle \rho } . Generally, the so-called Cole equation [55] (sometimes mistakenly referred to as the "Tait equation") is used in SPH. It reads
where ρ ρ --> 0 {\displaystyle \rho _{0}} is the reference density and c {\displaystyle c} the speed of sound. For water, γ γ --> = 7 {\displaystyle \gamma =7} is commonly used. The background pressure p 0 {\displaystyle p_{0}} is added to avoid negative pressure values.
Real nearly incompressible fluids such as water are characterized by very high speeds of sound of the order 10 3 m / s {\displaystyle 10^{3}\mathrm {m/s} } . Hence, pressure information travels fast compared to the actual bulk flow, which leads to very small Mach numbers M {\displaystyle M} . The momentum equation leads to the following relation:
where ρ ρ --> {\displaystyle \rho } is the density change and v {\displaystyle v} the velocity vector. In practice a value of c smaller than the real one is adopted to avoid time steps too small in the time integration scheme. Generally a numerical speed of sound is adopted such that density variation smaller than 1% are allowed. This is the so-called weak-compressibility assumption. This corresponds to a Mach number smaller than 0.1, which implies:
where the maximum velocity v max {\displaystyle v_{\text{max}}} needs to be estimated, for e.g. by Torricelli's law or an educated guess. Since only small density variations occur, a linear equation of state can be adopted:[56]
Usually the weakly-compressible schemes are affected by a high-frequency spurious noise on the pressure and density fields. [57] This phenomenon is caused by the nonlinear interaction of acoustic waves and by fact that the scheme is explicit in time and centered in space .[58]
Through the years, several techniques have been proposed to get rid of this problem. They can be classified in three different groups:
The schemes of the first group apply a filter directly on the density field to remove the spurious numerical noise. The most used filters are the MLS (moving least squares) and the Shepard filter[57] which can be applied at each time step or every n time steps. The more frequent is the use of the filtering procedure, the more regular density and pressure fields are obtained. On the other hand, this leads to an increase of the computational costs. In long time simulations, the use of the filtering procedure may lead to the disruption of the hydrostatic pressure component and to an inconsistency between the global volume of fluid and the density field. Further, it does not ensure the enforcement of the dynamic free-surface boundary condition.
A different way to smooth out the density and pressure field is to add a diffusive term inside the continuity equation (group 2) :
The first schemes that adopted such an approach were described in Ferrari [59] and in Molteni[56] where the diffusive term was modeled as a Laplacian of the density field. A similar approach was also used in Fatehi and Manzari .[60]
In Antuono et al. [61] a correction to the diffusive term of Molteni[56] was proposed to remove some inconsistencies close to the free-surface. In this case the adopted diffusive term is equivalent to a high-order differential operator on the density field.[62] The scheme is called δ-SPH and preserves all the conservation properties of the SPH without diffusion (e.g., linear and angular momenta, total energy, see [63] ) along with a smooth and regular representation of the density and pressure fields.
In the third group there are those SPH schemes which employ numerical fluxes obtained through Riemann solvers to model the particle interactions.[64][65][66]
For an SPH method based on Riemann solvers, an inter-particle Riemann problem is constructed along a unit vector e i j = − − --> r i j / r i j {\displaystyle \mathbf {e} _{ij}=-\mathbf {r} _{ij}/r_{ij}} pointing form particle i {\displaystyle i} to particle j {\displaystyle j} . In this Riemann problem the initial left and right states are on particles i {\displaystyle i} and j {\displaystyle j} , respectively. The L {\displaystyle L} and R {\displaystyle R} states are
{ ( ρ ρ --> L , U L , P L ) = ( ρ ρ --> i , v i ⋅ ⋅ --> e i j , P i ) ( ρ ρ --> R , U R , P R ) = ( ρ ρ --> j , v j ⋅ ⋅ --> e i j , P j ) . {\displaystyle {\begin{cases}(\rho _{L},U_{L},P_{L})=(\rho _{i},\mathbf {v} _{i}\cdot \mathbf {e} _{ij},P_{i})\\(\rho _{R},U_{R},P_{R})=(\rho _{j},\mathbf {v} _{j}\cdot \mathbf {e} _{ij},P_{j}).\end{cases}}}
The solution of the Riemann problem results in three waves emanating from the discontinuity. Two waves, which can be shock or rarefaction wave, traveling with the smallest or largest wave speed. The middle wave is always a contact discontinuity and separates two intermediate states, denoted by ( ρ ρ --> L ∗ ∗ --> , U L ∗ ∗ --> , P L ∗ ∗ --> ) {\displaystyle (\rho _{L}^{\ast },U_{L}^{\ast },P_{L}^{\ast })} and ( ρ ρ --> R ∗ ∗ --> , U R ∗ ∗ --> , P R ∗ ∗ --> ) {\displaystyle (\rho _{R}^{\ast },U_{R}^{\ast },P_{R}^{\ast })} . By assuming that the intermediate state satisfies U L ∗ ∗ --> = U R ∗ ∗ --> = U ∗ ∗ --> {\displaystyle U_{L}^{\ast }=U_{R}^{\ast }=U^{\ast }} and P L ∗ ∗ --> = P R ∗ ∗ --> = P ∗ ∗ --> {\displaystyle P_{L}^{\ast }=P_{R}^{\ast }=P^{\ast }} , a linearized Riemann solver for smooth flows or with only moderately strong shocks can be written as
{ U ∗ ∗ --> = U ¯ ¯ --> + 1 2 ( P L − − --> P R ) ρ ρ --> ¯ ¯ --> c 0 P ∗ ∗ --> = P ¯ ¯ --> + 1 2 ρ ρ --> ¯ ¯ --> c 0 ( U L − − --> U R ) , {\displaystyle {\begin{cases}U^{\ast }={\overline {U}}+{\frac {1}{2}}{\frac {(P_{L}-P_{R})}{{\bar {\rho }}c_{0}}}\\P^{\ast }={\overline {P}}+{\frac {1}{2}}{\bar {\rho }}c_{0}{(U_{L}-U_{R})},\end{cases}}}
where U ¯ ¯ --> = ( U L + U R ) / 2 {\displaystyle {\overline {U}}=(U_{L}+U_{R})/2} and P ¯ ¯ --> = ( P L + P R ) / 2 {\displaystyle {\overline {P}}=(P_{L}+P_{R})/2} are inter-particle averages. With the solution of the Riemann problem, i.e. U ∗ ∗ --> {\displaystyle U^{\ast }} and P ∗ ∗ --> {\displaystyle P^{\ast }} , the discretization of the SPH method is
d ρ ρ --> i d t = 2 ρ ρ --> i ∑ ∑ --> j m j ρ ρ --> j ( v i − − --> v ∗ ∗ --> ) ⋅ ⋅ --> ∇ ∇ --> i W i j , {\displaystyle {\frac {d\rho _{i}}{dt}}=2\rho _{i}\sum _{j}{\frac {m_{j}}{\rho _{j}}}(\mathbf {v} _{i}-\mathbf {v} ^{\ast })\cdot \nabla _{i}W_{ij},}
d v i d t = − − --> 2 ∑ ∑ --> j m j ( P ∗ ∗ --> ρ ρ --> i ρ ρ --> j ) ∇ ∇ --> i W i j . {\displaystyle {\frac {d\mathbf {v} _{i}}{dt}}=-2\sum _{j}m_{j}\left({\frac {P^{\ast }}{\rho _{i}\rho _{j}}}\right)\nabla _{i}W_{ij}.}
where v ∗ ∗ --> = U ∗ ∗ --> e i j + ( v ¯ ¯ --> i j − − --> U ¯ ¯ --> e i j ) {\displaystyle \mathbf {v} ^{\ast }=U^{\ast }\mathbf {e} _{ij}+({\overline {\mathbf {v} }}_{ij}-{\overline {U}}\mathbf {e} _{ij})} . This indicates that the inter-particle average velocity and pressure are simply replaced by the solution of the Riemann problem. By comparing both it can be seen that the intermediate velocity and pressure from the inter-particle averages amount to implicit dissipation, i.e. density regularization and numerical viscosity, respectively.
Since the above discretization is very dissipative a straightforward modification is to apply a limiter to decrease the implicit numerical dissipations introduced by limiting the intermediate pressure by [67]
P ∗ ∗ --> = P ¯ ¯ --> + 1 2 β β --> ρ ρ --> ¯ ¯ --> ( U L − − --> U R ) , {\displaystyle P^{\ast }={\overline {P}}+{\frac {1}{2}}\beta {\overline {\rho }}{(U_{L}-U_{R})},}
where the limiter is defined as
β β --> = min ( η η --> max ( U L − − --> U R , 0 ) , c ¯ ¯ --> ) . {\displaystyle \beta =\min {\big (}\eta \max(U_{L}-U_{R},0),{\overline {c}}{\big )}.}
Note that β β --> {\displaystyle \beta } ensures that there is no dissipation when the fluid is under the action of an expansion wave, i.e. U L < U R {\displaystyle U_{L}<U_{R}} , and that the parameter η η --> {\displaystyle \eta } , is used to modulate dissipation when the fluid is under the action of a compression wave, i.e. U L ≥ ≥ --> U R {\displaystyle U_{L}\geq U_{R}} . Numerical experiments found the η η --> = 3 {\displaystyle \eta =3} is generally effective. Also note that the dissipation introduced by the intermediate velocity is not limited.
In general, the description of hydrodynamic flows require a convenient treatment of diffusive processes to model the viscosity in the Navier–Stokes equations. It needs special consideration because it involves the Laplacian differential operator. Since the direct computation does not provide satisfactory results, several approaches to model the diffusion have been proposed.
Introduced by Monaghan and Gingold [68] the artificial viscosity was used to deal with high Mach number fluid flows. It reads
Here, α α --> {\displaystyle \alpha } is controlling a volume viscosity while β β --> {\displaystyle \beta } acts similar to the Neumann Richtmeyr artificial viscosity. The ϕ ϕ --> i j {\displaystyle \phi _{ij}} is defined by
where ηh is a small fraction of h (e.g. 0.01h) to prevent possible numerical infinities at close distances.
The artificial viscosity also has shown to improve the overall stability of general flow simulations. Therefore, it is applied to inviscid problems in the following form
It is possible to not only stabilize inviscid simulations but also to model the physical viscosity by this approach. To do so
is substituted in the equation above, where n {\displaystyle n} is the number of spartial dimensions of the model. This approach introduces the bulk viscosity ζ ζ --> = 5 3 μ μ --> {\displaystyle \zeta ={\frac {5}{3}}\mu } .
For low Reynolds numbers the viscosity model by Morris [69] was proposed.
Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code,[70] particle mesh, or particle-particle particle-mesh.
To discretize the governing equations of solid dynamics, a correction matrix B 0 {\displaystyle \mathbb {B} ^{0}} [71] [72] is first introduced to reproducing rigid-body rotation as
where
∇ ∇ --> a 0 W a = ∂ ∂ --> W ( | r a b 0 | , h ) ∂ ∂ --> | r a b 0 | e a b 0 {\displaystyle \nabla _{a}^{0}W_{a}={\frac {\partial W\left(|\mathbf {r} _{ab}^{0}|,h\right)}{\partial |\mathbf {r} _{ab}^{0}|}}\mathbf {e} _{ab}^{0}}
stands for the gradient of the kernel function evaluated at the initial reference configuration. Note that subscripts a {\displaystyle a} and b {\displaystyle b} are used to denote solid particles, and smoothing length h {\displaystyle h} is identical to that in the discretization of fluid equations.
Using the initial configuration as the reference, the solid density is directly evaluated as
where J = det ( F ) {\displaystyle J=\det(\mathbb {F} )} is the Jacobian determinant of deformation tensor F {\displaystyle \mathbb {F} } .
We can now discretize the momentum equation in the following form
where inter-particle averaged first Piola-Kirchhoff stress P ~ ~ --> {\displaystyle {\tilde {\mathbb {P} }}} is defined as
Also f a F : p {\displaystyle \mathbf {f} _{a}^{F:p}} and f a F : v {\displaystyle \mathbf {f} _{a}^{F:v}} correspond to the fluid pressure and viscous forces acting on the solid particle a {\displaystyle a} , respectively.
In fluid-structure coupling, the surrounding solid structure is behaving as a moving boundary for fluid, and the no-slip boundary condition is imposed at the fluid-structure interface. The interaction forces f i S : p {\displaystyle \mathbf {f} _{i}^{S:p}} and f i S : v {\displaystyle \mathbf {f} _{i}^{S:v}} acting on a fluid particle i {\displaystyle i} , due to the presence of the neighboring solid particle a {\displaystyle a} , can be obtained as [73]
and
Here, the imaginary pressure p a d {\displaystyle p_{a}^{d}} and velocity v a d {\displaystyle \mathbf {v} _{a}^{d}} are defined by
where n S {\displaystyle \mathbf {n} ^{S}} denotes the surface normal direction of the solid structure, and the imaginary particle density ρ ρ --> a d {\displaystyle \rho _{a}^{d}} is calculated through the equation of state.
Accordingly, the interaction forces f a F : p {\displaystyle \mathbf {f} _{a}^{F:p}} and f a F : v {\displaystyle \mathbf {f} _{a}^{F:v}} acting on a solid particle a {\displaystyle a} are given by
The anti-symmetric property of the derivative of the kernel function will ensure the momentum conservation for each pair of interacting particles i {\displaystyle i} and a {\displaystyle a} .
The discrete element method, used for simulating granular materials, is related to SPH.
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Pour les articles homonymes, voir Wheatley. Cet article est une ébauche concernant une localité anglaise. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. WheatleyGéographiePays Royaume-UniNation constitutive AngleterreRégion Angleterre du Sud-EstComté cérémonial OxfordshireDistrict non métropolitain South OxfordshireSuperficie 4,39 km2Coordonnées 51° 44′ 49″ N, 1° 08′ …
This article needs to be updated. The reason given is: Lacking in almost all information from his 13 year tenure as president (2006-2019). Please help update this article to reflect recent events or newly available information. (February 2020)Evo Morales addressing the Bolivian congress in 2008 The domestic policy of the Evo Morales administration refers to the domestic policy initiatives of the former President of Bolivia, including past pre-presidential advocacies by Morales. Politics Morales …
IPA-Zeichen ʲ IPA-Nummer 421 IPA-Zeichen-Beschreibung hochgestellte Minuskel j Unicode U+2B2 X-SAMPA ' oder _t Kirshenbaum <pzd> Palatalisierung bezeichnet die stellungsbedingte Änderung eines Lautes durch Hebung des Zungenrückens in Richtung des harten Gaumens (lateinisch palatum). Der fragliche Laut wird damit weiter nach vorn verschoben (z. B. von /k/ nach /tʃ/). Die Palatalisierung wird in der IPA-Lautschrift mit dem Zeichen [ʲ] dargest…
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (August 2011) (Learn how and when to remove this template message) 2002 single by Alisa MizukiHitomi no ChikaraSingle by Alisa Mizukifrom the album History: Alisa Mizuki Complete Single Collection ReleasedFebruary 6, 2002GenrePopLength4:31LabelAvex TuneSongwriter(s)Hiromi Mori, Toshihiko TakamizawaProducer(s)Keiichi…
لمعانٍ أخرى، طالع الحسين (توضيح). يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (فبراير 2016) ميدان مسجد الحسين في القاهرة. حي الحسين، هو أحد أحياء القاهرة ويتبع…
Protein-coding gene in the species Homo sapiens EIF1BIdentifiersAliasesEIF1B, GC20, eukaryotic translation initiation factor 1BExternal IDsMGI: 1916219 HomoloGene: 22219 GeneCards: EIF1B Gene location (Human)Chr.Chromosome 3 (human)[1]Band3p22.1Start40,309,707 bp[1]End40,312,424 bp[1]Gene location (Mouse)Chr.Chromosome 9 (mouse)[2]Band9|9 F4Start120,321,298 bp[2]End120,324,396 bp[2]RNA expression patternBgeeHumanMouse (ortholog)Top expressed i…
Verbs in the Tigrinya language This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article or section should specify the language of its non-English content, using {{lang}}, {{transliteration}} for transliterated languages, and {{IPA}} for phonetic transcriptions, with an appropriate ISO 639 code. Wikipedia's multilingual support t…
1920 border treaty You can help expand this article with text translated from the corresponding article in Finnish. (June 2018) Click [show] for important translation instructions. Machine translation, like DeepL or Google Translate, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Wikipedia. Do not translate text that appears unr…
Novel by Toni Morrison The Bluest Eye First edition coverAuthorToni MorrisonCountryUnited StatesLanguageEnglishGenreAfrican-American literaturePublisherHolt, Rinehart and WinstonPublication date1970Media typePrint (hardback & paperback)Pages224 pp (hardcover edition)ISBN978-0-375-41155-7 (hardcover edition)OCLC30110136Followed bySula The Bluest Eye, published in 1970, is the first novel written by Toni Morrison. The novel takes place in Lorain, Ohio (Morrison's hometown), and…
1925 silent film California Straight AheadTheatrical release posterDirected byHarry A. PollardWritten byDwinelle BenthallByron MorganHarry A. PollardBeatrice VanStarringReginald DennyGertrude OlmsteadTom WilsonCinematographyVirgil MillerGilbert WarrentonEdited byDaniel MandellEdward SchroederProductioncompanyUniversal PicturesDistributed byUniversal PicturesRelease date September 13, 1925 (1925-09-13) Running time80 minutesCountryUnited StatesLanguageSilent (English intertitles) C…
National Security Advisor of India since 2014 Ajit DovalKC, PPMDoval in 20175th National Security AdvisorIncumbentAssumed office 30 May 2014Prime MinisterNarendra ModiPreceded byShivshankar MenonDirector of the Intelligence BureauIn office31 July 2004 – 31 January 2005Prime MinisterManmohan SinghPreceded byK. P. SinghSucceeded byE. S. L. Narasimhan Personal detailsBorn (1945-01-20) 20 January 1945 (age 78)Ghiri Banelsyun, Pauri Garhwal, United Provinces, British India (now in…
Pour un article plus général, voir Guerre de Gaza de 2008-2009. Manifestation pro-palestinienne à Dar es Salam, (Tanzanie) Manifestation pro-israélienne à Londres, (Royaume-Uni) Les réactions internationales à la guerre de Gaza de 2008-2009 sont venues de nombreuses organisations internationales et ont souvent pris la forme de manifestations de la société civile, qui, dans plusieurs cas, étaient différentes de la position officielle des gouvernements. Réactions officielles Parties en…
1963 British filmParanoiacTheatrical release posterDirected byFreddie FrancisWritten byJosephine TeyScreenplay byJimmy SangsterBased onBrat Farrarby Josephine TeyProduced byAnthony HindsStarring Janette Scott Oliver Reed Sheila Burrell Alexander Davion CinematographyArthur GrantEdited byJames NeedsMusic byElisabeth LutyensProductioncompanyHammer Film ProductionsDistributed byRank Film DistributorsRelease dates 15 May 1963 (1963-05-15) (San Francisco)[1] 26 January…
Circunscripción de CeutaCortes Generales* Congreso: 1 diputado (de 350)* Senado: 2 senadores (de 266)Asamblea de Ceuta* 25 diputados (de 25) Ceuta es una de las 52 circunscripciones electorales de la Cámara Baja de las Cortes Generales, el Congreso de los Diputados, y una de las 59 de la Cámara Alta, el Senado. En las elecciones a la Asamblea de Ceuta constituye también una única circunscripción electoral, en la que se eligen 25 representantes que son concejales y diputados autonómicos al…
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