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| '''Smoothed-particle hydrodynamics (SPH)''' is a computational method used for simulating [[fluid]] flows. It was developed by Gingold and Monaghan (1977) and Lucy (1977) initially for astrophysical problems. It has been used in many fields of research, including [[astrophysics]], [[ballistics]], [[volcanology]], and [[oceanography]]. It is a mesh-free [[Continuum mechanics#Lagrangian description|Lagrangian method]] (where the coordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as the [[density]].
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you liked this short article and you would certainly such as to receive even more information relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] kindly go to our own web site. |
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| == Method ==
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| The smoothed-particle hydrodynamics (SPH) method works by dividing the fluid into a set of discrete elements, referred to as particles. These particles have a spatial distance (known as the "smoothing length", typically represented in equations by <math> h </math>), over which their properties are "smoothed" by a ''[[Kernel smoother|kernel function]]''. This means that the physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. For example, using Monaghan's popular cubic spline kernel the [[temperature]] at position <math> \mathbf{r} </math> depends on the temperatures of all the particles within a radial distance <math>2h</math> of <math> \mathbf{r}</math>.
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| The contributions of each particle to a property are weighted according to their distance from the particle of interest, and their density. Mathematically, this is governed by the kernel function (symbol <math> W </math>). Kernel functions commonly used include the [[Gaussian function]] and the [[cubic spline]]. The latter function is exactly zero for particles further away than two smoothing lengths (unlike the Gaussian, where there is a small contribution at any finite distance away). This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles.
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| The equation for any quantity <math>A</math> at any point <math>\mathbf{r}</math> is given by the equation
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| :<math>
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| A(\mathbf{r}) = \sum_j m_j \frac{A_j}{\rho_j} W(| \mathbf{r}-\mathbf{r}_{j} |,h),
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| </math>
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| where <math> m_j </math> is the [[mass]] of particle <math> j </math>, <math> A_j </math> is the value of the quantity <math> A </math> for particle <math> j </math>, <math> \rho_j </math> is the [[density]] associated with particle <math> j </math>, <math>\mathbf{r}</math> denotes position and <math> W </math> is the kernel function mentioned above. For example, the density of particle <math> i </math> (<math> \rho_i </math>) can be expressed as:
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| :<math>
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| \rho_i = \rho(\mathbf{r}_i) = \sum_j m_j \frac{\rho_j}{\rho_j} W(| \mathbf{r}_i-\mathbf{r}_j |,h) = \sum_j m_j W(\mathbf{r}_i-\mathbf{r}_j,h),
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| </math> | |
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| where the summation over <math> j </math> includes all particles in the simulation.
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| Similarly, the spatial derivative of a quantity can be obtained easily by virtue of the linearity of the derivative ([[del]], <math> \nabla </math>).
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| :<math>
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| \nabla A(\mathbf{r}) = \sum_j m_j \frac{A_j}{\rho_j} \nabla W(| \mathbf{r}-\mathbf{r}_j |,h).
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| </math>
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| 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. Combined with an [[equation of state]] and an [[integrator]], SPH can simulate hydrodynamic flows efficiently. However, the traditional [[numerical diffusion|artificial viscosity]] formulation used in SPH tends to smear out [[shock wave|shocks]] and [[contact discontinuity|contact discontinuities]] to a much greater extent than state-of-the-art grid-based schemes.
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| The Lagrangian-based adaptivity of SPH is analogous to the adaptivity present in grid-based [[adaptive mesh refinement]] codes. In some ways it is actually simpler because SPH particles lack any explicit topology relating them, unlike the elements in [[Finite element method|FEM]]. Adaptivity in SPH can be introduced in two ways; either by changing the particle smoothing lengths or by splitting SPH particles into 'daughter' particles with smaller smoothing lengths. The first method is common in astrophysical simulations where the particles naturally evolve into states with large density differences.<ref>http://arxiv.org/abs/astro-ph/9512078</ref> However, in hydrodynamics simulations where the density is often (approximately) constant this is not a suitable method for adaptivity. For this reason particle splitting can be employed, with various conditions for splitting ranging from distance to a free surface <ref>http://dl.acm.org/citation.cfm?id=1568695</ref> through to material shear.<ref>http://www.ase.uc.edu/~liugr/Storage/Journal%20Papers/2006/JA_2006_09.pdf</ref>
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| 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]],<ref>
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| [http://hpcc.astro.washington.edu/faculty/marios/papers/perform/node2.html "The Parallel k-D Tree Gravity Code"];
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| [http://www-hpcc.astro.washington.edu/faculty/trq/brandon/pkdgrav.html "PKDGRAV (Parallel K-D tree GRAVity code"]
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| use a [[kd-tree]] gravity simulation.
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| </ref> [[Particle Mesh|particle mesh]], or [[P3M|particle-particle particle-mesh]].
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| == Uses in astrophysics ==
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| The adaptive resolution of smoothed-particle hydrodynamics, combined with its ability to simulate phenomena covering many [[order of magnitude|orders of magnitude]], make it ideal for computations in [[theoretical astrophysics]].
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| Simulations of [[galaxy formation]], [[star formation]], stellar collisions, [[supernovae]] and [[meteor]] impacts are some of the wide variety of astrophysical and cosmological uses of this method.
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| 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 [[Magnetohydrodynamics|magnetic fields]] is an active area of research in the astronomical community, and has had some limited success.<ref>http://www.astro.ex.ac.uk/people/mbate/Cluster/clusterRT.html</ref>
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| == Uses in fluid simulation ==
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| [[Image:fluids sim.jpg|thumb|200px|right|Fig. SPH simulation of ocean waves using FLUIDS v.1 (Hoetzlein)]]
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| 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-base technique 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]], [[Volume_rendering#Splatting|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).
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| 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 [[structured grid|uniform grids]] and SPH particle techniques, many [[voxels]] or particles will be used to fill water volumes which 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 [[incompressible flow#Difference between incompressible flow and material|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.
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| Recent work in SPH for Fluid simulation has increased both performance and accuracy:
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| * B. Solenthaler, 2009, develops Predictive-Corrective SPH (PCISPH) to allow for better incompressibility constraints<ref>
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| {{cite journal|last=Solenthaler|title=Predictive-Corrective Incompressible SPH|year=2009}}</ref>
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| * M. Ihmsen et al., 2010, introduce boundary handling and adaptive time-stepping for PCISPH for accurate rigid body interactions<ref>{{cite journal|last=Imhsen|title=Boundary handling and adaptive time-stepping for PCISPH|year=2010|journal=Workshop on Virtual Reality Interaction and Physical Simulation VRIPHYS}}</ref>
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| * K. Bodin et al., 2011, replace the standard equation of state pressure equation with a density constraint and applies a variational time integrator. <ref>{{cite journal|last=Bodin|title=Constraint Fluids. http://www.physics.umu.se/english/research/statistical-physics-and-networks/complex-mechanical-systems/fluids-and-solids/ |year=2011|journal=IEEE Transactions on Visualization and Computer Graphics}}</ref>
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| * R. Hoetzlein, 2012, develops efficient GPU-based SPH for large scenes in Fluids v.3<ref>{{cite journal|last=Hoetzlein|title=Fluids v.3, A Large scale, Open Source Fluid Simulator. http://fluids3.com|year=2012}}</ref>
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| * M. Macklin et al., 2013 simulates incompressible flows inside the Position Based Dynamics framework, for bigger timesteps <ref>{{cite journal|last=Macklin|title=Position Based Fluids http://blog.mmacklin.com/publications |year=2013|journal=SIGGRAPH proceedings}}</ref>
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| == Uses in solid mechanics ==
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| In 1990, Libersky and Petschek <ref name="libersky248">{{cite journal|last=Libersky|first=L.D.|coauthors=Petschek, A.G.|title=Smooth Particle Hydrodynamics with Strength of Materials, Advances in the Free Lagrange Method|journal=Lecture Notes in Physics|year=1990|volume=395|pages=248–257}}</ref><ref>{{cite journal|last=L.D. Libersky|coauthors=A.G. Petschek, A.G. Carney, T.C. Hipp, J.R. Allahdadi, F.A. High|title=Strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response|journal=J. Comput. Phys.|year=1993|volume=109|pages=67–75}}</ref> extended SPH to Solid Mechanics.
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| The main advantage of SPH 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 <ref>{{cite journal|last=J.W. Swegle|coauthors=D.A. Hicks, S.W. Attaway|title=Smooth particle hydrodynamics stability analysis|journal=J. Comput. Phys.|year=1995|volume=116|pages=123–134}}</ref> and lack of consistency.<ref>{{cite journal|last=T. Belytschko|coauthors=Y. Krongauz, J. Dolbow, C. Gerlach,|title=On the completeness of meshfree particle methods|journal=Int. J. Numer. Methods Eng.|year=1998|volume=43|pages=785–819}}</ref> Over the past years, different corrections have been introduced to improve the accuracy of the SPH solution. That is the case of Liu et al.,<ref>{{cite journal|last=W.K. Liu|coauthors=S. Jun, Y.F. Zhang|title=Reproducing kernel particle methods|journal=Int. J. Numer. Methods Eng.|year=1995|volume=20|issue=8–9|pages=1081–1106}}</ref> Randles and Libersky <ref name="autogenerated525">{{cite journal|last=P.W. Randles|coauthors=L.D. Libersky|title=Recent improvements in SPH modelling of hypervelocity impact|journal=Int. J. Impact Eng.|year=1997|volume=20|pages=525–532}}</ref> and Johnson and Beissel,<ref name="autogenerated2725">{{cite journal|last=G.R. Johnson|coauthors=S.R. Beissel|title=Normalized smoothing functions for SPH impact computations|journal=Int. J. Numer. Methods Eng.|year=1996|volume=39|pages=2725–2741}}</ref> who tried to solve the consistency problem. Dyka et al.<ref>{{cite journal|last=C.T. Dyka|coauthors=R.P. Ingel|title=An approach for tension instability in Smoothed Particle Hydrodynamics|journal=Comput. Struct.|year=1995|volume=57|pages=573–580}}</ref><ref>{{cite journal|last=C.T. Dyka|coauthors=P.W. Randles, R.P. Ingel|title=Stress points for tension instability in SPH|journal=Int. J. Numer. Methods Eng.|year=1997|volume=40|pages=2325–2341}}</ref> and Randles and Libersky <ref>{{cite journal|last=P.W. Randles|coauthors=L.D. Libersky|title=Normalized SPH with stress points|journal=Int. J. Numer. Methods Eng.|year=2000|volume=48|pages=1445–1462}}</ref> introduced the stress-point integration into SPH and Belytschko et al.<ref>{{cite journal|last=T. Belytschko|coauthors=Y. Guo, W.K. Liu, S.P. Xiao|title=A unified stability analysis of meshless particle methods|journal=Int. J. Numer. Methods Eng.|year=2000|volume=48|pages=1359–1400}}</ref> showed later 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. | |
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| The recent improvements on the convergence and stabilility of SPH have allowed the use of SPH in more and more applications concerning Solid Mechanics. Here there are some examples of its last applications and recent developments of the method:
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| * Libersky and Petschek <ref name="libersky248"/> modified the SPH method in order to solve Strength of Materials problems.
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| * Johnson and Beissel <ref name="autogenerated2725"/> and Randles and Libersky <ref name="autogenerated525"/> applied SPH to impact phenomena.
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| * Bonet and Kulasegaram applied SPH to metal forming simulations.<ref>{{cite journal|last=J. Bonet|coauthors=S. Kulasegaram|title=Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations|journal=Int. J. Numer. Methods Eng.|year=2000|volume=47|pages=1189–1214}}</ref>
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| * [[William G. Hoover]] developed a SPH-based method whose acronym is SPAM (smooth-particle applied mechanics) to study impact fracture in solids.<ref>{{cite journal|last=W. G. Hoover|coauthors=C. G. Hoover|journal=Computing in Science and Engineering|year=2001|volume=3|issue=2|pages=78–85}}</ref>
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| * Rabczuk and co-workers applied a modified SPH (SPH/MLSPH) to simulate fracture and fragmentation.<ref>{{cite journal|last=T. Rabczuk|coauthors=J. Eibl, L. Stempniewski|title=Simulation of high velocity concrete fragmentation using SPH/MLSPH|journal=Int. J. Numer. Methods Eng.|year=2003|volume=56|pages=1421–1444}}</ref>
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| * Herreros and Mabssout developed the Taylor-SPH (TSPH) method to overcome the problem of shock wave propagation in solids.<ref>{{cite journal|last=M.I. Herreros|coauthors=M. Mabssout|title=A two-steps time discretization scheme using the SPH method for shock wave propagation|journal=Comput. Methods Appl. Mech. Engrg.|year=2011|volume=200|pages=1833–1845}}</ref>
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| == Notes ==
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| {{Reflist|30em}}
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| == References ==
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| * [1] R.A. Gingold and J.J. Monaghan, “Smoothed particle hydrodynamics: theory and application to non-spherical stars,” Mon. Not. R. Astron. Soc., Vol 181, pp. 375–89, 1977.
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| * [2] L.B. Lucy, “A numerical approach to the testing of the fission hypothesis,” Astron. J., Vol 82, pp. 1013–1024, 1977.
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| * [3] Hoover, W. G. (2006). Smooth Particle Applied Mechanics: The State of the Art, World Scientific.
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| * [4] [http://adsabs.harvard.edu/abs/1994MmSAI..65.1117S Impact Modelling with SPH] Stellingwerf, R. F., Wingate, C. A., Memorie della Societa Astronomia Italiana, Vol. 65, p. 1117 (1994).
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| * [5] Amada, T., Imura, M., Yasumuro, Y., Manabe, Y. and Chihara, K. (2004) Particle-based fluid simulation on GPU, in proceedings of ACM Workshop on General-purpose Computing on Graphics Processors (August, 2004, Los Angeles, California).
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| * [6] Desbrun, M. and Cani, M-P. (1996). Smoothed Particles: a new paradigm for animating highly deformable bodies. In Proceedings of Eurographics Workshop on Computer Animation and Simulation (August 1996, Poitiers, France).
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| * [7] Harada, T., Koshizuka, S. and Kawaguchi, Y. Smoothed Particle Hydrodynamics on GPUs. In Proceedings of Computer Graphics International (June 2007, Petropolis Brazil).
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| * [8] Hegeman, K., Carr, N.A. and Miller, G.S.P. Particle-based fluid simulation on the GPU. In Proceedings of International Conference on Computational Science (Reading, UK, May 2006). Proceedings published as Lecture Notes in Computer Science v. 3994/2006 (Springer-Verlag).
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| * [9] M. Kelager. (2006) Lagrangian Fluid Dynamics Using Smoothed Particle Hydrodynamics, M. Kelagar (MS Thesis, Univ. Copenhagen).
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| * [10] Kolb, A. and Cuntz, N. (2005) ] Dynamic particle coupling for GPU-based fluid simulation, A. Kolb and N. Cuntz. In Proceedings of the 18th Symposium on Simulation Techniques (2005) pp. 722–727.
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| * [11] Liu, G.R. and Liu, M.B. Smoothed Particle Hydrodynamics: a meshfree particle method. Singapore: World Scientific (2003).
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| * [12] Monaghan, J.J. (1992). Smoothed Particle Hydrodynamics. Ann. Rev. Astron. Astrophys (1992). 30 : 543-74.
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| * [13] Muller, M., Charypar, D. and Gross, M. ] Particle-based Fluid Simulation for Interactive Applications, In Proceedings of Eurographics/SIGGRAPH Symposium on Computer Animation (2003), eds. D. Breen and M. Lin.
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| * [14] Vesterlund, M. Simulation and Rendering of a Viscous Fluid Using Smoothed Particle Hydrodynamics, (MS Thesis, Umea University, Sweden).
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| * [15] Violeau, D., Fluid Mechanics and the SPH method. Oxford University Press (2012).
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| == External links ==
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| * [http://www.ukaff.ac.uk/starcluster First large simulation of star formation using SPH]
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| * [http://wiki.manchester.ac.uk/spheric SPHERIC (SPH European Research Interest Community)]
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| * [http://ia2.oats.inaf.it/index.php?option=com_content&view=section&id=12&Itemid=71 ITVO] is the web-site of The Italian Theoretical Virtual Observatory created to query a database of numerical simulation archive.
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| * [http://www.stellingwerf.com/rfs-bin/index.cgi?action=PageView&id=33 SPHC Image Gallery] depicts a wide variety of test cases, experimental validations, and commercial applications of the SPH code SPHC.
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| === Software ===
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| * [http://www.algodoo.com/ Algodoo is a 2D simulation framework for education using SPH]
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| * [http://onezero.ca/ Fluidix] is a GPU-based particle simulation API available from OneZero Software
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| * [http://www.sph-flow.com/ SPH-flow]
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| * [http://www.rchoetzlein.com/eng/ FLUIDS v.1] is a simple, open source (Zlib), real-time 3D SPH implementation in C++ for liquids for CPU and GPU.
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| * [[GADGET]] [http://www.mpa-garching.mpg.de/galform/gadget/ ] is a freely available ([[General Public License|GPL]]) code for cosmological N-body/SPH simulations
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| * [http://isph.sourceforge.net/ ISPH] parallel C++/OpenCL open source truly incompressible SPH implementation
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| * [http://www.simpartix.com SimPARTIX] is a commercial simulation package for SPH and DEM simulations from Fraunhofer IWM
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| * [http://www.astro.ex.ac.uk/people/dprice/splash SPLASH] is an open source (GPL) visualisation tool for SPH simulations
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| * [http://www.sphysics.org/ SPHysics] is an open source SPH implementation in Fortran
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| * [http://www.dual.sphysics.org DualSPHysics] is an open source SPH code based on SPHysics and using GPU computing
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| * [http://pal.sourceforge.net/ Physics Abstraction Layer] is an open source abstraction system that supports real time physics engines with SPH support
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| * [http://www.itm.uni-stuttgart.de/research/pasimodo/pasimodo_en.php Pasimodo] is a program package for particle-based simulation methods, e.g. SPH
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| * [http://sourceforge.net/projects/punto/ Punto] is a freely available visualisation tool for particle simulations
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| * [http://www.ce.jhu.edu/dalrymple/GPU/GPUSPH/Home.html GPUSPH] SPH simulator with viscosity (GPLv3)
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| * [http://code.google.com/p/pysph/ pysph] Framework for Smoothed Particle Hydrodynamics in Python (New BSD License)
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| {{DEFAULTSORT:Smoothed-Particle Hydrodynamics}}
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| [[Category:Fluid dynamics]]
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| [[Category:Numerical differential equations]]
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| [[Category:Computational fluid dynamics]]
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