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김경종
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<!-- source-page: 1301 -->
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# Johnson-Kendall-Roberts adhesive normal contact between particles
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The JKR model relates contact force, F, to the contact area, a, as follows:
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$$
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F = \frac {4 E ^ {*} a ^ {3}}{3 R ^ {*}} - \sqrt {8 \pi \Gamma E ^ {*} a ^ {3}}.
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$$
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The approach distance, for two contacting spherical particles is related to the contact area, a, as follows:
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$$
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\delta = \frac {a ^ {2}}{R ^ {*}} - \sqrt {\frac {2 \pi \Gamma a}{E ^ {*}}}.
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$$
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In the above equations
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$$
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R = \frac {R _ {1} R _ {2}}{R _ {1} + R _ {2}}
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$$
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and
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$$
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\frac {1}{E ^ {*}} = \frac {1 - \nu_ {1} ^ {2}}{E _ {1}} + \frac {1 - \nu_ {2} ^ {2}}{E _ {2}}.
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$$
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In the above equations is the surface energy per unit area of the two contacting particles. Like nonadhesive Hertz contact $E _ { 1 } , \nu _ { 1 }$ and $E _ { 2 } , \nu _ { 2 }$ are the effective Young’s modulus and Poisson’s ratio of the two contacting particles, respectively. $R _ { 1 }$ and $R _ { 2 }$ are the radii of the two contacting particles, respectively. You must specify the effective Young’s modulus and Poisson’s ratio for a contacting particle. In addition, you must specify the JKR-type pressure overclosure.
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$K _ { m a x }$ is the limiting value of the normal contact stiffness, beyond which the normal contact force increases linearly.
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Figure 15.1.1–5 shows the force-displacement curve for the JKR adhesion model. Adhesion between particles is triggered upon first contact. The pull-off force, $\begin{array} { r } { F _ { c } = - \frac { 3 } { 2 } \pi \Gamma R ^ { * } } \end{array}$ , is the maximum tensile force. Particles continue to experience adhesion force even after physical separation occurs. The adhesion force between two particles goes to zero at a separation distance of
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$$
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\delta_ {s e p a r a t i o n} = - \frac {3}{4} \left(\frac {\pi^ {2} \Gamma^ {2} R ^ {*}}{E ^ {* 2}}\right) ^ {\frac {1}{3}}.
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$$
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It can be seen from Figure 15.1.1–5 that adhesive forces are nonzero at $\delta = 0$ and reduce to zero at $\delta = \delta _ { F 0 }$ . In some situations it may be desirable to have zero adhesive forces when $\delta = 0$ . You can achieve this by applying a horizontal shift of $\delta = \delta _ { F 0 }$ to the force-displacement curve shown in Figure 15.1.1–5. Figure 15.1.1–6 shows the shifted curve. The pull-off force remains unchanged due to the shift, whereas the new separation distance $\delta _ { s e p a r a t i o n \_ s h i f t e d } = \delta _ { s e p a r a t i o n } + \delta _ { F 0 }$ . Abaqus automatically computes the horizontal shift $\delta _ { F 0 }$ for the “shifted” JKR type adhesive behavior.
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<!-- source-page: 1302 -->
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<details>
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<summary>text_image</summary>
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F
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δseparation δFo
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0 δ
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Kmax
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Fc (pull-off force)
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</details>
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Figure 15.1.1–5 Force versus approach distance relation for the JKR model.
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<details>
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<summary>line</summary>
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| δ | F |
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| --- | --- |
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| 0 | 0 |
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| >0 | K_max |
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</details>
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Figure 15.1.1–6 Force versus approach distance relation for the shifted JKR model.
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<!-- source-page: 1303 -->
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Input File Usage: Use the following options to define JKR-type normal contact behavior:
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*DISCRETE SECTION, ELSET=element_set_name
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*DISCRETE ELASTICITY
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effective Young's modulus, effective Poisson's ratio
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...
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*SURFACE INTERACTION
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*SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=JKR
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Surface energy per unit area, $K_{max}$ Use the following options to define the shifted JKR-type normal contact behavior:
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*DISCRETE SECTION, ELSET=element_set_name
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*DISCRETE ELASTICITY
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effective Young's modulus, effective Poisson's ratio
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...
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*SURFACE INTERACTION
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*SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=JKR
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Surface energy per unit area, $K_{max}$ , SHIFTED
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# Output
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No element output is available for PD3D elements. The nodal output includes all output variables generally available in Abaqus/Explicit analyses (see “Abaqus/Explicit output variable identifiers,” Section 4.2.2).
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# Limitations
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Discrete element method analyses are subject to the following limitations:
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• Volume average output for stress, strain, and other similar continuum element output is not available for DEM analysis.
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• Only a spherical shape is supported for PD3D elements.
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• PD3D elements cannot be part of a rigid body definition.
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• It is not possible to specify cohesive or thermal contact between PD3D elements or between PD3D elements and other elements.
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• Rolling friction is ignored for contact between PD3D elements or between PD3D elements and other elements.
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• It is not possible to solve problems involving particle-fluid interaction with PD3D elements.
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• User-defined surface interaction is not supported for contact between PD3D elements.
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• Although supported in Abaqus/Viewer, the functionality is not supported in Abaqus/CAE. You can use the existing functionality in Abaqus/CAE to generate mass elements, write an input file, and then manually edit the input file to convert the mass elements to particles. Alternatively, you can create a mesh using C3D8R elements, write an input file, and then use a script to convert these elements to
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<!-- source-page: 1304 -->
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particles as described in “Generating particle elements from a solid mesh” in the Dassault Systèmes Knowledge Base at www.3ds.com/support/knowledge-base.
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DEM computations are distributed across parallel domains except when multiple discrete sections are defined with different alpha damping parameters (which will degrade parallel scalability). DEM analyses are subject to the following limitations if multiple CPUs are used:
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• Contact output is not supported for DEM slave nodes.
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• Energy history output other than for the whole model is not supported.
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• Dynamic load balancing cannot be activated.
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• If any DEM particles participate in general contact, all DEM particles must be included in the general contact definition.
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• At least 10,000 DEM particles per domain is suggested to achieve good scalability.
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• A significant increase in memory usage may be needed if a large number of CPUs is used.
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# Input file template
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The following example illustrates a discrete element method analysis:
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```txt
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*HEADING
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...
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*ELEMENT, TYPE=PD3D, ELSET=name
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Element number, node number
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...
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*DISCRETE SECTION, ELSET=name
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Particle radius
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**
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*INITIAL CONDITIONS, TYPE=VELOCITY
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Data lines to define velocity initial conditions
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*NSET, NSET=name, ELSET=name
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*SURFACE, NAME=name
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,
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**
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*SURFACE INTERACTION
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**
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*STEP
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*DYNAMIC, EXPLICIT
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*DLOAD
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Data lines to define gravity load
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**
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*CONTACT
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*CONTACT INCLUSIONS
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*CONTACT PROPERTY ASSIGNMENT
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**
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```
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<!-- source-page: 1305 -->
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\*CONTACT CONTROLS ASSIGNMENT
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\*OUTPUT, FIELD
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\*END STEP
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# Additional references
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• Cundall, P. A., and O. D. Strack, “A Distinct Element Method for Granular Assemblies,” Geotechnique, vol. 29, pp. 47–65, 1979.
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• Johnson, K. L., K. Kendall, and A. D. Roberts, “Surface Energy and the Contact of Elastic Solids,” Proceedings of the Royal Society of London, vol. 324, pp. 301–313, 1971.
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• Munjiza, A., and K. R. F. Andrews, “NBS Contact Detection Algorithm for Bodies of Similar Size,” International Journal for Numerical Methods in Engineering, vol. 43, pp. 131–149, 1998.
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• O’Sullivan, C., and J. D. Bray, “Selecting a Suitable Time Step for Discrete Element Simulations that Use the Central Difference Time Integration Scheme,” Engineering Computations, vol. 21(2/3/4), pp. 278–303, 2004.
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• Zhu, H. P., Z. Y. Zhou, R. Y. Yang, and A. B. Yu, “Discrete Particle Simulation of Particulate Systems: A Review of Major Applications and Findings,” Chemical Engineering Science, vol. 63, pp. 5728–5770, 2008.
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• Zhu, H. P., Z. Y. Zhou, R. Y. Yang, and A. B. Yu, “Discrete Particle Simulation of Particulate Systems: Theoretical Developments,” Chemical Engineering Science, vol. 62, pp. 3378–3396, 2007.
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<!-- source-page: 1306 -->
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<!-- source-page: 1307 -->
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# 15.2 Continuum particle analyses
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• “Smoothed particle hydrodynamics,” Section 15.2.1
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• “Finite element conversion to SPH particles,” Section 15.2.2
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<!-- source-page: 1308 -->
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<!-- source-page: 1309 -->
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# 15.2.1 SMOOTHED PARTICLE HYDRODYNAMICS
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# Product: Abaqus/Explicit
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# References
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• “Continuum particle elements,” Section 33.2.1
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• \*SOLID SECTION
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• \*SECTION CONTROLS
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• \*INITIAL CONDITIONS
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# Overview
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Smoothed particle hydrodynamics (SPH) is a numerical method that is part of the larger family of meshless (or mesh-free) methods. For these methods you do not define nodes and elements as you would normally define in a finite element analysis; instead, only a collection of points are necessary to represent a given body. In smoothed particle hydrodynamics these nodes are commonly referred to as particles or pseudo-particles.
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# Introduction
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The example shown in Figure 15.2.1–1 contrasts the two approaches. Both discrete representations model the initial configuration of a body of fluid inside a bottle, as described in detail in “Impact of a water-filled bottle,” Section 2.3.2 of the Abaqus Example Problems Guide. The model on the left is a traditional tetrahedron mesh of the volume occupied by the fluid. On the right, the same volume is represented by a collection of discrete points. Note that in the latter case there are no edges connecting the points as these points (pseudo-particles) do not require the definition of multiple-node element connectivity, as is the case in the traditional finite element representation on the left. An alternative to directly defining particle elements is to define conventional continuum finite elements and automatically convert them to particle elements at the beginning of the analysis or during the analysis, as discussed in “Finite element conversion to SPH particles,” Section 15.2.2.
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Smoothed particle hydrodynamics is a fully Lagrangian modeling scheme permitting the discretization of a prescribed set of continuum equations by interpolating the properties directly at a discrete set of points distributed over the solution domain without the need to define a spatial mesh. The method’s Lagrangian nature, associated with the absence of a fixed mesh, is its main strength. Difficulties associated with fluid flow and structural problems involving large deformations and free surfaces are resolved in a relatively natural way.
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At its core, the method is not based on discrete particles (spheres) colliding with each other in compression or exhibiting cohesive-like behavior in tension as the word particle might suggest. Rather, it is simply a clever discretization method of continuum partial differential equations. In that respect, smoothed particle hydrodynamics is quite similar to the finite element method. SPH uses an evolving interpolation scheme to approximate a field variable at any point in a domain. The value of a variable
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<!-- source-page: 1310 -->
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<details>
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<summary>natural_image</summary>
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3D mesh model of a green rectangular prism with triangular mesh pattern (no text or symbols)
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</details>
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<details>
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<summary>scatter</summary>
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| x | y |
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|---|---|
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| 0.0 | 0.0 |
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| 0.1 | 0.1 |
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| 0.2 | 0.2 |
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| 0.3 | 0.3 |
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| 0.4 | 0.4 |
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| 0.5 | 0.5 |
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| 0.6 | 0.6 |
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| 0.7 | 0.7 |
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| 0.8 | 0.8 |
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| 0.9 | 0.9 |
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| 1.0 | 1.0 |
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</details>
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Figure 15.2.1–1 Finite element mesh and SPH particle distribution.
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at a particle of interest can be approximated by summing the contributions from a set of neighboring particles, denoted by subscript j, for which the “kernel” function, W, is not zero
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$$
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\langle f (\mathbf {x}) \rangle \simeq \sum_ {j} \frac {m _ {j}}{\rho_ {j}} f _ {j} W (| \mathbf {x} - \mathbf {x} _ {j} |, h).
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$$
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An example kernel function is shown in Figure 15.2.1–2. The smoothing length, h, determines how many particles influence the interpolation for a particular point.
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The SPH method has received substantial theoretical support since its inception (Gingold and Monaghan, 1977), and the number of publications related to the method is now very large. A number of references are listed below.
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The method can use any of the materials available in Abaqus/Explicit (including user materials). You can specify initial conditions and boundary conditions as for any other Lagrangian model. Contact interactions with other Lagrangian bodies are also allowed, thus expanding the range of applications for which this method can be used.
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The method is less accurate in general than Lagrangian finite element analyses when the deformation is not too severe and than coupled Eulerian-Lagrangian analyses in higher deformation regimes. If a large percentage of all nodes in the model are associated with smoothed particle hydrodynamics, the analysis may not scale well if multiple CPUs are used.
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