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Force calculationThe index k is calculated through:k Sx x x x S Kr i j i j1 with2[x] is the largest integer smaller than x.Use Newton – Gregory interpolation: with 2f( z) F( k) ( kzS) F( k1) F( k) z xi xjc27Verlet tablesDefine around each particle i a neighborhoodof radius r l > r c .Store all the coordinatesof the particles in thisneighborhood in thevector LIST whichr lr chas a length N·N u where N u is the averagenumber of particles in the neighborhood.28
Verlet tablesThe Verlet table LIST is a one-dimensional vectorin which all neighborhoods are sequentially stored.A second vector POINT[i] contains the index ofthe first particle in the neighborhood of i in LIST.Therefore the particles in the neighborhood of i are:LIST[POINT[i]],…,LIST[POINT[i+1]-1].The force is calculated by just going throughthis list to find all the neighbors of i, (i.e. N).29Verlet tablesEverynrl r tvcmaxtime steps the Verlet tablemust be renewed (n ≈ 10 – 20) where v max is thelargest velocity of a particle, since otherwiseparticles beyond a distance r l could reach r c .Renewal requires N 2 operations so that theentire algorithm still goes like O(N 2 ).30
Page 1 and 2: Guidance on Impact AssessmentsDocum
Page 3 and 4: Potentialshard core interaction 0.3
Page 5 and 6: PotentialsLenard Jones potentialε
Page 7 and 8: Contact timeone dimensionenergy:122
Page 9 and 10: Verlet method• One needs to store
Page 11 and 12: Total energy21Small perturbationdiv
Page 13: TricksMost time consuming loop is c
Page 17 and 18: Linked cell methodexamplecell 1 cel
Page 19 and 20: MoleculesMolecules are composed of
Page 21 and 22: Lagrange multipliersConstraints tha
Page 23 and 24: Lagrange multipliersFirst solve ter
Page 25 and 26: Contact Dynamics (CD)P. Lötstedt (
Page 27 and 28: Example: two particles (1D)Matrix H
Page 29 and 30: Example: two particles (1D)No overl
Page 31 and 32: Contact DynamicsCoulomb-Graph:Frict
Page 33 and 34: Contact Dynamics ext1F 1 1 1
Page 35 and 36: Contact DynamicsPerform direct inve
Page 37 and 38: Non-spherical objectsUp to now: ato
Page 39 and 40: Two dimensionsTime evolution of the
Page 41 and 42: 3d rigid body rotationangular momen
Page 43 and 44: Euler anglesRotations in three dime
Page 45 and 46: AOne can show:Quaternions 2 2
Page 47 and 48: Long-range potentialsLong-range are
Page 49 and 50: Ewald summationThe Ewald sum is onl
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Page 55 and 56: Multipole expansion„Fast Multipol
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Nosé - Hoover thermostatCoupling t
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Nosé - Hoover thermostatnormalener
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Constant pressureKeep pressure pfix
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