SPH development at Cranfield University. Prof. Rade Vignjevic. IV SPHERIC Nantes, May 2009

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1 SPH development at Cranfield University Prof. Rade Vignevic IV SPHERIC Nantes, May 2009

2 SPH development at Cranfield University Presentation outline 1. Introduction (CU, Motivation) 2. Normalised Corrected SPH 3. Non-collocational SPH 4. Contact algorithm, FE SPH coupling 5. Damage modelling

3 Cranfield University: History Formed in 1946 as College of Aeronautics (centre of excellence in aerospace) Postgraduate University 1992 Centre of Applied Research Five Maor Schools Health Management Engineering Applied Science

4 Cranfield University: Entirely Post Graduate, 35% PhD MSc Courses of 45 Week Duration International Industrial & Government Funded Research The UK National Flying Laboratory Applied approach to teaching and research Strong Industrial Links Worldwide Reputation 240 Million Turnover - 60% From Research

5 Motivation for our SPH work Challenging problems of computational mechanics are often characterised by: Extremely large deformations Tracking of interfaces between solids and liquid/gas Propagation of cracks with arbitrary and complex paths Change of boundaries of material that has undergone extensive micro cracking or phase change and failure.

6 Motivation for the SPH work Problem solving Manufacturing: Extrusion, Moulding, Friction Stir Welding, Manufacture of composites, Safety and Crashworthiness: Hypervelocity impacts on spacecraft, Aircraft impacts on water/soft soil, Bird strike... Defence: Armour penetration (metal, ceramics), Warhead fragmentation, Shape charges

7 People that inspired and influenced SPH research at Cranfield: L. Libersky J. J. Monaghan

8 Corrected Normalised SPH (CNSPH) R. Vignevic, J. Campbell, L. Libersky; A treatment of zero-energy modes in the smoothed particle hydrodynamics method, Comput. Methods Appl. Mech. Engrg. 184 (2000), pp , (Received 28 October 1998) R. Vignevic, J. Reveles, J. Campbell; SPH in a Total Lagrangian Formalism, Computer Methods in Engineering and Science, vol.14, no.3, pp , 2006

9 Corrected Normalised SPH CNSPH Emmy Noether, "Invariante Variationsprobleme, " Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, pp Specifically: The invariance with respect to translational and rotational transformations

10 Corrected Normalised SPH CNSPH Space homogeneity x m x x x x (, h) J = JW I J J J is interpolated solution space x m x x x x ' = x x ( h) J ' ' ' ' ' =, ' JW I x = x J I J J mj J x ' = x x W ( xi x J, h ) ( xi xj h ) J J J m W J, = 1

11 Corrected Normalised SPH CNSPH Space isotropy m x x x x (, h) J = JW I J J J x is interpolated solution space x m x x x ( h) J ' ' ' ' ' =, ' JW I x = x J I J J x' = C x x' C x = C x = C x C = C nnbr mj xj W ( xi xj, h) = I J = 1 J

12 Interpolation should not violate the following properties of space: Homogeneity Anisotropy Condit. 1 ), ( 1 = = h x x W m nnbr i x = 1 = ), ( 1 h x x W m nnbr i Correct. ), ( ), ( ~ 1 h x x W m h x x W W nnbr i i i = = 1 1 ~ ~ ~ ~ = = nnbr i i i x m W W W Corrected Normalised SPH CNSPH

13 CNSPH Form of Governing Equations Conservation of: m = v v % W % nnbr & Mass ( ) i i i i = 1 n n b r i = + i Momentum 2 2 v& m σ σ = 1 i nnbr σ = v v % W % i e& m Energy ( ) 2 i = 1 i i % W % i W here: ~ W i nnbr = Wi = 1 m 1 ( ) ~ ~ ~ xi x W nnbr i, W i= W i ( ) xi x = 1 m ~ W i 1

14 CNSPH Conservation of Momentum and A-momentum Materials X -Velocity

15 Corrected Normalised SPH Summary First order consistency Conservation of linear and angular momentum Homogeneity and isotropy of space maintained in the SPH discretisation

16 SPH in a Total Lagrangian Formalism R. Vignevic, J. Reveles, J. Campbell; SPH in a Total Lagrangian Formalism, Computer Methods in Engineering and Science, Vol.14, No.3, pp , 2006

17 SPH in a Total Lagrangian Formalism Taylor test for OFHPC copper 180 m/s

18 SPH in a Total Lagrangian Formalism The mapping from material into spatial coordinates is x ( X,t) = φ. The displacement of a material point is given by the difference between its current position and its original position. u ( X, t) = φ ( X, t) φ( X,0) = φ( X, t) X = x X

19 SPH in a Total Lagrangian Formalism The deformation gradient F ( u + X) x u F = = = + I X X X Discretised form F ~ 0 I ( u u ) W V. B I B ( X Xi ) = J S Velocity gradient J I 0 IJ J + ( ) F& = v v W% V 0 I J I 0 IJ J J S m = S ~ 1 W

20 SPH in a Total Lagrangian Formalism Conservation Equations in the Total Lagrangian formalism Mass Momentum Energy Continuous Discretised = J 1 1 u& = 0 P + b 0 0 = F I & 0 a ( P P ) V B 1 & = F & 0 : P = I J I 0W IJ J : J S e m J ~ 0 ( ) e& I ~ = P J : v J v I 0WIJ VJ. B J S I J

21 SPH in a Total Lagrangian Formalism Taylor test for OFHPC copper 180 m/s

22 SPH in a Total Lagrangian Formalism 3-D Taylor test, effective plastic strain

23 SPH in a Total Lagrangian Formalism Total Lagrangian SPH vs. conventional SPH

24 SPH in a Total Lagrangian Formalism Summary Stable well behaved Applicable to finite deformations Combined with Eulerian SPH when modelling extremely large deformations and failure

25 Non-Collocational SPH R. Vignevic, J. Campbell, L. Libersky; A treatment of zero-energy modes in the smoothed particle hydrodynamics method, Comput. Methods Appl. Mech. Engrg. 184 (2000), pp , (Received 28 October 1998)

26 Non-Collocational SPH Two types of particles: Velocity Stress Easy to apply essential boundary conditions (similar to FE)

27 Non-Collocational SPH Collision of elastic hoops

28 Non-Collocational SPH Taylor test for OFHPC copper 180 m/s

29 Non-Collocational SPH Summary Easy to apply to 1D and 2D structural elements, especially advantageous when modelling failure (ongoing work) Increased difficulties to extend to 3D continuum (update of stress particle locations)

30 Contact Algorithm R. Vignevic; T. De Vuyst; and J. Campbell; A Frictionless Contact Algorithm for Meshless Methods, CMES, Vol. 13, No. 1, pp , 2006 T. De Vuyst, R. Vignevic and J. C. Campbell; Modelling of Fluid-Structure Impact Problems using a Coupled SPH-FE solver, Journal of Impact Engineering, Vol. 31, No. 8, pp , 2005 J. Campbell, R. Vignevic, L. Libersky; A Contact Algorithm for Smoothed Particle Hydrodynamics, Computer Methods in Applied Mechanics and Engineering, Vol. 184, No. 1, pp , 2000

31 Contact Initial Boundary Value Problem Traction on the contact surface Ω A Γ CA tn = nˆ σ Γ CB Γ A Γ CB Γ B The Hertz Signorini Moreau (Kuhn-Tucker) Conditions for frictionless contact: g 0, t 0, g = 0 n t n ( ) = nˆ x x g Γ CB B A Ω B In the variational rate form the contact constraint imposed by: δg = δg n g & ( t ) = 0 For a body in contact g& = 0

32 Contact Initial Boundary Value Problem t σ + b = a on Ω = Ω A U Ω B σ n = t on Γ t, Γ t Ω A Γ A u t Γ t = u on = on = Γ Γ A t B C N g t N 0 0 g A = 0 B Γ u Γ to Γ C Γ C Γ B Γ u Ω B Γ t t Where Γ t Γ to Γ u = Γ, and Γ t Γ to =, Γ to Γ u =, Γ t Γ u =.

33 Discretised Contact Initial Boundary Value Problem t wσ n dγ w σ dv = w( d&& b) dv wtdγ Γ Ω Ω Γ A weak form of the initial boundary value problem with contact. This equation discretised in space: T T T T N N d B σ d N b N b c Ω Ω Ω Ω Γ Γ t u dv&& + dv dv dv ( T N σ) n d 0 Γ = Where: w = Nd and N is a shape functions matrix CBL c

34 Discretised Contact Initial Boundary Value Problem In the SPH method N has the following form: N i = m W ( x ) np = 1 W i ( x ) i Where: I - particle at which shape function is evaluated, J - particle at which the shape function is centered, Np - number of neighbors for the i particle, d - nodal displacement vector, W - SPH kernel function (B-spline)

35 Contact Force Due to the diffused nature of boundary in the conventional SPH the contact force is defined as: f = T c b c Ω CBL N dv Where specific contact force is defined as gradient of contact potential: ( x x ) ( ) W A B φ x = K dv φ( x A i ) W ( p ) Ω CBL avg n = NCONT m W K W ( ri ) ( p ) avg n. m ( ) n 1 i ( p ) W r NCONT bc ( xi ) = φ( xi ) = Kn n xw x i i W avg ( x )

36 Normal Contact - SPH Ω A Γ A Γ CA Γ CB Γ B Ω B Γ CA Γ CB 2h

37 Example, Plate Impact (2D) The symmetrical block impact Each block was discretised with 50 by 20 particles Steel blocs modelled with an elastic-plastic material model

38 Example, Plate Impact (2D)

39 Example, Hypervelocity Impact (3D) 7km/s impact of aluminium proectile on aluminium bumper plate

40 Contact Algorithm Summary Well suited for meshless methods Numerically effective, approximately additional 10% CPU time in 3D simulations Easy to add friction (ongoing work)

41 Alternative Form of the SPH Equations R. Vignevic, J. Campbell, J. Jaric, S. Powell; Derivation of SPH equations in a moving referential coordinate system, Comput. Methods Appl. Mech. Engrg. 198 (2009), pp , (Received 8 September 2008)

42 Moving Referential Configuration x 2 x 2 J Particle I Particle Support Moving Control of particle I Volume x x xr x xr x x R J Particle x 1 J Particle Fixed Frame frame fixed on particle centred on particle Ion I particle I Df % Df r = + vr { Dt Dt referential f r r r r& v v v R = J I h x 1

43 Moving Referential Configuration Conservation of Mass D% r r D% r r r + v + vr = + v + ( v vcv ) = Dt Dt D% + ( v) vcv = 0 Dt D% r r W IJ d Ω + ( v) W IJ d Ω + vcv W IJ = 0 Dt Ω Ω Ω Integrating by parts and neglecting the boundary terms & v W ( x ' x, h) d v W ( x ' x, h) d = Ω I Ω Ω Ω & = m ( v v ) J W ( ) J J I IJ

44 Moving Referential Configuration Conservation of Momentum r r ( v) r r D% v r r + ( v v R ) = + R = σ t v v Dt D % r v r r 1 W d W d W d IJ Ω + v vr IJ Ω = σ IJ Ω Dt Ω Ω Ω Using the divergence theorem, dropping the boundary terms and linearizing the velocity in the second term: 1 v r & + v r v r r r R WIJ dω v ( vr WIJ ) dω = σwij dω Ω Ω Ω

45 Moving Referential Configuration Conservation of Momentum 1 v r & + v r v r r r R WIJ dω v ( vr WIJ ) dω = σwij dω Ω Ω Ω When discretised, the above equation becomes: r m 1 m v& r r r v v v ( ) J ( ) J W = ( σ + σ ) I J I R IJ I J x IJ J J I J Ω J W

46 Comparison Conv. SPH [1, 2] Continuity equation v v mj & I = I ( I J ) xwij J Ω J r r & = mj vj vi WIJ Moving C. S. ( ) Conv. SPH J Momentum equation 1 mj v & = σ + σ W I J Ω J r mj 1 v& r r r v v v W [1, 2] ( ) Moving C. S. I I J x IJ ( ) ( ) ( σ σ ) = + J I J I R IJ I J x IJ J J I J Ω J 1.- Larry D. Libersky at. al. High Strain Lagrangian Hydrodynamics a 3-DSPH code for dynamic material response,. Journal of Computational Physics, Morris, J.P., An Overview of the Method of Smoothed Particle Hydrodynamics, November 1995, Universitat Kaiserlautern, Arbeitsgruppe Technomathematik. m W

47 Numerical Example Shock tube problem 0<x< <x<1 Initial conditions 0<x< <x<1 Left Right Density Pressure Velocity

48 Numerical Example Shock tube problem 0.12 Velocity at time Velocity Exact Basic SPH, h constant Moving SPH Position

49 Numerical Example Shock tube problem 1.05 Pressure at time Pressure Exact Basic SPH, h constant Moving SPH Position

50 Numerical Example Shock tube problem 1.1 Density at time Exact Conventional SPH Moving SPH Density Position

51 Alternative Form of the SPH Equations Summary Part of the ongoing work Framework for a more rigorous treatment of variable h interpolation In the shock tube problem performance similar to the conventional SPH with constant h

52 Damage Modelling The fracture model capable of simulating: Initiation and growth of damage Crack formation and propagation in an arbitrary direction Crack branching and crack oining, leading to fragmentation

53 How to represent a Crack Cracked/failed particle concept: A failed particle is split into two particles, this approach has been demonstrated by Rabczuk and Belytschko (2007)

54 Continuum Damage Mechanics Compatible with the concept of effective stress introduced by Kachanov in 1958 The effective stress tensor is defined as: ~ σ = 1 D S S D = S σ 0 0 The concept of the interaction area relatively simple to apply within SPH

55 How to represent a Crack (Damage) Inter-particle failure Damage is evaluated for pairs of neighbour particles. Following fracture the particles cease being neighbours. The concept of particle-particle interaction area (Swegle, 2000), - damage affects interaction effective area - at failure the effective area is at a critical value

56 The force acting on a surface due to a stress is given by The SPH momentum equation could be rewritten in term of an interaction area: Swegle Interaction Area i i i i i i W m dt dv a + = = 2 2 σ σ ( ) ( ) ( ) ( ) + = = + = i i i i i i i i i i i i i A F W vol vol m a σ σ σ σ = F σ A

57 Inter-particle failure i Damage is evolved as an inter-particle value, D i,which reduces the inter-particle interaction area: F i i = ( σ ) + ( σ ) A ( 1 D ) i i i i When damage reaches a critical value the material is assumed to have failed and the particles cease to be neighbours.

58 Spall demonstration problem Normal stress between pairs of particles is compared to a spall criterion. Damage initiated once this criterion is exceeded. V = 450 m/s PMMA Copper plate 5mm thick Copper flyer 2.5mm thick 1D strain state Spall plane opens in middle of target plate

59 Representing fracture within a meshless method

60 Numerical Results Stress-xx (Gpa) Copper Plate Impact Test (with contact / artificial viscosity Q=2.0, L=0.2) Information displayed for Particle 195 (in PMMA) No damage D1 D Time (us)

61 The Mock-Holt problem Mock and Holt (1983), experimental results from a number of tests on explosive driven Fragmentation of metallic cylinders.

62 The Mock-Holt problem

63 Damage Modelling Summary Well suited for meshless methods Numerically effective Can deal with crack closure Ongoing work

64 Overall Summary All developments discussed were implemented into our 3D SPH code The SPH code coupled with DYNA3D The codes are routinely used to analyse real world problems Further development of the SPH method is of high priority to us (stabilisation of the Eulerian SPH, development of SPH structural elements, modelling damage, )

65 The End