Speeding up SPH models: local adaptivity and parallelization for soil simulations. Yaidel Reyes L´ opez Dirk Roose Carlos Recarey Morfa KU Leuven, Belgium UCLV, Santa Clara, Cuba June 30, 2014
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Simulation of real life problems Large deformations and post-failure flow: SPH
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Simulation of real life problems � Computationally demanding simulations! Large deformations and post-failure flow: SPH
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Simulation of real life problems � Computationally demanding simulations! � Adaptivity � Parallelization Large deformations and post-failure flow: SPH
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Outline Elastic-Plastic soil model in SPH Dynamic refinement General aspects Error due to refinement Numerical instabilities Refinement parameters Results Cohesive soil: tensile instability Artificial stress method 3D results Parallelization Shared Memory parallelization Domain decomposition Scalability Application Conclusions
Section Outline Elastic-Plastic soil model in SPH Dynamic refinement General aspects Error due to refinement Numerical instabilities Refinement parameters Results Cohesive soil: tensile instability Artificial stress method 3D results Parallelization Shared Memory parallelization Domain decomposition Scalability Application Conclusions
� Continuity equation (soil density) Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Elastic-Plastic soil model in SPH [Bui et al. 2008] � Momentum equation N Dρ i j ) ∂W ij � m j ( v α i − v α Dt = , ∂x α i j =1 ( α : Einstein notation) W : cubic spline kernel function σ αβ N � � σ αβ Dv α ∂W ij j � i i − Π ij δ αβ + F α Dt = m j + i ρ 2 ρ 2 ∂x β i j j =1 i σ : total stress tensor; F : gravity force Π ij : Artificial viscosity → reduces instabilities
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Elastic-plastic model Total deformation: strain rate tensor ε αβ = ˙ ε αβ ε αβ ˙ + ˙ e p ε e → elastic deformation (Hooke’s law) ˙ ε p → plastic deformation (plastic flow rule) ˙ Drucker-Prager yield condition. Stress-strain relationship: σ αβ = σ αγ ˙ ω βγ + σ γβ ˙ ω αγ + 2 G ˙ e αβ + K ˙ ε γγ δ αβ + P αβ ˙ G, K : elastic moduli e : deviatoric strain rate ˙ ω : spin rate ˙ P → associated or non-associated plastic flow rule
Section Outline Elastic-Plastic soil model in SPH Dynamic refinement General aspects Error due to refinement Numerical instabilities Refinement parameters Results Cohesive soil: tensile instability Artificial stress method 3D results Parallelization Shared Memory parallelization Domain decomposition Scalability Application Conclusions
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Dynamic refinement � coarse discretization: accurate enough for undeformed regions � fine discretization: required for deformed regions Simulation of the collapse of a block of non-cohesive soil coarse discretization fine discretization
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Dynamic refinement � coarse discretization: accurate enough for undeformed regions � fine discretization: required for deformed regions Simulation of the collapse of a block of non-cohesive soil coarse discretization fine discretization ⇒ dynamic refinement of initially “coarse”discretization reduces computational cost and memory requirements
� nearly as accurate as simulation using fine resolution � nearly as fast as simulation using the coarse resolution Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Dynamic refinement � local properties are only slightly changed and global properties Refinement criterion decides when particle is refined must ensure that dynamically refined simulation is Refinement procedure (how to refine particle) must ensure that are conserved Related work: astrophysics Bate et al., Kitsionas & Whitworth, Meglicki et al., Monaghan & Varnas fluid flows Feldman & Bonet, Lastiwka et al., Vacondio et al. solids Spreng et al.
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Refinement criterion Refinement criterion can be problem dependent. We use a criterion based on deformation: particle p i is refined if the trace of the strain tensor exceeds threshold i ) 2 + ( ε yy i ) 2 + ( ε zz i ) 2 > ε max ( ε xx
� h d = αh p � m d = λ p m p , λ = 1 Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions � v d = v p Refinement procedure particle p selected for refinement : replaced by daughter particles d = 1 , ..., M . daughter particles’ position daughter particles’ properties 4 Other properties are interpolated: ρ d , σ d , ε d Corrected SPM [Chen at al.] N m j � f ( x j ) W ( x − x j , h j ) ρ j j =1 ǫ separation parameter f ( x ) = , N m j α smoothing ratio � W ( x − x j , h j ) ρ j j =1
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Error due to refinement: density & kernel gradient N Dρ i j ) ∂W ij � m j ( v α i − v α Dt = , ∂x α i j =1 Local error: M � � p ( x ) = � Dρ ( x ) � − � Dρ ( x ) � ∗ = m p ( v ( x ) − v p ) · e ρ � ∇ W p ( x ) − λ d ∇ W d ( x ) Dt Dt d =1 Global error: �� 2 � � M � � E ρ p = m 2 ( v ( x ) − v p ) · ∇ W p ( x ) − λ d ∇ W d ( x ) d x . p Ω d =1 v ( x ) varies in space and time → difficult to analyze ⇒ we consider � 2 � M � ∂W p ( x ) ∂W d ( x ) E ∇ W � = − ← Kernel gradient error λ d d x , p ∂x α ∂x α Ω d =1
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Kernel gradient error due to refinement 0.8 � 2 0.75 0 0 M � ∂W p ( x ) ∂W d ( x ) � 0.7 E ∇ W � = − λ d d x , depends on ref. params ǫ & α p ∂x α ∂x α 0.65 -1 -1 Ω d =1 0.6 0.55 -2 -2 0.5 h ( x ) = h p 0.45 -3 -3 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -4 -4 For h = h p and h = αh p : 0.8 compute E ∇ W numerically for refinement parameters α r -5 -5 p 0.75 0.7 in region in ( ǫ, α )-plane around (0.5, 0.5) -6 -6 0.65 0.6 -7 -7 0.55 ǫ r ( ǫ , α ) = (0.5, 0.5) : 0.5 -8 -8 h ( x ) = α r h p 0.45 ‘natural’ values (‘uniform’ grid refinement) 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 select ( ǫ, α ) so that E ∇ W is minimal (or small) α r p ǫ r
unstable in tension unstable in ompression unstable in tension Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Numerical instabilities � stable in compression � unstable in tension regime W ′′ > 0 W ′′ < 0 W ′′ > 0 Swegle’s instability condition: W ′′ T > 0 , T : total stress Cubic spline kernel: ’standard’ particle configurations: 3 h ⇒ W ′′ > 0 | x i − x j | > 2 W − 2 2 3 h 3 h ⇒ SPH is: W ′ regime ( T < 0 ) ( T > 0 ) We should avoid instabilities in compression regime due to the refinement
Elastic-Plastic soil model in SPH Dynamic refinement Cohesive soil: tensile instability Parallelization Application Conclusions Numerical instabilities h p = r a d ref h d = αh p i j d 1 d 2 = ǫd ref i 0 i 1 j 0 j 1 r a = 1.0 i j 0.8 d ref = 1 . 0 ij i 0 i 1 ij 0.7 d ref = 0 . 85 Conditions for W ′′ < 0 : ( ǫ, α )-plane → 0.6 α r 0.5 i 0 i 1 < 2 Cond . 1 : 3 h i 0 i 1 0.4 0.3 0 < 2 3 αr a − ǫ 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ǫ r i 1 j 0 < 2 r a = 1.2 Cond . 2 : 3 h i 1 j 0 0.8 ij d ref = 1 . 0 0.7 d ref = 0 . 85 ij ij < 2 3 αr a + ǫ 0.6 d ref α r 0.5 i 1 j < 2 Cond . 3 : 3 h i 1 j 0.4 0.3 ij < (1 + α ) r a + ǫ 0.2 d ref 3 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ǫ r
Recommend
More recommend
