Meshless Modelling in Geoscience Introduction Generalized Laplace Operator For Anisotropic Media Meshless Method For Anisotropic Case Meshless Finite Difference Method Smoothed Particle Hydrodynamics (SPH) Method Smoothed Particle Hydrodynamics (SPH) Method Modified SPH Discretization Modified SPH Discretization Results Monotonicity Issue Take Away Meshless Discretization of Generalized Laplace Operator For Anisotropic Heterogeneous Media Alex Lukyanov ] and Kees Vuik ¶ ] Schlumberger-Doll Research, Cambridge, MA 02139, USA ([email protected]). ¶ Delft University of Technology, Delft Institute of Applied Mathematics, 2628CN Delft, the Netherlands ([email protected]). July 1, 2015
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MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Meshless Discretization of Generalized
Laplace Operator For Anisotropic
Heterogeneous Media
Alex Lukyanov] and Kees Vuik¶
]Schlumberger-Doll Research, Cambridge, MA 02139, USA([email protected]).
¶Delft University of Technology, Delft Institute of Applied Mathematics,2628CN Delft, the Netherlands ([email protected]).
July 1, 2015
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Outline
Introduction
Generalized Laplace Operator For Anisotropic Media
I Lucy, L. [1977] A numerical approach to testing thefission hypothesis. The Astronomical Journal, 82, 1013– 1024.
I Gingold, R.A., Monaghan, J.J. [1977]Smoothed particlehydrodynamics: theory and application to non-sphericalstars, Monthly Notices Royal Astronom. Soc., 181,135.
I Liu, M. B., Liu, G. R. [2003] Particle Hydrodynamics:A Meshfree Particle Method. World ScientificPublishing Co. Pte. Ltd. 449 p.
−[〈m (rI ) F (rI )〉α − F (rI ) 〈m (rI )〉α + m (rI ) 〈F (rI )〉α] Nα
〈F (rI )〉α =∑Ωr,h
VrJ[F (rJ)− F (rI )]∇αW (rI − rJ)
A =
∑Ωr,h
VrJ[rJ − rI ]∇αW (rI − rJ)
−1
, ∇∗αW = Aαβ∇βW
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Modified SPH Discretization∗
I Spherical Part:
−(
Γ−1kk
n
)−1
〈∇vS (r)〉 =∑Ωr,h
VrJ·Meff · [F (rJ)− F (rI )]
(rαJ − rαI ) · ∇αW (rJ − rI , h)
‖r′ − r‖2 −
−
∑Ωr,h
VrJ·MS
eff · [F (rJ)− F (rI )]∇αW (rJ − rI , h)
Nα
MSeff =
(MS (rJ) ·MS (rI )
MS (rJ) + MS (rI )
)
∗Lukyanov, Vuik, JCP, To be submitted.
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Modified SPH Discretization†
I Deviatoric Part:
Lukyanov(2010), Lukyanov (2012)
〈vDγ (rI )〉 = −MD
γα〈p (rI )〉,α
〈p (rI )〉 =∑Ωr,h
VrJ[〈p (rJ)〉 − 〈p (rI )〉]∇W (rJ − rI , h)
〈∇vDγ (r)〉,γ =
∑Ωr,h
VrJ
[〈vDγ (rJ)〉 − 〈vD
γ (rI )〉]∇γW (rJ − rI , h)
†Lukyanov, Vuik, JCP, To be submitted.
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Results
I For Deviatoric Scheme:
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Results
I For Full Scheme:
Tensor Particle Distribution Cp αp Cu αu
D Uniform 0.348 1.991 0.325 1.891
D Weakly Distorted 0.231 1.923 0.247 1.873
D Highly Distorted 0.257 1.732 0.257 1.638
N Uniform 0.391 1.990 0.301 1.872
N Weakly Distorted 0.272 1.919 0.216 1.803
N Highly Distorted 0.293 1.727 0.225 1.612
Table: Convergence rates for the relatively simpleDirichlet problem ‖p − ph‖ ≤ Cph
αp and‖u − uh‖ ≤ Cuh
αu .
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Monotonicity Issue
Figure: Monotonicity Issue (H. Hajibeygi, 2014).
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Monotonicity Issue: Bouchon (2006)TheoremLet A = [aij ] and A = [aij ] be two real square matricesof dimension n, with the following properties:
I A = [aij ] is an irreducibly diagonal dominantM-matrix
I A · IIf∥∥∥A∥∥∥∞< Cm (A) with C =
1
(η)M ·M · ethen the
matrix A + A is monotone. Moreover(
A + A)−1
>> 0
Where‖A‖∞ = sup
x 6=0
‖Ax‖‖x‖ = max
i=1,...,n
(∑j |aij |
)m (A) = min
i=1,...,n(|aii |) , η (A) = max
i ,j=1,...,n
(|aii ||aij |
)
MeshlessModelling inGeoscience
Introduction
GeneralizedLaplace OperatorFor AnisotropicMedia
Meshless MethodFor AnisotropicCase
Meshless FiniteDifference Method
Smoothed ParticleHydrodynamics(SPH) Method
Smoothed ParticleHydrodynamics(SPH) Method
Modified SPHDiscretization
Modified SPHDiscretization
Results
MonotonicityIssue
Take Away
Take Away
I It is required to have further numerical analysis ofthe method.
I Due to the meshfree particle nature of the method,it is not always straightforward to directly apply thetechniques that were developed for mesh-basedEulerian or Lagrangian methods.
I The issues related to the stability, accuracy andconvergence are understood for uniformlydistributed particles and some times for onlyone-dimensional cases.
I It is not yet very well clear how the particleirregularity affects the accuracy of the solution.