meshless discretization of generalized laplace operator...
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![Page 1: Meshless Discretization of Generalized Laplace Operator ...ta.twi.tudelft.nl/nw/users/vuik/workshop_meshfree/lukyanov.pdf · Generalized Laplace Operator For Anisotropic Media Meshless](https://reader030.vdocuments.site/reader030/viewer/2022040918/5e92eac244cc914ba12f9775/html5/thumbnails/1.jpg)
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
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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
Outline
Introduction
Generalized Laplace Operator For Anisotropic Media
Meshless Method For Anisotropic Case
Results
Monotonicity Issue
Take Away
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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
Introduction
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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
Introduction
Literature
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.
I OTHER:Belytschko, Li, Liu, Atluri, Liszka, Yagawa, Yamada,Nayroles, Johnson, Morris, Williams, Zhu, Fox, Chen,Chaniotis, Duarte, Oden, Brookshaw, Schwaiger
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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
Introduction
I Fundamental Relations r′, r ∈ R3:
A(r) = 〈A(r′), δ(r − r′)〉 =∫
Ω,r∈Ω
A(r′)δ(r − r′)dr′
〈1, δ(r − r′)〉 =∫
Ω,r∈Ω
δ(r − r′)dr′ = 1(1)
I Set of Kernel Functions W (r − r′, h) ∈ C 0(Ω):
limh→0
W W (r − r′, h) = weakly = δ(r − r′)∫Ω,r∈Ω
W (r − r′, h)dr′ = 1(2)
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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
Introduction
I Basic Equalities:
A(r) = limh→0
∫Ω,r∈Ω
A(r′)W (r − r′, h)dr′
A(r) =∫
Ω,r∈Ω
A(r′)W (r − r′, h)dr′ + O(h2) =
=∑
J∈Ωr,h
A(rJ)W (r − rJ , h)VJ + O(h2),∀h ∈ Ωh
(3)
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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
Introduction
Kernel Function
W (z , h) =Ξ
hD
1− 3
2z2 + 3
4z3, 0 ≤ z ≤ 1
14 (2− z)3, 1 ≤ z ≤ 20, z > 2
(4)
where: z = ‖r − r′‖2 /hΞ = 3
2 ,107π ,
1π in 1D, 2D and 3D respectively.
Figure: Neighboring particles of a Kernelsupport.
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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
Generalized Laplace Operator For
Anisotropic Heterogeneous Media
I Generalized Laplace Operator:
Lu = −∇ (M (r)∇u (r))− g (r) , r ∈ Ω ⊂ R3 (5)
I Anisotropic Heterogeneous Media: Mxx (r) Mxy (r) Mxz (r)Mxy (r) Myy (r) Myz (r)Mxz (r) Myz (r) Mzz (r)
(6)
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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 For Anisotropic
Heterogeneous Media
I Mobility Decomposition:
M (r) = MS (r) + MD (r) ,MS (r) = 1
3tr [M] · I, tr
[MD]
= 0(7)
I Velocity Decomposition:
V (r) = VS (r) + VD (r)VS (r) = −MS (r)∇p(r),VD (r) = −MD (r)∇p(r)
(8)
I Divergence Decomposition:
∇V (r) = ∇VS (r) +∇VD (r) (9)
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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 Finite Difference Method
Seibold (2006)
I Construct neighbors (choosemore neighbors thanconstraints)
I Select unique stencil satisfyingadditional requirements
I Solve optimization problem
I Compute monotone stencil
I Not a flexible way givenpre-existing geology
Special Case
Lp = −∇2p (r)− g (r)∑s
TMSI (rS − rI ) = 0
∑s
TMSI (rS − rI ) (rS − rI ) = I
General Case
2∇ (φ(r)∇p(r)) = ∇2 (φ(r)p(r))+φ(r)∇2p(r)−p(r)∇2φ(r)
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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
SPH Discretization
I Brookshaw (1985):
〈∇ (m (r)∇F (r))〉 =∑Ωr,h
VrJ[F (rJ)− F (rI )]
(rJ − rI ) · (mJ + mI )∇W (rJ − rI , h)
‖r′ − r‖2
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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
SPH DiscretizationI Schwaiger (2008):
(Γ−1
kk
n
)−1
〈∇ (m (rI )∇F (rI ))〉 =∑Ωr,h
VrJ[F (rJ)− F (rI )]
(rJ − rI ) · (mJ + mI )∇W (rJ − rI , h)
‖r′ − r‖2
−[〈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
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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
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.
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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
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.
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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
Results
I For Deviatoric Scheme:
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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
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 .
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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
Monotonicity Issue
Figure: Monotonicity Issue (H. Hajibeygi, 2014).
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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
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 |
)
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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
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.
I Monotonicity issue has to be investigated.