Verification Procedures for Simulation of Fully Coupled Behavior of Porous Media

Transcription

1 Verification Procedures for Simulation of Fully Coupled Behavior of Porous Media Boris, Panagiota Tasiopoulou, Mahdi Taiebat, Nima Tafazzoli, Mario Martinelli UCD, LBNL, NTUA, UBC, UCD, URoma USNCCM, Minneapolis, MN

2 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

3 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

4 Introduction Numerical analysts, designers need the best available tools for performance assessment (numerical predictions) Verification and validation process ensures accuracy of numerical predictions How much can (should) we trust model implementations (verification)? How much can (should) we trust numerical simulations (validation)? How good are our numerical predictions? The T experiments Focus on verification

5 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

6 Verification Verification, Validation, Prediction Verification: provides evidence that the model is solved correctly. Validation: provides evidence that the correct model is solved. Prediction: use of computational model to foretell the state of a physical system under consideration under conditions for which the computational model has not been validated

7 Verification Role of Verification and Validation Reality Analysis Model Discovery and Building Model Validation Computer Simulation Mathematical Model Continuum Mathematics Programming Computer Implementation Discrete Mathematics Code Verification Oberkampf et al. Oden et al.

8 Verification Verification Verification: the process of determining that a model implementation accurately represents the developer s conceptual description and specification. Mathematics issue. Verification provides evidence that the model is solved correctly. Real World Conceptual Model Highly accurate solution Analytical solution Benchmark ODE solution Benchmark PDE solution Computational Model Computational Solution Experimental Data Unit Problems Benchmark Cases Subsystem Cases Complete System Verification Validation

9 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

10 Fully Coupled Formulation Dynamic Equilibrium for Saturated, Coupled Systems Effective stress principle σ ij = σ ij + αδ ij p ; (p = /3σ kk ) Equilibrium of the mixture σ ij,j ρü i ρ f [ẅ i + ẇ j ẇ i,j ] + ρb i = ; (ρ = nρ f + ( n)ρ s ) Equilibrium of the fluid p,i R i ρ f ü i ρ f [ẅ i + ẇ j ẇ i,j ]/n + ρ f b i = ; (Darcy: nẇ j = Ki; i = h,j ; R i = k ij ẇ j ; k ij = K ij /ρ f g [m] 3 [s]/[kg]) Flow conservation ẇ i,i + α ε ii + ṗ/q + nρ f /ρ f + ṡ = ; /Q n/k f + ( n)/k s

11 Fully Coupled Formulation Fully Coupled u p U Formulation Formulation: fully coupled by Zienkiewicz and Shiomi 9), nonlinear dynamics by Argyris and Mlejnek (99) Physical, velocity proportional damping from solid fluid interaction (not using Rayleigh damping) Accelerations of pore fluid not neglected important for SFSI inertial forces of fluid allow liquefaction modeling Stable formulation for near incompressible pore fluid

12 Fully Coupled Formulation Finite Element Discretization + + (M s ) KijL (M f ) KijL (C ) KijL (C 2 ) KijL (C 2 ) LjiK (C 3 ) KijL ü Lj p N U Lj + u Lj ṗ N U Lj (K EP ) KijL (G ) KiM (G ) LjM P MN (G 2 ) LjM (G 2 ) KiL + u Lj p M U Lj = f solid Ki f fluid Ki

13 Fully Coupled Formulation Finite Element Discretization (M s ) KijL = NK u ( n)ρ sδ ij NL u dω ; (M f ) KijL = NK U nρ f δ ij NL U dω Ω Ω (C ) KijL = NK u n2 k ij NL u dω ; (C 2) KijL = NK u n2 k ij NL U dω Ω Ω (C 3 ) KijL = NK U n2 k ij NL U dω ; (K EP ) KijL = NK u,m D imjnnl,n u dω Ω Ω (G ) KiM = NK u,i (α n)np M dω ; (G 2) KiM = nnk U,i Np M dω Ω Ω P NM = N p N Q Np M dω Ω

14 Fully Coupled Formulation Finite Element Discretization f solid Ki = Γ t N u K n jσ ij dγ NK u (α n)n ipdγ Γ p + NK u ( n)ρ sb i dω f fluid Ki = nnk U n ipdγ Γ p + nnk U ρ f b i dω Ω Ω

15 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

16 Verification Suite Code Verification Memory Function call arguments Code coverage Argument bounds Compiler warnings Computational Solution Verification Drilling of a well [Coussy ] The Case of a Spherical Cavity [Coussy ] Consolidation of a Soil Layer [Coussy 9] Line Injection of a fluid in a Reservoir [Coussy 9] Wave propagation, step displacement [Gajo and Mongiovi 9] Wave propagation, step velocity loading [de Boer et al. 93] Wave propagation, step force loading [Hiremath et al. ]

17 Vertical Consolidation Stresses (kpa) depth (m) 2 Effective Stresses Hydrostatic Pressures 2 Total Stresses 3 γ =. kn/m m Normalized Excess Pore Pressures Quicker dissipation towards the surface z/h = z/h =. 2 time (sec) z/h =.2 Numerical Analysis Normalized depth Normalized Excess Pore Pressure.2... T v = T.2 v =. T v =.2 T v =.... Numerical Analysis m Settlement of soil skeleton (m) Numerical Analysis. z = 9 m. z = m.2 z = 2 m.3 z =. 2 2 time (sec)

18 Vertical Consolidation: Normalized Excess Pore Pressure Numerical Analysis Normalized Excess Pore Pressures z/h = z/h = Time time factor (sec) T v z/h =.2

19 Shock Wave Propagation, Step Displacement x = Boundary Condition: δ (, t) = x -3 cm u x = U x = x -3 cm x. cm Semi-finite Soil column cm u x, U x ( -3 cm) No lateral flow. No lateral displacements. t (sec) (a) (b) (c)

20 Introduction Verification and Validation Saturated Soils Verification Suite Summary Shock Wave Propagation: Step Displacement x.. FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.).2 FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.) FEM (k = cm3s/g) FEM (k = cm3s/g). FEM (k = cm3s/g) (k = cm3s/g).2 (k = cm3s/g) x...2. FEM (k = cm3s/g) FEM (k = cm3s/g). FEM (k = cm3s/g). 3 (k = cm s/g) (k = cm3s/g).2.. x 2.. FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.).2 2 t [ sec] t [ sec] x.. FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.) FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.).2. 3 k = cm s/g x.. FEM (k = cm3s/g) (k = cm3s/g).2 (k = cm3s/g) (k = cm3s/g). 2 t [ sec] 3 k = cm s/g x FEM (k = cm3s/g) FEM (k = cm3s/g). t [ sec] t [ sec] FEM Vs x... k = cm3s/g..2 FEM Vs.2. FEM (k = cm3s/g). FEM (k = cm3s/g). FEM (k = cm3s/g).2 (k = cm3s/g) (k = cm3s/g) (k = cm3s/g) k = cm3s/g x 2... FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2.. k = cm3s/g x 2... FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec] t [ sec] k = cm3s/g x (k = cm3s/g) t [ sec] k = cm3s/g x t [ sec] x FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.).. (k = cm3s/g). t [ sec]..... FEM Vs. k = cm3s/g t [ sec] t [ sec] FEM Vs..2 FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.7; beta=.) t [ sec] k = cm s/g.. t [ sec]..2.. k = cm3s/g x.2 x FEM (Newmark, gamma=.; beta=.2) FEM (Newmark, gamma=.; beta=.32) t [ sec] k = cm3s/g x. t [ sec] x.2. k = cm3s/g.. x.2... FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec] k = cm3s/g x. k = cm3s/g. FEM (Newmark, gamma=.; beta=.2) FEM (Newmark, gamma=.; beta=.32) FEM (Newmark, gamma=.; beta=.2) FEM (Newmark, gamma=.; beta=.32) x.2. k = cm3s/g... k = cm3s/g x FEM (Newmark, gamma=.; beta=.2) FEM (Newmark, gamma=.; beta=.32) x.2.2 k = cm3s/g. k = cm3s/g x. 2 t [ sec] k = cm3s/g x k = cm3s/g x. FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec]... FEM (HHT, alpha=.) FEM (HHT, alpha=.3) FEM (HHT, alpha=.) FEM (HHT, alpha=.3) FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec] t [ sec] k = cm3s/g x... FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec] k = cm3s/g x k = cm3s/g 2 x... FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec]... FEM (HHT, alpha=.) FEM (HHT, alpha=.3) 2.. FEM (HHT, alpha=.) FEM (HHT, alpha=.3). t [ sec] Jeremic t [ sec] t [ sec] FEM (HHT, alpha=.) FEM (HHT, alpha=.3).2. t [ sec] 2

21 Shock Wave Propagation: Porous Solid x FEM Vs.. FEM (k = cm 3 s/g) FEM (k = cm 3 s/g). FEM (k = cm 3 s/g) (k = cm 3 s/g).2 (k = cm 3 s/g) (k = cm 3 s/g) 2 t [ sec]

22 Shock Wave Propagation: Pore Fluid FEM Vs x FEM (k = cm 3 s/g) FEM (k = cm 3 s/g) FEM (k = cm 3 s/g) (k = cm 3 s/g).2 (k = cm 3 s/g) (k = cm 3 s/g) 2 t [ sec]

23 Outline Introduction Verification and Validation Verification Saturated Soils Fully Coupled Formulation Verification Suite Summary

24 Summary Importance of verification and validation for numerical predictions Numerical predictions under uncertainty Would you trust numerical simulations (for design/regulation/evaluation) if your program of choice (simulation tool) did not follow (extensive) verification and validation procedures?