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Transactions, SMiRT-24BEXCO, Busan, Korea- August 20-25, 2017

Division V

Verification for the Real ESSI Simulator

Boris Jeremić 1,2, José Antonio Abell3, Yuan Feng4, Maxime Lacour4, Han Yang4,Fatemah Behbehani4, Sumeet Kumar Sinha4, Hexiang Wang4, David B McCallen2, Chao Luo5

1 Professor, Department of Civil and Environmental Engineering, UC Davis, CA, USA2 Faculty Scientist, Earth Science Devision, LBNL, Berkeley, CA, USA3 Professor, School of Engineering and Applied Sciences, Universidad de los Andes, Santiago, Chile4 Graduate Student, Department of Civil and Environmental Engineering, UC Davis, CA, USA5 Graduate Student, College of Civil Engineering, Tongji University, Shanghai, China

ABSTRACT

With the development of Finite Element Analysis (FEA), the correctness and efficiency become themain concern in both academia and industry. Verification is the process of comparison between theanalytic solutions and numerical results. In this paper, the differences between solution verificationand code verification in FEA are discussed. The verification procedures for the FEA systemsRealistic Earthquake Soil-Structure-Interaction (Real-ESSI) is presented.

Firstly, mesh dependency is explored with the mesh refinement techniques. Secondly, Newmarkand Hilber-Hughes-Taylor algorithms are verified with the analytic damping ratios and periodshift. Thirdly, the integration algorithms of the elastoplastic materials at the constitutive levelsare verified. Finally, other related numerical and programming issues are reviewed.

INTRODUCTION

Verification and validation are the primary means of assessing accuracy in modeling and computa-tional simulations in order to build confidence and credibility in numerical predictions Oberkampfand Roy (2010). Verification is a process of determining that a model implementation accuratelyrepresents the developer’s conceptual description and specification Roache (1998). It is essentiallya mathematics issue and it provides evidence that the model is solved correctly Salari and Knupp(2000). Validation is a process of determining the degree to which a model is the accurate repre-sentation of the real world from the perspective of the intended uses of the model Roache (1998).It is a physics issue, and it provides evidence that the correct model is addressed. Verification isthe principal concern in this paper and validation is not presented.

The Real ESSI (Realistic Earthquake-Soil-Structure Interaction) Simulator Jeremić et al. (2017)is a software, hardware and documentation system for high performance nonlinear finite elementmodeling and simulation of earthquake-soil/rock-structure interaction of various infrastructure.

Compared to the conventional finite element analysis, Real ESSI focuses on the following facts.Firstly, simulate the 3D and 6D earthquake motions. In the traditional structural dynamics

analysis, the PGA (peak ground acceleration) is extracted to simulate the static push over. Actually,earthquake motions have six components with 3 translations and 3 rotations Liu et al. (2009),Oliveira and Bolt (1989), Huang (2003), Takeo (1998). Cosserat materials are adopted to simulatethe 6D motions.

24th Conference on Structural Mechanics in Reactor TechnologyBEXCO, Busan, Korea - August 20-25, 2017

Division V; Document Number 317; Paper ID 05-13-36

Secondly, inelastic material behavior. Soil is highly elastoplastic material, even for a smallstrain. Furthermore, soil volume changes significantly affect behavior. Real ESSI has a hierarchicalorganization of various elastoplastic materials, from the simple von-Mises model to the complexbounding surface based model with zero elastic regions.

Thirdly, uncertain material and loads. Material parameters and loads are always uncertainKundu and Adhikari (2014), Riera (2010). The question is how are those uncertainties reflected inthe results of simulation and how are those results influencing design and decision making. RealESSI is capable to analyze the propagation of uncertainty in finite element analysis, which can beverified by Monte Carlo simulation.

The Real ESSI program Jeremić et al. (2017) is continuously being verified by the developerswhen new capabilities are added. The verification is conducted in accordance with the developmentof Real ESSI at University of California, Davis and Lawrence Berkeley National Laboratory. Thetest cases in this article represent a small subset of the verification test cases conducted for thesoftware. Some other simulation examples are also available for educational purposes.

SOLUTION VERIFICATION VERSUS CODE VERIFICATION

Verification and validation are extensively employed to build confidence in computational sim-ulations. Verification is originally the comparison between the continuum equation and discreteequations in mathematics Oberkampf and Trucano (2002). Validation is the traditional comparisonstrategy against the experimental results. With the fast development of computer technique, com-putational simulations become an indispensable part of the critical projects. The relation betweenverification and validation is described in Fig. 1.

Figure 1: Verification Versus Validation

Inside the verification range, solution verification and code verification are divided into solutionverification and code verification to represent the efforts in different directions Roy (2005).

Solution VerificationSolution verification is the comparison process between the continuum solutions in mathematics

and the discrete results in numerical models. The ultimate goal is to produce the consistent andconvergence matched results when the grids are refined further.



Code VerificationCode verification represents the process to assess the correctness of the code. Namely, code

verification checks whether the algorithm is correctly implemented as intended. Code verificationis a part of the software quality engineering, which aims to minimize the programming bugs andcoding errors.

THE ASYMPOTOTIC REGIME OF CONVERGENCE

The algorithm uncertainty originates from the truncation error and round-off error. First, trun-cation error represents the discretization error. When we use a finite number of steps to simulatethe infinite process, the continuum equations become incremental equations. The error in thisdiscretization process is the truncation error. Second, round-off error represents the accumulationof machine epsilon. This is the consequence of using fixed number of digits on the computer torepresent the decimal number. The digits after the fixed number are rounded in the computationprocess.

When the mesh size4x is too big, the error oscillates because the mesh is too coarse to representthe actual continuum functions. In the middle region, the error decreases asymptotically with themesh size. This region is called the asymptotic regime of convergence. Ideally, the error shouldcontinue decreasing arbitrarily close to zero. However, the numerical computation with floatingnumber is not as perfect as the theoretical computation in math equations. When the mesh sizeis too small, the round-off error in floating numbers increases. The relation between the algorithmerror and the mesh size is illustrated in Fig 2.

Figure 2: Asymptotic Convergence Regime on the Mesh Size Hemez and Kamm (2008).

MESH REFINEMENT TECHNIQUE

Mesh generation is the first stage of starting a finite element analysis. Through mesh generationprocess, the continuum mathematical problems are transformed to a discrete numerical model,which can be solved by the computer. The correctness of the numerical results depends on theproper mesh discretization. Real ESSI is able to refine the mesh and provide better results. Acantilever beam example is illustrated in Fig. 3 to demonstrate the influence of mesh dependency.



Figure 3: Problem Description For Cantilever Beams

In this section, the beam was cut into smaller elements with the number of divison 1, 2, and4 respectively. And the element side length of the original models is 1.0m. The numerical modelswere shown in Figure 4.

Figure 4: 8NodeBrick Element With Mesh Refinement

The verification is based on the vertical displacement at the tip of the cantilever. Note that thetheoretical displacement should contains both the bending and shear deformation Timoshenko andWoinowsky-Krieger (1959). The comparison between analytical solutions and numerical results areshown in Table. 1.

Table 1: Displacement Results For 8NodeBrick Beams With Mesh Refinement

Element TypeNumber of Division

1 2 48NodeBrick 1.10E-05 m 1.47E-05 m 1.64E-05 m

Error 33.33% 11.09% 0.73%



HHT ALGORITHMS

The accuracy of dynamic analysis plays an important role in the analysis of earthquake soil-structure-interation (ESSI). This section verifies the correctness of the integration algorithm inthe dynamic analysis.

Numerical time-stepping methods are the principal methods of dynamics calculation. New-mark’s method is the common algorithm for dynamic analysis. When the two Newmark parametersγ = 12 and β =

14 , this algorithm is unconditional stable for arbitrary time steps Newmark (1959).

Hilber-Hughes-Taylor (HHT) methods introduces a third parameter α, which allows for energydissipation and second order accuracy Hilber et al. (1977). When the HHT parameter α = 0, HHTalgorithms coincide with Newmark’s method. Analytical solution of the single degree-of-freedom(DOF) system is usually not possible when the excitation varies arbitrarily with time. To verifythe implementation of Newmark and HHT algorithm, the numerical damping ratios and periodshifts are calculated analytically.

Analytical solutionTo calculate the analytical damping ratio ξ and analytical period ω̄, we need to construct an

amplification matrix A Hilber et al. (1977) first.The explicit definition of amplification matrix A for the HHT family of algorithms defined is

A = 1D

Ω2·1 + αβ 1/Ω2 (12 − β)/Ω

2

−γ 1− (1 + α)(γ − β) 1− γ − (1 + α)(12γ − β)−1 −(1 + α) −(1 + α)(12 − β)

(1)

whereD =1 + (1 + α)βΩ2

Ω =ω4t

ω =(K/M)12

(2)

The eigenvalue of the amplification matrix A will be two complex conjugate roots λ1,2 and aso-called spurious root λ3 which satisfy |λ3| < |λ1,2| ≤ 1. The roots λ1,2 will be

λ1,2 = A±Bi (3)

Then, the analytical damping ratio ξ and analytical period ω̄ will be

ξ̄ =− ln(A2 +B2)

ω̄ =Ω̄/4t

Ω̄ =arctan(B/A)

(4)



Verification example descriptionA one degree of freedom (DOF) example was made to verify the Newmark and HHT algorithm

for Real ESSI simulator. The example was plotted in Fig. 5. The beam stiffness and the mass weredesigned to make the natural period to be 1 second. In the first loading stage, the beam was givena horizontal force to generate an initial displacement. At the topmost node, all DOFs were fixedexcept the DOF along initial displacement. Then, in the second loading stage, the beam starts freevibration.

Figure 5: Verification Example Description.

(a) Damping Ratio. (b) Period Shift Comparison.

Figure 6: Comparison For Newmark Algorithm With γ = 0.6, β = 0.3025.

The verification results for HHT algorithm were plotted in Fig. 7, 8. Both damping ratios andperiod shifts match between numerical results and analytical solutions.

VERIFICATION EXPERIMENTS OF ELASTOPLASTIC ALGORITHMS

In the beginning of the test, the stress state is put on the yield surface at the Lode angle −30◦.The radius of the yield surface is 1 unit. Then, different elastic predictors are added to the yieldsurface. The plastic correctors should correct the stress states back exactly on the yield surface.However, when the direction of the elastic predictors deviates far from the normal to the yieldsurface, or when the elastic predictors are too big, the plastic correctors cannot return back to the




Figure 7: Comparison For HHT Algorithm With α = −0.10.


Figure 8: Comparison For HHT Algorithm With α = −0.20.



yield surface, as shown in Fig.9. The right side is the normal to the yield surface, so the error iszero along the right boundary. On the left side, the error gets greater with the magnitude of theelastic predictors.

Substepping technique is able to alleviate the problem and return the stress states on theyield surface, as shown in Fig.10. However, the algorithm still has some errors when the elasticpredictors deviates far from the normal to the yield surface. The two algorithms above are bothexplicit algorithms.

The advantage of explicit algorithms is the high speed, which is important for large-scale model-ing. The disadvantage of explicit algorithms is the accuracy and the numerical stability. There areno tolerance check in the explicit algorithms such that the explicit algorithms cannot be targetedto a designated tolerance. Therefore, backward Euler is applied as the return mapping algorithmsto correct the stress states, as shown in Figure.11. As an implicit algorithm, backward Euler checksboth the yield surface value and residual stress. After several steps of iterations, backward Eulerguarantees the desired accuracy.

-30°-15°0°15°

30°

24

68

10Relative Stress Norm0.00×10

0

1.50×100

3.00×100

4.50×100

6.00×100

7.50×100

Figure 9: von-Mises Perfectly Plastic Materials with Forward Euler Algorithm. The radius of theyield surface is 1 unit. The stress starts on the yield surface at Lode angle −30◦. All possible stresspredictors are tested and compared to the exact solution on the yield surface with the correspondingLode angles. When the predictors are perpendicular to the yield surface, the relative stress errorsare zeroes. When the predictors deviates from the normal line, the relative stress errors increase.

-30°-15°0°15°

30°

24

68

10Relative Stress Norm0.00×10

0

4.00×10−2

8.00×10−2

1.20×10−1

1.60×10−1

2.00×10−1

Figure 10: von-Mises Perfectly Plastic Materials with Substepping Technique, which reduces therelative stress error.



-30°-15°0°15°

30°

24

68

10Relative Stress Norm

3.35×10−13

3.35×10−13

3.36×10−13

3.36×10−13

3.36×10−13

3.36×10−13

Figure 11: von-Mises Perfectly Plastic Materials with Backward Euler Algorithm, which achievesaccurate stress results.

CONCLUSION

The verification procedures for Real ESSI are discussed in this paper. The solution and codeverification build trust in the code development from the following perspective.

1. Mesh dependencies of finite element analysis are discussed. The higher order integrationrule is capable to provide accurate solutions for irregular meshes and high Poisson’s ratios.Refined mesh is able to approach the analytic results. Stress-strain output at Gauss pointsand the rotation angles are also verified.

2. Newmark and HHT integration rules are verified with the analytical solutions for the dampingratios and period shift. Newmark algorithm is a special case of HHT algorithm, which isverified with various parameters.

3. The elastoplastic computation at the constitutive level is verified with Mathematica. Abun-dant elastic predictors are projected to indicate the error distribution outside the yield surface.Backward Euler algorithm is able to provide accurate stress results from relatively great strainincrement. The conservation of energy is verified to prove the correctness of the elastoplasticalgorithms.

ACKNOWLEDGEMENTS

This work has been supported by the US-DOE.

References

F. M. Hemez and J. R. Kamm. A brief overview of the state-of-the-practice and current challenges ofsolution verification. In F. Graziani, editor, Computational Methods in Transport: Verificationand Validation, pages 229–250. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008. ISBN978-3-540-77362-7. doi: 10.1007/978-3-540-77362-7 10. URL http://dx.doi.org/10.1007/978-3-540-77362-7_10.

http://dx.doi.org/10.1007/978-3-540-77362-7_10http://dx.doi.org/10.1007/978-3-540-77362-7_10



H. M. Hilber, T. J. R. Hughes, and R. L. Taylor. Improved numerical dissipation for time integrationalgorithms in structural dynamics. Earthquake Engineering and Structure Dynamics, 5(3):283–292, 1977.

B.-S. Huang. Ground rotational motions of the 1999 Chi-Chi, Taiwan earthquake as inferred fromdense array observations. Geophysical Research Letters, 30(6), 2003.

B. Jeremić, G. Jie, Z. Cheng, N. Tafazzoli, P. Tasiopoulou, F. P. J. A. Abell, K. Watanabe, Y. Feng,S. K. Sinha, F. Behbehani, H. Yang, and H. Wang. The Real ESSI Simulator System. University ofCalifornia, Davis and Lawrence Berkeley National Laboratory, 2017. http://real-essi.info/.

A. Kundu and S. Adhikari. Transient response of structural dynamic systems with parametricuncertainty. ASCE Journal of Engineering Mechanics, 140(2):315–331, February 2014.

C.-S. Liu, B.-S. Huang, W. H. K. Lee, and C.-J. Lin. Observing rotational and translational groundmotions at the hgsd station in taiwan from 2007 to 2008. Bulletin of the Seismological of America,99(2B):1228–1236, May 2009.

N. M. Newmark. A method of computation for structural dynamics. ASCE Journal of the Engi-neering Mechanics Division, 85:67–94, July 1959.

W. L. Oberkampf and C. J. Roy. Verification and Validation in Scientific Computing. CambridgeUniversity Press, 2010. ISBN 978-0-521-11360-1.

W. L. Oberkampf and T. G. Trucano. Verification and validation in computational fluid dynamics.Progress in Aerospace Sciences, 38(3):209–272, 2002. URL http://www.sciencedirect.com/science/article/pii/S0376042102000052.

C. S. Oliveira and B. A. Bolt. Rotational components of strong surface ground motions. EarthquakeEngineering and Structural Dynamics, 18:517–526, 1989.

J. D. Riera. Considerations on model uncertainty and analyst qualifications in soil-structure inter-action studies. In Proceedings of the OECD – NEA – IAGE – ISSC Workshop on Soil StructureInteraction Knowledge and Effect on the Seismic Assessment of NPPs Structures and Compo-nents, 2010.

P. J. Roache. Verification of codes and calculations. AIAA journal, 36(5):696–702, 1998. URLhttp://arc.aiaa.org/doi/pdf/10.2514/2.457.

C. J. Roy. Review of code and solution verification procedures for computational simulation.Journal of Computational Physics, 205(1):131 – 156, 2005. ISSN 0021-9991. doi: http://dx.doi.org/10.1016/j.jcp.2004.10.036. URL http://www.sciencedirect.com/science/article/pii/S0021999104004619.

K. Salari and P. Knupp. Code verification by the method of manufactured solutions. Technicalreport, Sandia National Labs., Albuquerque, NM (US); Sandia National Labs., Livermore, CA(US), 2000. URL http://www.osti.gov/scitech/servlets/purl/759450/.

M. Takeo. Ground rotational motions recorded in near-source region of earthquakes. Geophysicalresearch letters, 25(6):789–792, 1998.

S. P. Timoshenko and S. Woinowsky-Krieger. Theory of plates and shells. McGraw-hill, 1959.

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