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Amit Kumar Verma Department of civil Engineering University of California, Davis 993161315

Amit Kumar Verma Department of civil Engineering University of California, Davis 993161315. Aim of Term Project.

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Amit Kumar Verma Department of civil Engineering University of California, Davis 993161315

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  1. Amit Kumar VermaDepartment of civil EngineeringUniversity of California, Davis993161315

  2. Aim of Term Project Aim of this Term Project is to find out the types of tensor multiplication involved in nDarray and to calculate the time spent by different functions in OpenSees during executation of the Cam Clay Models. Since, nDarray is one of the major component of Open Source Finite Element Software “OpenSees”, if somehow, optimized it will save a lot of computational time. This part of Project is involved with the application on 3 material models for Cam Clay.

  3. Critical State • The critical state line (CSL) takes the form where ec is the critical void ratio at the critical mean effective stress pc, ec,r is the reference critical void ratio, λc is the normal consolidation slope. • The critical state soil mechanics assumes that the normal consolidation line (NRL) is parallel to the CSL, which is expressed by where eλ is the intercept on the NRL at p = 1. λ is the normal consolidation slope or the elasto-plastic slope of e - ln p relation, and λc = λ.

  4. Yield Function • The yield function of the Cam Clay model is defined by where Mc is the critical state stress ration in the q-p plane, and the p0 is the initial internal scalar variable, which is controlled by the change of the plasticvolumetric strain. • The gradient of the yield surface to the stress can be obtained as where are independent of yield function.

  5. Plastic Flow • The plastic flow of the Cam Clay model is associated with its yield function, in other words, the plastic flow is defined by the potential function, g, which is assumed the same as the yield function, f. • The stress gradient to the yield surface can be obtained as • It can define the plastic dilation angle β, which is related to the ratio of plastic volumetric and deviatoric strain, by

  6. OpenSees • OpenSees is an object-oriented framework for finite element analysis. A software framework for developing applications to simulate the performance of structural and geotechnical systems subjected to earthquakes. A key feature of OpenSees is the interchangeability of components and the ability to integrate existing libraries and new components into the framework (not just new element classes) without the need to change the existing code.

  7. nDarray • The so-called nDarray developed by Jeremic and Sture are proved very useful in the implementation, especially for tensor multiplication calculation. nDarray, which is designed using an Object Oriented Paradigm (OOP) and implemented in the C++ programming language is to facilitate algebraic manipulations with matrices,vectors and tensors. Finite element equations, represented in terms of multidimensional tensors are easily manipulated and programmed. • The shortcoming of nDarray is also evident. It needs more memories (e.g. a major-symmetric rank-4 tensor has 81components, while its matrix counterpart has only 36 components) and thus spends more computing time.

  8. Memory Check One specific single element model example has been used to profile the timing consumed by elemental level plastic computations. The material model used here is: • Elastic perfectly-plastic Cam Clay Normally Consolidated • Elastic perfectly-plastic Cam Clay Normally Consolidated • Elastic perfectly-plastic Cam Clay Drucker-Prager Table 1 shows the timing profile of a single run on a EightNodeBrick element with Template3Dep CC_NC material using OpenSees.Table 2 shows the timing profile of a single run on a EightNodeBrick element with Template3Dep CC_OC material while Table 3 shows the timing profile of a single run on a EightNodeBrick element with Template3Dep CC_DC material using OpenSees.

  9. Types of Tensors Multiplication • Cam Clay Normally Consolidated ij*kl • Cam Clay Overconsolidated ij*kl • Cam Clay Drucker Pager Model ij*kl ij*ij

  10. Table1: Timing Profile of OpenSees (Cam Clay Normally Consolidated Analysis)

  11. Table2: Timing Profile of OpenSees (Cam Clay Over Consolidated Analysis)

  12. Table3: Timing Profile of OpenSees (Cam Clay Ducker Pager Model)

  13. Conclusion Observing tables1,2 & 3, it can be concluded that major emphasis should be on operator* (tensor multiplication) and val function in nDarray.These two functions must be optimized in order to increase computational performance of the system. Since only two types of tensors are involved in multiplication, if these tensor multiplications are written explicitly (not using for loop), performance issue can be solved. One way to write tensor multiplication explicitly is to use Template Metaprogramming which is the aim of next term Project.

  14. THANK YOU

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