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1. 3 2 CHANGING PARAMETERS 37 lt param name wall gt left lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 5 0e 08 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt It is straightforward to apply boundary conditions for other components of velocity on the same wall One can simply put something like lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 1 0e 08 lt param gt lt struct gt between lt list name variables gt and lt list gt However to work on other walls than left one needs to replicate a higher level structure The other way of applying boundary conditions is to put func in the type member and the function s name in the value member as follows lt struct gt lt param name name gt vx lt param gt lt param name type gt func lt param gt lt param name value gt variableVx lt param gt lt struct gt This kind of function is called a condition function and should be defined in one of the loaded plugins One can find an example in TestCondition c in the SnacSpherical plugin s source code directory Using this function makes it possible to assign values varying according to the spatial positions or the indexes of eac
2. LIST OF FIGURES Part I Preface Preface About This Document This document is organized into three parts Part I consists of traditional book front matter including this preface Part II begins with an introduction to Pyre and the Pyre compatible version of SNAC and their capabilities and proceeds to the details of implementation Part III provides appendices and references The style of this publication is based on the Apple Publications Style Guide developer apple com docunentation UserExperience Conceptual APStyleGuide AppleStyleGuide2008 pdf as recommended by Python org www python org The documentation was produced using LyX www lyx org to facilitate the transformation of files from one format to another LyX is a document processor that encourages an approach to writing based on the structure of your documents not their appearance It is released under a Free Software Open Source license Errors and bug fixes in this manual should be directed to the CIG Long Term Tectonics Mailing List cig long geodynamics org Who Will Use This Document This documentation is aimed at scientists who prefer to use prepackaged and specialized analysis tools Users are likely to be experienced computational earth scientists and have familiarity with basic scripting software installation and programming but are not likely to be professional programmers Of those there are likely to be two classes of users those who just ru
3. 3 3 2 Using a newly defined condition function Continuing from the above example we have only one place to use an element condition function which is elementICs struct name elementICs gt lt list name vcList gt lt struct gt lt param name type gt AllElementsVC lt param gt lt list name variables gt lt struct gt lt param name name gt elementMaterial lt param gt lt param name type gt int lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt In the above code the type parameter for the variables list should be changed from int to func and the value parameter should be the string used when the condition function was registered It is SnacCF_MyTest in the current example The modified code would look like lt struct name elementICs gt lt list name vcList gt lt struct gt lt param name type gt AllElementsVC lt param gt lt list name variables gt lt struct gt lt param name name gt elementMaterial lt param gt lt param name type gt func lt param gt lt param name value gt SnacCF_MyTest lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt Finally load the SnacCondFunc plugin as follows lt list name extensions gt lt param gt SnacCondFunc lt param gt lt list gt Run SNAC and check if the condition function works as
4. master object in a SNAC simulation Its main purpose is twofold 1 Providing an interface to control and run a SNAC simulation This is done mainly through the Entry Points described in the next section 2 Packaging together all the high level data structures that a SNAC run requires such as the Mesh boundary conditions simulation parameters etc Most of these objects are extensible and or config urable using the XML input file The SNAC Context inherits from the StGermain Mesh and Abstract Context which provide much of the default functionality 1 2 3 Call Tree of Entry Points An entry point EP is essentially a list of function pointers which can be dynamically modified When an EP is called it runs each of the function pointers in its list sequentially with the same arguments Since the SNAC Context calls the EPs in a specific hierarchy an application s set of EPs defines the call tree for an application The reason entry points are dynamic lists of function pointers or hooks is that a user e g a research scientist often wants to extend the default functionality of an application at only a few specific points in the code In SNAC this is simply a matter of adding pointers to their custom functions to the relevant entry points For instance to add a contribution from the Winkler foundation to the residual force at each node on the bottom surface one can simply include a plugin called SnacWinklerForce What this
5. CF 128x128x128 64 x 64 x 64 O 32 x 32x 32 1 256 512 1024 2048 4096 Number of cores 64 128 512 1024 Number of cores 1 128 256 2048 4096 64 13 256 512 1024 2048 4096 Number of cores 128 Figure 6 1 left Wall clock time as a function of the number of cores Speedup middle and efficiency right up to 4096 cores 55 56 CHAPTER 6 PARALLEL PERFORMANCE Chapter 7 Benchmark Problems 7 1 Odometer Test This problem concerns the determination of stresses in a Mohr Coulomb material subjected to an odometer test In this experiment two of the principal stress components are equal and during plastic flow the stress point evolves along a vertex of the Mohr Coulomb criterion representation in the principal stress space 7 1 1 Model Setup Bulk modulus 200 MPa Shear modulus 200 MPa Cohesion 1 MPa Friction angle 10 Dilation angle 10 Tension cut off 5 67 MPa Boundary Conditions V 10 m sec on the top surface All the other surfaces are confined to their normal directions dt 1 sec Mesh size 1m x 1m x 1 m and 5 x 5 x 5 nodes component of stress and ey component of strain are monitored These values were averaged over the top elements layer 7 1 2 Results Stress strain diagram is made using c and yy from both numerical and analytic solutions Relative error is about 0 8 57 58 CHAPTER
6. The usage is snac2vtk path to outputs timel time2 The only required argument is a path to output files including sim 0 which contains critical information to process data The last two optional arguments are used to set the range of time steps for data conversion snac2vtk automatically generates Parallel VTK Structured Grid files pvts for each time step so that there is no need for an extra step to combine data even for the parallel cases It is the group of pvts files that needs to be loaded in ParaView or another visualization tool for VTK However vts files should not be removed because pvts files only refer to vts files rather than contain actual data gt snac2vtk the last time step only or gt snac2vtk 10 from 10th to the last time step or gt snac2vtk 1 1001 from 1 to 1001th time step 41 42 CHAPTER 4 POSTPROCESSING AND GRAPHICS Example list of SNAC output files in the VTK format 8 processors and 51 outputs per processor snac 0 000001 vts snac 0 000003 vts snac 0 000005 vts snac 0 000101 vts snac 1 000001 vts snac 1 000003 vts snac 1 000005 vts snac 1 000101 vts snac 2 000001 vts snac 2 000003 vts snac 2 000005 vts snac 2 000101 vts snac 7 000001 vts snac 7 000003 vts snac 7 000005 vts snac 7 000101 vts snac 000001 pvts snac 000003 pvts snac 000005 pvts snac 000101 pvts 4 3 Converting to OpenDX files We recommend that you use VTK files for visualizing model result
7. lt struct gt lt param name type gt AllNodesVC lt param gt lt list name variables gt lt struct gt lt param name name gt temperature lt param gt lt param name type gt double lt param gt lt param name value gt 500 0 lt param gt lt in degrees Celsius gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt 36 CHAPTER 3 RUNNING SNAC lt element ICs initial conditions for elemental variables Currently material type is not used The block for hydrostatic pressure should be accompanied by SnacHydroStaticIC module in the above module list gt lt struct name elementICs gt lt list name vcList gt lt struct gt lt param name type gt AllElementsVC lt param gt lt list name variables gt lt struct gt lt param name name gt elementMaterial lt param gt lt param name type gt int lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt Note that the type of the data structure used for initial conditions in StGermain are AllNodesVC and AllElementsVC These types are used to assign values to all the nodes or elements see Section for other types of structure One way of assigning a value to all nodes or elements is to put a single value for the value member in the lowest level structure as in the next example lt struct gt lt param name name
8. An explicit inertial method for the simulation of viscoelastic flow An evaluation of elastic effects on diapiric flow in two and three layers models Flow and Creep in the Solar Systems Observations Modeling and Theory Kluwer Academic Publishers 175 195 Simo J and T Hughes 2004 Computational Inelasticity New York Springer Rudnicki J and J Rice 1975 Conditions for the localization of deformation in pressure sensitive dilatant materials J Mech Phys Solids 23 371 394 Cundall P A 1987 Distinct Element Models of Rock and Soil Structure in Analytical and Compu tational Methods in Engineering Rock Mechanics Chapter 4 pp 129 163 E T Brown Ed London George Allen amp Unwin 71 72 BIBLIOGRAPHY 17 Zienkiewicz O C and Zhu J Z 1992 The superconvergent patch recovery and a posteriori error estimates Part 1 The recovery technique Int J Num Meth Engng 33 1331 1364
9. Conventions for the notation A and n denote the face and the unit normal vector respectively associated with a local node 1 1 3 Approximation of Partial Derivatives The approximation of partial derivatives with respect to spatial variables follows the integral definitions e g 3 fidV nid 1 2 a aa where Q represent a tetrahedron as an integration domain OQ is the boundary surfaces of the tetrahedron fis the partial derivative of a variable f with respect to th spatial coordinate n is the i th component of the unit normal vector of the surface If the partial derivative is constant within a tetrahedron it is evaluated as 1 where V is the volume of the tetrahedron By further substituting an algebraic expression for the surface integral reordering terms and using Jao 0 when f 1 in Eq 1 3 se ae LOE pata V V 3 l l 1 m 1 l 1 4 4 A jm 5 nl A m 1 l 1 4m 1 4 oV imd where I is the local node index varying from 1 to 4 A and are the area and the unit normal vector of the triangular surface not having the node l as one of its apexes Fig 1 1b Hereafter we call such a face a corresponding face to node I f is the averaged f on the surface I 1 1 SNAC IMPLEMENTATION 15 1 1 4 Nodal Assemblage We can convert the differential equation for momentum balance Eq 1 1 the following description is applied to the heat equation in the same fashion to a principl
10. The other two side walls have free slip conditions T he bottom surface is supported by the Winkler restoring forces An initial temperature field is generated by a condition function that makes a horizontally uniform tem perature field with a constant vertical gradient The boundary conditions fix the top and bottom temperature at their initial values 0 and 700 C respectively to keep the steady state The Mohr Coulomb model is used with a two step strain weakening Cohesion decreases from 40 MPa to 0 4 MPa as the second invariant of plastic strain grows up to 2 but remains constant at 0 4 MPa thereafter Note that extremely large values of vis_min max are chosen such that viscosity is uniformly high and thus no viscous flow is allowed A finite value of plastic strain 2 in this problem is assigned to a set of elements These element have a lower cohesion according to the given strain weakening rule and thus strain localization is initiated from them The selection of elements is randomly made within the region at a certain distance from the left and right boundaries Fig 5 1p 5 1 1 Results Continued extension builds up strain and the seed elements start to yield first Initiated at the seed elements localization of plastic strain propagates in the direction normal to the applied velocities Eventually a long basin bounded by conjugate normal faults is created Fig 5 1p 5 1 2 Complete Listing of the Input XML File The problem
11. e dt 1 sec and results after 5000 steps were used for comparison to the analytic solutions Mesh size 31 x 3 x 31 nodes e The second invariant of stress in xz plane are monitored 7 3 2 Results e Relative error is about 0 1 7 4 Parallel Plate Viscometer Problem A parallel plate viscometer problem is simulated in which viscoelastic material is squeezed between two parallel plates The plates are moving at a constant velocity vo Each plate has the length of 2L and is at a distance 2h from the other No slip is assumed between the material and the plates The approximate analytical solution is given by Jaeger 9 62 CHAPTER 7 BENCHMARK PROBLEMS Resolution Mohr Coulomb Elastic 31 x 3 x 31 0 14 0 31 25 x 3 x 25 0 74 0 39 21 x 3 x 21 1 0596 0 5296 17x 3x17 1 0896 0 5696 13 x 3 x 13 0 8896 0 5996 9x3x9 1 12 0 54 Table 7 1 Relative errors for Thick Cylinder Benchmark Problem II x gt E A A L1 o Figure 7 7 The initial mesh blue with the velocity boundary condition red arrows 7 4 1 Model Setup Model Setup eL 10m eh 5m Viscosity 10 Pa sec Bulk modulus 1 5 GPa Shear modulus 500 MPa VO 107 m sec e dt 1 sec results compared after 500 time steps e Mesh size 20 m x 10 m x m each element is a 1 m cube Due to the assumption of the original
12. list gt lt struct gt lt list gt lt struct gt lt element ICs gt lt struct name elementICs gt lt list name vcList gt lt struct gt lt param name type gt AllElementsVC lt param gt lt list name variables gt lt struct gt lt param name name gt elementMaterial lt param gt lt param name type gt func lt param gt lt param name value gt SnacCF_DeadSea lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt 52 CHAPTER 5 lt Velocity BCs gt lt struct name velocityBCs gt lt list name vcList gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt left lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 1 5e 10 lt param gt lt struct gt lt struct gt lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 1 5e 10 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt right lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 1 5e 10 lt param gt lt struct gt lt struct gt
13. lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 1 5e 10 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt bottom lt param gt lt list name variables gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt top lt param gt lt list name variables gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt COOKBOOK EXAMPLES 5 2 COOKBOOK 2 CONDITION FUNCTIONS TO ASSIGN MULTIPLE MATERIAL TYPES lt struct gt lt list gt lt struct gt lt list gt lt struct gt lt StGermainData gt 53 54 CHAPTER 5 COOKBOOK EXAMPLES Chapter 6 Parallel Performance 6 1 Method We solve the problem of a 3D elastic bar with a given initial geometry deforming by its own weight to acquire SNAC s hard scaling data i e the size of problem is fixed while the number of processors is varied All the boundaries except the top surface is fixed such that the top surface is warped downward due to gravity The file I O has been red
14. m Dat pgi ppi in 0 gi on Po 1 1 ti hi on Ty where Q the domain boundary is the union of disjoint subsets and gi and v are components of the Cauchy stress tensor the gravitational acceleration and the velocity components respectively 5 is the material or total derivative and equal to the partial derivative with respect to time in SNAC since the updated Lagrangian viewpoint is taken 1 1 2 Spatial Descritization A 3D domain is discretized into hexahedral elements each of which is filled with two sets of 5 tetrahedra Fig 11 In this mesh hierarchy called the mixed discretization I hexahedral elements are used mostly as an averaging unit for volumetric strain The averaging is enforced at all times although needed only for incompressible viscoelastic or plastic constitutive laws This is a conservative choice and does influence the overall performance of SNAC The use of two equivalent sets of tetrahedra is required to ensure a symmetric response For a given loading responses of one set of tetrahedra can be different from those of the other set because of the differently orientated faces of tetrahedra in each set e g 10 13 14 CHAPTER 1 INTRODUCTION Figure 1 1 Configurations of tetrahedra and conventions for the notation a Two configurations of five tetrahedra in a hexahedral element used in the mixed discretization Numbers next to apexes indicate the local node numbering b
15. WallVC lt param gt lt param name wall gt bottom lt param gt lt list name variables gt lt struct gt lt param name name gt temperature lt param gt lt param name type gt double lt param gt lt param name value gt 700 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt lt StGermainData gt 47 48 CHAPTER 5 COOKBOOK EXAMPLES WAL A Za FF Viele Y GH IIA FF N SS N N N EN EN N B EN EN ES i 5 N N M Yj Figure 5 2 Phase index map of Cookbook 2 generated by the condition function SnacCF_DeadSea Arrows indicate the velocity boundary conditions at the initial time step 5 2 Cookbook 2 Condition functions to assign multiple material types This example shows how to introduce multiple material types and assign them spatially in a non trivial fashion The domain is 3006x300 km and discretized into 50x1x50 elements Three types of materials are all elasto plastic and have the same material properties except cohesions Each type has the initial value of 10 100 and 1 MPa respectively We use a condition function called SnacCF_DeadSea to assign these types to three different subdomains The boundary conditions set up the plane strain condition in the y direction while an oblique relative to the z axis spreading is applied on the left and right sides The map of phase index is shown in Fig 5 2 5 2 1 Complete
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17. attached to the point mass Now reducing the infinite series of mass and springs in one dimension to a single mass spring system the stiffness of that single system becomes 4k leading to an expression for the mass scaling m gt k Aty 1 11 For a given size of At the nodal mass is adjusted according to Eq to automatically satisfy the stability criterion Eq 1 10 The value of k is computed by equating internal force contribution at a node with ku I 2u ei dt n A k vidt nosum 1 12 3 where only the volumetric contribution from internal forces is taken into account By substituting the approximation Eq for the partial derivative e into the above equation and dividing both sides by vidt we obtain 2 io ARA 1 13 where l is the local index for apexes of a tetrahedron and the surface related quantities are computed on the corresponding face of the tetrahedron Finally a tetrahedron s contribution to the scaled mass is given as As in the standard FEM appropriate mappings between local and global indices are required max n A 1 3 1 14 1 1 7 Constitutive Update SNAC uses a general elasto visco plastic rheological model to update the Cauchy stress tensor e g 12 First the initial guess of stress is acquired by the Maxwell viscoelastic constitutive law 13 If this initial guess exceeds a given yield stress it is projected onto the yield surface
18. described above can be reproduced using the input file Snac examples Cookbook1 cookbook1 xml The full contents of this input file is listed below for reference lt xml version 1 0 gt lt DOCTYPE StGermainData SYSTEM stgermain dtd gt lt StGermain Snac input file gt lt StGermainData xmlns http www vpac org StGermain XML_I0_Handler Jun2003 gt lt StGermain simulation parameters gt lt param name start gt 0 lt param gt lt param name outputPath gt data lt param gt lt param name dumpEvery gt 1000 lt param gt lt param name maxTimeSteps gt 100000 lt param gt lt Snac variables gt lt param name density gt 2700 lt param gt 43 44 CHAPTER 5 COOKBOOK EXAMPLES Plastic Straln 2 50e 01 1 888 01 f 1 258 01 6 25e 02 0 008 00 0 008 00 Figure 5 1 a The mesh for the example problem with the locations of the seed elements b Plastic strain mapped on the deformed mesh after 2 km extension 0 1 My elapsed lt param name gravity gt 9 8 lt param gt lt param name demf gt 0 8 lt param gt lt param name dtType gt constant lt param gt lt param name timeStep gt 6 3072e 07 lt param gt lt param name forceCalcType gt complete lt param gt lt param name decomposedAxis gt 0 lt param gt lt param name storeForces gt no lt param gt lt param name forceCheckSum gt no lt param gt lt param name topo_kappa gt 0 0
19. gt lt param name plstrain0 gt 0 0 lt param gt lt param name plstrain1 gt 0 01 lt param gt lt param name plstrain2 gt 1000 0 lt param gt lt param name frictionAngle0 gt 0 0 lt param gt lt param name frictionAngle1 gt 0 0 lt param gt CHAPTER 5 COOKBOOK EXAMPLES lt param name frictionAngle2 gt 0 0 lt param gt lt param name dilationAngle0 gt 0 0 lt param gt lt param name dilationAngle1 gt 0 0 lt param gt lt param name dilationAngle2 gt 0 0 lt param gt lt param name cohesion0 gt 2 0e 07 lt param gt lt param name cohesion1 gt 1 0e 07 lt param gt lt param name cohesion2 gt 1 0e 07 lt param gt lt param name ten_off gt 1 0e 12 lt param gt lt struct gt lt struct name mat_strong gt lt param name density gt 2800 lt param gt lt param name alpha gt 0 0 lt param gt lt param name beta gt 0 0 lt param gt lt Elastic material parameters gt lt param name lambda gt 3 0e 10 lt param gt lt param name mu gt 3 0e 10 lt param gt lt Plastic material parameters gt lt param name yieldcriterion gt mohrcoulomb lt param gt lt param name nsegments gt 2 lt param gt lt param name plstrain0 gt 0 0 lt param gt lt param name plstrain1 gt 0 01 lt param gt lt param name plstrain2 gt 1000 0 lt param gt lt param name frictionAngle0 gt 0 0 lt param gt lt param name frictionAngle1 gt
20. gt temperature lt param gt lt param name type gt double lt param gt lt param name value gt 500 0 lt param gt lt struct gt The other way is to define a condition function and assign func to type and the function s name to value More details are given in the section 3 3 3 2 7 Boundary Conditions Structure The syntax to define a structure for boundary conditions is almost the same with the initial conditions As in the case of initial conditions a user can reuse most parts of the listing given below For example the name of the highest level structure velocityBCs need not be changed for every different problem Thus only a few notable differences are addressed here The value for the type of the second level structure is wa11VC This structure type requires specifying which wall to apply boundary conditions to For SNAC one of left right top bottom back or front can be assigned to a parameter named wall Those six values represent surfaces of which nodes have minimum x index maximum x index minimum y index maximum y index minimum z index and maximum z index respectively By including the following listing in an input XML file the x component of velocity of all the nodes belonging to the model s left wall will have the value 5 0e 08 m s as a double precision number lt struct name velocityBCs gt lt list name vcList gt lt struct gt lt param name type gt WallVC lt param gt
21. intended by visualizing the Phase variable 3 4 Checkpointing and Restarting 3 4 1 Checkpointing To restart an already finished run from a certain point in time it is necessary to record all the relevant information Generating such data set is called checkpointing It is possible to checkpoint as a response to an emergency like receiving a certain type of error or disk quota and wall clock limit soon to be exceeded However SNAC currently only allows a user to specify the frequency of checkpointing by assigning a non zero value to checkpointEvery in an input file The meaning of the assigned value is the same with dumpEvery the number of time steps between events of checkpointing The default value of this parameter 40 CHAPTER 3 RUNNING SNAC is 0 and SNAC checkpoints only at the last time step Since the size of checkpoint data files is much larger than that of the regular outputs it is advised to set checkpointEvery to be a sufficiently large value Just as dumping is logged in timeStep 0 checkpointing is logged in checkpointTimeStep 0 This file is used when restart files are generated Checkpoint data files have CP attached to their names e g velocityCP 0 stressTensorCP 2 etc The major difference between the checkpointed tetrahedral outputs and the regular hexahedral outputs is the storing format of element associated variables such as stress and plastic strain To restart with exactly the same state as before t
22. listing of the input XML file lt xml version 1 0 gt lt DOCTYPE StGermainData SYSTEM stgermain dtd gt lt StGermain Snac input file gt lt StGermainData xmlns http www vpac org StGermain XML IO Handler Jun2003 lt StGermain simulation parameters gt lt param name start gt 0 lt param gt lt param name outputPath gt data lt param gt lt param name dumpEvery gt 10 lt param gt lt param name checkpointEvery gt 0 lt param gt lt param name maxTimeSteps gt 2 lt param gt lt param name restartTimestep gt 0 lt param gt lt Snac variables gt lt param name gravity gt 0 0 lt param gt lt param name demf gt 0 8 lt param gt 5 2 COOKBOOK 2 CONDITION FUNCTIONS TO ASSIGN MULTIPLE MATERIAL TYPES 49 lt param name dtType gt constant lt param gt lt param name timeStep gt 3 1536e 08 lt param gt lt param name forceCalcType gt complete lt param gt lt param name storeForces gt no lt param gt lt param name forceCheckSum gt no lt param gt lt param name topo_kappa gt 0 0 lt param gt lt Extension modules gt lt 1 lt param gt SnacWinklerG3Force lt param gt lt param gt SnacRemesher lt param gt lt param gt SnacHydroStaticIC lt param gt lt param gt SnacCustomCartesian lt param gt lt param gt SnacViscoPlastic lt param gt lt param gt SnacVPSeeds lt param gt lt param gt SnacPlSeeds lt par
23. plugin does is to add or hook up a pointer to its own force computing routine to the entry point that already has a pointer to the function computing internal residual forces While being a part of the loop over all the nodes local to a processor this plugin s function takes care of deciding if a node is on the bottom and adding the computed restoring force In this way the core structure of the code does not need to be modified every time new operations on nodes or elements are added StGermain provides a flexible way of managing hooked up function pointers such that one can create a new entry point set a default function for it insert one function pointer before or after another within an Entry Point etc Fig 1 5 outlines the entry points set up in SNAC It is represented as a tree like hierarchy if a hook calls other entry points either in its own function or other normal functions it calls the called entry points are offset to the right of that hook 1 3 StGermain StGermain is a software framework tuned for the development of codes common to the physical sciences such as geodynamics It permits developers to build applications that are extensible adaptable cleanly designed and interoperable with one other Moreover StGermain answers the question of Now that I have written my code how do I share it with my collaborators with the optional inclusion of their additions This capability is inherent to StGermain and p
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25. using a return mapping method 14 otherwise the viscoelastic stress update is retained This elasto visco plastic model can deal with various constitutive laws that are typically used for the Earth s crustal and mantle material as its limiting cases For example elastic viscoelastic and elastoplastic material are realized in the following cases 1 Elastic material corresponds to the limit of infinite viscosity and yield stress 2 Viscoelastic material corresponds to the limit of infinite yield strength 3 Elasoplastic material corresponds to the inifinte viscosity 1 1 SNAC IMPLEMENTATION 17 Using the viscoplastic rheology is physically more realistic than using one of the limiting cases listed above since all materials have dissipative mechanisms and hence viscosity This viscosity also provides a length scale for the problem of localization which in turn enables a physically meaningful mesh independent solution when the mesh size is smaller than this length scale Since the nodal variables are velocities and whose spatial gradients are deformation rates we formulate the constitutive update in terms of strain rate The objective stress rate of choice is the Jaumann or the corotational stress rate Ac4 I5 Act SEN Ao As WT 1 15 where W 1 2 0v 0x 0v 0x are the components of spin tensor and Ac is the increment of stress tensor Correction to the stresses due to rotation can be given as g t gt Ag A
26. 0 0 lt param gt lt param name frictionAngle2 gt 0 0 lt param gt lt param name dilationAngle0 gt 0 0 lt param gt lt param name dilationAngle1 gt 0 0 lt param gt lt param name dilationAngle2 gt 0 0 lt param gt lt param name cohesion0 gt 2 0e 08 lt param gt lt param name cohesion1 gt 1 0e 08 lt param gt lt param name cohesion2 gt 1 0e 08 lt param gt lt param name ten_off gt 1 0e 12 lt param gt lt struct gt lt struct name mat_weak gt lt param name density gt 2800 lt param gt lt param name alpha gt 0 0 lt param gt lt param name beta gt 0 0 lt param gt lt Elastic material parameters gt lt param name lambda gt 3 0e 10 lt param gt lt param name mu gt 3 0e 10 lt param gt lt Plastic material parameters gt lt param name yieldcriterion gt mohrcoulomb lt param gt lt param name nsegments gt 2 lt param gt lt param name plstrain0 gt 0 0 lt param gt lt param name plstrain1 gt 0 01 lt param gt lt param name plstrain2 gt 1000 0 lt param gt lt param name frictionAngle0 gt 0 0 lt param gt lt param name frictionAngle1 gt 0 0 lt param gt lt param name frictionAngle2 gt 0 0 lt param gt lt param name dilationAngle0 gt 0 0 lt param gt lt param name dilationAnglel gt 0 0 lt param gt lt param name dilationAngle2 gt 0 0 lt param gt lt param name cohesion0 gt 2 0e 06 lt p
27. 1 19 in which 7 represents shear stress and ap represents normal stress gy is a function of the friction angle 9 and C is cohesive strength of the material In the case of Mohr Coulomb material gy is given as tan 1 1 1 T z 03 91 cos and 0 5 03 01 3 where al lt 02 lt are the principal stresses of stress tensor Figure 1 2 The actual form of the yield function for shear failure used in SNAC is 01 sind 1 20 fs 01 03 01 Ngo3 2CA 1 21 where Ng 182 and N 25 Figure 1 3 The tensile yield function is defined as 1 sin 1 sin 18 CHAPTER 1 INTRODUCTION Figure 1 2 A diagram showing a Mohr Coulomb yield envelope with tan 0 6 and a non zero cohesion as well as a Mohr circle corresponding to the principal stresses al and 01 1 22 where o is the tension cut off If the tension cut off is given as a parameter a smaller value between the theoretical limit 2C tan and the given value is assigned to To make sure a unique decision on the mode of yielding shear vs tensile we define additional function h o1 03 which bisects the obtuse angle made by two yield functions Figure 1 3 as fn o1 03 03 Ot YNg 1 No o1 Ngo 2c No 1 23 Once yielding is declared i e fs lt 0 or f gt 0 then shear or tensile failure is finally decided based on the value of h shear if h lt 0 and tensile ot
28. 3 Yield functions that are used to declare yielding in shear and tensile mode 20 CHAPTER 1 INTRODUCTION e In case of shear failure E a Ngaz E 2c Ns 8 5 5 1 28 52s e In case of tensile failure as Ot Ba 1 29 Hardening softening If strain hardening including softening as a negative hardening is considered a more general return mapping is required For completeness we review the cutting plane algorithm and then discuss the simplified version for piecewise linear hardening implemented in SNAC For more detailed discussion readers are referred to 14 Here are some preliminary definitions From 1 27 the plastic strain rate is given as 0G zy pem 8 We define an internal variable such that plastic parameters are defined as a function of that variable Our particular choice is 1 30 e y l eye ay a 1 31 where e amp The time rate of change of the internal variable is then gcc OE e dg Oc OG o eP AAA A 1 32 For the simpler notation a new function r o has been defined above We require that the updated stress stay on the yield surface discrete consistency condition i e F AB F o AP e A8B 0 1 33 where AG is the increment of the consistency parameter during a time interval between t and t 41 For later uses we also compute the derivati
29. 3 2 4 Mesh structure lt 9 aaa ws a a a YoX 33 3 2 5 Parameters for Material 33 3 2 6 Initial Conditions Structure lt lt lt lt cles 35 3 2 7 Boundary Conditions Structure 36 3 3 Using Condition Functions lt lt lt lt lt lt eee 38 3 3 1 Adding a new condition function 0 38 La ee ae 39 3 4 Checkpointing and Restarting lt lt lt 2 22 44 nn 39 34L1 Checkpointing lt 4 104 299 99 3 a a en 39 3 4 2 Rest rting MAR vov 4o eon a ae 40 A snac2restart ocs yea 00 ee hee ne 9 gw a bene 3 3 40 4 Postprocessing and Graphics 41 41 ELI Outputs trom x na aa a ana cda a a RO eee eng 41 A ee ee er ee Bd 41 A 42 a A A AA 42 43 5 1 Cookbook 1 Rifting of elasto visco plastic 43 5 1 1 Results 5 1 2 Complete Listing of the Input XML Filel 0 43 5 2 Cookbook 2 Condition functions to assign multiple material types 2 2 2 48 5 2 1 Complete listing of the input XML filef a aaa 48 55 6 1 Method 9 soria rs as a a a MES 55 0 2 Results o 25 222 nenne mueve Me be eh ee dd Soe 55 57 l1 Odometer 4345 4446 a 3X3 ox a a oo AS ROS Ox 57 57 L 57 7 2 Thick Cylinder with Pressure on the I
30. 7 BENCHMARK PROBLEMS AP L 7 m Y 2 D mu lc E AF a T NE TL ais E e 7 El NE 27 7 gt E je P ad erat E A Es 5 de Figure 7 1 Mesh plotted with velocity boundary conditions Stress Pa Se 06 Be 06 06 0 0 4 06 06 2 06 1e 06 of 0 000 Snac solution Analytic solution 0 005 0 010 0 015 0 020 0 025 0 0 Strain Figure 7 2 Stress strain plot 7 2 THICK CYLINDER WITH PRESSURE ON THE INNER WALL I 59 12 Figure 7 3 Mesh plotted with pressure boundary conditions Thick Cylinder with Pressure on the Inner Wall I A constant and uniform internal pressure is applied to the inner surface of a thick cylinder without pre stress Taking advantage of symmetry only a guadrant of the cylinder is modeled The outer surface has zero confining pressure Tresca yield criterion is used i e internal friction is ignored and only cohesion matters 7 2 1 Model Setup Bulk modulus 200 MPa Shear modulus 200 MPa Cohesion 1 MPa Friction angle 0 Dilation angle 0 Tension cut off 567 MPa Geometry of cylinder a inner ra
31. AM AS IS WITHOUT WARRANTY OF ANY KIND EITHER EX PRESSED OR IMPLIED INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU SHOULD THE PROGRAM PROVE DEFECTIVE YOU ASSUME THE COST OF ALL NECESSARY SER VICING REPAIR OR CORRECTION IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER OR ANY OTHER PARTY WHO MAY MODIFY AND OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE BE LIABLE TO YOU FOR DAM AGES INCLUDING ANY GENERAL SPECIAL INCIDENTAL OR CONSEQUENTIAL DAM AGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES 69 END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new program and you want it to be of the greatest possible use to the public the best way to achieve this is to make it free software which everyone can redistribute and change under these terms To do so attach the following notices to the program It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty and each file shoul
32. COMPUTATIONAL INFRASTRUCTURE FOR GEODYNAMICS CIG User Manual Version 1 2 Plastic Strain 0 20 0 15 0 10 Eun seo Choi Michael Gurnis Susan Kientz www geodynamics org Colin Stark SNAC User Manual California Institute of Technology Version 1 2 April 12 2010 About the cover Pictured is the plastic strain distribution over a deformed block of elasto plastic ma terial While similar to the example problem in Section 5 1 the domain is larger 100 x 20 x 100 km and a distributed loading is applied on the bottom surface to simulate an oblique rifting The warped surface on top of the domain represents 10 times exaggerated surface topography and the relief between the highest and the lowest point is about 1 km Contents II Chapters 1 1 3 Approximation of Partial Derivatives 1 1 6 Mass Scaling for Numerical Stability 1 1 8 Remeshing 1 2 1 Plugins 1 2 0 The SNAC Context 1 2 3 Call Tree of Entry Points 13 StGermain 2 Installation and Getting Help 2 2 Getting Help 2 3 System Requirements 2 3 1 Compiler 2 3 0 MPI Library 23 3 Libxml2 2 3 4 GNU Scientific Library 2 3 5 Other Environment Variables o 2 6 Installing from the Software Repository 2 6 1 Tools You WillNeedl 3 1 Using SN C Na pd pated 3 2 3 Plugins List 4 CONTENTS
33. C_INCDIR SNAC_DIR build include export SNAC_LIBDIR SNAC_DIR build lib export PATH SNAC_BINDIR PATH export LD_LIBRARY_PATH SNAC_LIBDIR LD_LIBRARY_PATH In tsch setenv SNAC_DIR HOME opt SNAC setenv SNAC_BINDIR SNAC_DIR build bin setenv SNAC_INCDIR SNAC_DIR build include setenv SNAC_LIBDIR SNAC_DIR build lib setenv PATH SNAC_BINDIR PATH setenv LD_LIBRARY_PATH SNAC_LIBDIR LD_LIBRARY_PATH 2 4 Downloading and Unpacking Source Download SNAC from the SNAC web page Save the SNAC tarball locally and unpack it using the tar command tar xzf SNAC 1 2 0 tar gz If you don t have GNU Tar try the following command instead gunzip c SNAC 1 2 0 tar gz tar xf 2 5 Installation Procedure After unpacking the source use the following procedure to install SNAC 1 Navigate i e cd to the directory containing the SNAC source cd SNAC 1 2 0 2 Type configure sh to configure the package for your system configure sh 30 CHAPTER 2 INSTALLATION AND GETTING HELP 3 Type make to build the package make All the newly created files during the building procedure are placed in the build subdirectory Upon suc cessful completion the make command executable Snac in the build bin subdirectory Currently installing SNAC into another location is not supported The following options can be included in the options argument during configuration in comma separated form optim
34. ac 0 000003 dx snac 0 000005 dx snac 0 000101 dx snac 1 000001 dx snac 1 000003 dx snac 1 000005 dx snac 1 000101 dx snac 2 000001 dx snac 2 000003 dx snac 2 000005 dx snac 2 000101 dx snac 7 000001 dx snac 7 000003 dx snac 7 000005 dx snac 7 000101 dx Let s assume that the global mesh has 51 x 31 x 41 nodes and was decomposed in 3D decompDims 3 by 8 processors Also we assume that the contents of sim 0 is 25 15 20 We can then infer that 2 processors were assigned in each dimension To get a combined dx file for time step 35 gt snac_combine py snac 35 51 31 41 2 2 2 The name of a combined file has the format of snac time step in 6 digits dx So snac 000035 dx should be the final product in the above example 4 3 1 Using OpenDX An OpenDX visual program snac_visualize net is provided for convenience and is placed in Snac snac2dx It can be opened in OpenDX to read data map node and element based variables on the domain visualize vector quantities as 3D glyphs create isosurfaces and make cross sections Chapter 5 Cookbook Examples 5 1 Cookbook 1 Rifting of elasto visco plastic lithosphere A 3D Cartesian block of elaso visco plastic material is extended by the velocities applied to two side walls The domain has a size of 40 x 10 x 80 km and is discretized into 20 x 5 x 40 elements Fig B 1h The two side walls perpendicular to the x axis are pulled at a constant velocity of 1 cm yr
35. am gt gt lt list name extensions gt lt param gt SnacPlastic lt param gt lt param gt SnacCondFunc lt param gt lt list gt lt struct name mesh gt lt param name shadowDepth gt 1 lt param gt lt param name decompDims gt 2 lt param gt lt Mesh size gt lt param name meshSizel gt 51 lt param gt lt param name meshSizeJ gt 2 lt param gt lt param name meshSizeK gt 51 lt param gt lt Initial geometry gt lt param name minX gt 0 lt param gt lt param name minY gt 6000 0 lt param gt lt param name minZ gt 0 lt param gt lt param name maxX gt 300000 0 lt param gt lt param name maxY gt 0 0 lt param gt lt param name maxZ gt 300000 0 lt param gt lt Remeshing gt lt param name meshType gt cartesian lt param gt lt param name buildNodeNeighbourTbl gt True lt param gt lt struct gt lt list name materials gt lt Three types of material with different cohesions gt lt struct name mat_normal gt lt param name density gt 2800 lt param gt lt param name alpha gt 0 0 lt param gt lt param name beta gt 0 0 lt param gt lt Elastic material parameters gt lt param name lambda gt 3 0e 10 lt param gt lt param name mu gt 3 0e 10 lt param gt lt Plastic material parameters gt lt param name yieldcriterion gt mohrcoulomb lt param gt lt param name nsegments gt 2 lt param
36. aram gt lt param name cohesion1 gt 1 0e 06 lt param gt 5 2 COOKBOOK 2 CONDITION FUNCTIONS TO ASSIGN MULTIPLE MATERIAL TYPES 51 lt param name cohesion2 gt 1 0e 06 lt param gt lt param name ten_off gt 1 0e 12 lt param gt lt struct gt lt list gt lt Remesher info gt 1 lt param name remeshCondition gt onBothTimeStepLength lt param gt lt param name remeshCondition gt onTimeStep lt param gt lt param name remeshCondition gt onMinLengthScale lt param gt gt lt param name remeshCondition gt onMinLengthScale lt param gt lt param name remeshTimeStepCriterion gt 15000 lt param gt lt param name remeshLengthCriterion gt 45 0 lt param gt lt param name bottomResotre gt on lt param gt lt node ICs gt lt struct name nodelCs gt lt list name vcList gt lt struct gt lt param name type gt AllNodesVC lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt
37. aram name name gt vz lt param gt lt param name type gt double lt param gt 5 1 COOKBOOK 1 RIFTING OF ELASTO VISCO PLASTIC LITHOSPHERE lt param name value gt 0 0 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt front lt param gt lt list name variables gt lt struct gt lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 0 0 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt bottom lt param gt lt list name variables gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt lt Temperature BCs gt lt struct name temperatureBCs gt lt list name vcList gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt top lt param gt lt list name variables gt lt struct gt lt param name name gt temperature lt param gt lt param name type gt double lt param gt lt param name value gt 0 0 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt
38. bution takes one quarter of a tetrahedron s volume dependent quantity While looping over the entire set of nodes mass and nodal forces are assembled by adding up the contributions from boundary conditions and all the tetrahedra sharing that node as one of their apexes The structured mesh of SNAC renders the assemblage step conveniently static The acquired net force or heat flux at each node is used to update velocities and node coordinates or temperature 1 1 5 Solution Scheme We seek static or quasi static solutions through a dynamic relaxation method Instead of adding a usual velocity dependent friction term we adopt a local non viscous damping scheme 16 pdamped F asgn vj F 1 7 where F is the i th component of the residual force vector the right hand side of Eq 1 6 a is a positive coefficient less than 1 sgn v returns the sign of the i th component of velocity vi Once net forces are assembled and damped velocity at that node is updated using a forward Euler method At At pene At x t At a t Atv t DE 1 9 Damping is irrelevant to the update of temperature field but the same forward Euler method is used There are a couple of SNAC specific numerical artifacts First as in the case of an under damped oscillator SNAC s solutions will exhibit artificial oscillations but these are only transient and do not affect the static equilibrium Another notable artifact is the randomness
39. d have at least the copyright line and a pointer to where the full notice is found For example One line to give the program s name and a brief idea of what it does Copyright year name of author This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WAR RANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program if not write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Also add information on how to contact you by electronic and paper mail If the program is interactive make it output a short notice like this when it starts in an interactive mode Gnomovision version 69 Copyright year name of author Gnomovision comes with ABSO LUTELY NO WARRANTY for details type show w This is free software and you are welcome to redistribute it under certain conditions type show c for details The hypothetical commands show w and show c should show the appropriate parts of the General Public License Of course the command
40. dius 3 0 m b outer radius 10 0 m Boundary Conditions P 20 MPa at r a and P 0 0 at r b Top and bottom V 0 0 free slip dt 1 sec and results after 5000 steps were used for comparison to the analytic solutions Mesh size 31 x 3 x 31 nodes The second invariant of stress in xz plane are monitored 7 2 2 Results Relative error is about 0 8 60 Square root of Ik Pa CHAPTER 7 BENCHMARK PROBLEMS Figure 7 4 Second invariant of stress field Snac solution o 700000 Anahrie solucion purely Elaste 600000 500000 400000 300000 3 4 5 Y 8 9 1C radius m Figure 7 5 Profile of the second invariant of stress along radial direction 7 3 THICK CYLINDER WITH PRESSURE ON THE INNER WALL II 61 Smaeslion Analytic solucion purely Elaste Tresca crierion Square root of Ik Pa e o radius m Figure 7 6 Profile of the second invariant of stress along radial direction 7 3 Thick Cylinder with Pressure on the Inner Wall II The same problem with the previous section but the Mohr Coulomb yield criterion is adopted here 7 3 1 Model Setup Bulk modulus 200 MPa Shear modulus 200 MPa Cohesion 1 MPa Friction angle 10 Dilation angle 10 Tension cut off 567 MPa Geometry of cylinder a inner radius 3 0 m b outer radius 10 0 m Boundary Conditions P 2 MPa at r a and P 0 0 at r b Top and bottom V 0 0 free slip
41. e of minimum work rate as in the standard finite element formulation o o Q where j are components of the strain rate tensor dv and 0 represent variations of velocity and strain rate and 2 here corresponds to the whole domain The local contribution to nodes corresponding to each term can be computed by following the standard finite element procedure for linear tetrahedral elements However our method does not need to construct coefficient matrices such as mass and stiffness matrices since it adopts an explicit time discretization The resultant momentum equation is Dv 1 1 Ui pi M Dt 3 4 pg inl 1 6 where the superscript n represents values evaluated at the global node n the superscript n means the sum of contributions from all the tetrahedra having the global node n as an apex T is the traction that is defined as o n and evaluated on a face of one of the contributing tetrahedra Any applied traction boundary conditions should be explicitly computed and added on the right hand side of Eq 1 6 The nodal mass M is not the actual inertial mass but an adjusted one to satisfy a local stability criterion discussed in Section The correspondence between an apex and a face for the traction calculation is determined as in the derivation of the expression Eq 1 4 Note that the factor of i in the traction term is inherited from Eq 1 4 and the factor of 1 in the body force term implies that the nodal contri
42. es are 2 0 W m K and 1000 0 J kg K respectively To be used for initial and boundary conditions temperature values in C can be assigned to topTemp and bottomTemp Remesher parameters Remeshing is triggered when the mesh is distorted too much The criteria to determine when to remesh are the number of time steps and the minimum length scale The former requires a priori knowledge on the rate of deformation because SNAC s remeshing algorithm would fail if the displacement is larger than one element size Using this criterion makes it convenient to force remeshing on a regular basis The latter criterion allows SNAC to have a dynamically determined frequency of remeshing The length scale is the global minimum of the ratio of a tetrahedron s volume to one of its surface area If this minimum length scale decreases below some fraction of the initial value remeshing is triggered The parameter name for the type of criterion is remeshCondition of which possible values are onTimeStep onMinLengthScale and onBothTimeStepLength The critical value for time step and minimum length scale should be provided remeshLoopCriterion and remeshLengthCriterion are the parameter names and they take a positive integer and a positive number less than 1 as values respectively lt param name remeshCondition gt onBothLoopLength lt param gt lt param name remeshLoopCriterion gt 10 lt param gt lt param name remeshLengthCriterion gt 0 5 lt param g
43. ews of the National Science Foundation Part II Chapters Chapter 1 Introduction SNAC StGermaiN Analysis of Continua is an updated Lagrangian explicit finite difference code for modeling a finitely deforming elasto visco plastic solid in 3D released under the GNU General Public License see Chapter 8 on page 65 In this code nodal velocities satisfying a weak form of the momentum balance are obtained as the nodal solution SNAC shares a mathematical foundation and thus major advantages with a standard finite element method FEM However it departs from the FEM by not making explicit use of shape functions A Cartesian mesh consisting of 4 node linear or constant strain tetrahedral elements is used to represent a discretized domain although a spherical domain can also be used On top of the tetrahedral discretization a coarser discretization is constructed by zones which are defined by eight nodes like a hexhedral element in the standard FEM and subdivided into two layouts of five tetrahedral elements for symmetric response To avoid the over stiff response of tetrahedrons in the incompressible limit mixed discretization is applied I The mixed discretization relieves over stiffness by replacing the first invariant of tetrahedral strain rate tensor with the one averaged over a zone 1 1 SNAC Implementation 1 1 1 Governing Equations Momentum balance for continuum in current or deformed configuration is as given below
44. h element For more details see Sec In most situations a user can keep the overall structure as it is in the above example and would have only to modify the lowest level structure The temperature boundary conditions take the same structure The following example shows how to apply a uniform temperature of 500 0 C to all the top surface nodes lt Temperature BCs gt lt struct name temperatureBCs gt lt list name vcList gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt top lt param gt lt list name variables gt 38 CHAPTER 3 RUNNING SNAC lt struct gt lt param name name gt temperature lt param gt lt param name type gt double lt param gt lt param name value gt 500 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt Again one can replace the double type of the lowest level structure with func to use a condition function In that case the function s name should be supplied as the structure s value 3 3 Using Condition Functions Condition functions CFs can be used to define both spatially and temporally non uniform initial and boundary conditions Two requirements should be met for a condition function to make effects 1 a CF must be found at run time and 2 an input file should specify where to apply it The first condition is satisfied by properly adding a function in the SnacCondFunc plugin and t
45. he second by setting the type of variables to be func see 3 2 6 and 3 2 7 3 3 1 Adding a new condition function Everything is done in Snac plugins conditionFunctions The example below shows the step by step procedure for adding a new element CF A new node CF can be added in the same way but by modifying counterparts for node 1 Go to Snac plugins conditionFunctions 2 Open SnacCondFunc_ElementCondFunc h a Copy and paste one of the existing function declarations and rename it e g void _SnacCondFunc_DeadSea void _SnacCondFunc_MyTest 3 Open SnacCondFunc_ElementCondFunc c a Copy and paste one of the existing function definitions and rename it e g void _SnacCondFunc_DeadSea void _SnacCondFunc_MyTest b Change the contents of the new function e g remove all the if statement blocks except the first one 4 Open Register c a Copy and paste one of the existing function calls to ConditionFunction_Register_Add b Note that the third argument of ConditionFunction_Register_Add is also a function call to ConditionFunction_New which takes two arguments c Change both arguments of ConditionFunction_New ConditionFunction_new _SnacCondFunc_DeadSea SnacCF_DeadSea 5 ConditionFunction new SnacCondFunc MyTest SnacCF_MyTest The string SnacCF_MyTest will be used in an input file 5 Rebuild the plugin i e run make 3 4 CHECKPOINTING AND RESTARTING 39
46. herwise In general flow rule for frictional materials is non associative i e flow direction differs from the normal of the yield surface normal As in the definitions of yield functions the plastic flow potential for the shear failure in the Mohr Coulomb model can be given as gs 01 03 01 Nyoz 1 24 where Ny 1587 andy is the dilation angle Likewise the tensile flow potential is given as gi 03 04 1 25 In the presence of plasticity the total strain Ae is given by Ae Ae 1 26 where is the elastic and Ae is the plastic strain increment We assume that if plastic yielding implies negligible viscous flow The plastic strain increment is Act 309 1 27 where 6 is a proportionality constant to be determined 6 is determined such that updated stress state is on the yield surface i e f o Ao 0 where Ao Ae Ac and C is the elastic moduli tensor This condition is called the consistency condition and the parameter is thus called the consistency parameter 4 In the principal component representation where and are principal stress and strain respectively and a is a corresponding elastic moduli matrix of which components are given in terms of Lame s constants a p 2u ap By expanding each yield function and using o C Ae we get the following formulae for 6 1 1 SNAC IMPLEMENTATION 19 Figure 1
47. hose values need to be recorded at the tetrahedra level That is because the regular outputs contain one single precision floating point number per hexahedral element 3 4 2 Restarting 1 Run snac2restart in the original outputPath see the next section for more details 2 Create a new directory for the restarting run This is mandatory to prevent accidental overwriting of the existing data files 3 Move all the restart files to the new directory 4 Prepare an input file that is identical with the one used for the original run except that a the new one has SnacRestart in the extensions plugins list AND b restartTimestep is set to be the time step to restart from AND c outputPath now points to where restart files are SNAC will abort if it detects existing outputs there 5 Start running SNAC as usual but with a new input file prepared as above 3 4 3 snac2restart The general usage of snac2restart is Usage snac2restart timeStep your outputPath path to write restart files timeStep is one of the checkpointed time steps and should be one of those recorded in checkpointTimeStep 0 The last two specify a path to where those output files are which need processing and where to write the processed restart files respectively All the arguments are optional and by default snac2restart tries to read and write in the current directory and process for the last time step recorded Files with the suffix
48. hy smoothing alpha Volumetric thermal expansion coefficient in the unit of K 3 2 3 Plugins List All the plugins to load should be listed under the extensions or plugins list as in the following example lt list name extensions gt lt param gt SnacSpherical lt param gt lt list gt Note the changed syntax used for the item of the list One can populate the list with other plugins The following is the list of plugins available for SNAC SnacSpherical needed to generate a spherical shell mesh SnacRemesher when using remeshing The related parameters should also be set SnacTemperature when solving for heat diffusion equation SnacElastic SnacPlastic SnacMaxwell SnacViscoPlastic for elastic elasto plastic Maxwell viscoelas tic and elasto visco plastic rheology respectively SnacHydroStaticIC generates an initial stress field assuming hydrostatic equilibrium Meaningful only when gravity is non zero SnacWinklerForce When gravity is non zero and there are no boundary conditions for the vertical velocity component on the bottom surface the Winkler foundation is applied SnacPlSeeds SnacVPSeeds can control the positions of elements that have the initial non zero plastic strain These elements play the role of seeds where strain localization initiates Used in tandem with SnacPlastic and SnacViscoPlastic respectively SnacRestart required to restart a run See for more details Other plugins found
49. ial boundary condition management 7 One of the advantages of using the StGermain framework is that a code can be easily extended through entry points and plugins By adding or removing entry points a problem specific algorithm can be implemented while keeping the modification of source codes limited Plugins are small pieces of codes that work as a part of the program but can be compiled separately and dynamically loaded at run time Thus their developments can be completely independent of the main program For example a new constitutive relation or time varying boundary conditions can be made available as plugins without modifying or compiling the whole program that already exists Chapter 2 Installation and Getting Help 2 1 Introduction To install SNAC follow the procedure that is commonly used with other open source software packages First download the source package in the form of a compressed tar file available at the SNAC web page geodynamics org cig software packages long snac After unpacking the source you run a prepack aged shell script to configure SNAC for your system 2 2 Getting Help For help send e mail to the CIG Long term Tectonics Mailing List cig long geodynamics org You can subscribe to the Mailing List and view archived discussion at the Geodynamics Mail Lists web page geodynamics org cig lists 2 3 System Requirements Installation of SNAC requires the following e A C compiler e An MPI lib
50. ilt MPI package available for their distribution On Mac OS X the Fink package manager offers a prepackaged version of so if you have Fink installed simply enter the following command from a Terminal window to install LAM MPI fink install lammpi lammpi dev We highly recommend that you set up the environment variables that allow StGermain to use MPI This is done by setting the following environment variables so the VMake build system knows where to find MPICH In bash export MPI_DIR HOME opt mpich 1 2 7p1 export MPI_BINDIR MPI_DIR bin export MPI_LIBDIR MPI_DIR lib export MPI_INCDIR MPI_DIR include export MPI_RUN MPI_BINDIR mpirun export PATH MPI_BINDIR PATH export LD_LIBRATY_PATH MPI_LIBDIR LD_LIBRATY_PATH In tcsh setenv MPI DIR HOME opt mpich 1 2 7p1 setenv MPI_BINDIR MPI_DIR bin setenv MPI_LIBDIR MPI_DIR lib setenv MPI_INCDIR MPI_DIR include setenv MPI_RUN MPI_BINDIR mpirun setenv PATH MPI_BINDIR PATH setenv LD_LIBRATY_PATH MPI_LIBDIR LD_LIBRATY_PATH MPI_DIR should be replaced with the actual path where MPICH is installed If you need to specify a machine file every time you run mpich then in order for VMake s make check command to work you must specify another environment variable MPI MACHINES Here are some ex amples In bash export MPI_MACHINES home raq machinefile In tcsh setenv MPI_MACHINES machinefile home raq machinefile 2 3 3 Lib
51. in the magnitude of residual forces Suppose the magnitude of assembled residual force is very small compared to those of the contributing internal and external forces It means that many significant figures in floating point numbers are lost during the assemblage The residual force ends up with having random numbers in the floating number corresponding to it This is related to a fundamental issue of representing real numbers with floating point numbers 16 CHAPTER 1 INTRODUCTION 1 1 6 Mass Scaling for Numerical Stability The conventional Courant Friedrichs Lewy CFL condition imposes a stringent upper limit for the time step size such that dynamic relaxation takes a long time to get the quasi static solution over a geological time scale To overcome this limit a mass scaling technique is applied This technique adjusts each nodal mass such that the stability condition for a user specified time step can be locally satisfied The stability condition to be satisfied however is not the same as in the CFL condition i e At lt Imin Up where At is the time step Imin is the minimum element size and vp is the P wave velocity Instead through an analogy of continuum to an infinite mass spring system we use a criterion that does not explicitly include length scale and P wave velocity see Chapter 9 in Bathe 11 At lt 1 10 ALA where T is the period of system 21 m k m is a point mass and k is the stiffness of the spring
52. in the plugins directory are experimental 3 2 CHANGING PARAMETERS 33 3 2 4 Mesh Structure Parameters needed for generating a mesh are grouped as members of mesh structure meshSize I J K specifies the node numbers in x y and z axis respectively while min max X Y Z defines the physical dimensions of the domain in meter In the case of using spherical geometry theta phi r Min Max should have appropriate values so that the SnacSpherical plugin can use them to initialize spherical node coordinates theta and phi are longitudes and latitudes in degrees and r Min Max are the radii in meters SNAC will not be confused even if both sets of parameters are present in one input file meshType is either cartesian or spherical If not specified it is cartesian by default However it is important to assign a correct value for remeshing Finally the two parameters for a parallel run are shadowDepth and decompDims shadowDepth determines how many elements in one local domain are made to transfer variables to neighbor domains It takes a non negative integer value corresponding to the number of layers of elements To run SNAC in parallel this parameter should be 1 or larger decompDims determines whether parallel domain decomposition is performed in 1 2 or 3 dimensions 3 2 5 Parameters for Material Property Four groups of parameters define elastic viscous plastic and thermal properties of material Assigned values should have MKS un
53. ised compiles optimized code with no debugging symbols cautious performs additional run time validation checks macro as func compiles some macros as functions instead for safety checks Will incur a performance penalty memory stats enable statistics recording of the memory module tau compiles the code ready for use with the tau profiler pdt automatically instruments the code for use with tau using For example to include cautious checks and memory statistics recording configure sh options cautious memory_stats Note that the build directory under SNAC_DIR is named according to the given extra options For example SNAC_DIR build optimised is created when optimised option is given 2 6 Installing from the Software Repository The SNAC source code is available via a Subversion server at the Geodynamics website geodynamics org This allows users to view the revision history of the code and check out the most recent development version of the software NOTE If you are content with the prepared source package you may skip this section 2 6 1 Tools You Will Need In addition to the usual system requirements you will need a handful of additional development tools installed in order to work with the source from the CIG software repository First you must have a Subversion client installed To check type svn it should return a usage message svn Type svn help for usage For more information on Subversio
54. its XML syntax is lt param name parameter gt value lt param gt e Elastic material parameters lambda mu Lame s constants e Viscous material parameters refvisc reftemp activationE srexponent vis_min and vis_max The following function form of viscosity is assumed by default E 1 1 n T mein D exp 3 7 5 gt 3 1 where is refvisc Pa n is srexponent dimensionless integer is activationE J To is reftemp C 2 is strain rate and R is the Gas constant vis_min and vis_max are used to set the range of viscosity variation Other viscosity models can be easily implemented by modifying Constitutive c in the relevant plugins SnacMaxwell and SnacViscoPlastic e Plastic material parameters Plastic parameters can dynamically vary and are approximated by piecewise linear function of accu mulated plastic strain So the number of linear segments is defined first by nsegments If nsegments is n n 1 values of parameters are required to define the n linear segments The working plasticity models in SNAC is Mohr Coulomb The Drucker Prager model has been imple mented but not verified The corresponding parameter name is yieldcriterion and the legitimate values it can assume are mohrcoulomb and druckerprager Three parameters are necessary to define each model s yield function friction angle dilation angle and cohesion The parameters should be named as
55. lt param gt lt param name alpha gt 0 lt param gt lt Extension modules gt lt list name extensions gt lt param gt SnacTemperature lt param gt lt param gt SnacViscoPlastic lt param gt lt param gt SnacHydroStaticIC lt param gt lt param gt SnacVPSeeds lt param gt lt param gt SnacWinklerForce lt param gt lt list gt lt struct name mesh gt lt param name shadowDepth gt 1 lt param gt lt param name decompDims gt 1 lt param gt lt Mesh size gt lt param name meshSizel gt 21 lt param gt lt param name meshSizeJ gt 6 lt param gt lt param name meshSizeK gt 41 lt param gt lt Initial geometry gt lt param name minX gt 0 lt param gt lt param name minY gt 0 lt param gt lt param name minZ gt 0 lt param gt lt param name maxX gt 40000 lt param gt lt param name maxY gt 10000 lt param gt lt param name maxZ gt 80000 lt param gt lt Remeshing gt lt param name meshType gt cartesian lt param gt lt param name buildNodeNeighbourTbl gt True lt param gt lt struct gt lt Elastic material parameters gt lt param name lambda gt 1 0e 10 lt param gt lt param name mu gt 1 0e 10 lt param gt lt Viscous material parameters gt lt param name refvisc gt 1 0e 20 lt param gt 5 1 COOKBOOK 1 RIFTING OF ELASTO VISCO PLASTIC LITHOSPHERE lt param name reftemp gt 1400 0 lt pa
56. modified by someone else and passed on we want its recipients to know that what they have is not the original so that any problems introduced by others will not reflect on the original authors reputations Finally any free program is threatened constantly by software patents We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses in effect making the program proprietary To prevent this we have made it clear that any patent must be licensed for everyone s free use or not licensed at all The precise terms and conditions for copying distribution and modification follow 65 66 CHAPTER 8 LICENSE GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING DISTRIBUTION AND MODIFICATION 0 This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License The Program be low refers to any such program or work and a work based on the Program means either the Program or any derivative work under copyright law that is to say a work containing the Program or a portion of it either verbatim or with modifications and or translated into another language Hereinafter translation is included without limitation in the term modification Each licensee is addressed as you Activities other than copying distribution and modification are not covered b
57. n visit the Subversion website subversion tigris org 2 6 2 Download Source from Subversion To check out the latest version of the software use the svn checkout command svn checkout http geodynamics org svn cig long 3D SNAC trunk SNAC This will create the local directory SNAC if it doesn t already exist and fill it with the latest SNAC source from the CIG software repository The SNAC directory thus created is called a working copy To merge the latest changes into an existing working copy use the svn update command cd SNAC svn update This will preserve any local changes you have made to your working copy Chapter 3 Running SNAC 3 1 Using SNAC The basic usage is Snac input xml Running SNAC in parallel depends on the system configuration but a typical example is mpirun np 8 which Snac input xml gt log stdout 3 2 Changing Parameters Input parameters of SNAC are defined using the XML syntax lt param name parameter gt value lt param gt Many of the parameters defined in this fashion can be conceptually grouped based on their functionality They can also be a member of a structure or itemized under a list which in turn can become a member of a structure 3 2 1 Simulation Control Parameters Parameters in this group control the time marching in SNAC startTime the starting time of simulation 0 sec by default stopTime the ending time of simulation 1 0e 20 sec by default outputPath the
58. n in the body of this License The Free Software Foundation may publish revised and or new versions of the General Public License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns Each version is given a distinguishing version number If the Program specifies a version number of this License which applies to it and any later version you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation If the Program does not specify a version number of this License you may choose any version ever published by the Free Software Foundation If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different write to the author to ask for permission For software which is copyrighted by the Free Software Foundation write to the Free Software Foundation we sometimes make exceptions for this Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally NO WARRANTY 11 12 BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE THERE IS NO WARRANTY FOR THE PROGRAM TO THE EXTENT PERMITTED BY APPLICABLE LAW EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND OR OTHER PARTIES PROVIDE THE PROGR
59. n models and those who modify the source code Citation Computational Infrastructure for Geodynamics CIG is making this source code available to you in the hope that the software will enhance your research in geophysics A number of individuals have contributed a significant portion of their careers toward the development of SNAC It is essential that you recognize these individuals in the normal scientific practice by citing the appropriate peer reviewed papers and making appropriate acknowledgements The SNAC development team asks that you cite e Choi E L Lavier and M Gurnis 2008 Thermomechanics of mid ocean ridge segmentation Phys Earth Planet Inter 171 374 386 dz doi org 10 1016 7 pepi 2008 08 010 The developers also request that in your oral presentations and in your paper acknowledgements that you indicate your use of this code the authors of the code and geodynamics org Support SNAC development is funded by the U S Dept of Energy s Advanced Simulation and Computing program www sandia gov NNSA ASC and the National Science Foundation s www nsf gov Information Technology Research ITR program grant 30205653 Continued support of SNAC is based upon work supported by the National Science Foundation under Grant No EAR 0406751 Any opinions findings and conclusions 9 10 or recommendations expressed in this material are those of the authors and do not necessarily reflect the vi
60. nheritance Having StGermain as a framework SNAC can directly use or extend many abilities of StGermain available in generic forms XML input extensible data structures classified output streams etc The three key design ideas relevant to SNAC are the plugins SNAC Context and Entry Points 1 2 1 Plugins SNAC is readily customizable through writing a plugin a shared object that can be loaded dynamically at run time Together with the ability to manipulate Entry Points use of plugins makes it simple and local to extend SNAC For instance if a mesh for a quadrant of a cylinder is desired one can simply include 1 2 SNAC DESIGN 23 TEE E ME A ho NE b Before remeshing 3 3 plstrain After remeshing o plstrain Figure 1 4 a Meshes and velocity fields before blue and after red remeshing b Plastic strain an element associated variable is shown on a deformed mesh upper panel and mapped onto a new mesh after remeshing lower panel 24 CHAPTER 1 INTRODUCTION SnacCylinderQuad in the list of plugins in an input file If a thermal problem needs to be solved in addition to a quasistatic momentum balance problem loading the plugin SnacTemperature is all that needs to be done Trying out different constitutive models can be done in the same fashion one can load one of the plugins such as SnacElastic SnacViscoElastic or SnacViscoPlastic 1 2 2 The SNAC Context The SNAC Context is the top level
61. nner Wall lt ees 59 C21 Model setup caera me y u wa eek a ud x 59 T22 Results ar A a a bd 59 7 3 Thick Cylinder with Pressure on the Inner Wall H 61 AAA ee dee AA a eM RRS s 61 RS 61 7 4_ParalleL Plate Viscometer Problem lt lt lt lt lt lt lt lt lt lt lt lt lt s a 61 A IUe 62 CR 62 65 List of Figures 1 1 Configurations of tetrahedra and conventions for the notation a Two configurations of five tetrahedra in a hexahedral element used in the mixed discretization Numbers next to apexes indicate the local node numbering b Conventions for the notation A and n denote the 1 4 a Meshes and velocity fields before blue and after red remeshing b Plastic strain an element associated variable is shown on a deformed mesh upper panel and mapped onto a right up to 4096 cores 7 1 Mesh plotted with velocity boundary conditions lt lt lt lt 4 4 4 eee 58 O R O e a 58 7 3 Mesh plotted with pressure boundary conditions lt lt lt lt lt 4 4 4 44 4 0 59 7 4 Second invariant of stress 60 7 5 Profile of the second invariant of stress along radial direction 60 7 6 Profile of the second invariant of stress along radial direction 7 7 The initial mesh blue with the velocity boundary condition red arrows
62. nt version of SNAC Although rounding off is forced for the continuation of calculation the mass of each phase is not guaranteed to be conserved e Domain decomposition the parallelism implemented in SNAC often makes it difficult to map variables between meshes when a parallel boundary virtual boundary between subdomains assigned to more than one processor of the old mesh has moved too far away from the corresponding one of the new mesh e This remeshing works best in the case of lateral extension because the nodes are moved back to their original positions If a mesh has shrunk laterally and then remeshing moves all the nodes back to their original horizontal positions it is impossible to remap variables for the regions of a new mesh that do not have overlapping parts in the deformed mesh In such a case one would have to assume states and properties of material for the unmappable regions e Note that Fig shows that remeshing cannot only generate a more regular mesh but also change the bottom boundary of the domain Although useful in some cases this boundary restoring technique should be used with caution because it also creates unmappable regions and thus requires us to assume the state and properties of the material SNAC retains the deformed boundary by default 1 2 SNAC Design SNAC follows StGermain s architecture see Section 1 3 Although primarily implemented in C it is object oriented and makes use of extensibility and i
63. o refrain entirely from distribution of the Program 68 10 CHAPTER 8 LICENSE If any portion of this section is held invalid or unenforceable under any particular circumstance the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system it is up to the author donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License If the distribution and or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries so that distribution is permitted only in or among countries not thus excluded In such case this License incorporates the limitation as if writte
64. of restart are created Chapter 4 Postprocessing and Graphics 4 1 Introduction SNAC writes binary output files which must be converted to a file format that external visualization tools can handle Two such conversion programs are provided One generates Visualization Tool Kit VTK files The other is for the open source Open Visualization Data Explorer better known as OpenDX There are several programs that can visualize VTK files ParaView www paraview org MayaVi2 https svn enthought com enthought wiki MayaVi or VisIt https wci llnl gov codes visit OpenDX with documentation is available from the OpenDX website www opendx org 4 1 1 Outputs from SNAC Once you have run SNAC you should have a series of files in the output directory as set in the input file Information on the parallel decompositon is not explicitly given in an input file by a user Instead StGermain computes an optimal configuration based on the total number of processors the global mesh size and the value of decompDims The resultant global and local element numbers in x y and z as well as decomposed processor numbers are written in the file sim 0 which is common to all the processors 4 2 Converting to VTK files A program called snac2vtk is provided to convert the binary outputs from SNAC to ascii files in the XML VTK Structured Grid format vts snac2vtk is compiled during the building procedure and installed in SNAC_BINDIR
65. ork complete source code means all the source code for all modules it contains plus any associated interface definition files plus the scripts used to control compilation and installation of the executable However as a special exception the source code distributed need not include anything that is normally distributed in either source or binary form with the major components compiler kernel and so on of the operating system on which the executable runs unless that component itself accompanies the executable If distribution of executable or object code is made by offering access to copy from a designated place then offering equivalent access to copy the source code from the same place counts as distribution of the source code even though third parties are not compelled to copy the source along with the object code 4 You may not copy modify sublicense or distribute the Program except as expressly provided under this License Any attempt otherwise to copy modify sublicense or distribute the Program is void and will automatically terminate your rights under this License However parties who have received copies or rights from you under this License will not have their licenses terminated so long as such parties remain in full compliance 5 You are not required to accept this License since you have not signed it However nothing else grants you permission to modify or distribute the Program or its derivative works These actions
66. othTimeStepLength lt param gt lt param name remeshCondition gt onTimeStep lt param gt lt param name remeshCondition gt onMinLengthScale lt param gt gt lt param name remeshCondition gt onMinLengthScale lt param gt lt param name remeshTimeStepCriterion gt 15000 lt param gt lt param name remeshLengthCriterion gt 0 7 lt param gt lt param name bottomResotre gt on lt param gt lt node ICs gt lt struct name nodelCs gt lt list name vcList gt lt struct gt lt param name type gt AllNodesVC lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt lt struct gt 45 46 CHAPTER 5 COOKBOOK EXAMPLES lt param name type gt AllNodesVC lt param gt lt list name variables gt lt struct gt lt param name name gt temperature lt param gt lt param name type gt func lt param gt lt param name value gt SnacTemperature_Top2BottomSweep lt param gt l
67. parameter 0 n So the input file should have frictionAngle0 frictionAnglel dilationAngle0 dilationAnglel and cohessionO cohesion1 To avoid a singularity of the yield surface at the tensional limit SNAC defines tension off a practical limit to tensional stress over which yielding occurs even if the stress state is not on the yield surface The following is an example of defining plastic material parameters lt param name nsegments gt 2 lt param gt lt param name plstrain0 gt 0 0 lt param gt 34 CHAPTER 3 RUNNING SNAC lt param name plstraini gt 0 1 lt param gt lt param name plstrain2 gt 1000 0 lt param gt lt param name frictionAngle0 gt 30 0 lt param gt lt param name frictionAnglel gt 20 0 lt param gt lt param name frictionAngle2 gt 20 0 lt param gt lt param name dilationAngle0 gt 5 0 lt param gt lt param name dilationAnglei gt 5 0 lt param gt lt param name dilationAngle2 gt 5 0 lt param gt lt param name cohesion0 gt 4 0e 07 lt param gt lt param name cohesion1 gt 1 0e 06 lt param gt lt param name cohesion2 gt 1 0e 06 lt param gt lt param name ten_off gt 1 0e 13 lt param gt e Thermal material parameters Values of thermal conductivity and heat capacity can be defined The parameter names are thermal _ conduct and heatCapacity and appropriate values in MKS unit should be assigned The default val u
68. problem setup artificial forces should be added to left and right surfaces 7 4 2 Results Color field represents stress Colored arrows are for SNAC s velocity black arrows for the analytic solution Over the most part of the domain the two sets of arrows show good agreement The relative error of velocity solution is 75 This might be acceptable because the analytic solution is only approximate and for viscous fluid Therefore refining the mesh does not help reduce the error 7 4 PARALLEL PLATE VISCOMETER PROBLEM 63 AE amp B OR Figure 7 8 The second invariant of stress and velocities plotted on the deformed mesh Colored arrows are for SNAC s solution black ones for the analytic solution 64 CHAPTER 7 BENCHMARK PROBLEMS Chapter 8 License GNU GENERAL PUBLIC LICENSE Version 2 June 1991 Copyright C 1989 1991 Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed Preamble The licenses for most software are designed to take away your freedom to share and change it By contrast the GNU General Public License is intended to guarantee your freedom to share and change free software to make sure the software is free for all its users This General Public License applies to most of the Free Software Foundation s software and to any other program who
69. ram gt lt param name activationE gt 45 0e 03 lt param gt lt param name vis_min gt 1 0e 30 lt param gt lt param name vis_max gt 1 0e 30 lt param gt lt param name srexponent gt 1 lt param gt lt Plastic material parameters gt lt param name yieldcriterion gt mohrcoulomb lt param gt lt param name nsegments gt 2 lt param gt lt param name plstrain0 gt 0 0 lt param gt lt param name plstrain1 gt 0 02 lt param gt lt param name plstrain2 gt 1000 0 lt param gt lt param name frictionAngle0 gt 30 0 lt param gt lt param name frictionAngle1 gt 30 0 lt param gt lt param name frictionAngle2 gt 30 0 lt param gt lt param name dilationAngle0 gt 5 0 lt param gt lt param name dilationAngle1 gt 5 0 lt param gt lt param name dilationAngle2 gt 5 0 lt param gt lt param name cohesion0 gt 4 0e 07 lt param gt lt param name cohesion1 gt 4 0e 05 lt param gt lt param name cohesion2 gt 0 0e 00 lt param gt lt param name ten_off gt 1 0e 13 lt param gt lt Temperature variables gt lt param name topTemp gt 0 0 lt param gt lt param name bottomTemp gt 700 0 lt param gt lt param name thermal_conduct gt 1 6 lt param gt lt param name heatCapacity gt 1000 0 lt param gt lt Remesher info gt lt Used only when the SnacRemesher plugin is loaded gt lt lt param name remeshCondition gt onB
70. rary e Libxml2 e GNU Scientific Library MPI installations are typically configured for a particular compiler and provide a special wrapper command to invoke the right compiler Therefore the choice of MPI implementation often determines which C compiler to use 2 3 1 C Compiler On Unix or Linux systems there is a high likelihood that a usable C compiler is already installed To check type cc at the shell prompt cc cc no input files On Linux if the cc command is not found install GCC using the package manager for your distribution The Mac OS X version of GCC is included in a software development suite called Xcode Xcode is available as a free download at the Apple Developer Connection developer apple com 27 28 CHAPTER 2 INSTALLATION AND GETTING HELP Warning If you are using an Intel compiler on an Itanium CPU do not use the 03 optimization flag as reports indicate that this optimization level will generate incorrect codes For any compiler you should always be careful about the correctness of the compiled codes when using an 03 or higher optimization level 2 3 2 MPI Library StGermain requires an implementation of the MPI 1 standard for parallel message passing even if you only intend to run the code in serial A popular choice is MPICH Installing MPICH from source involves walking through the standard GNU build procedure configure amp amp make amp amp make install Linux users may have a prebu
71. relative or absolute path to a directory where outputs are dumped dumpEvery the interval of time steps to dump outputs maxTimeSteps the maximum number of time steps The time marching ends either at stoptime or at maxTimeSteps whichever comes first checkpointEvery the interval of time steps to checkpoint see 3 4 restartTimestep the time step to restart see 3 4 31 32 CHAPTER 3 RUNNING SNAC 3 2 2 SNAC Specific Parameters Parameters in this group are specific to SNAC density the average density of the material SNAC is simulating Unit is kg m3 gravity the gravitational acceleration 9 8 m sec2 by default demf the dimensionless force damping factor 0 8 by default dtType the type of time marching method Either dynamic or constant timeStep the size of time step in seconds Used only when dt Type is constant forceCalcType the type of assembling forces on the parallel boundary nodes quick the least amount of computation Used only in serial runs Obsolete normal works in both serial and 1D parallel runs Obsolete complete Robust nodal force assembly decomposedAxis axis to decompose for parallel runs 0 1 or 2 for x y and z axis respectively Needed only when decompDims 1 storeForces yes to store residual forces for each node Otherwise no forceCheckSum If yes the sum of forces is checked if it is consistent topo_kappa Parameter for topograp
72. rovides an important feature lacking in most scientific codes As of the end of 2008 the development of StGermain in conjunction with SNAC has been finished and its interfaces have been frozen StGermain is funded primarily through VPAC and Caltech The project was born with the development of SNAC and the need to generalize the same abilities that had been incorporated into the development of Snark It was time to restart from scratch and was done late at night under a California moon aided by 1 3 STGERMAIN 25 AbstractContext_EP_Build defaultMeshBuild SnacBuild AbstractContext_EP_IC defaultMeshICs SnacIC AbstractContext_EP_Run default SnacTimeStepZero AbstractContext_EP_Dt SnacDt AbstractContext_EP_Step default AbstractContext_EP_BC SnacBC AbstractContext_EP_Solve SnacSolve AbstractContext_EP_Sync SnacSync Figure 1 5 A tree like diagram showing the hierarchy of entry points set up in SNAC together with hooked up functions in each entry point The functions defined by StGermain are marked in blue while those defined in SNAC are in red 26 CHAPTER 1 INTRODUCTION some fine Belgian beer and the finest Parisienne electro jazz However it is unclear whether StGermain is named for the French techno jazz group led by Ludovic Navarre or the famed Paris quartier StGermain provides a suite of libraries needed by general physical modeling software such as flow control domain discretization and init
73. rved Then both nodal velocity and temperature and elemental variables stress strain rate heat flux and plastic strain are transferred onto the new mesh from the deformed mesh The nodal transfer is accomplished by locating the tetrahedron in the old mesh which contains a node of the new mesh Then shape interpolation functions corresponding to the location of the new node with respect to the old element is calculated and used to interpolate nodal quantities of the old deformed mesh The element associated variables integrated by 1 point Gauss quadrature are first recovered as nodal fields by Superconvergent Patch Recovery method 17 Like other nodal fields these recovered fields are interpolated onto a new mesh Then element associated variables are evaluated at the new tetrahedron s barycenter coinciding with 1 point Gauss quadrature based on the interpolated recovered fields Fig demonstrates how remeshing actually works The current remeshing algorithm however has some limitations e The phase field is not accurately remapped This field is not a physical variable but a set of integers registered at each tetrahedron to represent the element s material type Although starting from integer values a phase field for more than one material types ends up including some real numbers after remeshing due to numerical diffusion during the recovery process Fractional material ID is not defined both physically and numerically in the curre
74. s The conversion tools for the DX file format will no longer be updated To convert binary files into ascii OpenDX files you need to run the program snac2dx The source code is Snac snac2dx snac2dx c compiled during SNAC s building procedure and placed in SNAC_BINDIR together with other executables Running snac2dx without any arguments converts data for all the time steps recorded in timeStep 0 Optionally a range of time steps can be set if two positive integers are given as arguments The converted output files are always written in the directory where snac2dx is run The naming convention for the ascii dx files is snac processor ID time step in 6 digits dx To visualize outputs from a parallel run one more step is needed to combine the dx files into a single dx file for each time step A Python script snac_combine py is provided for this purpose This script is under Snac snac2dx but not automatically installed in SNAC_BINDIR The following eight arguments are required e modelname Suffix for the combined data files snac unless snac2dx c is modified e timestep Time step of interest e gnodex gnodey gnodez Global node numbers in x y and z lon radius lat as set in the input file e nprocx nprocy nprocz Number of processors in x y and z lon radius lat calculated based on sim 0 Example list of SNAC output files in the OpenDX format 8 processors and 51 outputs per processor snac 0 000001 dx sn
75. s you use may be called something other than show w and show c they could even be mouse clicks or menu items whatever suits your program You should also get your employer if you work as a programmer or your school if any to sign a copyright disclaimer for the program if necessary Here is a sample alter the names Yoyodyne Inc hereby disclaims all copyright interest in the program Gnomovision which makes passes at compilers written by James Hacker signature of Ty Coon 1 April 1989 Ty Coon President of Vice This General Public License does not permit incorporating your program into proprietary programs If your program is a subroutine library you may consider it more useful to permit linking proprietary applications with the library If this is what you want to do use the GNU Library General Public License instead of this License 70 CHAPTER 8 LICENSE Bibliography 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Marti J and P Cundall 1982 Mixed discretization procecure for accurate modelling of plastic collapse Int J Numer Anal Methods Geomech 6 129 139 Cundall P 1989 Numerical experiments on localization in frictional materials Ingenieur Archiv 58 148 159 Wilkins M L 1964 Calculation of elastic plastic flow Meth Comput Phys 3 211 263 Needleman A 1988 Material rate dependence and mesh
76. se authors commit to using it Some other Free Software Foundation software is covered by the GNU Library General Public License instead You can apply it to your programs too When we speak of free software we are referring to freedom not price Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software and charge for this service if you wish that you receive source code or can get it if you want it that you can change the software or use pieces of it in new free programs and that you know you can do these things To protect your rights we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights These restrictions translate to certain responsibilities for you if you distribute copies of the software or if you modify it For example if you distribute copies of such a program whether gratis or for a fee you must give the recipients all the rights that you have You must make sure that they too receive or can get the source code And you must show them these terms so they know their rights We protect your rights with two steps 1 Copyright the software and 2 Offer you this license which gives you legal permission to copy distribute and or modify the software Also for each author s protection and ours we want to make certain that everyone understands that there is no warranty for this free software If the software is
77. sensitivity in localization problems Comp Meth App Mech Eng 67 69 85 Ortiz M and J C Simo 1986 An analysis of a new class of integration algorithms for elastoplastic constitutive relations Int J Num Meth Eng 23 353 366 Ortiz M and J J Quigley 1991 Adaptive mesh refinement in strain localization problems Comput Methods Appl Mech Engrg 90 781 804 Quenette S B Appelbe M Gurnis L Hodkinson L Moresi and P Sunter 2005 An investigation into design for performace and code maintainability in high performance computing ANZIAM J 46 E C101 C116 Tan E E Choi P Thoutireddy M Gurnis and M Aivazis 2004 GeoFramework Coupling mul tiple models of mantle convection within a computational framework Geochem Geophys Geosyst 7 06001 doi 10 1029 2005GC001155 Jaeger J C 1969 Elasticity Fracture and Flow 3rd Ed New York John Wiley amp Sons Inc Zienkiewicz O C M Huang and M Pastor 1995 Localization problems in plasticity using finite elements with adaptive remeshing Int J Numer Anal Methods Geomech 19 127 148 Bathe K J 1996 Finite Element Procedure Upper Saddle River NJ Prentice Hall Albert R R Phillips A Dombard and C Brown 2000 A test of the validity of yield strength envelope with an elastoviscoplastic finite element model Geophys J Int 140 399 409 Poliakov A N B P A Cundall Y Y Podladchikov and V A Lyakhovsky 1993
78. t 3 2 CHANGING PARAMETERS 35 3 2 6 Initial Conditions Structure Two structures need to be constructed for defining the initial conditions One is for variables associated with nodes the other for the element associated ones A name is assigned to the structure first This structure will have only a single list as a member This list in turn can have a series of structures of which members are the type of initial conditions and the list of variables Finally the list of variables has structures for each variable to be initialized The lowest level structure will have name type and value for the variable The following is an example of constructing such a structure lt node ICs initical conditions for nodal variables i e velocity and temperature gt lt struct name nodelCs gt lt list name vcList gt lt struct gt lt param name type gt AllNodesVC lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vy lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt struct gt lt param name name gt vz lt param gt lt param name type gt double lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt
79. t 1 16 During the viscoplastic constitutive update the initial guess of stress increment is first predicted by Maxwell viscoelastic constitutive law If the guessed stress increment exceeds a given yield criterion it is projected onto the yield surface using a return mapping method otherwise the viscoelastic stress update is kept The detailed algorithm of stress update at each quadrature point and each time step is as follows 1 Viscoelastic predictor Obtain viscoelastic stress increment predictor with Maxwell constitutive update corresponding to the strain increment Since viscous relaxation affects only the deviatoric part of stress the deviatoric stress update is given by Spt C157 2GAgf Cz 1 17 where S77 Gij 301504 is the deviatoric stress Es is the deviatoric part of the strain increment Esj es At 1 ni T and G is the shear modulus Volumetric elastic stress increment is given by Akk 3KAekk which gives viscoelastic predictor update at n 1 n 1 1 n 0i Sij Okk 1 18 2 Plastic correction If o is inside or on the yield surface i e f o t gt 0 where f is the yield function then it does not need plastic correction If o is outside the yield surface then yielding occurs and we project onto the yield surface using a return mapping algorithm I4 In general for frictional materials the yield function can be written as let qgop Cr
80. t struct gt lt list gt lt struct gt lt list gt lt struct gt lt element ICs gt lt struct name elementICs gt lt list name vcList gt lt struct gt lt param name type gt AllElementsVC lt param gt lt list name variables gt lt struct gt lt param name name gt elementMaterial lt param gt lt param name type gt int lt param gt lt param name value gt 0 lt param gt lt struct gt lt list gt lt struct gt lt list gt lt struct gt lt Velocity BCs gt lt struct name velocityBCs gt lt list name vcList gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt left lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 3 17e 10 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt right lt param gt lt list name variables gt lt struct gt lt param name name gt vx lt param gt lt param name type gt double lt param gt lt param name value gt 3 17e 10 lt param gt lt struct gt lt list gt lt struct gt lt struct gt lt struct gt lt param name type gt WallVC lt param gt lt param name wall gt back lt param gt lt list name variables gt lt struct gt lt p
81. ther derive the explicit form of the function r in terms of principal plastic strains and stresses by working out the partial derivative OeP From 1 31 gr dlho k 1 2 B 2228 e 3 3 Oc 2e 1 2 _ p p p p Od 2e E i 28 1 1 2 1 9 a eP z e eP 3 63 eb 1 1 1 _ 2 _ 53 dd With 0G 00 being 1 0 Ny the function r o eP is given by ea E HG Jep 3 Y 3 Y 2 3 Y 3 3 v Note that when W 0 N 1 and 0 Then the value of r becomes 1 Since the dilation angle is often set to be zero or a small value and the cohesion is always defined as a piecewise linear function F is approximately linear with regard to the consistency parameter The current consistency parameter computation implemented in SNAC is thus justified However if any non linearity is introduced such as friction angle varying with the internal variable and non linearly changing cohesion the interation presented above should be performed r 22 CHAPTER 1 INTRODUCTION 1 1 8 Remeshing Remeshing becomes necessary in SNAC when continued deformation distorts a mesh so severely that the accuracy of solutions is deteriorated In SNAC remeshing is done by moving the nodes back to their original positions under the constraint that physically meaningful features such as top and bottom topography are conse
82. uced to a minimal level The 32 element case problem was run until the 1000th step To save time however the 64 element and 128 element cases were run until 200th and 50fth step respectively These tests were performed on ranger at Texas Advanced Computing Center of University of Texas Austin by Dr Colin Stark at Lamont Doherty Earth Observatory of Columbia University 6 2 Results The total wall clock times taken for each case is shown in the left panel of Fig 6 1 as a function of the number of cores The plots of speedup and efficiency the middle and right panels of Fig show the acceptable performance of SNAC up to 4096 cores The performance of the 32 element case becomes notably poor for 2048 cores or more This is because the size of subdomain becomes too small to benefit from parallel processing These results suggest that the parallel overhead will dominate the computation time if the subdomains become too small and the empirical lower limit for the subdomain size is 3 elements Also a good scaling is expected to hold for even larger numbers of cores if the problem size is sufficiently large T T T T T T T T CF 128x128x128 E 64 x 64 x 64 CH 128x128x128 4 O 32 x 32 32 64 x 64 x 64 O 32 32 32 Ideal os E Wall time hours Speed up relative 10 64 cores Efficiency speed up relative to ideal 0 45 of
83. ve works based on the Program In addition mere aggregation of another work not based on the Program with the Program or with a work based on the Program on a volume of a storage or distribution medium does not bring the other work under the scope of this License 67 3 You may copy and distribute the Program or a work based on it under Section 2 in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following a Accompany it with the complete corresponding machine readable source code which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or b Accompany it with a written offer valid for at least three years to give any third party for a charge no more than your cost of physically performing source distribution a complete machine readable copy of the corresponding source code to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or c Accompany it with the information you received as to the offer to distribute corresponding source code This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer in accord with Subsection b above The source code for a work means the preferred form of the work for making modifications to it For an executable w
84. ve of F with respect to dF m BP 00 OF dAB OAB deP BAB Since on ny1 1 eng 480 6 1 1 and n41 and e is constant during a time interval 1 34 OOn 1 _ OG Dona 1 35 From Ea 1 32 ONer r 1 36 By substituting Eqs and into Eq we get OF aG 1 dAB lt Ve j OcP Now we present the cutting plane algorithm for updating the stress internal variable and consistency parameters iteratively in case of a general non linear hardening 1 1 SNAC IMPLEMENTATION 21 1 Initialization p 0 nil En AB 0 2 Update stresses and evaluate F cf a engi 68 FQ Flo If A lt 0 exit otherwise go to step 3 3 Incremental upate k k A g __ Fia EN Fia m OF OG OF r OAB do k 1 k OG k stk Piet d A n ery bx lk A Gr o eP n n n 1 gt n 1 k _ lk A24 2P OG k p k S EE Oe do Onti Engi k 1 k ABM 0 8 Go back to step 2 This algorithm is just a standard Newton method applied to the discrete consistency equation Thus it should be obvious that the plastic correction is accomplished by the single step update if the yield function is a linear function of the consistency variable In that case Eqs and are immediately retrieved For reference we fur
85. xml2 The Libxml2 library and toolkit are required for SNAC to handle the input files Although already included in most of the recent Linux systems it can be obtained from The XML C parser and toolkit of and installed manually Precompiled packages for various Linux distributions and OS X are also available through their own package management software 2 4 DOWNLOADING AND UNPACKING SOURCE 29 2 3 4 GNU Scientific Library GNU Scientific Library GSL is a numerical library for C and C programmers http www gnu org software gsl SNAC needs this library for remeshing see since the recovery method involves matrix inversion op erations GSL s linear algebra routines are utilized in SNAC s remesher plugin Although the remesher is a plugin it is expected to be used in most geodynamic applications Also the use of GSL will be expanded in future development For this reason GSL has become a requirement SNAC s build scripts will be able to find and link GSL if it is installed in a standard location But if it is installed locally one should define an environment variable GSL_DIR as follows In bash export GSL_DIR home raq opt gsl In tcsh setenv GSL DIR home rag opt gsl 2 3 5 Other Environment Variables It is handy although not necessary to define the following environment variables for both building and running SNAC In bash export SNAC_DIR HOME opt SNAC export SNAC_BINDIR SNAC_DIR build bin export SNA
86. y this License they are outside its scope The act of running the Program is not restricted and the output from the Program is covered only if its contents constitute a work based on the Program independent of having been made by running the Program Whether that is true depends on what the Program does 1 You may copy and distribute verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this License and to the absence of any warranty and give any other recipients of the Program a copy of this License along with the Program You may charge a fee for the physical act of transferring a copy and you may at your option offer warranty protection in exchange for a fee 2 You may modify your copy or copies of the Program or any portion of it thus forming a work based on the Program and copy and distribute such modifications or work under the terms of Section 1 above provided that you also meet all of these conditions a You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change b You must cause any work that you distribute or publish that in whole or in part contains or is derived from the Program or any part thereof to be licensed as a whole at no charge to all third parties under

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