6 AUTOMATIC REZONING AUTOMATIC REZONING Introduction

Transcription

1 AUTOMATIC REZONING AUTOMATIC REZONING 6.1 Introduction When FLAC is run in large strain mode, the grid may become distorted to such an extent that the simulation halts with a bad geometry error message. The objective of automatic rezoning logic is to map the existing stresses, velocities and displacements onto a new, more regular, grid, so that the run can continue. Because interpolation is used in the re-mapping, some smearing and accuracyloss is inevitable. In order to minimize these effects, the stresses are re-mapped for each subzone separately: thus, the overlap of a new subzone with each of the old subzones is performed in order to obtain a properly weighted estimate of the stress tensor in the new subzone. A progressively wider local search of possible overlaps is performed, in order to minimize search time and ensure that it is a linear process (time proportional to the number of zones). 6.2 Main Features In designing the automatic-rezoning logic, it is recognized that any FLAC model always uses some type of grid generator devised by the user. This may be as simple as a series of commands (e.g., GENERATE and INITIAL commands), or may be as complicated as a multi-line FISH function that produces a grid to conform to desired boundary shapes. Thus, the logic repeatedly invokes the user s grid generator within the rezone logic. When a REZONE command is issued, the following actions occur: 1. The model state is saved in a temporary area of memory. Specifically, subzone stresses, gridpoint velocities and displacements, material properties, fluid properties and pore pressure (if CONFIG gwflow is specified), temperature and thermal properties (if CONFIG thermal is specified) are saved. In addition, various conditions, such as fix flags, are saved. 2. A user-defined FISH function is called (from within the rezone logic). It is the task of this function to create a new grid. Even if the grid generator is a simple set of commands, these commands are embedded in the FISH function. (For technical reasons, a special form of embedding commands in a FISH function must be used; this is described below.) The form of the new grid is unrestricted, but normally it will be a version of the original grid that conforms to the new boundary geometry. 3. Stresses, velocities and displacements are mapped from the old grid to the new grid. In the rezoning logic, zone variables such as stress components are mapped, based on area overlapping fractions. For example, in the illustrative case shown in Figure 6.1, dash lines represent the new mesh, and solid lines define the old mesh. For the zone indicated by dash lines, its zone variables (e.g., stress components) are derived from the overlapping zones using area fractions (i.e., A1-A6) as the weighting factors. For nodal variables, such as velocity and displacement at a gridpoint in the new mesh, values are interpolated from the triangular subzone within which the gridpoint is located. The weighting factors are determined by the distance to each of the gridpoints of the host

2 6-2 Theory and Background subzone, as demonstrated in Figure 6.2. Such an interpolation scheme satisfies force and moment equilibrium. A detailed description of gridpoint weighting can be found in Section in Structural Elements. Figure 6.1 Zone variable mapping Figure 6.2 Gridpoint variable mapping 4. Advection of properties and state variables (e.g., plastic strains) is performed. Material properties are mapped as zone variables. This mapping is similar to that for stress components, except that stress components are re-mapped using subzone overlapping fractions, while material properties are re-mapped using full zone overlapping fractions.

3 AUTOMATIC REZONING 6-3 For integer-type properties, such as table number and plasticity indicator, a majority rule is adopted to determine the value for the new zone. 5. Histories used in rezoning are coordinate-based. Absolute locations (in x-y space) of gridpoint/zone histories are now stored and their indices in i-j space updated according to their absolute locations during the rezoning operation. 6. For dynamic simulations, the subzone masses are recalculated from the updated density. Simple tests show that momentum is conserved reasonably well. Normally a series of alternating STEP and REZONE commands will be issued (e.g., from a loop within a controlling FISH function). The rezoning logic can also be invoked automatically when a bad geometry error occurs by using the SET rez func command. At present, the rezone logic is not general: it does not recognize all features offered in FLAC. Support of such features will be added gradually over time. In particular, the logic excludes axisymmetry, interfaces, attach, multistepping and FISH models. Further, only one material model can be used in a rezoning run at present.

4 6-4 Theory and Background 6.3 Input Instructions for Automatic Rezoning FLAC Commands The following commands are provided to perform automatic rezoning during a FLAC run: REZONE <keyword value>... This command given without arguments invokes rezoning. The following keywords may also be given to set rezoning characteristics: set keyword history Location-based history advection is activated. method name The grid generator FISH function to be called within a rezone episode is specified. A function of the given name must exist. separate By default, each subzone receives weighted stresses from all old subzones that it overlaps. This results in some sharing (and smearing) of stress components between overlays (which are normally kept separate). This command forces subzones of a particular overlay to receive new stresses only from the same overlay in the old grid. smooth from i1,j1 to i2, j2 A surface range that will be subjected to a smoothing process is specified. The smoothing process consists of setting the coordinate of each surface gridpoint to the average of the three local gridpoints (the target gridpoint and its left and right neighbors). surface from i1,j1 to i2, j2 table n <seq ns> A surface range is specified. The table keyword specifies a table number to which the list of surface coordinates is written. This table may then be used by the user-defined FISH function to constrain the boundary of the new grid. If more than one surface is involved in the rezoning operation, the optional keyword seq is required to specify the sequence number (from 1 to 20) for each surface.

5 AUTOMATIC REZONING 6-5 SET rez func name The grid generator FISH function that is invoked automatically when a bad geometry error occurs is specified. A function of the given name must exist FISH Intrinsic Function The following FISH intrinsic function is provided to perform automatic rezoning during a FLAC run: rez exe(s) The intrinsic function rez exe (which always returns zero) causes the command contained in the given string, s, to be executed. Note that COMMAND...END COMMAND must not be used in a FISH function invoked within the rezoning logic (i.e., a function specified by the REZONE set method command). The commands that may be executed by rez exe( ) are limited to INITIAL, GENERATE, MODEL, PROPERTY and APPLY. If the string cannot fit on one line of FISH code, then it must be composed of separate concatenated sub-strings stored as a symbol (which is then given as the argument to rez exe( ) instead of being given directly as a quoted string).

6 6-6 Theory and Background 6.4 A Simple Rezoning Demonstration The automatic-rezoning capability is demonstrated for the case of a simple slope geometry. The slope is shown in Figure 6.3. The cohesive strength of the slope material is reduced to zero so that the slope fails. When run in large-strain mode, the run stops with a bad geometry error message after approximately 2000 calculation steps. JOB TITLE :. FLAC (Version 6.00) LEGEND Feb-08 13:10 step E+00 <x< 6.333E E+01 <y< 4.333E Grid plot 0 2E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.3 Simple slope initial geometry The data file for this model, including the rezoning facility, is listed in Example 6.1. The model is first stepped to an equilibrium state with a high cohesive strength assigned to the material. The cohesion is then set to zero, and the large-strain calculation mode is invoked. Automatic rezoning is implemented via the user-defined FISH function dostage slope. This function first calls the user-defined FISH function sloperezone with the command rezone set method sloperezone sloperezone is a simple grid generator composed of four commands: def sloperezone oo = rez exe( gen ) oo = rez exe( mod mohr ) oo = rez exe( gen tab 5 ) oo = rez exe( mod null reg 1 20 )

7 AUTOMATIC REZONING 6-7 The model surface is identified in dostage slope, and the table which will contain the list of surface coordinates is specified (id = 5) with the command rezone set surf from 1,11 to 61,21 tab 5 Then, a smoothing process is invoked to smooth the model surface with the command rezone set smooth from 1,11 to 61,21 The rezoning logic is implemented with the command rezone and then cycling is continued with the command step continue dostage slope is invoked automatically when a bad geometry error occurs by giving the command set rez func dostage slope Note that MOVIE commands are provided in the Example 6.1 data file in order to create a movie file of the distorting grid. The final shape of the slope after 10,000 steps is shown in Figure 6.4: JOB TITLE :. FLAC (Version 6.00) LEGEND Feb-08 13:10 step E+00 <x< 6.333E E+01 <y< 4.333E Grid plot 0 2E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.4 Simple slope geometry after 10,000 steps

8 6-8 Theory and Background Example 6.1 Automatic rezoning of a simple slope grid m mohr prop dens 2000 bulk 2e8 sh 1e8 fric 10 coh 1e20 tens 1e20 notnull table gen tab 1 mod null reg 1 20 fix x y j=1 fix x i=1 fix x i=61 set grav 10 solve save slope1.sav ; ini xv 0 yv 0 xdis 0 ydis 0 prop coh 0 set large def dostage slope command unmark rez set method=sloperezone rez set surf from 1,11 to 61,21 tab 5 rez set smooth from 1,11 to 61,21 rezone step continue Command def sloperezone oo = rez exe( gen ) oo = rez exe( mod mohr ) oo = rez exe( gen tab 5 ) oo = rez exe( mod null reg 1 20 ) set rez func dostage slope ; plot grid movie on movie step on 200 ; step save slope2.sav ret

9 AUTOMATIC REZONING Example Applications Failure of a Benched Slope A two-bench slope excavated in sensitive clay fails when the excavation level reaches 19 m depth. The initial geometry of the slope at this excavation depth is shown in Figure 6.5. A closeup view of the slope is shown in Figure 6.6. The properties of the clay are as follows: mass density (ρ) 2000 kg/m 3 bulk modulus (K) 1.7 MPa shear modulus (G) MPa initial cohesion (c) 50 kpa above 19 m depth 70 kpa below 19 m depth The clay exhibits a softening response under load. The behavior is simulated by using the strainsoftening constitutive model (MODEL ss) with the cohesion of the clay above the 19 m depth, degrading from 50 kpa to 5 kpa as a function of plastic shear strain. The data file for this model is listed in Example 6.2. An in-situ stress state is applied first, prior to excavation, with the vertical stress conforming to gravitational loading and the ratio of horizontal to vertical stress equal to Stresses are initialized using the FISH function inistress. The excavation is then made instantaneously (with the MODEL null command). The slope begins to fail. If the model is run without automatic rezoning, a bad geometry error occurs in zones at the toe of the slope after approximately 450 steps. Automatic rezoning is added to the model via the user-defined FISH function rezone, listed in Example 6.3. The rezoning operation involves four steps: 1. The gridpoints along the slope surface are stored in a table using the REZONE set surface from... to... table... command. Table 4 is designated to store the gridpoint data. 2. A new mesh is generated using the user-defined FISH function remesh. The slope surface is redefined in the new mesh by using the FISH function gentab4 to null zones above the surface defined by table 4. The slope surface is also smoothed with the REZONE set smooth command. 3. The rezoning logic is activated with the REZONE command to re-map zone and gridpoint values onto the new mesh. 4. The calculation stepping is continued. The rezone function is executed whenever the bad geometry error occurs by giving the command SET rez zone rezone before beginning the calculation.

10 6-10 Theory and Background JOB TITLE :. (*10^2) FLAC (Version 6.00) LEGEND 4-Mar-08 11:21 step E+01 <x< 5.389E E+01 <y< 1.159E+02 cohesion 5.000E E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.5 Benched slope initial slope geometry JOB TITLE :. FLAC (Version 6.00) LEGEND 4-Mar-08 12:25 step E+00 <x< 2.600E E+01 <y< 9.100E Grid plot 0 5E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.6 Benched slope closeup view

11 AUTOMATIC REZONING 6-11 Example 6.2 Benched slope grid 200,56 gen -20.0, , , ,70.0 model ss table 3-100,75 0,75 5,75 7.5, , , , ,84 100,84 gen tab 3 prop dens 2000 prop bulk 1.7e6 shear 0.625e6 prop cohesion = 45e3 ctab 99 j=21,56 table e e3 prop cohesion 70e3 ctab 100 j=1,20 table e e3 set gravity 10 def inistress loop i (1,izones) loop j (1,jzones) if model(i,j) # 1 depth = y(i,j) sy = -1. * * 10. * depth syy(i,j) = sy sxx(i,j) = sy * 0.65 szz(i,j) = sy * 0.65 if loop loop inistress fix x i 1 fix x y j 1 fix x i 201 hist 999 unbal solve save bench1.sav set large model null region 1,56 fix x y i 1 call rezone.fis save bench2.sav set rez func rezone set geom 0.21 step save bench3.sav

12 6-12 Theory and Background Example 6.3 Benched slope function def rezone command unmark rez set surfac from 1,21 to 201,57 table 4 rez set meth= remesh rez set smooth from 1,21 to 201,57 rezone step continue command ; def gentab4 loop ii (1,izones) loop jj (1,jzones) if model(ii,jj) # 1 x = (x(ii,jj)+x(ii+1,jj)+x(ii,jj+1)+x(ii+1,jj+1))/4. y = (y(ii,jj)+y(ii+1,jj)+y(ii,jj+1)+y(ii+1,jj+1))/4. yt = table(4, x) if y > yt oo = rez exe( model null i=ii j=jj ) if if loop loop ; def remesh oo=rez exe( gen -20.0, , , ,70.0 ) oo=rez exe( model ss ) gentab4 The failed slope geometry at the final equilibrium state is shown in Figure 6.7. A contour plot of cohesion at this state is given in Figure 6.8, and indicates the failure surface that has developed.

13 AUTOMATIC REZONING 6-13 JOB TITLE :. (*10^2) FLAC (Version 6.00) LEGEND 4-Mar-08 16:30 step E+01 <x< 5.389E E+01 <y< 1.159E Grid plot 0 2E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.7 Final state of failed bench slope JOB TITLE :. (*10^2) FLAC (Version 6.00) LEGEND 5-Mar-08 9:19 step E+01 <x< 5.389E E+01 <y< 1.159E+02 cohesion -1.00E E E E E E E E E+04 Contour interval= 1.00E+04 Boundary plot E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.8 Contour plot of cohesion for failed bench slope

14 6-14 Theory and Background Punch Problem A rigid punch is driven at a constant velocity into a cohesive, frictionless material. When run in large-strain calculation mode, a bad geometry error occurs after driving the punch 2.1 m into the material (approximately 10,000 steps). The automatic-rezoning logic allows the punch to be driven almost entirely through the material. The mesh is automatically re-created when a bad geometry condition is encountered, by invoking the rezoning logic. The initial grid is shown in Figure 6.9. The model is 55 m wide and 30 m deep. The punch area is 5 m wide at the top of the model. A constant velocity (denoted by the vectors in the figure) of m/step is specified to apply the loading over the punch area. The model has roller boundary conditions on the sides, and is fixed in both directions at the base. Note that the model is symmetric about a vertical line through the center of the punch, and the FLAC model could be reduced to half of the model shown in Figure 6.9. The whole model is run to illustrate the use of multiple tables to define the surface geometry when rezoning. JOB TITLE :. FLAC (Version 6.00) LEGEND 25-Feb-08 14:08 step E+00 <x< 5.806E E+01 <y< 4.556E Grid plot 0 1E Velocity vectors max vector = 2.000E E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.9 FLAC grid with constant-velocity punch

15 AUTOMATIC REZONING 6-15 The data file for this problem is listed in Example 6.4. The vertical velocity is applied and the model is first run for 4000 steps before the rezoning logic is invoked. The distorted grid at this stage is shown in Figure Example 6.4 Punch problem grid mod mohr prop dens 1000 bulk 2e8 sh 1e8 coh 1e5 tens 1e10 fix x i=1 fix x y j=1 fix x i=56 fix x y i=26,31 j=31 ini yvel -2e-4 i=26,31 j=31 save punch a.sav ; set large ncw=100 step 2000 save punch b.sav ; step 2000 save punch c.sav ; def inisetup width = 55.0 ntab1 = 1 ntab2 = 2 nz1 = jzones nz1p1 = nz1 + 1 irmax = igp jrmax = jgp jlmax = jgp inisetup ; call dostage punch.fis dostage punch save punch d.sav ; step save punch e.sav ret

16 6-16 Theory and Background JOB TITLE :. FLAC (Version 6.00) LEGEND 3-Mar-08 14:39 step E+00 <x< 5.806E E+01 <y< 4.567E+01 X-displacement contours -4.00E E E E E-01 Contour interval= 2.00E-01 Velocity vectors max vector = 2.013E E -4 Grid plot E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.10 x-displacement contours and velocity vectors after driving the punch 0.8 m before rezoning In order to continue the run when a bad geometry error occurs, a new mesh is generated by using the user-defined FISH function dostage punch, listed in Example 6.5. The rezoning process consists of four steps: 1. The gridpoints associated with the model surface are stored in tables using the REZONE set surface from... to... table... seq... command. Two tables are created: table 1 stores the gridpoints from the upper-left corner of the model to the left boundary gridpoint of the punch; and table 2 stores the gridpoints from the right boundary gridpoint of the punch to the upper-right corner. 2. A new mesh is generated using the user-defined FISH function footrezone with a new surface defined by tables 1 and 2. The mesh is also scanned to remove zones above the new surface. 3. The rezoning logic is activated with the REZONE command to re-map zone and gridpoint values onto the new mesh. 4. The punch velocity loading is fixed again and the calculation stepping is continued. The dostage punch function is executed whenever the bad geometry error occurs, by issuing the command set rez func dostage punch

17 AUTOMATIC REZONING 6-17 Example 6.5 Punch rezoning functions def dostage punch command unmark rez set surf from 1,jlmax to 26,nz1p1 tab ntab1 seq 1 rez set surf from 31,nz1p1 to irmax,jrmax tab ntab2 seq 2 rez set meth=footrezone rezone free x y i 2 55 j 2 31 fix x y i 26,31 j=nz1p1 step continue Command ; def footrezone h1 = y(26,nz1+1) tbs2 = table size(ntab2) h2 = 0.0 loop n (1,tbs2) h2 = max(h2,ytable(ntab2,n)) Loop nz1 = int(h1 * float(jzones) / h2) hinc = h1 / float(nz1) ytop = float(jzones) * hinc nz1p1 = nz1 + 1 nztop = jzones oo = rez exe( gen 0,0 0,ytop width,ytop width,0 ) oo = rez exe( mod mohr ) oo = rez exe( mod null i 26,30 j nz1p1,nztop ) oo = rez exe( gen tab ntab1 ) oo = rez exe( gen tab ntab2 ) loop i (1,26) loop j (nz1+1,jzones) if model(i,j) # 1 xcen = (x(i,j)+x(i+1,j)+x(i,j+1)+x(i+1,j+1))/4.0 ycen = (y(i,j)+y(i+1,j)+y(i,j+1)+y(i+1,j+1))/4.0 if ycen > table(ntab1,xcen) ii = i jj = j oo = rez exe( mod null i=ii j=jj ) if if Loop Loop

18 6-18 Theory and Background loop i (31,izones) loop j (nz1+1,jzones) if model(i,j) # 1 xcen = (x(i,j)+x(i+1,j)+x(i,j+1)+x(i+1,j+1))/4.0 ycen = (y(i,j)+y(i+1,j)+y(i,j+1)+y(i+1,j+1))/4.0 if ycen > table(ntab2,xcen) ii = i jj = j oo = rez exe( mod null i=ii j=jj ) if if Loop Loop jlmax = jgp section loop j (1,jzones) if model(1,j) = 1 jlmax = j exit section if Loop Section y(igp,jrmax) = table(ntab2,x(igp,jrmax)) jrmax = jgp section loop j (1,jzones) if model(izones,j) = 1 jrmax = j exit section if Loop Section y(igp,jrmax) = table(ntab2,x(igp,jrmax)) set rez func dostage punch Figures 6.11 and 6.12 display the distorted grid after driving the punch 13.8 m and 24.8 m, respectively, into the mesh.

19 AUTOMATIC REZONING 6-19 JOB TITLE :. FLAC (Version 6.00) LEGEND 25-Feb-08 15:08 step E+00 <x< 5.806E E+01 <y< 4.701E+01 X-displacement contours -2.00E E E E E E E E E+00 Contour interval= 5.00E-01 Grid plot 0 1E 1 Velocity vectors max vector = 2.063E E -4 Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.11 x-displacement contours and velocity vectors after driving the punch 13.8 m JOB TITLE :. FLAC (Version 6.00) LEGEND 25-Feb-08 15:07 step E+00 <x< 5.806E E+01 <y< 4.754E+01 X-displacement contours -3.00E E E E E E E+00 Contour interval= 1.00E+00 Grid plot 0 1E 1 Velocity vectors max vector = 2.648E E Itasca Consulting Group, Inc. Minneapolis, MN Figure 6.12 x-displacement contours and velocity vectors after driving the punch 24.8 m

20 6-20 Theory and Background 6.6 Accuracy and Zoning Sensitivity The accuracy of the rezoning logic and the sensitivity of the logic to zone density is tested in the following sections Accuracy A simple 2 2 uniform square grid is used to verify the correctness of implementation of the rezoning logic. As illustrated in Figure 6.13, each zone has a size of unity, but different density (e.g., 1000 at (1,1), 2000 at (1,2), 3000 at (2,1) and 4000 at (2,2)) before rezoning. The y-displacement of the top gridpoints (j = 3) and middle gridpoints (j = 2) are also initialized to different values (1e-3 at j = 2, 2e-3 at j = 3). In the rezoning operation, the line of i = 2 is translated in the x-direction by α (= 0.2), and the line of j = 2 is translated in the y-direction by β (= 0.3) to form a new grid. The re-meshing logic is then invoked to map zone quantities and interpolate gridpoint quantities from the old grid to the new grid. The data file is listed in Example 6.6. The comparison made in FISH function check shows that the expected mapped values of density in zones (1,1) and (2,1) and vertical displacement at gridpoint (2,2) exactly match the actual values provided by FLAC. Figure 6.13 Rezoning of a 2 2 grid: solid lines are the original grid, and dashed lines are the new grid. The original and new grids share the same boundary.

21 AUTOMATIC REZONING 6-21 Example 6.6 Accuracy test new def const ; prop d11 = d12 = d21 = d22 = yd2 = 1e-3 yd3 = 2e-3 ; geom x1 = 0. x2 = 2. y1 = 0. y2 = 2. alpha = 0.2 beta = 0.3 const grid 2 2 model elastic prop dens d11 i 1 j 1 prop dens d12 i 1 j 2 prop dens d21 i 2 j 1 prop dens d22 i 2 j 2 ini ydisp yd2 j 2 ini ydisp yd3 j 3 def remesh oo=rez exe( ini x add alpha i = 2 ) oo=rez exe( ini y add beta j = 2 ) def rezone command rez set method = remesh rezone command save accu grid.sav rezone ; note that the formulations for expected values may be different ; if alpha and/or beta are changed. def check area = x2 * y2 / 4.

22 6-22 Theory and Background ; check density (1,1) frac11 = 1./ area frac21 = alpha/ area frac12 = beta/ area frac22 = alpha * beta/ area val = d11* frac11+ d21* frac21+ d12* frac12+ d22* frac22 exp d11 = val/( frac11+ frac21+ frac12+ frac22) num d11 = density(1,1) ; check density (2,1) frac21 = (2.- alpha)/ area frac22 = (2.- alpha)* beta/ area val = d21* frac21+ d22* frac22 exp d21 = val/( frac21+ frac22) num d21 = density(2,1) ; check ydisp(2,2) exp yd22 = yd3* beta+ yd2*(1.- beta) num yd22 = ydisp(2,2) check print exp d11 num d11 print exp d21 num d21 print exp yd22 num yd Zoning Sensitivity As illustrated in Figure 6.14(a), a rectangular block filled with two different materials is fixed in both directions at all boundaries. The gridpoints at the material interface are forced to displace upward following a given sine-shaped profile. Once the system evolves to the position shown in Figure 6.14(b) (which takes 5000 steps), the model is stepped for another 5000 steps with the velocities at all gridpoints reversed. Figure 6.14(c) shows the contour plot of material property at step 10,000. If a re-meshing action is made just before reversing velocities, the material property contour will be shown in Figure 6.14(d); some smearing of material property is induced in the re-meshing process. The test is performed using a grid. The smearing is found to be significantly reduced in finer grids, for example, as shown in Figure 6.14(e) for a grid, and Figure 6.14(f) for a grid.

23 AUTOMATIC REZONING 6-23 Figure 6.14 Material advection testing; dark and light areas indicate the same material property with different values. State (c) is for no rezoning; states (d), (e) and (f) are for rezoning in progressively finer grids. Example 6.7 Zone density sensitivity new def const ; prop den1 = bul1 = 1e8 she1 = 3e7 den2 = bul2 = 2e8 she2 = 6e7 ; geom (dimensions) x1 = 0. x2 = 50. y1 = 0. y2 = 30. ; geom (IJ) iz = 25;50;100 jz = 15;30;600 ig = iz+1 jg = jz+1 jm = jz/2

24 6-24 Theory and Background jm1 = jm+1 const grid iz jz gen ( x1, y1) ( x1, y2) ( x2, y2) ( x2, y1) model elastic prop dens den1 bulk bul1 shear she1 j = 1, jm prop dens den2 bulk bul2 shear she2 j = jm1, jz def setsine j = jm1 loop i(2, iz) yvel(i,j) = amp1 * sin(pi * x(i,j)/x( ig,j)) loop def genmesh oo = rez exe( gen x1, y1 x1, y2 x2, y2 x2, y1 ) def rezone command rezone set method genmesh rezone command set amp1 = 1e-3 setsine fix x y j=1 fix x y j= jg fix x y i=1 fix x y i= ig set large ncw=100 fix y j= jm1 step 5000 plot hold bulk fill fix rezone plot hold bulk fill fix ini yvel mul -1 j= jm1 step 5000 plot hold bulk fill fix