afem.appg

GGraphicUtilities

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Appendix G: GRAPHIC UTILITIES

TABLE OF CONTENTS

PageG.1. Introduction G3G.2. Hex8 Element Plotting Module G3

G.2.1. Usage . . . . . . . . . . . . . . . . . . . . G4G.2.2. Source Code . . . . . . . . . . . . . . . . . G14G.2.3. Individual Element Example Plots . . . . . . . . . . G14G.2.4. Multiple Element Example Plots . . . . . . . . . . G14

G.3. Hex20 Element Plotting Module G15G.3.1. Usage . . . . . . . . . . . . . . . . . . . . G18G.3.2. Source Code . . . . . . . . . . . . . . . . . G28G.3.3. Individual Element Example Plots . . . . . . . . . . G28

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G.2 HEX8 ELEMENT PLOTTING MODULE

G.1. IntroductionThis Appendix collects a set of application-independent graphic utilities organized as Mathematicamodules. By application independent is meant that these modules are reusable across differentFEM application programs that do not necessarily simulate structures.

G.2. Hex8 Element Plotting ModuleThe Mathematica module PlotHex8Elem produces plots of the 8-node hexahedron finite elementdescribed in Chapter 11. The module aims to provide a comprehensive set of plotting options todisplay an individual element of that particular configuration.Although the module can display multiple elements, it is not recommended for showing a finiteelement mesh because PlotHex8Elem focuses on local (intrinsic) element properties. As a matterof fact, such multiple elements need not be connected in any way, as they would be in a mesh. Thatlack of restrictions has some practical uses, for example to illustrate initial and final configurationsof the same element. Another module, called PlotHex8Mesh and described in Section ?, is moresuitable for plotting a hexahedral mesh.

Returns x,y,z given,, using iso-Pshape functions ofHex8 element

Draws Hex8 facesas regular mesh of color filled polygonswith edge lines

Draws lines (e.g., hexahedron edges) joining two end pointsof given location

PlotHex8ElemDriver that generates plotstream by interpreting"plotwhat" commands, and displays or savesplot image

Draws points at givenlocations and places labels nearby (point orlabel may be omitted)

Injects plotting optionsthat affect downstreamcolors, point size, line thickness and labels,into the plot stream.

Expands a macro command keyword intosequence of low-levelinstructions

PlotHex8Faces PlotHex8Lines PlotHex8Points PlotHex8Options PlotHex8Macro Hex8CarInterp

Figure G.1. Organization of PlotHex8 graphics module.

The overall organization is diagrammed in Figure ?. Module PlotHex8Elem is actually a driverthat interprets an array of plot what commands, and calls subordinate modules PlotHex8Faces,PlotHex8Lines, PlotHex8Points, PlotHex8Options, and PlotHex8Macro as appropriateto construct the plot stream. Another subordinate module, Hex8CarInterp, is called byPlotHex8Faces, PlotHex8Lines, and PlotHex8Points to map natural to Cartesian coordi-nates using the Hex8 shape functions. Once all instructions are processed, the plot image is createdusing the built in Show function. The image is either output to the screen, or returned as a list,according to input specifications.In this Section the abbreviation pt denotes a printer point, which is a length dimension of0.35146 mm = 0.013837 inches = (1/72.27) inches. This is a unit traditionally used in typog-raphy as well as (currently) in computer graphics. Typical dimensions specified in pt are linethicknesses, point sizes and label font sizes.1

1 Do not confuse point which is a distinguished element location such as a nodal point (a corner in Hex8) or a Gauss

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G.2.1. UsageThe module may be invoked in either of two ways:

PlotHex8[xyzhex,plotwhat,plotspecs]phex= PlotHex8[xyzhex,plotwhat,plotspecs]

(G.1)

The first form outputs a plot image to the screen. The second form displays no image but places theplot object into phex as function return. A logical flag provided in argument plotspecs determinesthe appropriate calling form, as described below.The arguments are:

xyzhex To plot an individual element: nodal coordinates stored node-by-node as a 83matrix: { { x1,y1,z1 },{ x2,y2,z2 },{ x3,y3,z3 }, . . . { x8,y8,z8 } }.To plot m > 1 elements: stack their nodal coordinate matrices in a m 8 3three dimensional list: { xyzhex1,. . .,xyzhexm }. Elements stored in the listmay be totally independent of each other.

plotwhat A list of commands specifying what is to be plotted. The list may includecommands of three types.Macro commands. A character string that collectively represents a sequenceof lower level commands. For example: "faces" produces a plot of allelement faces and is equivalent to a list of six more detailed commands:{ "",-1 },{ "",1 },{ "",-1 },{ "",1 },{ "",-1 },{ "",1 }. Availablemacro commands are listed in Table ?.Low-level commands. These specify only individual action, such as : plot apoint, plot a line, insert a label, etc. Low level commands are listed in Table ?.Options commands. These are used to override default settings such as colors,line thicknesses, point sizes and label styles. For example, by default pointsare show as black disks of 6-pt diameter. To have them displayed in red with4-pt diameter, say { "pcolor","red" } and { "psize",4 } before entering thecommands to draw those points. Option commands are listed in Table ?. Thelast column of this Table gives the default settings.

plotspecs A list containing the following items:{ Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave }which can be used to control the overall appearance of the plot. These itemsare described in Table ?.

point, and pt, which is a length dimension.

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G.2 HEX8 ELEMENT PLOTTING MODULETable G.1. Macro Commands in plotwhat Argument of PlotHex8Elem

Command What to plot Equivalent to low-level command sequence

"corners"

or "nodes"

Corner points (element nodes) withoutlabels

{ 1 },{ 2 },{ 3 },{ 3 },{ 4 },{ 5 },{ 6 },{ 7 },{ 8 }

"diagonals" Diagonal lines (from each corner toopposite corner)

{ 1,7 },{ 2,8 },{ 3,5 },{ 4,6 }

"edges" or

"sides"

Edges as straight lines { 1,2 },{ 1,4 },{ 1,5 },{ 2,3 },{ 2,6 },{ 3,4 },{ 3,7 },{ 4,8 },{ 5,6 },{ 5,8 },{ 6,7 },{ 7,8 }

"faces" All faces using filled polygons, withNsub subdivisions along each direction.

{ "",-1 },{ "",1 },{ "",-1 },{ "",1 },{ "",-1 },{ "",1 }.

"fcenters" Face center points without labels { -1,0,0 },{ 1,0,0 },{ 0,-1,0 },{ 0,1,0 },{ 0,0,-1 },{ 0,0,1 }

"gp111" Gauss pt for 111 rule, labeled G1 { { 0,0,0 },"G1" }"gp222" Gauss pts for 222 rule, labeled

G1 through G8. (In the next column, = 1/3)

{ { - ,- ,- },"G1" },{ { ,- ,- },"G2" },{ { , ,- },"G3" },{ { - , ,- },"G4" },{ { - ,- , },"G5" },{ { ,- , },"G6" },{ { , , },"G7" },{ { - , , },"G8" }

"labcorners"

or"labnodes"

Corner points (element nodes) withlabels

{ 1,"1" },{ 2,"2" },{ 3,"3" },{ 4,"4" },{ 5,"5" },{ 6,"6" },{ 7,"7" },{ 8,"8" }

"labfcent" Face center points with face labels { { -1,0,0 },"C1485" },{ { 1,0,0 },"C2376" },{ { 0,-1,0 },"C1265" },{ { 0,1,0 },"C3487" },{ { 0,0,-1 },"C1234" },{ { 0,0,1 },"C5678" }

"labnaxes"

or"labhaxes"

Natural axes labels {, , } placednear tips. Axis tips are not plottedas points.

{ "psize",0 },{ { ,0,0 },"" },{ { 0,,0 },"" },{ { 0,0, },"" },in which = 1.5 by default.

"medians" Median lines (from each face centerto opposite face center)

{ -1,0,0 },{ 1,0,0 } },{ { 0,-1,0 },{ 0,1,0 } },{ { 0,0,-1 },{ 0,0,1 }

"medsurf"or

"medplanes"

Median surfaces = 0, = 0,and = 0

{ { "",0 },{ "",0 },{ "",0 } }

"naxes" or

"haxes"

Natural axes {, , }without labels.Tips usually stick outside element.

{ { -1,0,0 },{ ,0,0 } },{ { 0,-1,0 },{ 0,,0 } },{ { 0,0,-1 },{ 0,0, } },in which = 1.5 by default.

Axis-tip distances from faces may be adjusted with the { "tips", } option; see Table ?

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Table G.2. Low-level Commands in plotwhat Argument of PlotHex8Elem

Command Format What to plot

{ n } n: integer in range 18 Corner point n, without label{ n,label } n: integer in range 18; label: a

character stringCorner point n with label

{ n1,n2 } n1,n2: integers in range 18 Line joining corners n1 and n2{ m } m: integer of form 100*n1+n2, with

both n1,n2 integers in range 18.Same effect as { n1,n2 }.

{ ,, } ,,: numeric values (real, integeror fraction) of natural coordinates.Values outside range [1, 1] are OK;if so point will fall out of element.

Point at location { ,, }, without label

{ { ,, },label } ,,: as in previous case; label:a character string

Point at location { ,, }, with label.To suppress point plotting, preceed withoption command { "psize",0 }

{ { 1,1,1 },{ 2,2,2 } }

{ 1,1,1 } and { 2,2,2 }: :natural coordinates of two points

Line joining the two points

{ "",c } c: coordinate of surface =c Plot surface =c within element


{ "",c } c: coordinate of surface =c Plot surface =c within element If c is outside the range [1, 1], the plotted plane is outside the element, but will be bounded

by the [1, 1] range of of the other two coordinates.

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G.2 HEX8 ELEMENT PLOTTING MODULETable G.3. Option Commands in plotwhat Argument of PlotHex8Elem

Command Description Defaults

{ "fcolor",color } Set face color for subsequent face plots. Here coloris either a list { R,G,B } of 3 numbers, all in range[0, 1], which defines a RGB specification, or one ofthe following strings: "black", "red", "green",or "blue". For example: { "fcolor","blue" }and { "fcolor",{ 0,0,1 } } are equivalent

Mathematica defaultfor 3D graphics (thesedefaulst may vary withversion).

{ "lcolor",color } Set face color for subsequent line plots. Here coloris as described above.

Black

{ "pcolor",color } Set face color for subsequent point plots. Herecolor is as described above.

Black

{ "style",{ fontfamily,fontsize,fontslant } }

Set font style for subsequent labels. Here fontfamis a string that defines the font family, e.g., "Times","Helvetica", or"Courier"; fontsize is a numberdefining the font size in printer points; fontslant isone of the strings "Plain", "Bold" or "Italic".Strings for fontfamily and fontslant must becapitalized as per Mathematica naming rules.

Font family: Times,font size: 14 pt,font slant: Plain

{ "psize",size } Value psize specifies the size, in printer pt, ofsubsequent point to be plotted. If psize is 0, thosepoints are not shown. This is handy to show labels-only at specified locations, omitting points.

Point size: 6 pt

{ "ethick",thickness } Value thickness specifies the thickness, in printerpoints, of edges of subsequent face polygons.

Polygon edgethickness: 1.25 pt

{ "lthick",thickness } Value thickness specifies the thickness, in printerpoints, of subsequent lines to be plotted; e.g.,{ "lthick",1.5 }. By line it is meant thoseproduced by jpining two specified points; for exampleedges, medians and diagonals.

Line thickness: 2 pt

{ "offset",{ x,y,z } }

The list { x,y,z }defines the offset, in {x, y, z}coordinate units, from a point (actually the pointcenter) to its respective label. This setting applies toall subsequent labels until reset.

x=x=z=0.04.

{ "tips", } Defines ends (tips) of natural axes, as well as labelspositions, drawn with macro commands "naxes"and "labnaxes" Tips placed at = , = and = , for axes {, , }, respectively.

= 1.5, whenceaxes tips stick 50%outside element.

Note: The label offset specification in terms of coordinate values is awkward, but unfortunatelyMathematica lacks the ability to specify an absolute distance between graphic objects in points.

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Table G.4. Items in plotspecs Argument of PlotHex8Elem

Item Description Recommended specifications

Nsub A positive integer that specifies the subdivisions alongeach direction when plotting faces or natural-coordinatesurfaces. For example, ifNsub=4, the face or surface willbe rendered with 42 = 16 filled polygons. Since elementfaces or surfaces are (possibly warped) quadrilaterals,polygons are also quadrilaterals.

2 to 4, but may be adjusted forappearance. If face or surfaceis highly warped, a larger Nsub(e.g., 6 or 8) may be appropriate.

boxrat A list of three positive numbers that specifies the {x, y, z}side ratios of the plot bounding box. Set throughthe Mathematica 3D graphics option BoxRatios ->boxrat. If an empty list: { }, the module usesBoxRatios -> Automatic to specify default settings.

The default { }, which triggersBoxRatios -> Automatic, of-ten works OK, but plots with highaspect ratios may require fiddlingwith 3 numbers.

view A list of 3 numeric items that specifies the location in{x, y, z} space from which the objects plotted are tobe viewed. Set through the Mathematica 3D graphicsoption ViewPoint -> view.

Start with a reasonable guess, e.g.,{ 1,1,1 }, but experimentation willbe usually needed.

light A specification, usually stated in terms of one of thebuilt-in functions GrayLevel, Hue or RGBColor, whichsets a level of diffuse isotropic lighting of the plot object.Set through the 3D graphics option AmbientLight ->light. For other types of lighting, such as point lightsources, the Show statement within PlotHex8Elem hasto be modified; cf. Remark ?.

GrayLevel[0], but some experi-mentation may be required. Hue[0]can also be a useful start. As h inHue[h] changes from 0 to 1, theambient light color ranges fromred, yellow, green, cyan, blue,magenta, and back to red.

imgsiz Width of the plot in printer points. Specified through thegraphics option ImageSize -> imgsiz

300, which is about 4 inches or 10cm.

showbox A logical flag. If True, the plot bounding box is drawnvia the 3D graphics option Boxed -> True

False

showxyz A logical flag. If True, the labeled Cartesian axes aredrawn along the edges of the plot bounding box via the3D graphics options Axes -> True and AxesLabel->{ "x","y","z" }.

False

plotsave A logical flag. If True, do not display plot onscreen, but return the plot object as function value fromPlotHex8Elem; see invokation format in (?). Useful fordownstream combining plots from various sources.

False, unless plot is to be mergeddownstream with other plots.

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PlotHex8Elem[xyzhex_,plotwhat_,plotspecs_]:=Module[{i,j,k,m,n,p,e, ip,n1,n2,np,ne=1,what,w,whead,wlen,kw,w1,w2,w1head,w1len,w2head,w2len, F,L,P,expMC,Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave, boxdim,ps,black,red,blue,psizdef=6,ethidef=1.25,lthidef=2,tipsdef=1.5, styledef={TextStyle->{FontFamily->"Times",FontSize->14, FontWeight->"Plain"}},offsetdef={0.04,0.04,0.04},plotopt, axespec=False,axeslab=" ",polyfill,polyedge,lines,points,labels, optkey,key,dfun,xyzc=xyzhex,mulele,phex,modname="PlotHex8Elem:"}, optkey[key_]:=StringLength[key]>=4&&Length[StringPosition[ validoptkeys,StringTake[key,4]]]>0; validoptkeys="ethi fcol pcol lcol lthi psiz styl offs tips"; polyfill=polyedge=lines=points=labels={}; plotopt={styledef,psizdef,ethidef,lthidef,offsetdef,tipsdef}; {Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave}=plotspecs; If [showxyz, axespec=True; axeslab={"x","y","z"}]; {black,red,blue}={RGBColor[0,0,0],RGBColor[1,0,0],RGBColor[0,0,1]}; mulele=Length[Dimensions[xyzhex]]==3; If [mulele, ne=Length[xyzhex]]; np=Length[plotwhat]; If [np0, {points,labels}=PlotHex8Points[ xyzc,expMC,plotopt,points,labels]; p=True]; If [P=1&&w1=102&&w1


If [wlen==3, If [whead==List, {points,labels}=PlotHex8Points[ xyzc,{what},plotopt,points,labels]; p=True]; ]; If [!p, Print[modname," Illegal command ",what]]; ]];

boxdim=Automatic; If [Length[boxrat]==3, boxdim=boxrat]; If [plotsave, phex=Show[Graphics3D[red],polyfill,

Graphics3D[AbsoluteThickness[ethidef]],polyedge, Graphics3D[black],Graphics3D[AbsoluteThickness[lthidef]], lines,Graphics3D[black],Graphics3D[AbsolutePointSize[psizdef]], points,Graphics3D[black],labels,AmbientLight->light, ViewPoint->view,Axes->axespec,AxesLabel->axeslab, BoxRatios->boxdim,ImageSize->imgsiz, PlotRange->All,Boxed->showbox,DisplayFunction->Identity]; ClearAll[xyzc,polyfill,polyedge,sides,points,labels]; Return[phex]];

dfun=$DisplayFunction; If [$VersionNumber>=6.0, dfun=Print]; Show[Graphics3D[red],polyfill,

Graphics3D[AbsoluteThickness[ethidef]],polyedge, Graphics3D[black],Graphics3D[AbsoluteThickness[lthidef]], lines,Graphics3D[black],Graphics3D[AbsolutePointSize[psizdef]], points,Graphics3D[black],labels,AmbientLight->light, ViewPoint->view,Axes->axespec,AxesLabel->axeslab, BoxRatios->boxdim,ImageSize->imgsiz,

PlotRange->All,Boxed->showbox,DisplayFunction->dfun]; ClearAll[xyzc,polyfill,polyedge,sides,points,labels]; Return[]];

Figure G.3. Plot driver module PlotHex8Elem, Part 2.

PlotHex8Faces[xyzc_,facelist_,Nsub_,polyfinc_,polyeinc_]:= Module[ {i,j,kf=Length[polyfinc],ke=Length[polyeinc], n=Nsub,nf=Length[facelist],if,ic,cval, am,ap,bm,bp,xyz1,xyz2,xyz3,xyz4,polyfill,polyedge}, If [n


PlotHex8Lines[xyzc_,linelist_,Nsub_,linesinc_]:=Module[ {i,k,nl=Length[linelist],lk,nn,1,2,d,beg,end, xyz1,xyz2,nbeg,nend,n1,n2,lindef,node,lines=linesinc}, node={{-1,-1,-1},{1,-1,-1},{1,1,-1},{-1,1,-1},

{-1,-1,1}, {1,-1,1}, {1,1,1}, {-1,1,1}}; For [k=1,k=1&&n1=1&&n2


PlotHex8Options[opt_,graphicsinc_]:=Module[{w1,w2,key14,key24, w2head,w2len,R,G,B,color,style,psize,ethik,lthik,offset,tips, plotopt,polyfill,polyedge,lines,points,labels}, {plotopt,polyfill,polyedge,lines,points,labels}=graphicsinc; {style,psize,ethik,lthik,offset,tips}=plotopt; {w1,w2}=opt; w2head=Head[w2]; w2len=Length[w2]; key14=StringTake[w1,4]; key24=StringTake[w1,{2,4}]; If [key24=="col", R=G=B=0; color=False; If [w2head==String,

If [w2=="black", color=True]; If [w2=="red", color=True; R=1]; If [w2=="green", color=True; G=1]; If [w2=="blue", color=True; B=1]];

If [w2head==List&&w2len==3, color=True; {R,G,B}=w2]; If [key14=="pcol"&&color,

AppendTo[points,Graphics3D[RGBColor[R,G,B]]]]; If [key14=="lcol"&&color,

AppendTo[lines,Graphics3D[RGBColor[R,G,B]]]]; If [key14=="fcol"&&color,

AppendTo[polyfill,Graphics3D[RGBColor[R,G,B]]]]]; If [key14=="psiz", psize=6; If [w2head==Integer||w2head==Real, psize=w2]; AppendTo[points,Graphics3D[AbsolutePointSize[psize]]]]; If [key14=="ethi", ethik=1.25; If [w2head==Integer||w2head==Real, ethik=w2]; AppendTo[polyedge,Graphics3D[AbsoluteThickness[ethik]]]]; If [key14=="lthi", lthik=2; If [w2head==Integer||w2head==Real, lthik=w2]; AppendTo[lines,Graphics3D[AbsoluteThickness[lthik]]]]; If [key14=="tips", tips=1.5; If [w2head==Integer||w2head==Real, tips=w2]]; If [key14=="styl", If [w2len==3, style={TextStyle->{FontFamily->w2[[1]],

FontSize->w2[[2]],FontWeight->w2[[3]]}}]]; If [key14=="offs", offset={0.04,0.04,0.04}; If [w2len==3, offset=w2]]; plotopt={style,psize,ethik,lthik,offset,tips}; Return[{plotopt,polyfill,polyedge,lines,points,labels}]];

Figure G.8. Module PlotHex8Options that injects non default options in the plot command stream.

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PlotHex8Macro[keyw_,plotopt_,plotspecs_]:=Module[{i,node, t=plotopt[[6]],Nsub=plotspecs[[1]],FLP={0,0,0}, keylen=StringLength[keyw],kw=Null}, If [keylen


G.2.2. Source CodeA complete listing of PlotHex8 and its subordinate modules is given in Figures ? through ?.Because of its length, the driver module listing is split into Figures ? and ?. The module notebookcan be downloaded from the Index of Appendix G.

Remark G.1. Implementation Considerations. Some plot options and details are wired in the module. Thosecould be externally controlled with additional arguments in the plotspecs argument, but it would be just aseasy (in fact, faster) to go in and change the source code. For example, to pass from the AmbientLight diffuse-isotropic lighting to light point sources it would be necessary to modify theShow command options, which is rel-atively easy. Note that the secondShow statement contains a specification, namely, DisplayFunction->dfun,that is Mathematica version dependent; it was changed in version 6.

Remark G.2. Line Plotting Logic. PlotHex8Elem draws the line joining two specified points as a straightline in the Cartesian (physical) space if and only if it knows that to be the case. That happens for edges andmedians since mapping from {, , } to {x, y, z} is linear along lines where two hexahedral coordinates areconstant. Aside from that PlotHex8Elem assumes a curved line, and draws it in piecewise linear manner withNsub segments. To illustrate, observe that the diagonals displayed in Figure ?(e) are curved. For example,the diagonal joining corner 1 to the opposite corner 7 has the equation = = in natural (hexahedral)coordinates; plugging into the iso-P map gives a quadratic parametric form in {x, y, z}.

G.2.3. Individual Element Example PlotsFigure ? shows six sample plots of an individual hexahedron. Its geometry is defined by the node(corner) coordinate matrix

xyzhex= { { 0,0,0 }, { 2,0,0 }, { 2,1,0 }, { 0,1,0 },{ 0,0,0 }*c, { 2,0,0 }*c, { 2,1,0 }*c, { 0,1,0 }*c }; (G.2)

in which c is a parameter. This hexahedron has a 21 plane rectangular faces at z = 0 and at z = c.By setting c=0.4 to 0.4 the upper face is shrunk so the element resembles a truncated pyramid.The plots of Figure ? are produced by the script listed in Figure ?. Salient features are brieflydescribed in the figure legend. They are chosen to illustrate some commonly used options.

G.2.4. Multiple Element Example PlotsFigure ? shows two sample plots that comprise multiple elements. That of Figure ?(a) shows 7configurations of a hexahedron that is rigidly rotated about the z axis by angular increments of30, which are applied 6 times for a final total rotation of 180. The plot is produced by the scriptlisted in Figure ?. The node coordinates of the 7 configurations are stacked into a 7 8 3 listcalled xyzhex supplied as first argument. The following plots specifications are chosen: yields apoorly scaled plot), view={ 0,6,1.5 }, imgsiz=576, light=GrayLevel[1] (so ambient light iswhite), showbox=False, showxyz=True (for this plot type it is convenient to display axes), andplotsave=False. Only faces and edges are drawn; more features would clutter the display.The right plot in Figure ?(b) shows three configurations of a element undergoing large translationsand visible warping deformations. In the initial configuration (lower one in plot) the hexahedronis actually a rectanguar box. This is moved rigidy in the z direction while the element is warped.The plot is produced by the script listed in Figure ?. In this case the number of face subdivisionsis boosted to Nsub=6 to help in visualizing highly warped faces.

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(d) (e) (f)

(a) (b) (c)

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Figure G.10. Sample plots of an individual Hex8 element, with Nsub=4, boxrat={ 1,1,1 }, view={ 3,2,1.5 },and imgsiz=300 chosen for all: (a) shows faces, edges and labeled corners; (b) same as (a) except for whitelighting set via light=GrayLevel[1] and corner labels are omitted; (c) shows median faces and labeled naturalaxes , , (axis labels slightly moved with Illustrator for better view); (d) shows four natural coordinate surfaces = c, c = 1, 1/3, 1/3, 1 plus the labeled axis; (e) shows green-colored diagonals (curved because elementhas variable metric) and red-colored labeled sample points of the 222 Gauss rule note that Gauss point fallon the diagonals; (f) shows all lines (edges and face polygon edges) plotted in black with same thickness of 2.5pt, ambient light set to yellow via light=Hue[0.2], and labeled {x, y, z} axes drawn along plotting box edges.

xyzhex={{0,0,0}, {2,0,0}, {2,1,0}, {0,1,0}, {0,0,1}*c,{2,0,1}*c,{2,1,1}*c,{0,1,1}*c}/.c->.4;

etaaxis={{0,-1,0},{0,1.5,0}};Nsub=4; boxrat={1,1,1}; view={3,2,1.5}; showbox=False; imgsiz=300;light=GrayLevel[0]; showbox=False; showxyz=False; plotsave=False; plotsp={Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave};plotspwhite={Nsub,boxrat,view,GrayLevel[1],imgsiz,showbox,showxyz,plotsave};plotspaxes={Nsub,boxrat,view,Hue[.15],imgsiz,showbox,True,plotsave};PlotHex8Elem[xyzhex,{"faces","edges","labcorners"},plotsp];PlotHex8Elem[xyzhex,{"faces","edges","corners"},plotspwhite];PlotHex8Elem[xyzhex,{"medsurf","edges","naxes","labnaxes"},plotsp];PlotHex8Elem[xyzhex,{"edges","corners",etaaxis,{"offset",{0.0,0.05,0.0}}, {"psize",0},{{0,1.5,0},""},{"",-1},{"",-1/3},{"",1/3},{"",1}},plotsp];PlotHex8Elem[xyzhex,{"edges","labcorners",{"lcolor","green"},"diagonals", {"pcolor","red"},"gp222"},plotsp]; PlotHex8Elem[xyzhex,{{"fcolor","black"},{"ethick",2.5},"faces", {"lthick",2.5},"edges"},plotspaxes];

Figure G.11. Script to produce the individual element plots of Figure ?.

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(a) (b)

Figure G.12. Sample plots that contain multiple Hex8 elements.

RotateHex8[xyzhex_,_]:=Module[{c=Cos[],s=Sin[], xn,yn,zn,xyzrot=xyzhex}, For [n=1,n4,c->2,b->1/4,f->.4};xyzhex2=RotateHex8[xyzhex1, Pi/6];xyzhex3=RotateHex8[xyzhex1, Pi/3];xyzhex4=RotateHex8[xyzhex1, Pi/2];xyzhex5=RotateHex8[xyzhex1,2*Pi/3];xyzhex6=RotateHex8[xyzhex1,5*Pi/6];xyzhex7=RotateHex8[xyzhex1, Pi];xyzhex={xyzhex1,xyzhex2,xyzhex3,xyzhex4,xyzhex5,xyzhex6,xyzhex7};Nsub=3; boxrat={1,1,1/3}; view={0,6,1.5}; showbox=False; imgsiz=576;light=GrayLevel[1]; showbox=False; showxyz=True; plotsave=False; plotsp={Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave};PlotHex8Elem[xyzhex,{{"ethi",1.5},"faces","edges"},plotsp];

Figure G.13. Script to produce the multiple-element plot of Figure ?(a).

WarpHex8[xyzhex_,s_,h_]:=Module[{warp,xyzwrp=xyzhex}, warp={1,-1,1,-1,1,-1,1,-1}; For [n=1,n4,c->2,b->1/2,f->1};xyzhex2=WarpHex8[xyzhex1, .3,2];xyzhex3=WarpHex8[xyzhex1, .6,4];xyzhex={xyzhex1,xyzhex2,xyzhex3};Nsub=6; boxrat={1,1,1}; view={1,2,1}; showbox=False; imgsiz=400;light=GrayLevel[.2]; showbox=False; showxyz=True; plotsave=False; plotsp={Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave};PlotHex8Elem[xyzhex,{"faces","edges"},plotsp];

Figure G.14. Script to produce the multiple-element plot of Figure ?(b).

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G.3. Hex20 Element Plotting ModuleThe Mathematica modulePlotHex20Elem produces plots of the 20-node hexahedron finite elementdescribed in Chapter 11. The module aims to provide a comprehensive set of plotting options todisplay an individual element of that particular configuration.Although the module can display multiple elements, it is not recommended for showing a finiteelement mesh because PlotHex20Elem focuses on local (intrinsic) element properties. As a matterof fact, such multiple elements need not be connected in any way, as they would be in a mesh. Thatlack of restrictions has some practical uses, for example to illustrate initial and final configurationsof the same element. Another module, called PlotHex20Mesh and described in Section ?, is moresuitable for plotting a hexahedral mesh.

Returns x,y,z given,, using iso-Pshape functions ofHex20 element

Draws Hex8 facesas regular mesh of color filled polygonswith edge lines

Draws lines (e.g., hexahedron edges) joining two end pointsof given location

PlotHex20ElemDriver that generates plotstream by interpreting"plotwhat" commands, and displays or savesplot image

Draws points at givenlocations and places labels nearby (point orlabel may be omitted)

Injects plotting optionsthat affect downstreamcolors, point size, line thickness and labels,into the plot stream.

Expands a macro command keyword intosequence of low-levelinstructions

PlotHex20Faces PlotHex20Lines PlotHex20Points PlotHex20Options PlotHex20Macro Hex20CarInterp

Figure G.15. Organization of PlotHex20 graphics module.

The overall organization is diagrammed in Figure ?. On comparing Figures ? and ?, it is plain theorganization mimics that of 8-node tetrahedron plotting module PlotHex8Elem, which is describedin ?. In fact over 95% of the code is identical. In addition PlotHex20Elem is invoked with thesame arguments, but has some additional plotting commands because of the presence of midnodes.Module PlotHex20Elem is actually a driver that interprets an array of plot what com-mands, and calls subordinate modules PlotHex20Faces, PlotHex20Lines, PlotHex20Points,PlotHex20Options, and PlotHex20Macro as appropriate to construct the plot stream. Anothersubordinate module, Hex20CarInterp, is called by PlotHex20Faces, PlotHex20Lines, andPlotHex20Points to map natural to Cartesian coordinates using the Hex20 shape functions.Once all instructions are processed, the plot image is created using the built in Show function. Theimage is either output to the screen, or returned as a list, according to input specifications.As in the other Sections, the abbreviation pt denotes a printer point, which is a length dimensionof 0.35146 mm = 0.013837 inches = (1/72.27) inches.

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G.3.1. UsageThe module may be invoked in either of two ways:

PlotHex20[xyzhex,plotwhat,plotspecs]phex= PlotHex20[xyzhex,plotwhat,plotspecs]

(G.3)

The first form outputs a plot image to the screen. The second form displays no image but places theplot object into phex as function return. A logical flag provided in argument plotspecs determinesthe appropriate calling form, as described below.The arguments are:

xyzhex To plot an individual element: nodal coordinates stored node-by-node as a 203matrix: { { x1,y1,z1 },{ x2,y2,z2 },{ x3,y3,z3 }, . . . { x20,y20,z20 } }.Corner nodes are 18 whereasa midside nodes (ofthen called simplymidnodes) are not necessarily collocated at the midpoints of the sides.To plot m > 1 elements: stack their nodal coordinate matrices in a m 20 3three dimensional list: { xyzhex1,. . .,xyzhexm }. Elements stored in the listmay be totally independent of each other.

plotwhat A list of commands specifying what is to be plotted. The list may includecommands of three types.Macro commands. A character string that collectively represents a sequenceof lower level commands. For example: "faces" produces a plot of allelement faces and is equivalent to a list of six more detailed commands:{ "",-1 },{ "",1 },{ "",-1 },{ "",1 },{ "",-1 },{ "",1 }. Availablemacro commands are listed in Table ?.Low-level commands. These specify only individual action, such as : plot apoint, plot a line, insert a label, etc. Low level commands are listed in Table ?.Options commands. These are used to override default settings such as colors,line thicknesses, point sizes and label styles. For example, by default pointsare show as black disks of 6-pt diameter. To have them displayed in red with4-pt diameter, say { "pcolor","red" } and { "psize",4 } before entering thecommands to draw those points. Option commands are listed in Table ?. Thelast column of this Table gives the default settings.

plotspecs A list containing the following items:{ Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave }which can be used to control the overall appearance of the plot. These itemsare described in Table ?.

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G.3 HEX20 ELEMENT PLOTTING MODULETable G.5. Macro Commands in plotwhat Argument of PlotHex20Elem

Command What to plot Equivalent to low-level command sequence

"corners" Corner nodes without labels { 1 },{ 2 },{ 3 },{ 3 }, . . ., { 8 }"midpoints" Midpoint nodes without labels { 9 },{ 10 },{ 11 },{ 12 }, . . ., { 20 }"nodes" All node points without labels { 1 },{ 2 },{ 3 },{ 3 },{ 4 },. . .,{ 20 }"diagonals" Diagonal lines (from each corner to

opposite corner){ 1,7 },{ 2,8 },{ 3,5 },{ 4,6 }

"edges" or

"sides"

Edges (generally curved) { 1,2 },{ 1,4 },{ 1,5 },{ 2,3 },{ 2,6 },{ 3,4 },{ 3,7 },{ 4,8 },{ 5,6 },{ 5,8 },{ 6,7 },{ 7,8 }

"faces" All faces using filled polygons, withNsub subdivisions along each direction.

{ "",-1 },{ "",1 },{ "",-1 },{ "",1 },{ "",-1 },{ "",1 }.

"fcenters" Face center points without labels { -1,0,0 },{ 1,0,0 },{ 0,-1,0 },{ 0,1,0 },{ 0,0,-1 },{ 0,0,1 }

"gp111" Labeled Gauss pt for 111 rule,labeled G1

{ { 0,0,0 },"G1" }

"gp222" Labeled Gauss pts for 222 rule See Table ?."gp333" Labeled Gauss pts for 333 rule See source code in Figure ?."labcor" Labeled corner nodes { 1,"1" },{ 2,"2" }, . . . ,{ 8,"8" }"labmid" Labeled midnodes { 9,"9" },{ 10,"10" }, . . . ,{ 20,"20" }"labnodes" Labeled nodes { 1,"1" },{ 2,"2" }, . . . ,{ 20,"20" }"labfcent" Labeled face center points See Table ?.

"labnaxes"

or"labhaxes"

Natural axes labels {, , } placednear tips. Axis tips are not plotted

See Table ?.

"medians" Median lines (from each face centerto opposite face center)

{ -1,0,0 },{ 1,0,0 } },{ { 0,-1,0 },{ 0,1,0 } },{ { 0,0,-1 },{ 0,0,1 }

"medsurf"or

"medplanes"

Median surfaces = 0, = 0,and = 0

{ { "",0 },{ "",0 },{ "",0 } }

"naxes" or

"haxes"

Natural axes {, , }without labels.Tips usually stick outside element.

{ { -1,0,0 },{ ,0,0 } },{ { 0,-1,0 },{ 0,,0 } },{ { 0,0,-1 },{ 0,0, } },in which = 1.5 by default.

Axis-tip distances from faces may be adjusted with the { "tips", } option; see Table ?

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Table G.6. Low-level Commands in plotwhat Argument of PlotHex20Elem

Command Format What to plot

{ n } n: integer in range 18 Corner point n, without label{ n,label } n: integer in range 18; label: a

character stringCorner point n with label

{ n1,n2 } n1,n2: integers in range 18 Line joining corners n1 and n2{ m } m: integer of form 100*n1+n2, with

both n1,n2 integers in range 18.Same effect as { n1,n2 }.

{ ,, } ,,: numeric values (real, integeror fraction) of natural coordinates.Values outside range [1, 1] are OK;if so point will fall out of element.

Point at location { ,, }, without label

{ { ,, },label } ,,: as in previous case; label:a character string

Point at location { ,, }, with label.To suppress point plotting, preceed withoption command { "psize",0 }

{ { 1,1,1 },{ 2,2,2 } }

{ 1,1,1 } and { 2,2,2 }: :natural coordinates of two points

Line joining the two points



{ "",c } c: coordinate of surface =c Plot surface =c within element If c is outside the range [1, 1], the plotted plane is outside the element, but will be bounded

by the [1, 1] range of of the other two coordinates.

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G.3 HEX20 ELEMENT PLOTTING MODULETable G.7. Option Commands in plotwhat Argument of PlotHex20Elem

Command Description Defaults

{ "fcolor",color } Set face color for subsequent face plots. Here coloris either a list { R,G,B } of 3 numbers, all in range[0, 1], which defines a RGB specification, or one ofthe following strings: "black", "red", "green",or "blue". For example: { "fcolor","blue" }and { "fcolor",{ 0,0,1 } } are equivalent

Mathematica defaultfor 3D graphics (thesedefaulst may vary withversion).

{ "lcolor",color } Set face color for subsequent line plots. Here coloris as described above.

Black

{ "pcolor",color } Set face color for subsequent point plots. Herecolor is as described above.

Black

{ "style",{ fontfamily,fontsize,fontslant } }

Set font style for subsequent labels. Here fontfamis a string that defines the font family, e.g., "Times","Helvetica", or"Courier"; fontsize is a numberdefining the font size in printer points; fontslant isone of the strings "Plain", "Bold" or "Italic".Strings for fontfamily and fontslant must becapitalized as per Mathematica naming rules.

Font family: Times,font size: 14 pt,font slant: Plain

{ "psize",size } Value psize specifies the size, in printer pt, ofsubsequent point to be plotted. If psize is 0, thosepoints are not shown. This is handy to show labels-only at specified locations, omitting points.

Point size: 6 pt

{ "ethick",thickness } Value thickness specifies the thickness, in printerpoints, of edges of subsequent face polygons.

Polygon edgethickness: 1.25 pt

{ "lthick",thickness } Value thickness specifies the thickness, in printerpoints, of subsequent lines to be plotted; e.g.,{ "lthick",1.5 }. By line it is meant thoseproduced by jpining two specified points; for exampleedges, medians and diagonals.

Line thickness: 2 pt

{ "offset",{ x,y,z } }

The list { x,y,z }defines the offset, in {x, y, z}coordinate units, from a point (actually the pointcenter) to its respective label. This setting applies toall subsequent labels until reset.

x=x=z=0.04.

{ "tips", } Defines ends (tips) of natural axes, as well as labelspositions, drawn with macro commands "naxes"and "labnaxes" (Table ?). Tips placed at = , = and = , for axes {, , }, respectively.

= 1.5, whenceaxes tips stick 50%outside element.

Note: The label offset specification in terms of coordinate values is awkward, but unfortunatelyMathematica lacks the ability to specify an absolute distance between graphic objects in points.

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Table G.8. Items in plotspecs Argument of PlotHex20Elem

Item Description Recommended specifications

Nsub A positive integer that specifies the subdivisions whenplotting faces, natural-coordinate surfaces, or lines. Forexample, if Nsub=4, the face or surface will be renderedwith 42 = 16 filled polygons. Since element faces orsurfaces are (possibly warped) quadrilaterals, polygonsare also quadrilaterals.

2 to 4, but may be adjusted forappearance. If face or surface ishighly warped or doubly curved,a larger Nsub (e.g., 6 or 8) may beappropriate.

boxrat A list of three positive numbers that specifies the {x, y, z}side ratios of the plot bounding box. Set throughthe Mathematica 3D graphics option BoxRatios ->boxrat. If an empty list: { }, the module usesBoxRatios -> Automatic to specify default settings.

The default { }, which triggersBoxRatios -> Automatic, of-ten works OK, but plots with highaspect ratios may require fiddlingwith 3 numbers.

view A list of 3 numeric items that specifies the location in{x, y, z} space from which the objects plotted are tobe viewed. Set through the Mathematica 3D graphicsoption ViewPoint -> view.

Start with a reasonable guess, e.g.,{ 1,1,1 }, but experimentation willbe usually needed.

light A specification, usually stated in terms of one of thebuilt-in functions GrayLevel, Hue or RGBColor, whichsets a level of diffuse isotropic lighting of the plot object.Set through the 3D graphics option AmbientLight ->light. For other types of lighting, such as point lightsources, the Show statement within PlotHex20Elemhas to be modified; cf. Remark ?.

GrayLevel[0], but some experi-mentation may be required. Hue[0]can also be a useful start. As h inHue[h] changes from 0 to 1, theambient light color ranges fromred, yellow, green, cyan, blue,magenta, and back to red.

imgsiz Width of the plot in printer points. Specified through thegraphics option ImageSize -> imgsiz

300, which is about 4 inches or 10cm.

showbox A logical flag. If True, the plot bounding box is drawnvia the 3D graphics option Boxed -> True

False

showxyz A logical flag. If True, the labeled Cartesian axes aredrawn along the edges of the plot bounding box via the3D graphics options Axes -> True and AxesLabel->{ "x","y","z" }.

False

plotsave A logical flag. If True, do not display plot onscreen, but return the plot object as function value fromPlotHex20Elem; see invokation format in (?). Usefulfor downstream combining plots from various sources.

False, unless plot is to be mergeddownstream with other plots.

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PlotHex20Elem[xyzhex_,plotwhat_,plotspecs_]:=Module[{i,j,k,m,n,p,e, ip,n1,n2,np,ne=1,what,w,whead,wlen,kw,w1,w2,w1head,w1len,w2head,w2len, F,L,P,expMC,Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave, boxdim,ps,black,red,blue,psizdef=6,ethidef=1.25,lthidef=2,tipsdef=1.5, styledef={TextStyle->{FontFamily->"Times",FontSize->14, FontWeight->"Plain"}},offsetdef={0.04,0.04,0.04},plotopt, axespec=False,axeslab=" ",polyfill,polyedge,lines,points,labels, optkey,key,dfun,xyzc=xyzhex,mulele,phex,modname="PlotHex20Elem:"}, optkey[key_]:=StringLength[key]>=4&&Length[StringPosition[

validoptkeys,StringTake[key,4]]]>0; validoptkeys="ethi fcol pcol lcol lthi psiz styl offs tips"; polyfill=polyedge=lines=points=labels={}; plotopt={styledef,psizdef,ethidef,lthidef,offsetdef,tipsdef}; {Nsub,boxrat,view,light,imgsiz,showbox,showxyz,plotsave}=plotspecs; If [showxyz, axespec=True; axeslab={"x","y","z"}]; {black,red,blue}={RGBColor[0,0,0],RGBColor[1,0,0],RGBColor[0,0,1]}; mulele=Length[Dimensions[xyzhex]]==3; If [mulele, ne=Length[xyzhex]]; np=Length[plotwhat]; If [np0, {points,labels}=PlotHex20Points[

xyzc,expMC,plotopt,points,labels]; p=True]; If [P=1&&w1=102&&w1


If [wlen==3, If [whead==List, {points,labels}=PlotHex20Points[ xyzc,{what},plotopt,points,labels]; p=True];

]; If [!p, Print[modname," Illegal command ",what]]; ]]; boxdim=Automatic; If [Length[boxrat]==3, boxdim=boxrat]; If [plotsave, phex=Show[Graphics3D[red],polyfill,

Graphics3D[AbsoluteThickness[ethidef]],polyedge, Graphics3D[black],Graphics3D[AbsoluteThickness[lthidef]], lines,Graphics3D[black],Graphics3D[AbsolutePointSize[psizdef]], points,Graphics3D[black],labels,AmbientLight->light, ViewPoint->view,Axes->axespec,AxesLabel->axeslab, BoxRatios->boxdim,ImageSize->imgsiz, PlotRange->All,Boxed->showbox,DisplayFunction->Identity]; ClearAll[xyzc,polyfill,polyedge,sides,points,labels]; Return[phex]];

dfun=$DisplayFunction; If [$VersionNumber>=6.0, dfun=Print]; Show[Graphics3D[red],polyfill,

Graphics3D[AbsoluteThickness[ethidef]],polyedge, Graphics3D[black],Graphics3D[AbsoluteThickness[lthidef]], lines,Graphics3D[black],Graphics3D[AbsolutePointSize[psizdef]], points,Graphics3D[black],labels,AmbientLight->light, ViewPoint->view,Axes->axespec,AxesLabel->axeslab, BoxRatios->boxdim,ImageSize->imgsiz, PlotRange->All,Boxed->showbox,DisplayFunction->dfun];

ClearAll[xyzc,polyfill,polyedge,sides,points,labels]; Return[]];

Figure G.17. Plot driver module PlotHex20Elem, Part 2.

PlotHex20Faces[xyzc_,facelist_,Nsub_,polyfinc_,polyeinc_]:= Module[ {i,j,kf=Length[polyfinc],ke=Length[polyeinc], n=Nsub,nf=Length[facelist],if,ic,cval, am,ap,bm,bp,xyz1,xyz2,xyz3,xyz4,polyfill,polyedge}, If [n


PlotHex20Lines[xyzc_,linelist_,Nsub_,linesinc_]:=Module[ {i,k,nl=Length[linelist],lk,nn,1,2,d,beg,end, xyz1,xyz2,nbeg,nend,n1,n2,lindef,nod,lines=linesinc}, nod={{-1,-1,-1},{1,-1,-1},{1,1,-1},{-1,1,-1}, {-1,-1,1}, {1,-1,1}, {1,1,1}, {-1,1,1}, {0,-1,-1}, {1,0,-1}, {0,1,-1},{-1,0,-1}, {-1,-1,0}, {1,-1,0}, {1,1,0}, {-1,1,0}, {0,-1,1}, {1,0,1}, {0,1,1}, {-1,0,1}}; For [k=1,k=1&&n1=1&&n2


Hex20CartInterp[xyzc_,_]:=Module[{Ne,v,v,v}, Ne[{v_,v_,v_}]:= {-(1-v)*(1-v)*(1-v)*(2+v+v+v)/8, -(1+v)*(1-v)*(1-v)*(2-v+v+v)/8, -(1+v)*(1+v)*(1-v)*(2-v-v+v)/8, -(1-v)*(1+v)*(1-v)*(2+v-v+v)/8, -(1-v)*(1-v)*(1+v)*(2+v+v-v)/8, -(1+v)*(1-v)*(1+v)*(2-v+v-v)/8, -(1+v)*(1+v)*(1+v)*(2-v-v-v)/8, -(1-v)*(1+v)*(1+v)*(2+v-v-v)/8,

(1-v^2)*(1-v)*(1-v)/4,(1+v)*(1-v^2)*(1-v)/4, (1-v^2)*(1+v)*(1-v)/4,(1-v)*(1-v^2)*(1-v)/4, (1-v)*(1-v)*(1-v^2)/4,(1+v)*(1-v)*(1-v^2)/4, (1+v)*(1+v)*(1-v^2)/4,(1-v)*(1+v)*(1-v^2)/4, (1-v^2)*(1-v)*(1+v)/4,(1+v)*(1-v^2)*(1+v)/4, (1-v^2)*(1+v)*(1+v)/4,(1-v)*(1-v^2)*(1+v)/4};

Return[Simplify[Ne[].xyzc]]];

Figure G.21. Module Hex20CartInterp that returns {x, y, z} of a point, given its natural(hexahedral) coordinates.

PlotHex20Options[opt_,graphicsinc_]:=Module[{w1,w2,key14,key24, w2head,w2len,R,G,B,color,style,psize,ethik,lthik,offset,tips, plotopt,polyfill,polyedge,lines,points,labels}, {plotopt,polyfill,polyedge,lines,points,labels}=graphicsinc; {style,psize,ethik,lthik,offset,tips}=plotopt; {w1,w2}=opt; w2head=Head[w2]; w2len=Length[w2]; key14=StringTake[w1,4]; key24=StringTake[w1,{2,4}]; If [key24=="col", R=G=B=0; color=False; If [w2head==String, If [w2=="black", color=True]; If [w2=="red", color=True; R=1]; If [w2=="green", color=True; G=1]; If [w2=="blue", color=True; B=1]]; If [w2head==List&&w2len==3, color=True; {R,G,B}=w2]; If [key14=="pcol"&&color, AppendTo[points,Graphics3D[RGBColor[R,G,B]]]]; If [key14=="lcol"&&color, AppendTo[lines,Graphics3D[RGBColor[R,G,B]]]]; If [key14=="fcol"&&color, AppendTo[polyfill,Graphics3D[RGBColor[R,G,B]]]]]; If [key14=="psiz", psize=6; If [w2head==Integer||w2head==Real, psize=w2]; AppendTo[points,Graphics3D[AbsolutePointSize[psize]]]]; If [key14=="ethi", ethik=1.25; If [w2head==Integer||w2head==Real, ethik=w2]; AppendTo[polyedge,Graphics3D[AbsoluteThickness[ethik]]]]; If [key14=="lthi", lthik=2; If [w2head==Integer||w2head==Real, lthik=w2]; AppendTo[lines,Graphics3D[AbsoluteThickness[lthik]]]]; If [key14=="tips", tips=1.5; If [w2head==Integer||w2head==Real, tips=w2]]; If [key14=="styl", If [w2len==3, style={TextStyle->{FontFamily->w2[[1]], FontSize->w2[[2]],FontWeight->w2[[3]]}}]]; If [key14=="offs", offset={0.04,0.04,0.04}; If [w2len==3, offset=w2]]; plotopt={style,psize,ethik,lthik,offset,tips}; Return[{plotopt,polyfill,polyedge,lines,points,labels}]];

Figure G.22. Module PlotHex20Options that injects non default options in the plot command stream.

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PlotHex20Macro[keyw_,plotopt_,plotspecs_]:=Module[{i,nod, t=plotopt[[6]],Nsub=plotspecs[[1]],FLP={0,0,0},gp333, g=Sqrt[3/5],keylen=StringLength[keyw],kw=Null}, If [keylen


G.3.2. Source CodeA complete listing of PlotHex20 and its subordinate modules is given in Figures ? through ?.Because of its length, the driver module listing is split into Figures ? and ?. The module notebookcan be downloaded from the Index of Appendix G.

Remark G.3. Implementation Considerations. The same considerations expressed in Remark ? hold: if anew plot feature needs to be inserted, the quickest way is to modify the source code.

Remark G.4. Line Plotting Logic. Unlike the Hex8 plotting module, any line that joins two points is assumedto be curved, and is drawn with Nsub piecewise linear segments. The straight-line versus curved-line switchlogic described in Remark ? does not apply.

G.3.3. Individual Element Example PlotsFigure ? shows six sample plots of an individual 20-node hexahedron.The geometry is roughlysimilar to the 8-node hexahedrom shown in Figure ?. One starts the same way by defining thecorner nodes as given in (?). This 8-node hexahedron has a 2 1 plane rectangular faces at z = 0and at z = c. By setting c=0.4 to 0.4 the upper face is shrunk so the element resembles a truncatedpyramid. Then 12 midpoints are placed at the midpoints of the sides. Finally two midnodes: 11and 12, are moved down and up by 1/8 along the z directions, recpectively, to produce some visiblycurved faces.The plots of Figure ? are produced by the script listed in Figure ?. Salient features are brieflydescribed in the figure legend. They are chosen to illustrate some commonly used options.

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1718 1920

00.511.52

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(d) (e) (f)

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G1

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G5G6 G7

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Figure G.24. Sample plots of an individual Hex20 element, with Nsub=8, boxrat={ 1,1,1 }, view={ 4,3,1.5 },and imgsiz=300 chosen for all: (a) shows faces, edges and labeled corners; (b) same as (a) except for whitelighting set via light=GrayLevel[1] and node labels are omitted; (c) shows median faces and labeled naturalaxes , , (axis labels slightly moved with Illustrator for better view); (d) shows four natural coordinate surfaces = c, c = 1, 1/3, 1/3, 1 plus the labeled axis; (e) shows corner and midnodes with different colors (blackand blue, repectively, while point size is boosted to 8 pt), green-colored diagonals (curved because element hasvariable metric) and red-colored labeled sample points of the 2 2 2 Gauss rule (note that Gauss point fall onthe diagonals); (f) shows all lines (edges and face polygon edges) plotted in black with same thickness of 1.25 pt,

ambient light set to yellow via light=Hue[0.2], and labeled {x, y, z} axes drawn along plotting box edges.

xyzcor={{0,0,0}, {2,0,0}, {2,1,0}, {0,1,0}, {0,0,1}*c,{2,0,1}*c,{2,1,1}*c,{0,1,1}*c}/.c->.4;mij={{1,2},{2,3},{3,4},{4,1},{1,5},{2,6},{3,7},{4,8},{5,6},{6,7},{7,8},{8,5}};xyzhex=Table[0,{20}]; For [n=1,n

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