(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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As it happenned, we used the interactive front-end to NAG for prototyping only: the real runs were done using stand-alone code calling NAG (because of conservatism of our engineering clients: see section below on "Remarks on the software"). Here we merely describe the InterCall prototyping phase. We remark that readers who only have the Foundation Library subset of NAG, not the full library, can run the essential parts of this Notebook as d02raf and d02bbf are both in the Foundation Library. At sites where IMSL is available and NAG is not, just substitute the corresponding IMSL routines. There are other Fortran codes that could do an equivalent (or better) job with our engineering application: Keller's AUTO branch-following package is an example. We solve an engineering problem involving inflation of a flat circular rubber membrane. This arises in determining rubber properties through comparison of experimental observations and numerics. The whole process - including the experiments, and the reason for them [H2] - occured in engineering project work of Evatt Hawkes. By measurements of strain and curvature at the pole of the inflated disk, and inflating pressure, it is possible to obtain an estimate of the parameters (C1 and C2 below) defining the strain energy function of the rubber. In order to check the determined parameters it is useful to solve the full differential equations describing the inflation. This solution may then be compared with the experimentally obtained inflated profiles to check that the model and improve our estimates of the parameters. The equations are highly nonlinear due to nonlinear nature of rubber as a material, and due to the large deformations involved. Numerical solution is appropriate. The numerical problem, which we solve in this Notebook, is a two-point boundary-value problem for a system of nonlinear d.e.s. For nearly two decades the NAG f77 library has been a reliable workhorse for numerics at UWA. Its documentation, its on-line help, its example programs and its widespread availability have been major factors for continuing to use it. The same factors, and a certain coherence of style through the library, have also lent its use to various 'links', of which InterCall is a major example: see [BKRRD], [HK]. f77 and the NAG f77 library are old, but still work. (NAG f77 library usage at UWA today is being squeezed from various directions: see section on "Remarks on software".) Returning to the application in this Notebook, the NAG routine d02raf is the essential tool and is ideal for our rubber-properties problem. InterCall is a marvellously convenient tool for linking out to the established Fortran codes (or to C which is 'Fortran-like'). In references [HK], [H1] we described - using exactly this rubber properties problem - our use of genmex, an analogous tool but for Matlab rather than Mathematica . Users of genmex and of InterCall can share databases: more on this in our section on "Remarks on the software". ;[s] 3:0,0;3024,1;3036,0;3151,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Analytical phase: Mathematica description ;[s] 3:0,0;18,1;29,0;42,-1; 2:2,19,14,Times,1,18,0,0,0;1,19,14,Times,3,18,0,0,0; :[font = text; inactive; preserveAspect] The original interest in this circular-disk problem stemmed from extensive experimental investigations by Treloar [Trel].Theory for the problem has been treated before by Green and Adkins [GA], Adkins and Rivlin [AR], Klingbiel and Shield [KS] , and many others. The work below follows similar lines to these. They, however, consider a wider range of problems while we proceed directly with the case of the inflation of a flat circular membrane composed of incompressible homogeneous isotropic material. The first few paragraphs should be regarded as defining the physical significance of variables occuring in the Mathematica code. For a coherent account of the mathematical modelling for this problem, see Hawkes [H1]. There is also no claim that this is necessarily the only way to solve the problem (use of variational formulations may be better). However, we wish to keep our account similar to that we have used [HK], [H1], in previous numerical solution in other systems. The geometry is as follows. The undeformed configuration is a flat circular disk of rubber, in a plane z=constant. When inflated the rubber deforms into a curved axisymmetric shape, so (r,z) coordinates are appropriate. We adopt a 'particle-following' approach, and will seek functions r(rho), z(rho) for a point Q which in the undeformed state is at a distance rho from the axis of symmetry (the z-axis). The arc-length distance along the deformed meridian from the deformed centre of the disk to the point Q is denoted by s. The origin of our (r,z) coordinates is at the deformed centre of the disk. Thus z is the distance of the point Q along a line parallel to the axis of symmetry from the z=0 plane containing the deformed centre point. We also define omega to be the angle between the tangent at Q to the deformed meridian and the axis of symmetry. Our problem is to find the functions of rho: z=z(rho), omega=omega(rho), r=r(rho) and s=s(rho). The first two (with r(0)=0, s(0)=0) determine the last two through the geometrical relations, written in several equivalent ways: D[r,rho] = Tan[ omega ] D[z,rho] ; D[s,rho]= Sqrt[ D[r,rho]^2 + D[z,rho]^2 ]; D[r,rho] = Sin[omega] D[s,rho] ; D[z,rho] = Cos[omega] D[s,rho] ; In local calculations of elastic stresses and strains and tensions, the principal curvatures kappa1 and kappa2 occur, and these are related to previous variables by: kappa1 = -D[omega,rho]/D[s,rho] ; kappa2 = Cos[omega]/r ; The four d.e.s we will solve are for lambda1(rho), lambda2(rho), omega(rho) and z(rho). lambda1, lambda2 denote the principal extension ratios: lambda1 = D[s,rho] ; lambda2 = r/rho ; and the third principal extension ratio, lambda3, the ratio of the deformed to the undeformed thickness of the membrane, must satisfy lambda3 = 1/(lambda1 lambda2) ; due to the incompressibility condition. We are already well on the way to our 4 d.e.s. The rhs of the j-th d.e., when completed, is denoted (* j *). Steps on the way are denoted (* ja *), (* jb *), etc. ;[s] 3:0,0;620,1;631,0;3153,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect] rho =.; lambda1 =.; lambda2 =.; omega =.; z =.; domegadrho= -Kappa1*lambda1; (* 3a *) dzdrho=lambda1*Cos[omega]; (* 4 *) drdrho=lambda1*Sin[omega]; dlambda2drho= (drdrho - lambda2)/rho; (* 1 *) r = rho*lambda2; kappa2 = Cos[omega]/r; :[font = text; inactive; preserveAspect] Considering force balances we have the following. Here P is the internal pressure, another quantity measured in the experiments. T1 corresponds to the meridional stress resultant per unit length of the deformed circumference and T2 the hoop stress resultant per unit length of the deformed meridian. We have kappa1 T1 + kappa2 T2 = P ; D[T1 r,rho]/D[r,rho] = T2; :[font = input; preserveAspect] P=.; kappa1 = (P - T2*kappa2)/T1; domegadrho = domegadrho /. {Kappa1 -> kappa1}; (* 3b *) dT1drho = (T2-T1)*drdrho/r; :[font = text; inactive; preserveAspect] Finally, we incorporate some specific constitutive relation for the rubber. The rubber's behaviour is determined through its strain energy function W. Here we use the Mooney strain energy. In the numerics we take C1 and C2 as given. (The real engineering problem involved using the numerics in combination with experiments to determine the C1 and C2 for pieces of rubber.) :[font = input; preserveAspect] C1 = .; C2 = .; w = C1*(lambda1^2+lambda2^2+(lambda1*lambda2)^(-2)-3)+ C2*(lambda1^(-2)+lambda2^(-2)+(lambda1*lambda2)^2-3); :[font = text; inactive; preserveAspect] Relating the strain energy function to the stresses and thence the geometry proceeds as follows. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Derivation of next equations, for t1,t2 (omit on first reading) :[font = text; inactive; preserveAspect; endGroup] The principal stresses are denoted sigma1, sigma2, sigma3. The Cauchy stress relations, see [Ogd], relate these to partial derivatives of W with respect to the corresponding lambda. In a membrane, the direct normal stress, sigma3 is considered negligible in comparison with sigma1 and sigma2, so sigma1= lambda1 D[W,lambda1]; sigma2= lambda2 D[W,lambda2]; Now if h is the deformed half thickness, and h0 is the undeformed half thickness of the sheet, then the following equation applies: T_i=2 sigma_i h = 2 lambda3 h0 sigma_i= (2 h0/(lambda1 lambda)) sigma_i The applicable relations for the stress resultants are therefore T1 = (2 h0/lambda2) D[W,lambda1]; T2 = (2 h0/lambda1) D[W,lambda2] ; as we enter, as t1,t2 into Mathematica . ;[s] 3:0,0;784,1;795,0;798,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = text; inactive; preserveAspect] Continue now with essential items. The stress resultants T1 and T2 are related to the principal extension ratios via the strain energy function W by, ;[s] 2:0,1;34,0;151,-1; 2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect] h0 =.; t1= 2*h0*D[w,lambda1]/lambda2; t2= 2*h0*D[w,lambda2]/lambda1; domegadrho = domegadrho /. {T1->t1, T2->t2}; (* 3 *) :[font = text; inactive; preserveAspect] (h0 is the undeformed half-thickness of the rubber sheet, another quantity we are given.) We now have the rhs of 3 of the 4 d.e.s - those for z(rho), omega(rho) and lambda2(rho). The rhs of the last differential equation, for lambda1(rho), involves derivatives of T1. :[font = input; preserveAspect] dlambda1drho = (dT1drho - dT1dlambda2*dlambda2drho)/ dT1dlambda1; (* 2a *) dt11=D[t1,lambda1]; dt12=D[t1,lambda2]; dlambda1drho = dlambda1drho /. { T1 -> t1, T2 -> t2, dT1dlambda1 -> dt11, dT1dlambda2 -> dt12}; (* 2 *) :[font = text; inactive; preserveAspect] Equations (1), (2), (3) and (4) above form a set of first order nonlinear coupled ordinary differential equations. Note it is convenient, in equation (4), to keep track of the z-variable for the purpose of obtaining the final deformed profile, even though it is not necessary for solution of the other variables. Thus we are left with four nonlinear o.d.e.'s for the variables z, omega, lambda1, lambda2 as functions of the independent variable rho . The appropriate boundary conditions are: lambda1=lambda2 at rho=0, due to the polar symmetry, omega=Pi /2 at rho = 0, z=0 at rho=0, and lambda2=1 at rho=r0 since the edge of the disk is clamped. Adkins and Rivlin [AR] employed a Taylor series approach to calculate the deformation, and Klingbiel and Shield [KS] effected a solution to the differential equations using a central difference method. Both converted the two-point boundary value problem to an initial value problem by using a scaling technique, with the unfortunate consequence that the pressure term was also scaled. Hence the solution could not be developed for a particular pressure, which makes a comparison with experimental results more cumbersome. InterCall/d02raf solves the problem as posed. The next step on the way to do this is to ask Mathematica to write the right-hand side of the equations in form suitable to provide as an ASP for the NAG routine d02raf. ;[s] 3:0,0;1288,1;1299,0;1415,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; preserveAspect; endGroup] :[font = section; inactive; preserveAspect; startGroup] ASP code generation for d02??f in Mathematica ;[s] 2:0,0;34,1;46,-1; 2:1,19,14,Times,1,18,0,0,0;1,19,14,Times,3,18,0,0,0; :[font = text; inactive; preserveAspect] In this Notebook we solve the two-point BVP - as formulated in the preceding section - directly using the NAG Fortran library routine D02RAF, see [NAG]. D02RAF solves two-point BVPs with general boundary conditions for a system of ordinary differential equations. (D02RAF uses a finite difference technique with deferred correction allied to a Newton iteration.) - Specifically D02RAF solves a two-point boundary-value problem for a system of n ordinary differential equations on the interval (a,b). In this NAG routine, the independent variable (our rho) is denoted x, and the dependent variables by y. The right-hand side of the i-th d.e. is denoted by f_i: thus D[y_i,x]=f_i. The n nonlinear boundary conditions, g_i(y(a),y(b))=0. Both the function values f_i and the boundary condition functions, g_i are evaluated by Argument SubPrograms, ASPs, supplied by the user. - The user may supply an initial mesh for the finite difference equations and an initial approximate solution. Alternatively a default mesh of zeros is used (though this alternative won't work for our rubber-properties application). The solution and mesh is refined by the program, with additional points being added to the mesh, until the error is less than the user specified tolerance everywhere. - The Newton iteration requires Jacobian matrices for f and g with respect to y. These may be supplied by the user ASP's, or calculated by numerical differentiation using a routine supplied by NAG. In this Notebook, NAG's numerical differentiation scheme was employed. Concerning the initial mesh and guess for D02RAF, we use the NAG initial-value problem solver D02BBF to find these. The notation is very similar. Partly to illustrate the process of developing the code we also include the D02BAF example. (Terry Robb's NAG Example's booklet works the Example as in the NAG documentation for D02BAF. All we had to do for our function f, FCN in the NAG code, is to adapt Terry's example to our purpose.) :[font = input; preserveAspect] rho = x; f =.; (* lambda1 = y[[2]]; lambda2 = y[[1]]; omega = y[[3]]; z = y[[4]]; f={dlambda2drho,dlambda1drho,domegadrho,dzdrho} *) :[font = input; preserveAspect] h0data = 0.0015; Pdata = 0.13; C1data = 0.5; C2data = 0.05; (* as in [H]. Also given there is Pdata = 0.23 *) :[font = input; preserveAspect; startGroup] !!d02bbFCN.mm :[font = print; inactive; preserveAspect; endGroup] (* FILE d02bbFCN.mm For producing FCN SUBROUTINE ASP for d02bbf and d02baf Frequently used test values include h0=0.0015, P=0.13, C1=0.5, C2=0 or C2=0.05 *) fcn=Function[{x,y,f}, Block[{ h0= <* ToString[InputForm[h0data]] *>, P= <* ToString[InputForm[Pdata]] *>, C1= <* ToString[InputForm[C1data]] *>, C2= <* ToString[InputForm[C2data]] *>, lambda2,lambda1,omega,z, f1,f2,f3,f4}, {lambda2,lambda1,omega,z}=y; f1= <* ToString[InputForm[dlambda2drho]] *>; f2= <* ToString[InputForm[dlambda1drho]] *>; f3= <* ToString[InputForm[domegadrho]] *>; f4= <* ToString[InputForm[dzdrho]] *>; f = {f1,f2,f3,f4}; ] ]; :[font = input; preserveAspect; startGroup] Splice["d02bbFCN.mm",FormatType -> OutputForm] :[font = output; output; inactive; preserveAspect; endGroup] "d02bbFCN.mm" ;[o] d02bbFCN.mm :[font = input; Cclosed; preserveAspect; startGroup] (* This cell closed in printed form *) !!d02bbFCN.m :[font = print; inactive; preserveAspect; endGroup] (* FILE d02bbFCN.mm For producing FCN SUBROUTINE ASP for d02bbf and d02baf Frequently used test values include h0=0.0015, P=0.13, C1=0.5, C2=0 or C2=0.05 *) fcn=Function[{x,y,f}, Block[{ h0= 0.0015, P= 0.13, C1= 0.5, C2= 0.05, lambda2,lambda1,omega,z, f1,f2,f3,f4}, {lambda2,lambda1,omega,z}=y; f1= (-lambda2 + lambda1*Sin[omega])/x; f2= (lambda2*((lambda1*((2*h0*(C1*(-2/(lambda1^2*lambda2^3) + 2*lambda2) + C2*(-2/lambda2^3 + 2*lambda1^2*lambda2)))/lambda1 - (2*h0*(C1*(2*lambda1 - 2/(lambda1^3*lambda2^2)) + C2*(-2/lambda1^3 + 2*lambda1*lambda2^2)))/lambda2)*Sin[omega])/(lambda2*x) - (((2*h0*((4*C1)/(lambda1^3*lambda2^3) + 4*C2*lambda1*lambda2))/lambda2 - (2*h0*(C1*(2*lambda1 - 2/(lambda1^3*lambda2^2)) + C2*(-2/lambda1^3 + 2*lambda1*lambda2^2)))/lambda2^2)*(-lambda2 + lambda1*Sin[omega]))/x))/(2*h0*(C1*(2 + 6/(lambda1^4*lambda2^2)) + C2*(6/lambda1^4 + 2*lambda2^2))); f3= -(lambda1*lambda2*(P - (2*h0*(C1*(-2/(lambda1^2*lambda2^3) + 2*lambda2) + C2*(-2/lambda2^3 + 2*lambda1^2*lambda2))*Cos[omega])/(lambda1*lambda2*x)))/(2*h0*(C1*(2*lambda1 - 2/(lambda1^3*lambda2^2)) + C2*(-2/lambda1^3 + 2*lambda1*lambda2^2))); f4= lambda1*Cos[omega]; f = {f1,f2,f3,f4}; ] ]; :[font = text; inactive; preserveAspect] The expressions for f are fairly messy. It would not be hard to do some optimization of the code. Indeed various common subexpressions are visible even in our derivation of the rhs. See also the subsection on ASPs in our section on remarks on software. Return now to D02RAF. The subroutine FCN differs only in its parameter list. (Of course, since we have called them the same name, this means that we should be careful not to get confused about which is loaded. As we need d02bbf first, we haven't found this a problem.) In our application, the boundary conditions are so simple that they will just be coded directly. :[font = input; preserveAspect; startGroup] Splice["d02raFCN.mm",FormatType -> OutputForm] :[font = output; output; inactive; preserveAspect; endGroup] "d02raFCN.mm" ;[o] d02raFCN.mm :[font = text; inactive; preserveAspect; endGroup] This concludes the code generation part. :[font = section; inactive; preserveAspect; startGroup] InterCall to d02raf :[font = subsection; inactive; preserveAspect; startGroup] Introduction and InterCall initialization :[font = input; preserveAspect; startGroup] < In, (* DATA=R *) $XEND -> In, (* DATA=R *) $N -> ROWS[$Y], (* DATA=I *) $Y -> In :> Out, (* DATA=R[$N] *) $TOL -> 10^(-6), (* DATA=R *) $IRELAB -> 0, (* DATA=I *) $FCN -> In, (* DATA=S[R, R[$N], R[$N]] *) $OUTPUT -> In, (* DATA=S[R, R[$N]] *) $W :> Null, (* DATA=R[$N, 7] *) $IFAIL -> -1 (* DATA=I *) ] (* CODE="LIBRARY" *) :[font = output; output; inactive; preserveAspect; endGroup] DefaultEntry[d02bbf, D02BBF, {{$X, Hold[In], None}, {$XEND, Hold[In], None}, {$N, Hold[ROWS[$Y]], None}, {$Y, Hold[In], Keep[Out, #1 & ]}, {$TOL, Hold[10^(-6)], None}, {$IRELAB, Hold[0], None}, {$FCN, Hold[In], None}, {$OUTPUT, Hold[In], None}, {$W, None, Keep[Null, #1 & ]}, {$IFAIL, Hold[-1], None}}, S, {R, R, I, R[$N], R, I, S[R, R[$N], R[$N]], S[R, R[$N]], R[$N, 7], I}, Null, {4}] ;[o] d02bbf[$X_, $XEND_, $Y_, $FCN_, $OUTPUT_] -> $Y :[font = text; inactive; preserveAspect] The initial point is x0, the final point xend=r0. r0 is the radius of the experimental rig. :[font = input; preserveAspect] r0=0.036; x0=0.0000001; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Exploratory use of NDSolve (Can be omitted) :[font = text; inactive; preserveAspect] In this subsection the notation is a rather unpleasant hybrid of notation related to the names of the variables in the real problem, and NAG's x=rho for independent, and y for dependent variables. :[font = input; preserveAspect; startGroup] (* n=4;r0=0.036;h0=0.0015;P=0.13; C1=0.5;C2=0;x0=0.0000001; lambda = 1.4323 *) lambda = 1.31 y ={lambda, lambda, N[Pi/2], 0}; C2data x= .; rho = x; xend = r0 x0 y0 = y; :[font = output; output; inactive; preserveAspect] 1.31 ;[o] 1.31 :[font = output; output; inactive; preserveAspect] 0.05 ;[o] 0.05 :[font = output; output; inactive; preserveAspect] 0.036 ;[o] 0.036 :[font = output; output; inactive; preserveAspect; endGroup] 1.*10^-7 ;[o] -7 1. 10 :[font = input; preserveAspect] paramRules = {h0->h0data, P->Pdata, C1->C1data, C2->C2data}; varRules = {lambda2 -> lambda2y[x], lambda1 -> lambda1y[x], omega -> omegay[x], z -> zy[x]}; varList ={lambda2y[x],lambda1y[x],omegay[x],zy[x]}; ics = {lambda2y[x0]==lambda, lambda1y[x0]==lambda, omegay[x0]==N[Pi/2], zy[x0]==0}; des = ({D[lambda2y[x],x]==dlambda2drho, D[lambda1y[x],x]==dlambda1drho, D[omegay[x],x]==domegadrho, D[zy[x],x]==dzdrho} /. paramRules) /. varRules; deic = Join[des,ics]; :[font = input; preserveAspect; startGroup] ansNDS = NDSolve[deic,varList,{x,x0,xend}, MaxSteps -> 2500] :[font = output; output; inactive; preserveAspect; endGroup] {{lambda2y[x] -> InterpolatingFunction[{1.*10^-7, 0.036}, {{1.*10^-7, 1.*10^-7, {1.31, 0}, {0}}, {1.000094802511519*10^-7, 1.*10^-7, {1.31, 0.}, {0}}, {1.000189605023039*10^-7, 1.000094802511519*10^-7, {1.31, 0.}, {0}}, {1.000284407534559*10^-7, 1.000189605023039*10^-7, {1.31, 0.}, {0}}, {1.000379210046078*10^-7, 1.000284407534559*10^-7, {1.31, 0.}, {0}}, {1.000474012557598*10^-7, 1.000379210046078*10^-7, {1.31, 0.}, {0}}, {1.000663617580637*10^-7, 1.000474012557598*10^-7, {1.31, 0., 0.}, {0, 0}}, {1.000853222603676*10^-7, 1.000663617580637*10^-7, {1.31, 0., 0.}, {0, 0}}, {1.001042827626715*10^-7, 1.000853222603676*10^-7, {1.31, 0., 0.}, {0, 0}}, {1.002938877857105*10^-7, 1.001042827626715*10^-7, {1.31, -(2.213939551325951*10^-9), -5.838293511005602}, {0, 0}}, {1.004834928087496*10^-7, 1.002938877857105*10^-7, {1.31, -(6.629286026541169*10^-9), -11.64353771974113}, {0, 0}}, {1.006730978317887*10^-7, 1.004834928087496*10^-7, {1.31, -(1.102800101055986*10^-8), -11.59967946395772}, {0, 0}}, {1.025691480621792*10^-7, 1.006730978317887*10^-7, {1.31, -(1.493731597589097*10^-7), -36.48246141660875}, {0, 0}}, {1.044651982925697*10^-7, 1.025691480621792*10^-7, {1.309999999999999, -(4.314839347123261*10^-7), -74.39433049600922}, {0, 0}}, {1.063612485229603*10^-7, 1.044651982925697*10^-7, {1.309999999999998, -(8.39233578201121*10^-7), -107.5260657532285}, {0, 0}}, {1.253217508268656*10^-7, 1.063612485229603*10^-7, {1.309999999999902, -(9.32141993551022*10^-6), -223.6804231595168}, {0, 0}}, {1.442822531307711*10^-7, 1.253217508268656*10^-7, {1.309999999999611, -0.00002132998450940178, -316.6731656528453}, {0, 0}}, {1.632427554346764*10^-7, 1.442822531307711*10^-7, {1.309999999999087, -0.00003396175028414467, -333.1073610887687}, {0, 0}}, {2.353027624971892*10^-7, 1.632427554346764*10^-7, {1.309999999994888, -0.0000825811276500906, -337.3534041133813}, {0, 0}}, {3.073627695597019*10^-7, 2.353027624971892*10^-7, {1.309999999987532, -0.0001215686147359704, -270.5209774130176}, {0, 0}}, {3.794227766222147*10^-7, 3.073627695597019*10^-7, {1.309999999977459, -0.0001580073810388074, -252.836266524551}, {0, 0}}, {5.147218650103638*10^-7, 3.794227766222147*10^-7, {1.309999999951448, -0.0002264978768495014, -253.1077504905534}, {0, 0}}, {6.50020953398513*10^-7, 5.147218650103638*10^-7, {1.30999999991638, -0.0002918753980280651, -241.6037016857319}, {0, 0}}, {7.853200417866622*10^-7, 6.50020953398513*10^-7, {1.309999999872537, -0.0003562193899285073, -237.7842772888858}, {0, 0}}, {1.081285113949899*10^-6, 7.853200417866622*10^-7, {1.309999999746366, -0.0004963850529493393, -236.7942642629205}, {0, 0}}, {1.377250186113137*10^-6, 1.081285113949899*10^-6, {1.309999999578966, -0.0006348333799340761, -233.8930164509015}, {0, 0}}, {1.673215258276374*10^-6, 1.377250186113137*10^-6, {1.30999999937065, -0.000772869946457636, -233.197393048168}, {0, 0}}, {2.294954199228002*10^-6, 1.673215258276374*10^-6, {1.309999998800032, -0.001062686148084712, -233.0690443673078}, {0, 0}}, {2.91669314017963*10^-6, 2.294954199228002*10^-6, {1.309999998049451, -0.001351770264006073, -232.480303935034}, {0, 0}}, {3.538432081131257*10^-6, 2.91669314017963*10^-6, {1.309999997119195, -0.001640662883522673, -232.3263032828728}, {0, 0}}, {4.858030745939019*10^-6, 3.538432081131257*10^-6, {1.309999994549672, -0.002253736967915922, -232.2956595604621}, {0, 0}}, {6.177629410746781*10^-6, 4.858030745939019*10^-6, {1.309999991171362, -0.002866471281026113, -232.1669191744198}, {0, 0}}, {7.497228075554542*10^-6, 6.177629410746781*10^-6, {1.309999986984546, -0.003479118624516248, -232.1339661174121}, {0, 0}}, {0.00002069321472363216, 7.497228075554542*10^-6, {1.309999900649492, -0.00960584454494631, -232.1403439537308}, {0, 0}}, {0.00003388920137170977, 0.00002069321472363216, {1.30999973348214, -0.01573080554959339, -232.097887763959}, {0, 0}}, {0.0000470851880197874, 0.00003388920137170977, {1.309999485490959, -0.02185536428717909, -232.0735743521008}, {0, 0}}, {0.00012382955063484, 0.0000470851880197874, {1.309995291715733, -0.05464603631871653}, {0}}, {0.0002005739132498925, 0.00012382955063484, {1.309988553219192, -0.0878044498774283}, {0}}, {0.0004296318052370952, 0.0002005739132498925, {1.30995519103977, -0.199313973988548, -234.2832251142489}, {0, 0}}, {0.0006586896972242976, 0.0004296318052370952, {1.309897356846628, -0.3059060720120135, -233.2110866613757}, {0, 0}}, {0.0008877475892115, 0.0006586896972242976, {1.309815134196973, -0.4121965721990427, -232.4148110915873}, {0, 0}}, {0.001560026261887281, 0.0008877475892115, {1.309433272417503, -0.724042193639204, -232.0928116946491}, {0, 0}}, {0.002232304934563062, 0.001560026261887281, {1.308841693031279, -1.035936759423262, -232.0097464470128}, {0, 0}}, {0.002904583607238842, 0.002232304934563062, {1.308040445352947, -1.347777136495544, -231.9551898518217}, {0, 0}}, {0.003704934873267823, 0.002904583607238842, {1.306813330035899, -1.718678157968852, -231.659712409346, 75.05607618473998}, {0, 0, 0}}, {0.004505286139296805, 0.003704934873267823, {1.305289368624471, -2.089492418563528, -231.5750521923289, 55.15784236506964}, {0, 0, 0}}, {0.005305637405325787, 0.004505286139296805, {1.303468722693319, -2.460087734431861, -231.4552502935553, 52.52669596151001}, {0, 0, 0}}, {0.006105988671354768, 0.005305637405325787, {1.301351612793212, -2.830342705522271, -231.2370615652496, 71.69938287500834}, {0, 0, 0}}, {0.007203611473507748, 0.006105988671354768, {1.297966517786364, -3.337626048574333, -230.9613811136583, 77.7099205087008}, {0, 0, 0}}, {0.00830123427566073, 0.007203611473507748, {1.294025033472364, -3.84415491238402, -230.6025144501997, 93.3464581160078}, {0, 0, 0}}, {0.00939885707781371, 0.00830123427566073, {1.28952792332641, -4.350008546498602, -230.2761847786884, 96.2242027427359}, {0, 0, 0}}, {0.0104964798799667, 0.00939885707781371, {1.28447587188301, -4.855305395899452, -230.0248542798116, 86.2749533067419}, {0, 0, 0}}, {0.01251270980666422, 0.0104964798799667, {1.273751473505923, -5.782475114354713, -229.7601991283288, 39.38519733934341, -3685.269679489805}, {0, 0, 0, 0}}, {0.01452893973336175, 0.01251270980666422, {1.26115827375123, -6.709280868160302, -229.8432469797758, -23.12568071220556, -5311.54620452398}, {0, 0, 0, 0}}, {0.01686450245954363, 0.01452893973336175, {1.24423261117137, -7.785272031859707, -230.8523880918134, -177.5716529167236, -10921.7666930765}, {0, 0, 0, 0}}, {0.01920006518572552, 0.01686450245954363, {1.224786243503611, -8.86881424200346, -233.1366744638516, -385.8890323640797, -16610.09040444008}, {0, 0, 0, 0}}, {0.02153562791190741, 0.01920006518572552, {1.202793181679749, -9.96755864113955, -237.3937257284373, -698.256839722519, -25023.06158520804}, {0, 0, 0, 0}}, {0.0238711906380893, 0.02153562791190741, {1.178206131436561, -11.09212885067047, -244.271821319876, -1113.669795217152, -34744.55363319254}, {0, 0, 0, 0}}, {0.02620675336427119, 0.0238711906380893, {1.150948741811746, -12.2569595112843, -254.7078739769689, -1671.144614559253, -47208.49214774289}, {0, 0, 0, 0}}, {0.02620942528117464, 0.02620675336427119, {1.150915990434257, -12.25830427841575, -250.1219849572774, 380849.4422091184}, {0, 0, 0}}, {0.02621209719807807, 0.02620942528117464, {1.150883235447584, -12.25966675054604, -256.6181283472708, -413331.741145513}, {0, 0, 0}}, {0.02621476911498152, 0.02621209719807807, {1.150850476816471, -12.2610292993703, -254.9830740389356, -1790.457420448886}, {0, 0, 0}}, {0.02621744103188497, 0.02621476911498152, {1.150817714544638, -12.26239192530236, -254.9975420656865, -1800.118932741744}, {0, 0, 0}}, {0.02622023439955675, 0.02621744103188497, {1.150783459185676, -12.26381657126064, -255.0050916449448}, {0, 0}}, {0.02622302776722853, 0.02622023439955675, {1.150749199847035, -12.26524130156896, -255.0201899133221}, {0, 0}}, {0.02622582113490032, 0.02622302776722853, {1.150714936528481, -12.26666611626426, -255.0352947978222}, {0, 0}}, {0.0262286145025721, 0.02622582113490032, {1.150680669229777, -12.26809101538361, -255.0504063136427}, {0, 0}}, {0.02623140787024388, 0.0262286145025721, {1.150646397950689, -12.26951599896402, -255.0655244586124}, {0, 0}}, {0.02623420123791567, 0.02623140787024388, {1.15061212269098, -12.27094106704255, -255.0806492331655}, {0, 0}}, {0.02623699460558745, 0.02623420123791567, {1.150577843450413, -12.27236621965624, -255.0957806377362}, {0, 0}}, {0.02623978797325923, 0.02623699460558745, {1.150543560228753, -12.27379145684216, -255.1109186805747}, {0, 0}}, {0.02624258134093101, 0.02623978797325923, {1.150509273025763, -12.2752167786374, -255.1260633595098}, {0, 0}}, {0.0262453747086028, 0.02624258134093101, {1.150474981841207, -12.27664218507904, -255.1412146788837}, {0, 0}}, {0.02624816807627458, 0.0262453747086028, {1.150440686674849, -12.27806767620419, -255.1563726373938}, {0, 0}}, {0.02625096144394636, 0.02624816807627458, {1.150406387526452, -12.27949325204996, -255.1715372398163}, {0, 0}}, {0.02625654817928993, 0.02625096144394636, {1.150337777282237, -12.28234465810799, -255.1867140512878}, {0, 0}}, {0.02626213491463349, 0.02625654817928993, {1.150269151106865, -12.28519640351907, -255.2120205349622}, {0, 0}}, {0.02626772164997706, 0.02626213491463349, {1.150200508998545, -12.28804848856393, -255.2407203261024}, {0, 0}}, {0.02632358900341271, 0.02626772164997706, {1.149513210843292, -12.31658813789767, -255.3624507299279}, {0, 0}}, {0.02637945635684835, 0.02632358900341271, {1.14882431710173, -12.34516210815092, -255.6078037699549}, {0, 0}}, {0.02643532371028399, 0.02637945635684835, {1.1481338259746, -12.37377068004311, -255.8960392716522}, {0, 0}}, {0.0267724675262981, 0.02643532371028399, {1.143932805469363, -12.54717944595763, -256.7475369686028}, {0, 0}}, {0.0271096113423122, 0.0267724675262981, {1.139673051242491, -12.72193314531972, -258.3611022118174}, {0, 0}}, {0.02744675515832631, 0.0271096113423122, {1.135354123362236, -12.89810015902607, -260.2962980093497}, {0, 0}}, {0.02790891107508611, 0.02744675515832631, {1.129336886697087, -13.14197717336353, -265.375721668028, -2788.447590586859}, {0, 0, 0}}, {0.0283710669918459, 0.02790891107508611, {1.123206339922485, -13.38884724049129, -268.9979883065229, -2700.516941226652}, {0, 0, 0}}, {0.02883322290860569, 0.0283710669918459, {1.116960964862213, -13.63889335451257, -272.3617714896377, -2563.335075690168}, {0, 0, 0}}, {0.02929537882536548, 0.02883322290860569, {1.110599228220942, -13.8923430326905, -275.9865392314936, -2588.862653319468}, {0, 0, 0}}, {0.03025565766315391, 0.02929537882536548, {1.09700122600171, -14.43086983298022, -284.1926438202903, -2718.688578108307}, {0, 0, 0}}, {0.03102571181511527, 0.03025565766315391, {1.085718015037563, -14.87552873693643, -291.9855760003747, -3046.007527874305}, {0, 0, 0}}, {0.03179576596707662, 0.03102571181511527, {1.074087585335604, -15.33285940170662, -300.5966593847785, -3386.743554719211}, {0, 0, 0}}, {0.03256582011903798, 0.03179576596707662, {1.06209967474769, -15.80417651919734, -310.0786935916048, -3745.615805193043}, {0, 0, 0}}, {0.03333587427099935, 0.03256582011903798, {1.049742984237639, -16.29086316145752, -320.4802699996007, -4124.073042306204}, {0, 0, 0}}, {0.03422219899884302, 0.03333587427099935, {1.035048037234532, -16.87194943767385, -334.3358072064114, -5304.517702927108, -206904.4695881317}, {0, 0, 0, 0}}, {0.03510852372668669, 0.03422219899884302, {1.019827559101246, -17.47765509256615, -349.3903295052321, -6029.905741416392, -205984.9248610078}, {0, 0, 0, 0}}, {0.03599484845453037, 0.03510852372668669, {1.004058443825407, -18.11047151761382, -365.3563422670659, -6425.49638577506, -168223.6535124222}, {0, 0, 0, 0}}, {0.036, 0.03599484845453037, {1.003965137187231, -18.11424596124247, -366.7645313713375, -79020.69367559913, -(1.409293529264805*10^9)}, {0, 0, 0, 0}}}][x], lambda1y[x] -> InterpolatingFunction[{1.*10^-7, 0.036}, {{1.*10^-7, 1.*10^-7, {1.31, 0}, {0}}, {1.000094802511519*10^-7, 1.*10^-7, {1.31, -(1.436903937847917*10^-10)}, {0}}, {1.000189605023039*10^-7, 1.000094802511519*10^-7, {1.31, -(1.436767741569295*10^-10)}, {0}}, {1.000284407534559*10^-7, 1.000189605023039*10^-7, {1.31, -(1.43663157110683*10^-10)}, {0}}, {1.000379210046078*10^-7, 1.000284407534559*10^-7, {1.31, -(1.436495426453181*10^-10)}, {0}}, {1.000474012557598*10^-7, 1.000379210046078*10^-7, {1.31, -(1.436359307601012*10^-10)}, {0}}, {1.000663617580637*10^-7, 1.000474012557598*10^-7, {1.31, -(1.436087147271781*10^-10), 0.0007177033732241913}, {0, 0}}, {1.000853222603676*10^-7, 1.000663617580637*10^-7, {1.31, -(1.435815090060499*10^-10), 0.0007174314449112035}, {0, 0}}, {1.001042827626715*10^-7, 1.000853222603676*10^-7, {1.31, -(1.435543135908571*10^-10), 0.0007171596711115851}, {0, 0}}, {1.002938877857105*10^-7, 1.001042827626715*10^-7, {1.31, 0., 0.3785614729238701}, {0, 0}}, 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] {Evaluate[lambda2y[x] /. ansNDS] /. {x->xend}, lambda} :[font = output; output; inactive; preserveAspect; endGroup] {{1.003965137187231}, 1.31} ;[o] {{1.00397}, 1.31} :[font = input; preserveAspect; startGroup] zend = (Evaluate[zy[x] /. ansNDS] /. {x->xend})[[1]] :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "zend" is similar to existing symbol "xend". :[font = output; output; inactive; preserveAspect; endGroup] 0.02358438523563815 ;[o] 0.0235844 :[font = input; preserveAspect; startGroup] ParametricPlot[ Evaluate[ {x*lambda1y[x],zend-zy[x]} /. ansNDS ], {x,x0,xend}, PlotLabel -> " r,z-profile: solution with NDSolve "] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 21.147 0.0147151 24.9574 [ [(0.01)] .23528 .01472 0 2 Msboxa [(0.02)] .44675 .01472 0 2 Msboxa [(0.03)] .65822 .01472 0 2 Msboxa [(0.04)] .86969 .01472 0 2 Msboxa [( r,z-profile: solution with NDSolve )] .5 .61803 0 -2 Msboxa [(0.005)] .01131 .1395 1 0 Msboxa [(0.01)] .01131 .26429 1 0 Msboxa [(0.015)] .01131 .38908 1 0 Msboxa [(0.02)] .01131 .51386 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .23528 .01472 m .23528 .02097 L s P [(0.01)] .23528 .01472 0 2 Mshowa p .002 w .44675 .01472 m .44675 .02097 L s P [(0.02)] .44675 .01472 0 2 Mshowa p .002 w .65822 .01472 m .65822 .02097 L s P [(0.03)] .65822 .01472 0 2 Mshowa p .002 w .86969 .01472 m .86969 .02097 L s P [(0.04)] .86969 .01472 0 2 Mshowa p .001 w .0661 .01472 m .0661 .01847 L s P p .001 w .1084 .01472 m .1084 .01847 L s P p .001 w .15069 .01472 m .15069 .01847 L s P p .001 w .19299 .01472 m .19299 .01847 L s P p .001 w .27757 .01472 m .27757 .01847 L s P p .001 w .31987 .01472 m .31987 .01847 L s P p .001 w .36216 .01472 m .36216 .01847 L s P p .001 w .40446 .01472 m .40446 .01847 L s P p .001 w .48904 .01472 m .48904 .01847 L s P p .001 w .53134 .01472 m .53134 .01847 L s P p .001 w .57363 .01472 m .57363 .01847 L s P p .001 w .61593 .01472 m .61593 .01847 L s P p .001 w .70051 .01472 m .70051 .01847 L s P p .001 w .74281 .01472 m .74281 .01847 L s P p .001 w .7851 .01472 m .7851 .01847 L s P p .001 w .8274 .01472 m .8274 .01847 L s P p .001 w .91198 .01472 m .91198 .01847 L s P p .001 w .95428 .01472 m .95428 .01847 L s P p .001 w .99657 .01472 m .99657 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P [( r,z-profile: solution with NDSolve )] .5 .61803 0 -2 Mshowa p .002 w .02381 .1395 m .03006 .1395 L s P [(0.005)] .01131 .1395 1 0 Mshowa p .002 w .02381 .26429 m .03006 .26429 L s P [(0.01)] .01131 .26429 1 0 Mshowa p .002 w .02381 .38908 m .03006 .38908 L s P [(0.015)] .01131 .38908 1 0 Mshowa p .002 w .02381 .51386 m .03006 .51386 L s P [(0.02)] .01131 .51386 1 0 Mshowa p .001 w .02381 .03967 m .02756 .03967 L s P p .001 w .02381 .06463 m .02756 .06463 L s P p .001 w .02381 .08959 m .02756 .08959 L s P p .001 w .02381 .11454 m .02756 .11454 L s P p .001 w .02381 .16446 m .02756 .16446 L s P p .001 w .02381 .18942 m .02756 .18942 L s P p .001 w .02381 .21437 m .02756 .21437 L s P p .001 w .02381 .23933 m .02756 .23933 L s P p .001 w .02381 .28925 m .02756 .28925 L s P p .001 w .02381 .3142 m .02756 .3142 L s P p .001 w .02381 .33916 m .02756 .33916 L s P p .001 w .02381 .36412 m .02756 .36412 L s P p .001 w .02381 .41403 m .02756 .41403 L s P p .001 w .02381 .43899 m .02756 .43899 L s P p .001 w .02381 .46395 m .02756 .46395 L s P p .001 w .02381 .4889 m .02756 .4889 L s P p .001 w .02381 .53882 m .02756 .53882 L s P p .001 w .02381 .56378 m .02756 .56378 L s P p .001 w .02381 .58873 m .02756 .58873 L s P p .001 w .02381 .61369 m .02756 .61369 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w .02381 .60332 m .02511 .60332 L .02641 .60331 L .02901 .6033 L .0316 .60327 L .0342 .60324 L .0368 .6032 L .03939 .60316 L .04459 .60304 L .04978 .60288 L .05497 .60269 L .06536 .6022 L .07574 .60158 L .08612 .60081 L .10686 .59887 L .12759 .59638 L .14828 .59333 L .18958 .58558 L .23072 .57565 L .27166 .56355 L .31237 .5493 L .35282 .53293 L .393 .51446 L .43288 .49393 L .47246 .47138 L .51173 .44684 L .55071 .42035 L .5894 .39196 L .62785 .3617 L .66608 .32964 L .70417 .2958 L .74218 .26026 L .78022 .22306 L .8184 .18427 L .85688 .14396 L .89584 .10219 L .93551 .05908 L .97619 .01472 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect] :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Testing fcn with d02baf (Can be omitted) :[font = input; preserveAspect; startGroup] (* n=4;r0=0.036;h0=0.0015;P=0.13; C1=0.5;C2=0;x0=0.0000001; lambda = 1.4323 *) lambda = 1.31 y ={lambda, lambda, N[Pi/2], 0}; C2data x= .; rho = x; xend = r0 x0 y0 = y; :[font = output; output; inactive; preserveAspect] 1.31 ;[o] 1.31 :[font = output; output; inactive; preserveAspect] 0.05 ;[o] 0.05 :[font = output; output; inactive; preserveAspect] 0.036 ;[o] 0.036 :[font = output; output; inactive; preserveAspect; endGroup] 1.*10^-7 ;[o] -7 1. 10 :[font = input; preserveAspect; startGroup] GetDefault[d02baf] :[font = output; output; inactive; preserveAspect] DefaultEntry[d02baf, D02BAF, {{$X, Hold[In], None}, {$XEND, Hold[In], None}, {$N, Hold[ROWS[$Y]], None}, {$Y, Hold[In], Keep[Out, #1 & ]}, {$TOL, Hold[10^(-6)], None}, {$FCN, Hold[In], None}, {$W, None, Keep[Null, #1 & ]}, {$IFAIL, Hold[-1], None}}, S, {R, R, I, R[$N], R, S[R, R[$N], R[$N]], R[$N, 7], I}, Null, {4}] ;[o] d02baf[$X_, $XEND_, $Y_, $FCN_] -> $Y :[font = print; inactive; preserveAspect; endGroup] D02BAF[ (* TYPE=S *) $X -> In, (* DATA=R *) $XEND -> In, (* DATA=R *) $N -> ROWS[$Y], (* DATA=I *) $Y -> In :> Out, (* DATA=R[$N] *) $TOL -> 10^(-6), (* DATA=R *) $FCN -> In, (* DATA=S[R, R[$N], R[$N]] *) $W :> Null, (* DATA=R[$N, 7] *) $IFAIL -> -1 (* DATA=I *) ] (* CODE="LIBRARY" *) :[font = input; preserveAspect; startGroup] {x0,xend,y} :[font = output; output; inactive; preserveAspect; endGroup] {1.*10^-7, 0.036, {1.31, 1.31, 1.570796326794897, 0}} ;[o] -7 {1. 10 , 0.036, {1.31, 1.31, 1.5708, 0}} :[font = input; preserveAspect; startGroup] ansba = d02baf[x0,xend,y,fcn] :[font = output; output; inactive; preserveAspect; endGroup] {1.00397255166395, 1.25100276131569, 0.285117117964169, 0.0235841076955946} ;[o] {1.00397255166395, 1.25100276131569, 0.285117117964169, 0.0235841076955946} :[font = text; inactive; preserveAspect] Compare the result with NDSolve. :[font = input; preserveAspect; startGroup] (varList /. ansNDS) /. {x->xend} :[font = output; output; inactive; preserveAspect; endGroup] {{1.003965137187231, 1.251005840403049, 0.2851021794669445, 0.02358438523563815}} ;[o] {{1.00397, 1.25101, 0.285102, 0.0235844}} :[font = input; preserveAspect] :[font = input; preserveAspect; endGroup] :[font = text; inactive; preserveAspect] Continue now with essential items. Use of d02bbf. We next initialise for calling d02bbf with the ASP "out" preparing the initial guess for d02raf. The NAG documentation and example indicates that, when out is called with an x which is larger than xend, the tabulation of its y for the corresponding x will stop. ;[s] 5:0,0;1,1;35,0;36,1;49,0;313,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect] np=33; n=4; xbbList=Table[x0+(r0-x0)/(np-1)*(i-1),{i,np}]; ybbList={}; count=1; x=x0; (* making the above global *) x =.; out=Function[{x,y}, count=count+1; (* Print[x]; Print[y]; *) ybbList=Append[ybbList,y]; If[count<=np,x=xbbList[[count]], x=xbbList[[np]]+1]; ]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Testing ASP out for d02bbf (Can be omitted) :[font = input; preserveAspect; startGroup] {x0,xend,y} :[font = output; output; inactive; preserveAspect; endGroup] {1.*10^-7, 0.036, {1.31, 1.31, 1.570796326794897, 0}} ;[o] -7 {1. 10 , 0.036, {1.31, 1.31, 1.5708, 0}} :[font = input; preserveAspect; startGroup] ansbb = d02bbf[x0,xend,y,fcn,out,$TOL -> 0.00001] :[font = output; output; inactive; preserveAspect; endGroup] {1.00397335203683, 1.25100622033897, 0.28511222215846, 0.0235839555778674} ;[o] {1.00397335203683, 1.25100622033897, 0.28511222215846, 0.0235839555778674} :[font = input; preserveAspect; startGroup] (varList /. ansNDS) /. {x->xend} :[font = output; output; inactive; preserveAspect; endGroup] {{1.003965137187231, 1.251005840403049, 0.2851021794669445, 0.02358438523563815}} ;[o] {{1.00397, 1.25101, 0.285102, 0.0235844}} :[font = input; preserveAspect; startGroup] lambda x x xbbList Length[xbbList] y ybbList // MatrixForm :[font = output; output; inactive; preserveAspect] 1.31 ;[o] 1.31 :[font = output; output; inactive; preserveAspect] x ;[o] x :[font = output; output; inactive; preserveAspect] x ;[o] x :[font = output; output; inactive; preserveAspect] {1.*10^-7, 0.001125096875, 0.00225009375, 0.003375090624999999, 0.0045000875, 0.005625084375, 0.00675008125, 0.007875078124999999, 0.009000075, 0.010125071875, 0.01125006875, 0.012375065625, 0.0135000625, 0.014625059375, 0.01575005625, 0.016875053125, 0.01800005, 0.019125046875, 0.02025004374999999, 0.021375040625, 0.0225000375, 0.023625034375, 0.02475003125, 0.02587502812499999, 0.02700002499999999, 0.02812502187499999, 0.02925001874999999, 0.030375015625, 0.0315000125, 0.032625009375, 0.03375000625, 0.034875003125, 0.036} ;[o] -7 {1. 10 , 0.0011251, 0.00225009, 0.00337509, 0.00450009, 0.00562508, 0.00675008, 0.00787508, 0.00900007, 0.0101251, 0.0112501, 0.0123751, 0.0135001, 0.0146251, 0.0157501, 0.0168751, 0.018, 0.019125, 0.02025, 0.021375, 0.0225, 0.023625, 0.02475, 0.025875, 0.027, 0.028125, 0.02925, 0.030375, 0.0315, 0.032625, 0.03375, 0.034875, 0.036} :[font = output; output; inactive; preserveAspect] 33 ;[o] 33 :[font = output; output; inactive; preserveAspect] {1.31, 1.31, 1.570796326794897, 0} ;[o] {1.31, 1.31, 1.5708, 0} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] MatrixForm[{{1.31, 1.31, 1.5707963267949, 0}, {1.30970626635537, 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{1.25262451084882, 1.28772504955978, 1.08394054232225, 0.004824489155711501}, {1.24415268668346, 1.28481728078534, 1.04721267419568, 0.005524769135046071}, {1.23509620737551, 1.28182277742257, 1.01002165704293, 0.0062696407117595}, {1.22545276878677, 1.27876593250978, 0.972314474058952, 0.007058415519365951}, {1.21521906571357, 1.27567325384583, 0.934032991352588, 0.0078903779764902}, {1.2043906351278, 1.27257347558805, 0.895113392194437, 0.00876478745136001}, {1.19296168183649, 1.26949765461228, 0.855485660889058, 0.00968088121087526}, {1.18092503680715, 1.26647980600117, 0.81507179258881, 0.0106378569838732}, {1.16827181410619, 1.26355739478712, 0.773784403418437, 0.0116348671067533}, {1.15499095320661, 1.26077118836134, 0.731526850558335, 0.0126710358125891}, {1.14106898716862, 1.25816566849836, 0.688192002944807, 0.0137454533570899}, {1.12648974262017, 1.25578930363121, 0.643661351545876, 0.0148571762019201}, {1.11123421247711, 1.25369538808789, 0.597802932424529, 0.0160052034009648}, {1.09528054369956, 1.25194463370764, 0.5504663934587301, 0.017188391264349}, {1.07860278017025, 1.25060443380653, 0.501485477604171, 0.0184055142187118}, {1.06117060812727, 1.24974999811529, 0.450675748746661, 0.019655234921562}, {1.04294892983595, 1.24946531843439, 0.397832978627089, 0.0209360849762267}, {1.02389815935868, 1.24984701509397, 0.342728962739513, 0.0222463337031443}, {1.00397335203683, 1.25100622033897, 0.28511222215846, 0.0235839555778674}}] ;[o] 1.31 1.31 1.5707963267949 0 1.30970626635537 1.30987497694405 1.53681257508733 0.00002503895362760971 1.30882524240625 1.30950064690403 1.50280939268842 0.000100120240473947 1.30735699199763 1.30887924868378 1.46876111503397 0.000225163809694973 1.30530221337574 1.30801391412446 1.43464470102723 0.000400036892369355 1.30266107476615 1.30690982863396 1.40043650929102 0.000624554351377064 1.29943459218201 1.30557283535381 1.36611209097804 0.000898480089660987 1.29562317880514 1.30401088104885 1.33164596349031 0.00122152760257508 1.29122790558148 1.30223272449239 1.29701135815567 0.00159336254377628 1.28624928355418 1.30024913874558 1.26218005406199 0.00201360282824266 1.28068808588823 1.29807208344795 1.22712193951571 0.0024818217535925 1.274544911347 1.29571507948417 1.1918048633003 0.00299754984864102 1.26781996627902 1.2931933897197 1.15619439247938 0.003560276700732739 1.26051328868524 1.29052386364976 1.12025313267972 0.004169451938114671 1.25262451084882 1.28772504955978 1.08394054232225 0.004824489155711501 1.24415268668346 1.28481728078534 1.04721267419568 0.005524769135046071 1.23509620737551 1.28182277742257 1.01002165704293 0.0062696407117595 1.22545276878677 1.27876593250978 0.972314474058952 0.007058415519365951 1.21521906571357 1.27567325384583 0.934032991352588 0.0078903779764902 1.2043906351278 1.27257347558805 0.895113392194437 0.00876478745136001 1.19296168183649 1.26949765461228 0.855485660889058 0.00968088121087526 1.18092503680715 1.26647980600117 0.81507179258881 0.0106378569838732 1.16827181410619 1.26355739478712 0.773784403418437 0.0116348671067533 1.15499095320661 1.26077118836134 0.731526850558335 0.0126710358125891 1.14106898716862 1.25816566849836 0.688192002944807 0.0137454533570899 1.12648974262017 1.25578930363121 0.643661351545876 0.0148571762019201 1.11123421247711 1.25369538808789 0.597802932424529 0.0160052034009648 1.09528054369956 1.25194463370764 0.5504663934587301 0.017188391264349 1.07860278017025 1.25060443380653 0.501485477604171 0.0184055142187118 1.06117060812727 1.24974999811529 0.450675748746661 0.019655234921562 1.04294892983595 1.24946531843439 0.397832978627089 0.0209360849762267 1.02389815935868 1.24984701509397 0.342728962739513 0.0222463337031443 1.00397335203683 1.25100622033897 0.28511222215846 0.0235839555778674 :[font = text; inactive; preserveAspect] Continue now with essential items. Code for production runs with d02bbf. We organise this into a loop, with different initial data each time we run the loop. If we were, through our choice of the initial-value lambda to have obtained the lambda2(xend) entry of ybbList to be exactly 1, we would have solved the two-point boundary-value problem. In practice we just get reasonably near to 1, then switch to using d02raf. ;[s] 4:0,1;34,0;35,1;72,0;421,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = input; preserveAspect] d02bbLoop[anotherRun_]:= Module[{localTorF=anotherRun,ans}, While[localTorF, y={}; lambda=Input["lambda at pole: float data, please"]; Print["lambda entered was"]; Print[lambda]; y ={lambda, lambda, Pi/2, 0}; ansbb = d02bbf[x0,xend,y,fcn,out]; Print[ansbb]; localTorF=Input["Another Run? reply True or False"]; ] ]; :[font = input; preserveAspect; startGroup] xbbList :[font = output; output; inactive; preserveAspect; endGroup] {1.*10^-7, 0.001125096875, 0.00225009375, 0.003375090624999999, 0.0045000875, 0.005625084375, 0.00675008125, 0.007875078124999999, 0.009000075, 0.010125071875, 0.01125006875, 0.012375065625, 0.0135000625, 0.014625059375, 0.01575005625, 0.016875053125, 0.01800005, 0.019125046875, 0.02025004374999999, 0.021375040625, 0.0225000375, 0.023625034375, 0.02475003125, 0.02587502812499999, 0.02700002499999999, 0.02812502187499999, 0.02925001874999999, 0.030375015625, 0.0315000125, 0.032625009375, 0.03375000625, 0.034875003125, 0.036} ;[o] -7 {1. 10 , 0.0011251, 0.00225009, 0.00337509, 0.00450009, 0.00562508, 0.00675008, 0.00787508, 0.00900007, 0.0101251, 0.0112501, 0.0123751, 0.0135001, 0.0146251, 0.0157501, 0.0168751, 0.018, 0.019125, 0.02025, 0.021375, 0.0225, 0.023625, 0.02475, 0.025875, 0.027, 0.028125, 0.02925, 0.030375, 0.0315, 0.032625, 0.03375, 0.034875, 0.036} :[font = input; preserveAspect; startGroup] ybbList={}; count=1; x=x0; (* making the above global *) x =.; anotherRun = True; d02bbLoop[anotherRun] (* C2=0, try 1.4323 *) (* C2=0.05, try 1.31 *) :[font = print; inactive; preserveAspect; endGroup] lambda entered was 1.31 {1.00397255166395, 1.25100276131569, 0.285117117964169, 0.0235841076955946} :[font = input; Cclosed; preserveAspect; startGroup] Dimensions[ybbList] (* to be {np,n} *) ybbList // MatrixForm :[font = output; output; inactive; preserveAspect] {33, 4} ;[o] {33, 4} :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1.31, 1.31, 1.5707963267949, 0}, {1.30970625181262, 1.30987499153649, 1.53681257511644, 0.0000250389496366257}, {1.30882513287586, 1.30950075696198, 1.5028093929471, 0.000100120194735504}, {1.30735691581556, 1.30887932573201, 1.46876111574495, 0.000225163735524388}, {1.30530199573401, 1.30801413383877, 1.43464470312409, 0.0004000367115547171}, {1.30266092146906, 1.30690998435121, 1.40043651560047, 0.000624554096627527}, {1.29943435282022, 1.30557307855237, 1.3661121028513, 0.000898479667905488}, {1.29562301978424, 1.30401104237206, 1.33164598059627, 0.00122152720600898}, {1.29122767807859, 1.302232952048, 1.29701140220862, 0.00159336192586732}, {1.28624909037304, 1.30024933226676, 1.26218009762984, 0.0020136022048247}, {1.28068792616769, 1.29807223307773, 1.22712201080937, 0.00248182131730676}, {1.27454472383591, 1.29571523717735, 1.19180500592758, 0.00299754960114467}, {1.26781979457394, 1.29319353450882, 1.15619453694459, 0.0035602763203427}, {1.26051314470332, 1.29052395964436, 1.12025330276925, 0.0041694521412015}, {1.25262436780781, 1.28772507307632, 1.08394086535908, 0.004824491099561571}, {1.24415253770872, 1.28481725196895, 1.04721315418104, 0.00552477239230478}, {1.23509607742776, 1.28182273698775, 1.01002212402947, 0.006269643762320481}, {1.22545262967152, 1.27876584279997, 0.972314987155543, 0.00705842025464234}, {1.2152188808269, 1.27567300450129, 0.934033800753819, 0.0078903886594457}, {1.20439040548774, 1.27257294013859, 0.895114803184165, 0.00876480794376226}, {1.19296148698993, 1.26949701112808, 0.855487296208121, 0.00968090403967034}, {1.18092485023426, 1.26647927382389, 0.81507318432319, 0.010637876244041}, {1.16827145527513, 1.26355675546386, 0.7737860408067791, 0.0116348950186514}, {1.15499030549685, 1.26077012773634, 0.731529368352872, 0.0126710861830733}, {1.1410680394515, 1.25816385986382, 0.688195988379447, 0.0137455376679381}, {1.12648867189403, 1.25578676314192, 0.6436666310114871, 0.0148572894272379}, {1.11123331446955, 1.25369295600676, 0.5978078947413151, 0.0160053098281261}, {1.09527965702005, 1.25194233802972, 0.5504712755655011, 0.0171884949877823}, {1.07860171822891, 1.25060184902234, 0.5014909162381381, 0.0184056347071942}, {1.06116933604956, 1.2497466790814, 0.4506820216032821, 0.0196553877324622}, {1.04294773658375, 1.24946167752677, 0.397839146596068, 0.0209362486898592}, {1.02389720998834, 1.24984354589291, 0.342734459668752, 0.0222464879165906}, {1.00397255166395, 1.25100276131569, 0.285117117964169, 0.0235841076955946}}] ;[o] 1.31 1.31 1.5707963267949 0 1.30970625181262 1.30987499153649 1.53681257511644 0.0000250389496366257 1.30882513287586 1.30950075696198 1.5028093929471 0.000100120194735504 1.30735691581556 1.30887932573201 1.46876111574495 0.000225163735524388 1.30530199573401 1.30801413383877 1.43464470312409 0.0004000367115547171 1.30266092146906 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preserveAspect; endGroup] :[font = subsection; inactive; preserveAspect; startGroup] ASP for the bcs for use with d02raf :[font = input; preserveAspect] g=Function[{eps,ya,yb,bc,n}, bc={ya[[1]]-ya[[2]], ya[[3]]-Pi/2, ya[[4]], yb[[1]]-1};]; :[font = input; preserveAspect; endGroup] :[font = subsection; inactive; preserveAspect; startGroup] Use of d02raf :[font = text; inactive; preserveAspect] The following cell is done before. It is repeated here only in case in our numerical experimentation we have changed C1, C2, etc. parameters in which case the Splice may need to be repeated. :[font = input; preserveAspect; startGroup] Import[{D02RAF}] Splice["d02raFCN.mm",FormatType -> OutputForm] :[font = output; output; inactive; preserveAspect] {D02RAF} ;[o] {D02RAF} :[font = output; output; inactive; preserveAspect; endGroup] "d02raFCN.mm" ;[o] d02raFCN.mm :[font = input; preserveAspect] fcn =.; < 1] :[font = output; output; inactive; preserveAspect; endGroup] {55, {1.*10^-7, 0.000017678076171875, 0.00003525615234375, 0.00005283422851562501, 0.00007041230468750002, 0.000087990380859375, 0.00010556845703125, 0.000123146533203125, 0.000140724609375, 0.00017588076171875, 0.0002110369140625, 0.00024619306640625, 0.00028134921875, 0.0003516615234375, 0.0004219738281250001, 0.0005625984375, 0.0007032230468749999, 0.00084384765625, 0.000984472265625, 0.001125096875, 0.00140634609375, 0.0016875953125, 0.00196884453125, 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1.30662060043632, 1.30650483672954, 1.30622744273886, 1.30588919595118, 1.30503239170436, 1.30393901444476, 1.30261530979779, 1.30106893944472, 1.29930901137286, 1.29734611140072, 1.29519233984502, 1.29286135483313, 1.29036842353413, 1.28773048277118, 1.28496621083037, 1.28209611275252, 1.2791426219909, 1.27613022205904, 1.2730855927089, 1.27003778630938, 1.26701844148584, 1.26406204279389, 1.26120623730918, 1.25849222161196, 1.25596521584318, 1.25367504544496, 1.25167685603091, 1.25003199274219, 1.24880908262019, 1.24808536713486, 1.24794834214393, 1.24849777414962, 1.24984817436253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1.5707963267949, 1.57008585745933, 1.56965893051029, 1.56915226910388, 1.56863219684442, 1.56810723653163, 1.56757992330382, 1.56705129127192, 1.5665218440225, 1.56546155025176, 1.56440008844202, 1.56333797007372, 1.56227544379034, 1.56014968240028, 1.55802331716763, 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2.28673224721385 10 , -7 -7 3.086409056216091 10 , 4.00762899859436 10 , -7 -7 6.215298865381782 10 , 8.91042178609177 10 , -6 -6 1.2093412413418 10 , 1.57645099412079 10 , -6 -6 2.45716032577055 10 , 3.533234210381 10 , -6 -6 6.271607999379661 10 , 9.79168389943826 10 , 0.0000140934836455954, 0.000019176976732386, 0.0000250421028378365, 0.00003911685800068719, 0.00005631693520264721, 0.00007664124052382921, 0.000100088458587913, 0.000156344450606435, 0.000225070960819667, 0.000399849846001938, 0.0006242515863040411, 0.000898042821419033, 0.00122094097229876, 0.00159261562042162, 0.00201269013486843, 0.00248074349409731, 0.0029963122604617, 0.00355889266253072, 0.0041679427327334, 0.004822884437558591, 0.005523105724631421, 0.00626796239504447, 0.007056779689633459, 0.007888853453465412, 0.0087634507123491, 0.00967980945698935, 0.0106371373823051, 0.0116346092686182, 0.0126713626143479, 0.0137464910320709, 0.0148590347958907, 0.0160079677715225, 0.0171921797640681, 0.0184104530746592, 0.0196614317598672, 0.0209435817353248, 0.0222551394659402, 0.0235940465687372, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}} :[font = input; preserveAspect] iAns = Length[ansra[[2]] ]-ansra[[1]] xAns = Drop[ ansra[[2]], -iAns]; yAns = Transpose[Drop[Transpose[ ansra[[3]] ], -iAns]]; :[font = input; preserveAspect; endGroup; endGroup] :[font = section; inactive; preserveAspect; startGroup] Results: Mathematica Plots ;[s] 3:0,0;9,1;20,0;27,-1; 2:2,19,14,Times,1,18,0,0,0;1,19,14,Times,3,18,0,0,0; :[font = text; inactive; preserveAspect] First remind ourselves of parameter settings: :[font = input; preserveAspect] {h0data, Pdata, C1data, C2data} :[font = text; inactive; preserveAspect] In [H1] we show two sets of results from the program for pressures P of .13 Mpa (dashed line), and .23 Mpa (solid line). The Mooney constants were C1=.5 Mpa, C2=.05 Mpa. The P=0.13 case of this is what you will get here, unless you have changed the data. Our first figure shows lambda1 and lambda2 versus rho, with, for our usual test data, lambda2 the lower line. :[font = input; preserveAspect; startGroup] Dimensions[xAns] Dimensions[yAns] :[font = output; output; inactive; preserveAspect] {55} ;[o] {55} :[font = output; output; inactive; preserveAspect; endGroup] {4, 55} ;[o] {4, 55} :[font = input; preserveAspect] dataPairs[j_]:= MapThread[List,{xAns,yAns[[j]]} ]; xy1 = dataPairs[1]; xy2 = dataPairs[2]; xy3 = dataPairs[3]; xy4 = dataPairs[4]; :[font = input; preserveAspect; startGroup] g1 = ListPlot[xy1, PlotJoined -> True, PlotRange -> All, DisplayFunction -> Identity] g2 = ListPlot[xy2, PlotJoined -> True, PlotRange -> All, DisplayFunction -> Identity] Show[{g1,g2}, DisplayFunction -> $DisplayFunction] :[font = output; output; inactive; preserveAspect] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = output; output; inactive; preserveAspect] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238069 26.4551 -1.90256 1.91728 [ [(0.005)] .15608 .01472 0 2 Msboxa [(0.01)] .28836 .01472 0 2 Msboxa [(0.015)] .42063 .01472 0 2 Msboxa [(0.02)] .55291 .01472 0 2 Msboxa [(0.025)] .68518 .01472 0 2 Msboxa [(0.03)] .81746 .01472 0 2 Msboxa [(0.035)] .94974 .01472 0 2 Msboxa [(1.05)] .01131 .11058 1 0 Msboxa [(1.1)] .01131 .20644 1 0 Msboxa [(1.15)] .01131 .30231 1 0 Msboxa [(1.2)] .01131 .39817 1 0 Msboxa [(1.25)] .01131 .49403 1 0 Msboxa [(1.3)] .01131 .5899 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 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.01847 L s P p .001 w .36772 .01472 m .36772 .01847 L s P p .001 w .39418 .01472 m .39418 .01847 L s P p .001 w .44709 .01472 m .44709 .01847 L s P p .001 w .47354 .01472 m .47354 .01847 L s P p .001 w .5 .01472 m .5 .01847 L s P p .001 w .52645 .01472 m .52645 .01847 L s P p .001 w .57936 .01472 m .57936 .01847 L s P p .001 w .60582 .01472 m .60582 .01847 L s P p .001 w .63227 .01472 m .63227 .01847 L s P p .001 w .65873 .01472 m .65873 .01847 L s P p .001 w .71164 .01472 m .71164 .01847 L s P p .001 w .73809 .01472 m .73809 .01847 L s P p .001 w .76455 .01472 m .76455 .01847 L s P p .001 w .791 .01472 m .791 .01847 L s P p .001 w .84391 .01472 m .84391 .01847 L s P p .001 w .87037 .01472 m .87037 .01847 L s P p .001 w .89683 .01472 m .89683 .01847 L s P p .001 w .92328 .01472 m .92328 .01847 L s P p .001 w .97619 .01472 m .97619 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .11058 m .03006 .11058 L s P [(1.05)] .01131 .11058 1 0 Mshowa p .002 w .02381 .20644 m .03006 .20644 L s P [(1.1)] .01131 .20644 1 0 Mshowa p .002 w .02381 .30231 m .03006 .30231 L s P [(1.15)] .01131 .30231 1 0 Mshowa p .002 w .02381 .39817 m .03006 .39817 L s P [(1.2)] .01131 .39817 1 0 Mshowa p .002 w .02381 .49403 m .03006 .49403 L s P [(1.25)] .01131 .49403 1 0 Mshowa p .002 w .02381 .5899 m .03006 .5899 L s P [(1.3)] .01131 .5899 1 0 Mshowa p .001 w .02381 .03389 m .02756 .03389 L s P p .001 w .02381 .05306 m .02756 .05306 L s P p .001 w .02381 .07223 m .02756 .07223 L s P p .001 w .02381 .09141 m .02756 .09141 L s P p .001 w .02381 .12975 m .02756 .12975 L s P p .001 w .02381 .14892 m .02756 .14892 L s P p .001 w .02381 .1681 m .02756 .1681 L s P p .001 w .02381 .18727 m .02756 .18727 L s P p .001 w .02381 .22562 m .02756 .22562 L s P p .001 w .02381 .24479 m .02756 .24479 L s P p .001 w .02381 .26396 m .02756 .26396 L s P p .001 w .02381 .28313 m .02756 .28313 L s P p .001 w .02381 .32148 m .02756 .32148 L s P p .001 w .02381 .34065 m .02756 .34065 L s P p .001 w .02381 .35983 m .02756 .35983 L s P p .001 w .02381 .379 m .02756 .379 L s P p .001 w .02381 .41734 m .02756 .41734 L s P p .001 w .02381 .43652 m .02756 .43652 L s P p .001 w .02381 .45569 m .02756 .45569 L s P p .001 w .02381 .47486 m .02756 .47486 L s P p .001 w .02381 .51321 m .02756 .51321 L s P p .001 w .02381 .53238 m .02756 .53238 L s P p .001 w .02381 .55155 m .02756 .55155 L s P p .001 w .02381 .57073 m .02756 .57073 L s P p .001 w .02381 .60907 m .02756 .60907 L s P p .002 w .02381 0 m .02381 .61803 L s P P p .004 w .02381 .60332 m .02427 .60332 L .02474 .60332 L .0252 .60332 L .02567 .60332 L .02613 .60332 L .0266 .60331 L .02706 .60331 L .02753 .60331 L .02846 .6033 L .02939 .6033 L .03032 .60329 L .03125 .60328 L .03311 .60326 L .03497 .60324 L .03869 .60318 L .04241 .6031 L .04613 .603 L .04985 .60289 L .05357 .60276 L .06101 .60244 L .06845 .60205 L .07589 .60159 L .08333 .60107 L .09821 .5998 L .1131 .59825 L .14286 .59431 L .17262 .58925 L .20238 .58306 L .23214 .57576 L .2619 .56733 L .29167 .55778 L .32143 .54712 L .35119 .53533 L .38095 .52243 L .41071 .50842 L .44048 .49328 L .47024 .47702 L .5 .45964 L .52976 .44113 L .55952 .42148 L .58929 .40068 L .61905 .37873 L .64881 .3556 L .67857 .33128 L .70833 .30574 L .7381 .27896 L .76786 .25091 L .79762 .22155 L .82738 .19083 L Mistroke .85714 .15869 L .8869 .12509 L .91667 .08995 L .94643 .05319 L .97619 .01472 L Mfstroke .02381 .60332 m .02427 .60332 L .02474 .60332 L .0252 .60332 L .02567 .60332 L .02613 .60332 L .0266 .60332 L .02706 .60332 L .02753 .60331 L .02846 .60331 L .02939 .60331 L .03032 .60331 L .03125 .6033 L .03311 .6033 L .03497 .60329 L .03869 .60326 L .04241 .60323 L .04613 .60318 L .04985 .60314 L .05357 .60308 L .06101 .60295 L .06845 .60278 L .07589 .60259 L .08333 .60237 L .09821 .60184 L .1131 .60119 L .14286 .59955 L .17262 .59745 L .20238 .59491 L .23214 .59195 L .2619 .58857 L .29167 .58481 L .32143 .58068 L .35119 .57621 L .38095 .57143 L .41071 .56637 L .44048 .56107 L .47024 .55557 L .5 .54991 L .52976 .54413 L .55952 .5383 L .58929 .53245 L .61905 .52666 L .64881 .521 L .67857 .51552 L .70833 .51032 L .7381 .50547 L .76786 .50108 L .79762 .49725 L .82738 .4941 L Mistroke .85714 .49175 L .8869 .49036 L .91667 .4901 L .94643 .49115 L .97619 .49374 L Mfstroke P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] lambda2Interp = Interpolation[xy1]; lambda1Interp = Interpolation[xy2]; omegaInterp = Interpolation[xy3]; zInterp = Interpolation[xy4]; zend = Last[ yAns[[4]] ] rAns = xAns*(yAns[[1]]); rInterp = Interpolation[ MapThread[List,{xAns,rAns}] ]; :[font = output; output; inactive; preserveAspect; endGroup] 0.0235940465687372 ;[o] 0.0235940465687372 :[font = text; inactive; preserveAspect] The next figure shows the inflated profile. :[font = input; preserveAspect; startGroup] ParametricPlot[{rInterp[rho],zend - zInterp[rho]}, {rho,x0,r0}, PlotLabel -> " r,z Profile "] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 26.455 0.0147151 24.9471 [ [(0.005)] .15608 .01472 0 2 Msboxa [(0.01)] .28836 .01472 0 2 Msboxa [(0.015)] .42063 .01472 0 2 Msboxa [(0.02)] .55291 .01472 0 2 Msboxa [(0.025)] .68519 .01472 0 2 Msboxa [(0.03)] .81746 .01472 0 2 Msboxa [(0.035)] .94974 .01472 0 2 Msboxa [( r,z Profile )] .5 .61803 0 -2 Msboxa [(0.005)] .01131 .13945 1 0 Msboxa [(0.01)] .01131 .26419 1 0 Msboxa [(0.015)] .01131 .38892 1 0 Msboxa [(0.02)] .01131 .51366 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .15608 .01472 m .15608 .02097 L s P [(0.005)] .15608 .01472 0 2 Mshowa p .002 w .28836 .01472 m .28836 .02097 L s P [(0.01)] .28836 .01472 0 2 Mshowa p .002 w .42063 .01472 m .42063 .02097 L s P [(0.015)] .42063 .01472 0 2 Mshowa p .002 w .55291 .01472 m .55291 .02097 L s P [(0.02)] .55291 .01472 0 2 Mshowa p .002 w .68519 .01472 m .68519 .02097 L s P [(0.025)] .68519 .01472 0 2 Mshowa p .002 w .81746 .01472 m .81746 .02097 L s P [(0.03)] .81746 .01472 0 2 Mshowa p .002 w .94974 .01472 m .94974 .02097 L s P [(0.035)] .94974 .01472 0 2 Mshowa p .001 w .05026 .01472 m .05026 .01847 L s P p .001 w .07672 .01472 m .07672 .01847 L s P p .001 w .10317 .01472 m .10317 .01847 L s P p .001 w .12963 .01472 m .12963 .01847 L s P p .001 w .18254 .01472 m .18254 .01847 L s P p .001 w .20899 .01472 m .20899 .01847 L s P p .001 w .23545 .01472 m .23545 .01847 L s P p .001 w .2619 .01472 m .2619 .01847 L s P p .001 w .31481 .01472 m .31481 .01847 L s P p .001 w .34127 .01472 m .34127 .01847 L s P p .001 w .36772 .01472 m .36772 .01847 L s P p .001 w .39418 .01472 m .39418 .01847 L s P p .001 w .44709 .01472 m .44709 .01847 L s P p .001 w .47354 .01472 m .47354 .01847 L s P p .001 w .5 .01472 m .5 .01847 L s P p .001 w .52646 .01472 m .52646 .01847 L s P p .001 w .57937 .01472 m .57937 .01847 L s P p .001 w .60582 .01472 m .60582 .01847 L s P p .001 w .63228 .01472 m .63228 .01847 L s P p .001 w .65873 .01472 m .65873 .01847 L s P p .001 w .71164 .01472 m .71164 .01847 L s P p .001 w .7381 .01472 m .7381 .01847 L s P p .001 w .76455 .01472 m .76455 .01847 L s P p .001 w .79101 .01472 m .79101 .01847 L s P p .001 w .84392 .01472 m .84392 .01847 L s P p .001 w .87037 .01472 m .87037 .01847 L s P p .001 w .89683 .01472 m .89683 .01847 L s P p .001 w .92328 .01472 m .92328 .01847 L s P p .001 w .97619 .01472 m .97619 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P [( r,z Profile )] .5 .61803 0 -2 Mshowa p .002 w .02381 .13945 m .03006 .13945 L s P [(0.005)] .01131 .13945 1 0 Mshowa p .002 w .02381 .26419 m .03006 .26419 L s P [(0.01)] .01131 .26419 1 0 Mshowa p .002 w .02381 .38892 m .03006 .38892 L s P [(0.015)] .01131 .38892 1 0 Mshowa p .002 w .02381 .51366 m .03006 .51366 L s P [(0.02)] .01131 .51366 1 0 Mshowa p .001 w .02381 .03966 m .02756 .03966 L s P p .001 w .02381 .06461 m .02756 .06461 L s P p .001 w .02381 .08956 m .02756 .08956 L s P p .001 w .02381 .1145 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L .05622 .60288 L .0627 .60269 L .07566 .60221 L .0886 .60158 L .10154 .60082 L .12738 .59888 L .15315 .59639 L .17885 .59334 L .22995 .58561 L .28055 .57568 L .33053 .56359 L .37977 .54935 L .42815 .53299 L .47553 .51453 L .5218 .49402 L .56684 .47148 L .61051 .44695 L .65269 .42047 L .69325 .39209 L .73206 .36184 L .76898 .32977 L .80386 .29593 L .83654 .26038 L .86687 .22317 L .89466 .18437 L .91972 .14403 L .94183 .10224 L .96074 .0591 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; preserveAspect] Other interesting figures are given in [H1]. One shows two different solutions for the same h0 as above, also the same pressure of .13 Mpa, and Mooney constants of C1=.5 Mpa, C2=0 Mpa. This is possible since uniqueness is not guaranteed in this problem. :[font = input; preserveAspect; endGroup] :[font = section; inactive; preserveAspect; startGroup] More that might be done :[font = text; inactive; preserveAspect; endGroup] d02raf allows users to provide codes for jacobians. In this Notebook we merely followed Terry Robb's default settings and did not provide jacobians. (In the treatment of this problem with the AXIOM-NAG link, described in [HK], the CAS generated the jacobian codes.) It probably won't be done by us, but we remark that more calculus might be done - and Mathematica would facilitate this. In particular, an analytical study of the behaviour at r=0 would allow, for our subsequent numerics, a more scientific approach to choice of x0 and related matters. We suggest, though, that more careful consideration of the mathematical modelling, to explore possible use of variational structure, would seem appropriate before investing any more effort into tidying the present approach. Related to this, though closer to the existing d.e. formulation would be to explore whether there is any way the system, by choice of appropriate combinations for independent variables, can be transformed into some useful special structure (gradient system?). At P=0, lambda1=lambda2=1 and omega=Pi/2 (so z=0) solve the d.e.s and b.c.s. Mathematica could be used in the manipulations to do a perturbation expansion solution for P small, based on a series expansion of the rhs of the d.e.s. (At P small and h small, it appears that the surface z(r) is approximately paraboloidal.) As we reported above, in the section on numerical results, at fixed h0, C1, C2 there can be, at a given P, more than one solution. It would be possible to adapt d02raf to do 'branch following' (as in Keller's AUTO code, for example). Starting at the solutions for small P we could follow that branch. A (P,solution)-space is the usual way of "visualising" structure of the solution set. Our experimental purpose did not need us to elucidate the structure of the solution set, whether there is secondary bifurcation, pitchfork bifurcations or whatever, stability of branches of solutions, etc.. Nevertheless, the numerical codes, InterCall and Mathematica graphics would be excellent tools to use should a bifurcation picture be needed. ;[s] 7:0,0;353,1;364,0;1117,1;1128,0;2005,1;2016,0;2099,-1; 2:4,13,9,Times,0,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup] Remarks and speculations on the software :[font = text; inactive; preserveAspect] This section is Grant Keady's. From Mathematica 3, we understand that NDSolve may be enhanced to solve two-point b.v. problems. The particular task we have solved in this Notebook with d02raf will probably be within its capabilities. So, why use NAG, InterCall? Of course, there is much more in the NAG library (integral equations, p.d.e., etc.) which isn't in the numerical capability of Mathematica 3. As both NAG and Mathematica are in the business of selling useful numerical codes to engineers and so on, there are bound to be points of overlap. Furthermore, at any given time, by virtue of mass-market considerations, it will often be the case that one will find in NAG numerical items which might well be included in a future release of Mathematica . InterCall to NAG remains important. Less obvious, but we feel, still true, is that there is still a case for efficient numerical solution via D02RAF, and stand-alone C/Fortran programs. For more on this, and related matters of ASPs in Fortran, see a supplement to this paper available through the URL given before. NAG distribute numerical libraries in f90 and C as well as the f77 library used here. Our long-standing interest in links from Mathematica to NAG's C library are also documented in the supplement available through the Web. ;[s] 15:0,0;37,1;48,0;52,2;129,0;236,2;263,0;391,1;402,0;423,1;434,0;748,1;759,0;1205,1;1216,0;1302,-1; 3:8,13,9,Times,0,12,0,0,0;5,13,9,Times,2,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] ESC and genmex :[font = text; inactive; preserveAspect; endGroup] genmex for Matlab, and InterCall for Mathematica have many points of similarity. In particular, the database entries for genmex and InterCall are very similar. Thus, before adding a new large database to InterCall, it is probably worth checking if the corresponding genmex database is available. Of course, for individual Fortran subroutines, one may as well write the entry from scratch: the chances of finding its genmex entry, if one, faster than just writing a single database entry are remote. As we remarked before, the database entry for d02raf in InterCall, and in genmex, was wrong! d02raf is one of the more elaborate entries in the NAG library, and it is understandable how small errors can creep in. We first noted the error - and corrected it - for the genmex database. See the result of the GetDefault if you run our Mathematica Notebook. For further comments on lessons from genmex relevant to InterCall, see our supplement available through the Web. ;[s] 5:0,0;37,1;49,0;832,1;844,0;970,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] Conclusions :[font = text; inactive; preserveAspect; endGroup; endGroup] Mathematica and InterCall are good products. InterCall is very important for re-use of established numeric codes. InterCall is easier to use than raw MathLink . While InterCall to NAG is but one small use of InterCall, the breadth of the NAG library makes it an impressive example with which to demonstrate InterCall to engineering users. ;[s] 4:0,1;11,0;151,1;159,0;340,-1; 2:2,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup] References :[font = text; inactive; preserveAspect; endGroup] Papers [H1], [HK], [LK] and various related items, as well as this paper, are available from http://maths.uwa.edu.au/~keady/homepage.html (Follow links to electronically available reports, to Matlab/genmex, to Mathematica as appropriate.) [AR] J.E. Adkins and R.S. Rivlin, Large Elastic Deformations of Isotropic Materials, Part IX; Phil. Trans. A 244 (1951-1952), 505. [BKRRD] K. Broughan, G. Keady, T. Robb, M. Richardson and M. Dewar, Some symbolic computing links to the NAG numeric library. SIGSAM Bulletin 25 (July 1991), 28-37. [GA] A.E. Green and J.E. Adkins, Large Elastic Deformations and Nonlinear Continuum Mechanics, Oxford University Press, Oxford (1960). [HK] E. Hawkes and G. Keady, Two more links to NAG numerics involving CA systems. In the Electronic Proceedings of the IMACS Applied Computer Algebra Conference, University of New Mexico, May 1995. S. Steinberg and M. Wester, eds., http://math.unm.edu/aca.html [H1] E. Hawkes, Inflation of a rubber disk. Project report for 3A2 Numerical Analysis course. (May 1995). [H2] E. Hawkes, Modelling of inflatable packers. Honours Thesis (Oct. 1995). Available from Dept. of Mechanical Engineering, University of Western Australia. [KS] W.W. Klingbiel and R.T. Shield, Some Numerical Investigations on Empirical Strain Energy Functions in the Large Axi-Symmetric Extensions of Rubber Membranes, Z.A.M.P. 15, (1964) pp 608-629. [LK] M. Laine and G. Keady, Using Genmex - examples of calls to NAG, Jan 1995, Available via Web, or anon ftp to 130.95.16.1 in /pub/keady . Also update Apr 1995, on d02raf. [NAG] NAG Fortran Library Documentation, Numerical Algorithms Group Ltd. (1990-) [Ogd] R.W. Ogden, Elastic Deformations of Rubberlike Solids, Mechanics of Solids, The Rodney Hill 60th anniversary volume, Pergamon Press, Oxford, (1981) pp 499-537 (eds: H.G. Hopkins and M.J. Sewell). [Trel] L.R.G. Treloar, Strains in an inflated rubber sheet, Inst. Rubber Ind. Trans. 19 (1944), 201. ;[s] 27:0,0;212,2;223,0;337,2;349,0;352,1;355,0;502,2;518,0;519,1;521,0;576,2;636,0;769,2;840,0;1371,2;1378,0;1380,1;1382,0;1588,2;1621,0;1724,2;1744,0;1928,2;1953,0;1954,1;1956,0;1971,-1; 3:14,13,9,Times,0,12,0,0,0;4,13,9,Times,1,12,0,0,0;9,13,9,Times,2,12,0,0,0; ^*)