A Symbolic-Numeric Approach to Tube Modeling in CAD Systems

Abstract

In this note we present a symbolic-numeric method to the problem of tube modeling in CAD systems. Our approach is based on the Kirchhoff kinetic analogy which allows us to find analytic solutions to the static Kirchhoff equa- tions for rods under given boundary conditions. Keywords: Kirchhoff rod; boundary value problems; automatic differentiation

Full text

A Symbolic-Numeric Approach to Tube Modeling in
CAD Systems
Gerrit Sobottka and Andreas Weber
Institut f¨
ur Informatik II, Universit¨
at Bonn, R¨
omerstr. 164, Bonn, Germany
E-mail: {sobottka,weber}@cs.uni-bonn.de
Abstract. In this note we present a symbolic-numeric method to the problem of
tube modeling in CAD systems. Our approach is based on the Kirchhoff kinetic
analogy which allows us to find analytic solutions to the static Kirchhoff equa-
tions for rods under given boundary conditions.
Keywords: Kirchhoff rod; boundary value problems; automatic differentiation
1 Introduction
In this short note we address the problem of physics based tube modeling which fre-
quently occurs in computer aided design (CAD) applications. The task at hand is to
connect a tube of given length and with certain material properties to connectors fixed
in space. In particular, we aim to predict the internal forces and torques along the tube
for dimensioning purposes as well as the final configuration. The equilibrium shape
of the tube is governed by the well known static Kirchhoff equations. Along with the
boundary conditions at both ends defined by the position and orientation of the con-
nectors we have to solve a two-point boundary value problem (BVP). Such BVPs can
be solved employing standard shooting techniques which usually perform at slow con-
vergence rates. Our approach is based on the analytic solution to the static Kirchhoff
equations and is a continuation of our work described in [1], to which we refer for more
details on the basis of the method. In addition to adding a new case of boundary con-
ditions we will also show how the “symbolic-numeric method” introduced in [1] can
be made “more symbolic” by using the Jacobian Matrix in symbolic form within the
numerical part.
2 Related Work
While the number of publications on solution methods for Kirchhoff equations is large,
the task of tube modeling based on these equations has rarely been addressed before.
Gr´
egoire and Sch¨
omer [2] use an extended spring-mass system that is solved with an
energy minimizing algorithm. In [3] Healey and Metha present a method to solve asso-
ciated BVPs by augmenting the system of boundary conditions by a constraint on the
magnitude of the quaternions used for the parameterization of the rotations. In particu-
lar, they show that if these constraints are met at the end points they are also met on the

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whole domain. In [4] a geometrically exact approach is proposed, which is based on the
explicit solution of the kinematic relation based on Rodriguez’ formula. Henderson and
Neukirch [5] study spatial equilibria of clamped elastica based on Kirchhoff rods. In
contrast to their work we do not restrict ourselves to the case where the tangents at the
end points are collinear. In [6] Nizette and Goriely study explicit solution of the static
Kirchhoff equations in terms of Euler-Kirchhoff filaments.
3 Physics Based Tube Modeling
3.1 The symbolic part
Let r(s):[a1, a2]∈R7→ R3be the centerline of the tube. The centerline is furnished
with a set of right-handed orthonormal triads {d1,d2,d3},such that d3=r0is the
tangent of the centerline and d1,d2span the cross section plane at each point of the
rod. Further, we assume the tube to be inextensible, unshearable, and initially straight.
The equilibrium state is given by the static Kirchhoff equations:
F0=0,(1)
M0+d3×F=0,(2)
M=u1·d1+u2·d2+b·u3·d3,(3)
where Fand Mare the internal force and the torque of the rod. Note that since we as-
sume the tube to have a circular cross section we use the scaled form of the constitutive
equation for M,where b= 1/(1 + ν)with νbeing Poisson’s ratio.
Further, we have the kinematic relation di=u×di,where u={u1, u2, u3}is
the twist vector. The components of the twist vector as well as the local directors {di}
are conveniently expressed in terms of Euler angles (ϕ, θ, ψ)w.r.t. to the global frame
{ex,ey,ez}.With
F=F·ez,(4)
M0
z=M0
z·ez= 0,(5)
M0
3=M0
3·d3= 0,(6)
H=1
2·M·u+F·d3,(7)
being first integrals, which do not depend on the arc length parameter, we obtain the
following equations for the Euler angles:
ϕ0=Mz−M3·z
1−z2,(8)
ψ0=M3−Mz·z
1−z2+M3·1
b−1,(9)
z02= 2F·(h−z)·1−z2−(Mz−M3·z)2,(10)
where z= cos θand h= 1/F ·H−M2
3/2b.The right hand side of z02is a cubic
polynomial in zwith the roots z1, z2, z3,such that −1≤z1≤z2≤1≤z3. The

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solution is given by
z=z1+ (z2−z1)·sn2[λ(s+s0), k],(11)
where λ=p1/2·F·(z3−z1), k =p(z2−z1)/(z3−z1),and sn is one of the Ja-
cobi elliptic function. With the function zat hand the solutions for ϕand ψare obtained
by directly integrating the above equations [1]:
ϕ=Zs
0
Mz−M3·z(σ)
1−z2(σ)dσ+ϕ0,(12)
ψ=Zs
0
M3−Mz·z(σ)
1−z2(σ)dσ+ψ0+M3·s·1
b−1.(13)
Since the system integral His a function of θand θ0[1] the set of quanti-
ties determining the configurations of the centerline of the tube is given by η=
{F, Mz, M3, θ(0), θ(0)0}.
Since the tube is clamped at s= 0 and can freely rotate at the other end (s=L) the
boundary conditions imposed by the underlying problem are thus given as r(0) = x0,
d1(0) = d10,d2(0) = d20 ,r(L) = xLand hd3(L),tLi= 1,where xLand tLare the
coordinates of the point and the tangent to be matched at s=L.
3.2 Numerical computations
Thus the solution of this two point boundary value problem is reduced to the solution
of the following set F(η) = 0 of non-linear equations:
r(L)−xL= 0,
1− hd3,tLi= 0,
θ0−θ0
0= 0.
(14)
Standard numeric solution techniques [7,8] require that the Jacobian matrix is
known numerically at every iteration point. These methods use numeric approxima-
tions to the partial derivatives at the iteration points, if those are not given as program
code. As we have derived a symbolic expression for the function F, we will show how
we can come up with rather efficient code for the Jacobian, too.
4 Using the Jacobian Matrix in Symbolic Form
Using the common subexpression elimination algorithm of Maple the function Fcan
be described by a computation sequence involving the following number of commands:
43 assignments + 29 additions + 65multiplications + 5 divisions
+19 functions + 5integrals (15)
The 19 function evaluations consist of 9 trigonometric and square root functions and
10 instances of the Jacobi elliptic functions sn. The 5 integrals come from the necessity
to have the centerline of the space curve available in a form that allows for boundary

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conditions at two distinct points, i.e. to explicitly carry the integration of the tangent
vector [1,6].
Standard tools for automatic differentiation and also the automatic differentiation
procedure available in Maple cannot handle integral operators in their inputs. Thus we
could not use a straight-forward automatic differentiation approach on the computation
sequence of Fto obtain a computation sequence for the Jacobian matrix.
Whereas the symbolic differentiation algorithm of Maple can handle occurring in-
tegrals, the symbolic expression representing Fwas too large for a straight-forward
symbolic differentiation.
However, with the following method we successfully derived symbolic computation
sequences for the Jacobian matrix.
– We used auxiliary symbolic functions for the roots z1,z2, and z3of the cubic poly-
nomial occurring on the right-hand-side of (10) and its partial derivatives.
– Using these auxiliary functions in the expression of Fwe could successfully com-
pute the Jacobian matrix in symbolic form using Maple.
This computation required several minutes of computation time.
– A computation sequence could be obtained by Maple generated from the expres-
sions of the Jacobian and assignment of the expressions of the roots z1,z2, and z3
and its partial derivatives to the auxiliary symbolic functions.
Notice that all symbolic partial derivatives of the expressions of the roots z1,z2,
and z3could be obtained by Maple easily.
Using the optimize function on computational sequences the result for computing
the Jacobian required the following number of commands:
260 assignments + 174 additions + 419 multiplications
+31 divisions + 76 functions + 24 integrals (16)
Notice that because of the symbolic differentiation rules for “special functions”
used by Maple the computation sequence contains calls to various of Jacobi’s elliptic
function and also to incomplete elliptic integrals of the second kind.
Remark. If equation (14) is solved via a corresponding minimization problem of a
real valued function mF, then the gradient function of mFcan be expressed in symbolic
form similarly. The computational costs for this gradient after common subexpression
elimination is almost identical to the one for the Jacobian after common subexpression
evaluation, i.e. is also given by the number of commands stated in (16).
References
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