Particular algorithms Numerical continuation

1 particular algorithms

1.1 natural parameter continuation
1.2 simplicial or piecewise linear continuation
1.3 pseudo-arclength continuation
1.4 gauss–newton continuation

particular algorithms
natural parameter continuation

most methods of solution of nonlinear systems of equations iterative methods. particular parameter value




λ

0




{\displaystyle \lambda _{0}}

mapping repeatedly applied initial guess





u


0




{\displaystyle \mathbf {u} _{0}}

. if method converges, , consistent, in limit iteration approaches solution of



f
(

u

,

λ

0


)
=
0


{\displaystyle f(\mathbf {u} ,\lambda _{0})=0}

.

natural parameter continuation simple adaptation of iterative solver parametrized problem. solution @ 1 value of



λ


{\displaystyle \lambda }

used initial guess solution @



λ
+
Δ
λ


{\displaystyle \lambda +\delta \lambda }

.



Δ
λ


{\displaystyle \delta \lambda }

sufficiently small iteration applied initial guess should converge.

one advantage of natural parameter continuation uses solution method problem black box. required initial solution can given (some solvers used start @ fixed initial guess). there has been lot of work in area of large scale continuation on applying more sophisticated algorithms black box solvers (see e.g. loca).

however, natural parameter continuation fails @ turning points, branch of solutions turns round. problems turning points, more sophisticated method such pseudo-arclength continuation must used (see below).

simplicial or piecewise linear continuation

simplicial continuation, or piecewise linear continuation (allgower , georg) based on 3 basic results.

the first is

the second result is:

please see article on piecewise linear continuation details.

with these 2 operations continuation algorithm easy state (although of course efficient implementation requires more sophisticated approach. see [b1]). initial simplex assumed given, reference simplicial decomposition of ir^n. initial simplex must have @ least 1 face contains 0 of unique linear interpolant on face. other faces of simplex tested, , typically there 1 additional face interior zero. initial simplex replaced simplex lies across either face containing zero, , process repeated.

references: allgower , georg [b1] provides crisp, clear description of algotihm.

pseudo-arclength continuation

this method based on observation ideal parameterization of curve arclength. pseudo-arclength approximation of arclength in tangent space of curve. resulting modified natural continuation method makes step in pseudo-arclength (rather



λ


{\displaystyle \lambda }

). iterative solver required find point @ given pseudo-arclength, requires appending additional constraint (the pseudo-arclength constraint) n n+1 jacobian. produces square jacobian, , if stepsize sufficiently small modified jacobian full rank.

pseudo-arclength continuation independently developed edward riks , gerald wempner finite element applications in late 1960s, , published in journals in 1970s h.b. keller. detailed account of these developments provided in textbook m. a. crisfield: nonlinear finite element analysis of solids , structures, vol 1: basic concepts, wiley, 1991. crisfield 1 of active developers of class of methods, standard procedures of commercial nonlinear finite element programs.

the algorithm predictor-corrector method. prediction step finds point (in ir^(n+1) ) step



Δ
s


{\displaystyle \delta s}

along tangent vector @ current pointer. corrector newton s method, or variant, solve nonlinear system

f
(
u
,
λ
)
=
0








u
˙




0


∗


(
u
−

u

0


)
+




λ
˙




0


(
λ
−

λ

0


)
=
Δ
s






{\displaystyle {\begin{array}{l}f(u,\lambda )=0\\{\dot {u}}_{0}^{*}(u-u_{0})+{\dot {\lambda }}_{0}(\lambda -\lambda _{0})=\delta s\\\end{array}}}

where



(




u
˙




0


,




λ
˙




0


)


{\displaystyle ({\dot {u}}_{0},{\dot {\lambda }}_{0})}

tangent vector @



(

u

0


,

λ

0


)


{\displaystyle (u_{0},\lambda _{0})}

. jacobian of system bordered matrix

[





f

u





f

λ










u
˙




∗







λ
˙







]



{\displaystyle \left[{\begin{array}{cc}f_{u}&f_{\lambda }\\{\dot {u}}^{*}&{\dot {\lambda }}\\\end{array}}\right]}

at regular points, unmodified jacobian full rank, tangent vector spans null space of top row of new jacobian. appending tangent vector last row can seen determining coefficient of null vector in general solution of newton system (particular solution plus arbitrary multiple of null vector).

gauss–newton continuation

this method variant of pseudo-arclength continuation. instead of using tangent @ initial point in arclength constraint, tangent @ current solution used. equivalent using pseudo-inverse of jacobian in newton s method, , allows longer steps made. [b17]