Belytschko T. - Introduction (779635), страница 60

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Viewing an equilibrium solution as the determination ofthe stationary points of a potential provides substantial insight, particularly whenthe stability of a solution is of interest. This is pursued further in Section ??. Ascan be seen from a comparison of Eqs. (6.3.1) and (6.3.27) , the equations for astationary point are identical to the discrete equations derived previously.

Thesemethods are not applicable to dynamic problems.6.3.8Convergence Criteria. The termination of the iterative procedure inimplicit and equilibrium solutions by the Newton method is determined byconvergence criteria. These criteria pertain to the convergence of the discretesolution to the equations r dn ,t n = 0 , not the convergence of the discretesolution to the solution of the partial differential equations. Three types ofconvergence criteria are used to control the iterations:1. criteria based on the magnitude of the residual r ;2. criteria based on the magnitude of the displacement increments ∆d ;3. energy error criteria.()Usually an l 2 norm of the vectors is used for the first two criteria.

The criteriathen are:residual error criterion:1rl2 n DOF  2=  ∑ ra2  ≤ ε max  f ext a =1 l2, fintl2, Ma l2 (6.3.28)displacement increment error criterion:1∆d l2 nDOF 2  2=  ∑ ∆da  ≤ ε d l2 a=1(6.3.29)The l 2 norm, which has been indicated in the above, is the probably mostsuitable when the mean error over all degrees of freedom is to be controlled, but amaximum norm can also be used. A maximum norm would limit the maximumerror at any node.

The terms on the right-hand side of Eqs. (6.3.28) and (6.3.29)are scaling factors. Without these, the criterion would depend on the parameters6-19T. Belytschko & B. Moran, Solution Methods, December 16, 1998of the problem. The error toleranceε determines the precision with which thedisplacements are calculated before terminating the iterative procedure; whenε =10−3 , the mean accuracy of the nodal displacements is in the third significantdigit when the l 2 norm is used. The convergence tolerance determines the speedand accuracy of a calculation. If the criterion is too coarse, the solution may bequite inaccurate. On the other hand, a criterion which is too tight results inunnecessary computations.The energy convergence criterion measures the energy flow to the systemresulting from the residual, which is like an error in energy.

It is given by(∆dT r =∆dara ≤ ε max W ext ,W int , W kin)(6.3.30)where the computation of the energies used for scaling the criterion is described inSection 6.?. The left hand side in the above represents an error in the energy,since a nonzero residual is an error in the forces on the system.6.3.7.Convergence and Robustness of Newton Iteration. The rateof the convergence of the iterations in the Newton method is quadratic when theJacobian matrix A satisfies certain conditions. These conditions may roughly bedescribed as follows:1. the Jacobian A should be a sufficiently smooth function of d;2.

the Jacobian A should be regular (invertable) and well-conditioned in theentire domain in the displacement space that the iterative procedure traverses.Quadratic convergence means that the l 2 norm of the difference between thesolution and the iterate dυ decreases quadratically in each iteration:dυ +1 −d ≤ c dυ − d2(6.3.31)where c is a constant that depends on the nonlinearity of the problem and d is thesolution to the nonlinear algebraic equations.

Thus the convergence of theNewton algorithm is quite rapid when A meets the above conditions. The abovegives the requirements for convergence only in broad terms and convergence hasbeen proven for various conditions on A. One set of conditions for quadraticconvergence are: the residual must be continuously differentiable and the inverseof the Jacobian matrix must exist and be uniformly bounded in the neighborhoodof the solution, Dennis and Schnabel (1983, p 90).These conditions are usually not satisfied by nonlinear finite elementproblems. For example, in an elastic-plastic material, the residual is notcontinuously differentiable when a discrete point changes from elastic to plastic orvice versa; therefore, the Jacobian is discontinuous. In a two degree of freedomproblem, the discontinuities in the Jacobian appear as kinks in the contour plotsfor the residual components.

This is illustrated in Example X. In the solution ofcontact-impact problems with Lagrange multiplier methods, the residual oftenlacks smoothness, as illustrated by Chapter 10. Thus the conditions for quadraticconvergence of the Newton method are often not satisfied in engineering6-20T. Belytschko & B. Moran, Solution Methods, December 16, 1998problems. Yet, Newton’s method is remarkably effective in engineering problems,although the rate of convergence often deteriorates. At this time, more robustmethods are not available. In many problems, the conditions for quadraticconvergence are satisfied; for example, the above conditions are satisfied in theresponse of a model with a Mooney-Rivlin material when the load is smallenough so that the equilibirium solutions are stable.Newton’s method fails particularly often when applied to equilibriumproblems. Since Eq.

(6.3.3) are nonlinear algebraic equations, they can havemultiple solutions and solutions in which are unstable. When the equilibrium pathis unstable, the inverse of the Jacobian matrix is no longer regular at all points andthe proof of quadratic convergence does not apply. The convergence of theNewton method often fails in the vicinity of unstable states. These types ofproblems are considered in the next Section.In summary, Newton’s method sometimes lacks robustness when appliedto engineering problems. The robustness decreases as we increase the time stepand appears more often in equilibrium solutions, since in the latter we lose theeffect of the mass matrix. The mass matrix improves the conditioning of theJacobian matrix because it is always positive definite, see Exercise X.

As thetime step increases, the beneficial effects of the mass matrix decrease since thecoefficient of the mass matrix is inversely proportional to the square of the timestep, as can be seen from Eq. (6.3.9). For many problems, a straightforwardapplication of the Newton method will sometimes fail completely, andenhancements of the Newton method such as the arc length method, line search,and augmented Lagrangian, which are described in Section ?, are needed to solvethe nonlinear algebraic equations.6.3.8.Line Search.

An effective way to increase the robustness of Newtonmethods when convergence is slow is to use the line search technique. Therationale behind line search is that the direction ∆d found by the Newton methodis often a good direction, but the step size is not optimal. It is cheaper to find thebest point along this direction by several computations of the residual than to get anew direction by using a new Jacobian. Therefore, before proceeding to the nextdirection, the residual is minimized along the line dold +ξ∆d where dold is thelast iterate and ξ > 0 is a parameter.

In other words, we find the parameter ξ sothat dold +ξ∆d minimizes some measure of the residual. We can use as ameasure of the measure of the residual its l 2 norm, as defined in Eq. (6.3.28), themaximum norm, i.e. the maximum absolute value of any component of theresidual, or some other measure. Line search then involves the calculation of twoor more residuals along the line and an interpolation of a measure of the residual.For example, if the l 2 norm is used, thenA measure for the residual which is frequently used in line search is basedon the existence of a potential for the problem, i.e. on the solution by thestationary energy principle, Sections 4.9.3 and 6.3.6. For a conservative problem,the minimizer of the potential W (d) , along the line ∆d is the point where thegradient of the function is orthogonal to the line.

The residual is given in terms ofa potential by6-21T. Belytschko & B. Moran, Solution Methods, December 16, 1998∂W ∂W int ∂W ext=−= f int − f ext = r∂d∂d∂d(6.3.32)where the above follows from Eqs. (4.9.34) and (6.3.3). When the residual isorthogonal to the incremental displacement∆dT r = 0 ⇒ ∆dT∂W=0∂d(6.3.33)the potential must be minimum (or be stationary) at that point. This is illustratedin Fig.

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