Belytschko T. - Introduction (779635), страница 61
Текст из файла (страница 61)
6.1, which shows the contours of the potential energy for a two degree-offreedom system and the residual of the nodal forces for several points along theline dold +ξ∆d . As can be seen, the potential is minimum when the residual, i.e.the gradient of the potential, is normal to the line. The line search can then beconducted by minimizing ∆dT r .This criterion can also be used for systems that are not conservative, since∆d r does not involve the potential. Note that this measure of the residual isequivalent to the criterion for error in energy, Eq. (6.3.30).TEquation (6.3.33a) can also be derived directly by using the chain rule toexpand the potential energy in the parameter ξ .
This givesdW (ξ ) ∂W dd=⋅= 0 ⇒ rT ∆d = 0dξ∂d dξ(6.3.34)where we have set the derivative of the potential energy with respect to theparameter ξ equal to zero, since we are looking for the minimum of the potentialalong the line ∆d parametrized by ξ . The second equation follows from (6.3.32)anddd d (d old + ξ∆d )==∆ ddξdξ(6.3.35)Once a measure of the residual has been chosen, the line search can bemade with any of the methods for minimizing a function of a single parameter.The method of bisection or searches based on interpolation or combinationsthereof can be used.
Once the residual has been evaluated at two points, aquadratic fit can be made to the residual measure, since its value at ξ = 0 isknown to vanish. This quadratic fit can then be used to estimate the position ofthe minimum. The iteration along the line is terminated when the measure hasbeen minimized to a suitable precision. Note that when the orthogonalitycondition (6.3.29) is used, it should be normalized like the error energy criterionis in Eq. (6.3.26).6.3.9.
Secant Methods to be inserted6-22T. Belytschko & B. Moran, Solution Methods, December 16, 19986.3.10.Stability of Implicit methods. The advantage of an implicitmethod over an explicit method is that for linear transient, problems, suitableimplicit integrators are unconditionally stable. The unconditional stability ofimplicit integrators has not been proven for all nonlinear systems, although resultswhich deal with specific situations indicate that unconditional stability holds atleast for certain nonlinear systems. In any case, experience indicates that the timesteps which can be used with implicit integrators are much larger than those forexplicit integration in many problems.The major restrictions on the size of time steps in implicit methods arisefrom accuracy requirements and the decreasing robustness of the Newtonprocedure as the time step increases.
The latter is particularly pronounced inproblems with very rough response, such as contact-impact. With a large timestep, the starting iterate may be far from the solution, so the possibility of failureof the Newton method to converge increases. Therefore small time steps are oftenused to improve the robustness of the Newton algorithm.In return for their enhanced stability, implicit methods exact a significantprice: implicit methods require the solution of nonlinear algebraic equations ineach time step. The construction of the linearized algebraic equations for theNewton procedure is often quite involved.
Furthermore, the storage of theseequations requires significant amounts of memory. The memory requirementscan be reduced substantially by iterative linear equation solvers (an iterativemethod within an iterative Newton method). In recent research, iterative solvershave been improved dramatically, so implicit solutions are feasible in manyproblems where they were prohibitive before, see Section ?. The robustness andspeed of Newton methods has increased markedly over the past two decades, andwe are certain that further improvements are imminent.
Nevertheless, high costand lack of robustness are still plague many implicit and equilibrium solutions.6.4 LINEARIZATIONThere are several different ways to linearize the discrete equations. Indiscussing the various linearization procedures, it is useful to keep in mind thatthe order in which linearization and spatial discretization are carried out does notmatter (in mathematical terminology, the operations of linearization and spatialdiscretization are said to commute). This means that linearization of the semidiscrete equations of motion (6.2.x) gives rise to the same finite element equationsas does the semi-discretization of a linearized weak form (we have not yetdeveloped such forms, but they appear frequently in the literature).
The choicebetween these two approaches is a matter of style. For completeness, we willconsider both approaches.In the linearization procedure, there are several possibilities:1. Linearization is carried out before the stress-update algorithm(integration algorithm for the constitutive equation) is introduced; thisgives rise to the so-called continuum tangent moduli which will bediscussed below.2. Linearization is carried out after the stress-update algorithm isintroduced; this gives rise to the so-called algorithmic moduli.These two distinct approaches yield different tangent stiffness matrices.The choice of which approach to use rests on practical considerations related toease of implementation and on convergence of the iterative scheme. The first6-23T.
Belytschko & B. Moran, Solution Methods, December 16, 1998approach, based on the continuum tangent modulus, is straightforward toimplement. However, it can run into convergence difficulties, especially forelastic-plastic materials where the slope discontinuity at the yield point on thestress-strain curve requires small steps to assure convergence and to preserveaccuracy.The second approach, based on the algorithmic moduli, exhibits betterconvergence because, through linearization of the stress-update algorithm, itaccounts for the change in slope associated with a finite increment of strain.
Onedrawback of the method is that it is not always possible to derive explicit formsfor the algorithmic moduli for complex constitutive relations. Numericaldifferentiation schemes are sometimes used to obtain the algorithmic moduli, andthey introduce additional inaccuracies.We first consider linearization of the discrete equations based on thecontinuum tangent moduli, which relate a stress rate to a strain rate.
The resultingmaterial tangent stiffness matrix is called the continuum tangent stiffness matrix.A somewhat more mathematical approach to linearization based ondirectional derivatives is then presented and it is shown how the resultingexpressions are equivalent to those obtained by using the procedure based on thematerial time derivative. This linearization procedure based on the directionalderivative is then used to develop the linearized equations for the second approachdiscussed above, i.e, linearization of the weak form after introduction of thestress-update algorithm.
Because the stress-update algorithm is introduced priorto linearization, the expression for the stress increment that appears in thelinearization of the weak form is based on the linearized constitutive integrationscheme and not on the continuum rate form of the constitutive relation. As aresult, the material tangent stiffness differs from the continuum tangent stiffnessand is referred to as the algorithmic modulus (sometimes referred to as theconsistent tangent modulus because of the consistent linearization of the weakform and the stress-update algorithm). Examples of the algorithmic modulus forthe 2-node bar element and the 3-node triangle are also given.6.4.1 Linearization of the Discrete EquationsIn the following, we derive expressions for the continuum tangent stiffnessmatrix K int .
As will be seen, part of the expression can be derived independentlyof the material response. These expressions are completed upon introduction ofthe constitutive relation. The continuum rate form of the constitutive relation willbe used, i.e., linearization is carried out prior to introduction of the stress-updatealgorithm. Specific examples for the continuum tangent matrices for hyperelasticmaterials and elastic-plastic materials are presented in Section 6.4.2.For notational convenience, we will develop the tangential stiffness matrixby relating rates of the internal nodal forces ˙f int to the nodal velocities ˙d . Thusthe stiffness matrices K int can be derived by taking the material time-derivative ofthe nodal internal forces.
The procedure is identical to relating an infinitesmalincrement of nodal displacements df int to an infinitesmal increment of nodaldisplacements dd , and we will occasionally recast the equations in that form; thedot notation is chosen for convenience. The derivation is perfectly rigorous for6-24T. Belytschko & B. Moran, Solution Methods, December 16, 1998any continuously differentiable residual; for rougher residuals, directionalderivatives are needed and are described later.By (4.9.10-11), the internal nodal forces in the total Lagrangian form aregiven by,f int = ∫ΩοB0T PdΩ0 ,fIiint = ∫Ωο∂NIP dΩ∂X j ji 0(6.4.1)where P is the nominal stress tensor with components Pji , N I are the nodal shape( ) jI = ∂N I / ∂X j .functions and B0TWe have chosen the total Lagrangian formbecause this leads to the simplest derivation.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
