Belytschko T. - Introduction (779635), страница 71
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Belytschko & B. Moran, Solution Methods, December 16, 1998We now ascertain the conditions under which the eigenvalues µi fall within theunit circle, which corresponds to a stable numerical integration. Using again thefact that the eigenvectors y i span the space, expand the initial solution vector att = 0 in terms of the eigenvectors bynDd0 = ∑ r0iy i(6.5.45)i=1where r0i is determined by the initial conditions. Substituting the above into ()and using the fact that y i are also eigenvectors of () with eigenvalues µi , weobtain thatnDd1 = ∑i =1µi r0iyinD,d2 = ∑ ()2µi r0iy i ,i=1nDdn = ∑ ( µi ) r0iy in(6.5.46)i=1where the second equation follows by repeating the process and the last equationcan be obtained by induction.
We can see immediately from the above that if anyof the eigenvalues of the generalized amplification matrix µi is greater than one,the solution will grow exponentially. Since we are examining the behavior of thedifference of two solutions, this indicates that the procedure is unstable. Althoughsome readers will advance the counterargument that this unstable growth willoccur only if the initial data contains the eigenvector associated with µi , in fact,due to roundoff error, the constant ri 0 will be initially be nonzero or becomenonzero later in the calculation.
No matter how small the constant, theexponenetial growth will dominate ina very few time steps.Using Eqs. (6.5.42) and (6.5.36) it follows thatµi =1−α∆tλ i1+α∆tλ i(6.5.47)Since this eigenvalue is always real, the stability condition can be written asµi ≤1 . We consider eigenvalues µi =1 to lead to stable solutions at this point,but this is not always the case. From the preceding we deduce the conditions onthe time step necessary for numerical stability as follows:µi ≤1 →1− (1 −α) ∆tλi≤1 → always met1−α∆tλiµi ≥−1 →1− (1−α )∆tλ i≥−1 → (1−2α )∆tλi ≤ 21−α∆tλ i(6.5.48)(6.5.49)There are two distinct consequences of Eq.().
If 1− 2α ≥ 0, i.e. α ≥ 0.5, then thecondition of stability is met regardless of the size of the time step. The method is6-71T. Belytschko & B. Moran, Solution Methods, December 16, 1998then called unconditionally stable.yields the requirement that∆t ≤When 1− 2α <0, i.e. α < 0.5, Eq (6.5.49)2∀i(1−2α )λi(6.5.50)where we have indicated that the condition on the eigenvalue µi must be met forall i .
The maximum eigenvalue then sets the time step, so the critical time step isgiven by∆t ≤ maxi22or ∆tcrit =(1− 2α )λ i(1− 2α )λmax(6.5.51)A method which is stable only for time steps below a critical value is calledconditionally stable. If we consider the explicit form of this generalized updateequation, i.e. α = 0 , then the above gives∆tcrit =2(6.5.52)λmaxThus the stable time step is inversely proportional to the maximum eigenvalue ofthe system.
The stiffer the system, the smaller the stable time step. For thetrapezoidal rule, α = 0.5 , and for any 0.5 < α ≤ 1 the method is unconditionallystable. For 0 ≤α < 0.5 , the integrator is implicit but conditionally stable, so thesevalues of α are of little practical value.To give the reader a appreciation of the explosive growth of an exponentialinstability, Table ? shows the results for exponential growth for several values ofthe eigenvalue µi .
Exponential growth is truly startling. It is also the reason whycompound interest can make you very rich if you live long enough and startsaving early.In summary, we have shown that the determination of the stability of anintegration formula for the semidiscrete initial value problem () can be reduced toexamining the eigenvalues of the generalized amplification matrix (). If anyeigenvalue lies outside the unit circle in the complex plane, the perturbation growsexponentially so the solution is numerically unstable. Otherwise, the method isstable.Stability of thhe Central Difference Method.
We now use the same techniques toexamine the stability of the central difference method for the equations of motion.MATERIAL STABILITYAn important issue in modern computational mechanics is the stability of thematerial models. The issue has already been discussed on several occasions inChapter 5, cf...In this Section, we examine the implications of material instabilityon computational procedures and provide some remedies for the majordifficulties.6-72T. Belytschko & B. Moran, Solution Methods, December 16, 1998As pointed out in Chapter 5, material instability results from the loss of positivedefiniteness in the tangent modulus tensor relating the Truesdell rate of theCauchy stress to the rate of deformation.
The name material instability is a slightmisnomer because the occurrence of this phenomenon does not lead automaticallyto the violation of stability definitions such as (6.5.1). Instead, an unstablematerial is characterized by the possibility of unbounded spectral growth for abody in a homogeneous state of stress. When a material fails to meet the stabilitycriteria for a subdomain of the problem, unbounded growth of the solution doesnot necessarily occur.Nevertheless, the consequences in a computation of the failure to meet materialstability criteria are dramatic: for rate-independent materials, loss of materialstability changes the PDE locally from hyperbolic to elliptic in dynamic problemsand vice versa in static problems. Furthermore, in rate indenpendent materialsthis is accompanied by a phenomenon called localization to a set of measure zero:the domain in which material instability occurs in a three dimensional problemwill localize to a surface.
On that surface in the domain, the strains will beinfinite and the motion will be discontinuous. Although this ostensibly looks likea good way to model fracture and failure of materials, because of the localizationto a set of measure zero, the dissipation associated with this process vanishes, sothat the model is inappropriate for any realistic physical model of fracture or shearbanding.The literature on material instability goes back at least as far as Hadamard (1906).I haven't read the literature of that time, and even my knowledge of Hadamard issecond-hand, so there could be earlier studies.
Hadamard examined the questionof what happens when the tangent modulus in a small deformation problem isnegative. He concluded that according to the wave equation and the formula forthe wavespeed, (???), that the wavespeed is then imaginary (the square root of anegative number), so such materials could not exist.The next major milestone in the study of unstable materials is the work of Hill(??), who examined the conditions under which materials are unstable. Hismethodology was to consider the momentum equation for a homogeneous state ofinitial stress in terms of the displacements.
The momentum equation is thenCijklvk ,l = ρ˙v˙ i wrong eqn unless v=displUsing the technique of linear stability analysis, he examined the growth and decayof solutions of the form( )κ x − ctui = AieThe solution grows exponentially if any of the eigenvalues of the problem( )κ x − ctui = Aieare negative. He also showed that equivalently one could examine the materialinstability through the possibility of acceleration waves.
This technique is nowclassical and is used in finite elements to detect the possibility of materialinstability> It goes as follows:6-73T. Belytschko & B. Moran, Solution Methods, December 16, 1998Hill() also examined material instabilities for large deformation problems and thequestion of which rate is appropriate for ascertaining unstable behavior. heconcluded thatAnother milestone paper in this stream is the work of Rudnicki and Rice(??), whoshowed that material instabilities can occur even in the presence of strainhardening when the plasticity is nonassociative.
The argument has been given inSection 5.?Thus when computers came on the scene for nonlinear analysis in the 1970's therewere two known causes of material instability: a negative modulus (or a negativeeigenvalue of the tangent modulus matrix) and a nonassociative plasticity law.Computational analysts soon began to include material models which includedeither or both of these and they discovered many difficulties.
In fact it wasargued by many, including Drucker and Sandler(), that material models thatviolate the stability postulates should never be used in computational methods.Their arguments proved fruitless since there is no way to replicate observedphenomena such as shear banding without a model that exhibits strain softening,although the models which were first used to examine shear bands, Clifton andMilliner(), are viscoplastic and satisfy the stability postulates.Zdenek Bazant and I started studying the problem in 197? and based on somecomputational results of Hyun we surmised that the closed form solution for arate-independent material model must exhibit an infinite strain.
We were able toconstruct a one-dimensional solution of this behavior, albeit quite inelegant inretrospect, and learned that for these materials the unstable behavio must localizeto a set of measure zero and that the dissipation would then vanish.This led to the search for a regularization of the governing equations, which wecalled a localization limiter at the time.
We soon discovered that both gradientmodels and nonlocal models regularize the solution, Bazant, Chang andBelytschko and Lasry and Belytschko(). This solution of remedying thedifficulties associated with negative moduli had already occurred in anothercontext, the heat equation, where Kahn and Hilliard() circumvented the difficultyby a gradient theory, which came to be known as the Kahn-Hilliard theory.Hilliard was incidentally also at Northwestern but we were unaware of his workuntil later.
Aifantis(??) had proposed gradient regularization in solid mechaincsbefore us.Subsequently a plethora of work emerged in this area, with two goals: to obtainphysical ustifications for the regularization procedure and to simplify thetreatment of nonlocal and gradient models. Schreyer et al (), introduced gradienttheories based on the gradient of the plasticity parameter lambda in Eq.(5.??),Pijaudier-Cabot and Bazant(??) introduced the gradient on the damage parameter.These are important because introducing nonlocality in the 6 strain components isawkward indeed. Mulhaus and Vardoulakis showed that a coupled stress theoryalso regularizes the equations, and Needleman showed that viscoplasticityregularizes the equations.
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