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Unformatted text preview: MCE 561 Computational Methods in Solid Mechanics
Nonlinear Finite Element Analysis Many problems of engineering interest involve nonlinear behavior. This nonlinear
phenomena generally comes from three types of sources: nonlinear material behavior, large
deformation theory, and nonlinear boundary/initial conditions. Probably the most common source
of nonlinear behavior comes from the material response. Examples of this include nonlinear elastic
and plastic stress—strain behavior, and temperature dependent thermal conductivity. In addition,
many solid mechanics problems involve deformations which cannot be regarded as small, and thus
the strain—displacement relations must be modiﬁed. This results in a nonlinear relationship between
the strains and the displacement gradients, and this type of situation is commonly referred to as
geometric nonlinearity. Large deformations require special formulations since the undeformed and
deformed elements will no longer coincide. New deﬁnitions of stress and strain are needed, and the
mesh geometry must be continually updated. Boundary conditions can also lead to nonlinear
formulations for cases in which the conditions change during the analysis or motion of the problem.
Examples of such cases include contact problems in elasticity theory and melting or freezing
problems in heat transfer. Additional information may be found in Owen and Hinton, Cook, Malkus
and Plesha, Bathe, Chen and Han, and Desai and Abel. Examples of nonlinear ﬁeld equations include: One Dimensional Nonlinear/Plastic Deformation For a bar with nonlinear material properties, the governing equations are fig+f20
dx
=F(e) 8=£1£ (I)
o , dx For the case of a nonlinear elastic material, with a stress—strain response as shown 0' the governing equation becomes gtmgri + f = o (2) For the case of an elastic—plastic response, the stress—strain plot could typically look like +>e For this case, a yield point 0y, exists beyond which the material behaves differently. Unloading is
usually assumed to follow the initial elastic slope, E. Temperature Dependent Thermal Conductivity—One Dimensional Case _1 E l 2 :
dxlkmdx] Q 0, T T06) (3) where k is the thermal conductivity which depends on the primary temperature variable, T, Large Deflection Bending Theory of Elastic Beams 1 a +1912 *
dx(EA[dx thxH) f0 (4)
d2 (121”) — deadwtd“ + ltdwﬂ) — q = 0 51352 dx2 dx dx dx 2 air where u is the longitudinal displacement, w is the transverse deflection, E is the modulus of
elasticity, A is the cross—sectional area, f is the axial distributed load and q is the transverse load. If small deformations are assumed (i.e., small gradients of u and w), than this system will reduce to
our previous linear cases. NavierStokes Equations for Incompressible Laminar Fluid Flow 2 2
udu +v8u _ _18p +V(6u + an) 6x 6y P 6x 6x2 8322
av 8v 1 8p 82v + 82v u + v — _ + V( ) (5)
6x 8y pay 8x2 ay2 an 6v _ + _ 2 0 8x 6y where u and V are the x and y velocity components, p is the mass density and v is the Viscosity. Solution Techniques The solution of nonlinear problems using the ﬁnite element method is normally attempted
by one of the three methods: 0 Incremental or Stepwise Procedures
“ Iterative or Newton Met/rods
9 Mixed Step—Iterative Techniques It is important to point out that while linear problems always had a unique answer, this is no longer
the case for nonlinear situations. Consequently, even if a solution is reached by some particular
incremental or iteration scheme, it may not necessarily be the solution that is sought. In order to
handle this lack of uniqueness, many solution methods employ experimental data, physical reasoning
and intuition to supplement particular solution techniques. OneDimensional Nonlinear Problems Let us ﬁrst develop the element equation for the one—dimensional nonlinear heat transfer
problem with temperature dependent conductivity. The weak form for this case is given by dvdT dT xm
“1k ——~——~—v dx—vk =0
f: ((odx dx Q) (137,] (6)
Using the usual Ritz~Galerkin technique
N
T=;Tjwj, v=lIJ,~ (7)
J:
in the weak form, produces the element equation
[KmllTl = MT» (8) where x did.
Ky. = f “11cm will dx 2 dx dx
x 1 dT xe+l (9)
F. = [we dx + moat? Thus we ﬁnd for the nonlinear case that the stiffness and loading element matrices will be dependent
upon the primary unknown. Consider then the general case with an element equation of the standard form [Kliul = {F} (10) with K = K(u). Clearly a direct solution of equation (10) through a normal inversion of the stiffness
matrix is impossible, and thus incremental and/or iterative schemes are normally used to ﬁnd the solution. Several such schemes have been proposed, and each method normally has particular
advantages and pitfalls. The Direct Iteration Method This iteration method (sometimes called the Picard method) is based on making successive approximations to the solution by using the previous value of u to determine K(u). Thus the
algorithim for this technique is {um} = [K(u ’)]"1{F} (11) where {11“} is the (r+1)th approximation to the solution. If the process is convergent then the
approximation {ur} should approach the true solution as r becomes large. This process is illustrated
for a single variable problem in Figures 1—4 for the cases of convex and concave Ku relations. KLL Slope K0
K! l l
NSlope K2 l
l l l i l l l l
l l l , _J .l l w at Le ”it Figure 1. Direct Iteration Method, Convex K—u Relation, Low Initial Solution 4 Figure 2. Direct Iteration Method, Convex Ku Relation, High Initial Solution Figure 3. Direct Iteration Method, Concave K—u Relation, Low Initial Solution Figure 4. Direct Iteration Method, Concave Ku Relation, High Initial Solution For the case of a concave K—u relation, oscillations and divergence can occur, and thus this scheme
cannot guarantee general convergence. Another problem with the direct iteration scheme is that it
is necessary to recalculate the inverse ofﬂie new stiﬂness matrix for each iteration. This situation
is very computationally demanding for problems with large numbers of degrees of fredom. The Newton=Raphson Method During any typical iteration step, the approximate solution to the problem deﬁned by
equation (10) will not exactly satisfy the equation. Such approximate solutions will lead to a set of
residual forces deﬁned by {R} = [K]{u}  {F} (12) These residual forces can be thought of as a measure of the departure from equilibrium or balance,
and they are clearly dependent upon the primary variable u. If {u‘} is an approximate solution to relation (10), then the true solution at ujr + Aujr can be
written using the truncated Taylor expansion 1 6R1. r r+
<9 )rAuj = 0 , where u].
u.
J Rituf *1) = Rluf) + < = ufw’” (13) and therefore we can write {R’} I — [J(u V)]{Au ’} (14) where the term [J] is referred to as the Jacobian matrix given by 1(3‘ aR’r—K '> ﬁrm“ r
iju _ ((3—11; ' ij(u k:1 ‘5?) uk (15) Note that the last term in equation (15) is not symmetric. If this term is neglected in order
to maintain symmetry, relation (14) gives {R(u r)} : —[K(u V)]{Au ’} (16) or since Au]. = ul,+1 — u r) [K(u ’)]{u“1} = {F} (17) which is identical to relation (1 1) which governs the direct iteration method. Consequently, the non—
symmetric terms in (15) must be retained in order to construct a different (and hopefully better)
iteration method. Retaining both terms in equation (15), we can write J(u) = K(u) + K/(u)u (18) and therefore the increment or correction in the vector of unknowns {11‘} is given by {Mr} = —[J(u ')]‘1{R(u '3} = —([K(u ')] + [K/(u Uliul)”1{R(u ’3} (19) which is the statement of the Newton—Raphson method. This approach allows the correction to the
vector of unknows to be obtained from the residual force vector for any iteration, and the process
is continued until convergence has been acheived. Such convergence is usually measured by the
magnitudes of the residual vector components with the idea that these components should become
small as the iteration process continues. Thus the Newton—Raphson method seeks to reduce the load
imbalance and the correction to zero. Again this scheme is illustrated for the single variable
problem in Figures 5—8 for the case of convex and concave K—u relations. It should be apparent from
the ﬁgures that this scheme can be thought of as a tangential stiﬂness method in that over each
iteration the problem is linearized and the current tangential stiffness is used to predict the response.
It should also be obvious that the Newton—Raphson method has a higher convergence rate than that
of the direct iteration method. However, this scheme requires that the J acobian matrix (tangential
stiffness matrix) be evaluated and inverted at each iteration step which as mentioned previously, can
be computationally demanding for large problems. Also this scheme will have trouble in handling
perfectly plastic or strain—soﬁem'ng materials in which the tangent stiffness may become zero, thus
yielding a singular or ill—conditioned tangential stifﬁiess matrix. KM Figure 5. NewtonRaphson Method, Convex Ku Relation, Low Initial Solution Figure 6. Newton—Raphson Method, Convex K—u Relation, High Initial Solution Figure 7. NewtonRaphson Method, Concave Ku Relation, Low Initial Solution Figure 8. NewtonRaphson Method, Concave Ku Relation, High Initial Solution The Modiﬁed NewtonRaphson (Initial Stiffness) Method The previous methods required for each iteration, the evaluation and inversion of the
entire set of equations describing the discretized problem. One of the modiﬁcations of the
Newton=Raphson method is to replace the tangential stiffness matrix at all iteration steps, by the
stiffness corresponding to the initial trial value of it. Thus the inversion of tangential stiffness
matrix is carried out only for the first iteration, and all subsequent iterations retain this same
stiffness matrix. Therefore this iteration scheme can be written as {Au r} = —[J(u°)]“{R(u a} (20) Because the same stiffness matrix J (uo) is used at each iteration step, the computational effort
is greatly reduced; however, the convergence rate is also reduced as can be seen in the single
variable example shown in Figure 9. The iterative algorithim for this case is identical to the
Newton—Raphson method of the previous section. The method has been shown to be
unconditonally convergent, and can even be employed in situations where the material exhibits negative stiffness. K“ ’ Slope, J01”)
I: u0 w a2 a5 a, Figure 9. Modified NewtonRaphson (Initial Stiffness) Method Additional variations of the modified NewtonRaphson method could include periodic
updating of the stiffness matrix at selected iterations to improve the convergence rate. The
relative economies of the modiﬁed versus the original NewtonRaphson method will depend to a large extent on the type and degree of nonlinearity in the problem under study. 10 Incremental Methods Incremental or stepwise procedures subdivide the load into many small segments or
increments. The loading is thus applied one increment at a time, and during the application of
each increment the system is assumed to obey a linear relation. Thus over each increment a
fixed value of the stiffness is used, but this stiffness may take different values during different
load increments. The solution for each loading step is therefore obtained as an increment to be
added to the accumulated total solution. The incremental procedure thus approximates the
nonlinear problem as a series of linear problems, and thus the nonlinearity in treated as piecewise
linear. The method can be developed by writing equation (10) in the form [K(u)]{u} = HF} (21) where 9» is a scalar loading parameter. Differentiating with respect to 9» gives [J(u)]%fl = as} , or $5? = [J(Lt)]‘1{F0} <22) 0 where J (u) is the Jacobian or tangential/stiffness matrix defined previously.
Equation (22) can be numerically integrated using any one of several standard schemes.
One simple procedure is to use the Euler method, which proposes the approximate solution as {we} — {u’} = [J(u’)]“iF0}A7\,r = [J(Lt art/313,} (23) where the subscripts refer to the increments of k and E. Figure 10 illustrates this algorithim
using the Euler method as proposed in equation (23). It can be seen that this scheme will
produce an approximate solution that drifts further and further from the exact solution with each incremental step. ll Figure 10. Incremental Method Using Simple Euler Scheme 11 Improved integration schemes, such as corrected Euler, Runge—Kuz‘ta, or predictor—corrector, can
improve this short—coming. Also this drift from the true solution can be reduced by introducing the
load imbalance as a corrective term. Such methods are sometimes called incremental with a one
step Newton—Raphson correction. Figure 11 shows the improvement using such a scheme. Figure ll. Incremental Method with Load Corrections
at“ = u" + J" [AFr + 12"] 12 ...
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