This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MCE 561 Computational Methods in Solid Mechanics
Time Dependent Problems Time dependent problems occur in many areas of engineering science, e. g. transient diffusion
(heat transfer), vibrations and wave propagation. In these types of problems the governing ﬁeld
equations contain ﬁrst and/or second order time derivatives, see for example the transient diffusion
and wave equation in the basic ﬁeld equations class handout. For these types of problems we have
both spatial and temporal (time) variations to approximate, and therefore this normally leads to two
stages of solution. The first stage considers the spatial approximation (i.e. the ﬁnite element
discretization), and this is then followed by a time approximation scheme to handle or generate a
time—stepping method. Such procedures are commonly classiﬁed as semidiscrete approximations
in the spatial sense, and these methods reduce the governing ﬁeld equations to a set of ordinary
differential equations in time. The ﬁnal step to incorporate a time marching scheme ultimately
involves the reduction of the problem to a set of algebraic equations. Approximation of FirstOrder Time Derivatives As mentioned for ﬁeld equations involving ﬁrst order time derivatives, the spatial
discretization will typically produce the following matrix differential equation [AM + [B]{u} : {P} (1) In order to develop a time—stepping scheme to handle the ﬁrst order derivative, we introduce
the socalled oc — family of approximation. In this method the weighted time average of the time
derivative of the dependent variable at two consecutive time steps is approximated by a linear
interpolation of the values of the variable at the two steps, i.e. 1 H {u}"+1 — {uin
+ — : —————————————
( 0‘) u n At (2) n+1 odd} 'n+l where O s 06 s l, {}n denotes to the value at time tn, and Atn = tn — tn‘l. From the above method,
several well—known difference schemes can be obtained by choosing various values of the oc
parameter, i.e. Forward Difference, Euler — Conditionally Stable, Crank—Nicolson, — Stable , 0(At)2 a: (3)
Galerkin ,  Stable, 0(At)2 0
l
2
2
3
1 Backward DiﬁerenceStable,0(At) Using the approximation (2) for times tn and th in equation (1) yields [A]{u}n+1 : + “At11+l({‘P}n+l _ [B]{u}n+1)
+ — 00Atn+l({‘P}n _ Rearranging the terms to express {u}n+1 in terms of {u}n, yields ([A] + amnﬂwnmlﬂ : ([A] — (1 — oc)Atn+1[B]){u}n
+ Ar,,+,(a{1)},+, + (1 — am") (5)
01‘
Wu)“, = [é]{u}n + {Plum1 (6)
where A [A] = [A] + 06Af,,.1lB]
[B] = [A]  (1‘06)Af,,.1[B] (7) {P} + (1 —oc){P}n] n , n+1 _ Thus the solution at time tn+1 is obtained in terms of the values at time tn. Note that the column
vector {P} is known at all times. In order to start a time—stepping scheme, the value a t = O
(presumed to be given) is ﬁrst used to calculate the solution at t =t1 = At. This procedure is then
repeated to obtain successive values of {u} at each time step. Time stepping schemes such as these have inherent problems with accuracy and stability.
Generally one can expect more accurate results if smaller time steps are used. However, in actual
applications it is desired to take as large a time step as possible to minimize the computational
expense. In addition to accuracy, larger time steps also can introduce some unwanted numerical
oscillations into the solution. This is related to the stability of the numerical routine. Stability is
concerned with the behavior of the solution as t we while keeping the time step constant. In some
case the numerical scheme is conditionally stable , i.e. is stable only for certain values of time step.
Examples of typical behaviors are shown below Unstable Stable (Oscillatory Decay) Stable (Monotonic Decay) Consequently, an estimate for the largest time step for accurate and stable solutions is highly
desirable. For all numerical routines in which 06 < 1/2, the time stepping approximations are stable
only if the time step satisﬁes the following stability relation At < Arc, = ————2——— (g)
(l—Za)/l where it is the largest eigenvalue of the ﬁnite element equation (1). This is found by using a solution dammmiman):LnannnanothBy—apqxo}={P} Approximations for Second—Order Time Derivatives For second order time derivative problems, the spatial discretization will produce the system [A]{iil+[B]{u}=iF} (9) There exists several approximation schemes available for this type of equation, and Bathe
outlines several these techniques. These schemes are normally described as direct integration
methods, in that the equations are integrated using a numerical stepping procedure without any
transformation of the equations themselves. The direct integration procedure in based on two ideas.
First, instead of attempting to satisfy the equations at any time t, the method satisﬁes the equation
only at discrete time intervals. Secondly, the variation of the primary unknown along with its ﬁrst
and second time derivatives are assumed to be known over each time interval. Some of the more
common techniques within this general scheme are 1. Central Difference Method
2. Houbolt Method 3. Wilson 6 Method 4. Newmark Method The most commonly used scheme is the Newmark direct integration method. In the
Newmark direct integration method the ﬁrst time derivative {u} and the function {u} itself are
approximated at the (n+1)th step AtI = At2 = = At) by the following a %%+mmWMaMMm n+1— an“: {u}n+ {mum + [(én [3)(2'1}n+ [3{ti}n+l](At)2 (10) where on and [3 are parameters that control the accuracy and the stability of the scheme, and the subscript n indicates that the solution is evaluated at the nth time step (i.e., at time t = tn). For linear problems, the choice on = 1/2 and [3 = 1/4 is known to given an unconditionally stable scheme, which corresponds to a constant—averageacceleration method. The case on = 1/2 and [3 = 1/6 corresponds to a linear acceleration method. Other particular schemes are given in the text (see p 325).
Combining equations (9) and (10) yields [A‘]{u}n+l= {an , ",1 (11) where M=m+nm n
m=anﬁnmppamﬁam) () Once the solution is known at tn“, the ﬁrst and second derivatives of {u} at tn+1 can be computed
from equation (10). {ii}n+1= a0({zt}n+l {u}n) a1{12}"— 61251)" ham: {u}n+ a3iii}n+ a4{ii}n+1 where ( 1 3) 1 1
, a=aAt, a=~——
BAH 1 0 2 2B a0: 1, a3: (1—06)At, a4: on For a given set of initial conditions {u}0, {u}0, and {u}0, we can solve (l3) repeatedly,
marching forward in time, for the solution {u} and its time derivatives at any time t > 0. The various schemes within the Newmark method are generally stable. For all schemes in
which 26 < 06, the stability requirement is given by the relation At 3 Arm. = [l/zmﬁmx (a — zenV2 (14) where wmaxis the maximum natural frequency of the system ([B] — (02 = {F} Initial values of {u} are generally not known from the problem description. For this case one
can make use of equation (9) at the time t = O to compute {1'1}. One~Dimensional Time~Dependent Problems Consider the general one—dimensional time dependent equation, 824 821! a (Bu 62
cl»— + (:2 — (a ) +
6: ar2 6x 6x 6x2 2
(bgiwwo, 0<x<L (15)
x Clearly both second and fourth order problems are contained in this general equation. Following the
standard technique, we construct the weak form +c — c
e l at 2 at? 8x 6x 3x2 . 2 2 2
fits“ Q au 6(a6u) + a )baaz) UM :
x x 2 2 2
fxe+1(clv___ + czvﬂ + 6159.39.11 + + Vﬁdx =
xe at 3% ax 8x 6x2 8x2 2 = 2 x=x
+ {v 8“ 3(ba—‘i>1>:=:“‘ — [byits“ = 0
8x 8x 5x2 9 8x 3x2 9 which can be written as 2 2 2
fxmwlvﬂlf + czvo‘u + advau + bavau + vf) dx at at2 6x 6x 6x2 6x2
A A A d A
— leee) — ngrxetl) — ta—ngrxen ~ gilt—gem] = 0 E (16) where A 611 6) 6214 *(e) azu
: "a — + —_ b— _. p : b_ .w
Q1 [ ax éktaxznM 2 [ékaﬂ
Q : [aau _ a (bdzuﬂ A(e) : [be/ﬂu] (17)
3 6x 8x 6x2 ﬁx“ ’ 4 6x2 ﬁx?” Assuming that the time and spatial variations of u are separable, we follow the Ritz scheme
and let r u=Z%m%m, v:wn> an f=1 Equations (18) with (16) then give . r du. " dzu. dill. dd].
xe+l _~L ____J __.l_ .._.._J_.
fxe (clwi dl‘ wj + 62% alt2 j + a dx1=1 uj dx
6121‘;~ r t J _ _ “ _
+bdngujdx241pﬂdx Qi~0
or
[Ml]{zz} + [M2]{ii} + ([Kl] + [K2]){u} = {F} (20)
where 1 3+ 2 — 2+
M], = f: lcllllilllj dx , My “ f: chqu‘ljf dx d .d . . d2 .d2 4
K1} 2 fxmaiji 01x, K3. = f’““‘b 1'" w! dx (21)
J xe 6156 056 J Xe dx2 alxz F, = f dx + Q
Consider the special case of the ﬁrst order time derivative example {MW} + [Iq{u} = {F} (22) Using the previous time stepping scheme yields the matrix equation [Kl]{u}n+1 = {F1} (23) where [Kt]= M4H+uAaK1 {F1} = ([M1]—(1—oc)At [K]){u}n (24)
+Ai (0607} +(1—oc){F}n) n+1 Likewise the special case of a second order time derivative example would be [M2]{ii} + [K]{u} = {F} (25) Using the Newmark time stepping scheme produces the system [1'32]{u}n+1 = {F2} (26)
where
Wﬁ=M+%Wﬁ
{#2) = {F}n+l+ [M 2](aO{u}n+ al{i2}n+ a2lii}n) (27)
{ii}0 = [M2]"1({F} — [K]{u}0)
Mass Lumping For time dependent problems considered here, the matrices that multiply either the ﬁrst or second
order time derivatives of the primary nodal unknowns are generally called mass matrices, see for
example relations (22) or (25). These matrices were developed from the standard variational/weak
form procedure, and are commonly called consistent mass matrices. In general they are symmetric,
positive deﬁnite and nondiagonal. Since solution procedures normally require inverting these
matrices, it is computationally advantageous to have these matrices be diagonal. Several methods
have been developed to construct such diagonal mass matrices (see text pp 326328). One method
is to use a row—sum lumping scheme whereby the sum of elements in each row in the consistent
matrix are used as the diagonal elements. Considering the one—dimensional consistent mass matrix X
[M] = I b pix/{1,11 idx , the linear and quadratic consistent and lumped mass matrices are given by
x" A h2l th
mn=pr[ JstML=peL J 6 l 2 2
4 2 —l 1 0 0 (28)
pk, pkg
[M]: 30 2 l6 2 :> [M]L = 6 O 4 0
—1 2 4 O 0 1 Use of mass lumping will also effect the critical time step required for conditional stability and in
fact will generally lead to a larger critical time step (see text pp 327328). ...
View
Full
Document
This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.
 Spring '11
 Sadd

Click to edit the document details